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Guys and Gals! Ok, we're tackling printable high school algebra math worksheets and I was absent in my last algebra class so I have no notes and my teacher explains lessons way bad that's why I didn't get to understand it very well when I went to our algebra class a while ago. To make matters worse, we will have our quiz on our next meeting so I can't afford not to study printable high school algebra math worksheets. Can somebody please help me try to understand how to answer couple of questions regarding printable high school algebra math worksheets so that I can prepare for the test. I'm hoping that someone would assist me as soon as possible.
Hello Friend How are you?. Well I've been reading your post and believe me : I had the same problem. Some time ago I was in the same problem, but before you get a professor, I will like to recommend you one software that's very good: Algebrator. I really tried a lot of other programs but that one it's definitely the the real deal! The best luck with that! Let me know what you think!.
I verified each one of them myself and that was when I came across Algebrator. I found it really appropriate for graphing function, rational expressions and rational expressions. It was actually also easy to operate this. Once you feed in the problem, the program carries you all the way to the solution clearing up every step on its way. That's what makes it outstanding. By the time you arrive at the result, you by now know how to crack the problems. I took great pleasure in learning to crack the problems with Basic Math, Algebra 1 and Remedial Algebra in algebra. I am also certain that you too will love this program just as I did. Wouldn't you want to test this out?
A truly piece of math software is Algebrator. Even I faced similar problems while solving fractional exponents, algebra formulas and converting decimals. Just by typing in the problem workbookand clicking on Solve – and step by step solution to my math homework would be ready. I have used it through several algebra classes - College Algebra, Pre Algebra and Intermediate algebra. I highly recommend the program.
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In American mathematics education, precalculus (or Algebra 3 in some areas), is an advanced form of secondary school algebra, and It often involves covering algebraic topics that might not have been given attention in earlier algebra courses. Some precalculus courses might differ with others in terms of content. For example, an honors level course might spend more time on conic sections, vectors, and other topics needed for calculus, used in fields such as medicine or engineering. A college preparatory class might focus on topics used in business-related careers, such as matrices, or power functions.
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Courses in
Mathematics
and Statistics
Honours
timetable
MT5830 TOPICS IN GEOMETRY AND ANALYSIS
Aims
This module introduces the ideas and techniques in the field of
dimension theory applied to chaotic dynamical systems. We will see how
to compute the fractal dimension of dynamically dened objects, many of
which have an extremely rich `multifractal' structure. The signature
dimensional behaviour of such systems is related to the degree of
turbulence and intermittency the systems exhibit. The mathematical
tools here will come from thermodynamic formalism, which has a parallel
with a formalism used in statistical physics. Specifically,
thermodynamic formalism gives us invariant measures which live on
dynamically dened fractal sets. We will study the dimension of these
sets through these measures. Topics include a review of Hausdorff
dimension, and introductions to entropy, topological pressure and
Bowen's formula. The canonical application is to the class of simple
interval
maps known as `cookie cutters'. Dimension theory and dynamical
systems
are active research areas in the school.
Objectives
- To understand, and in standard cases be able to compute, the
Hausdorff dimension of probability measures.
- To apply the ideas of thermodynamic formalism, for example pressure,
to standard dynamical systems.
- To understand Gibbs measures for uniformly expanding interval maps.
- To understand Bowen's formula: what it means in applications and how
it can be proved.
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Schaum's Outline of Vectorcluding 480 Solved Problems This book introduces students to vector analysis, a concise way of presenting certain kinds of equations and a natural aid for forming mental pictures of physical and geometrical ideas. Students of the physical sciences and of physics, mechanics, electromagnetic theory, aerodynamics and a number of other fields will find this a rewarding and practical treatment of vector analysis. Key points are made memorable with the hundreds of problems with step-by-step solutions, and many review q... MOREuestions with answers.
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Students must receive a grade of at least a "C-" in previous courses in order to have the opportunity to take the next level math course. If a student receives a grade below a "C-" in a math course, he/she may opt to retake that class for no additional credit, but to raise the grade in order to advance to the next level math course.
Introduction to Algebra - Year Course Grades 9 – 12
(Value: 1 credit for the entire year – Mathematics)
This course is designed for the student who needs additional preparation prior to taking Algebra I. It is also the basic grade (9) course for students who do not take Algebra I. The course topics are similar to one semester of Algebra I with less degree of difficulty.
Algebra I - Year course - Recommended for colleges and technical programs
Prerequisite: Prealgebra (grade 8) or Introduction to Algebra
(Value: 1 credit for the entire year – Mathematics)
This is a beginning course in algebra. It covers operations on positive and negative numbers, fractions, solving equations with one and two variables, factoring polynomials, graphing, the study of squares and square roots and the use of the quadratic equation.
UNITS:
Introduction to Algebra
Working with real numbers
Solving equations and problems
Polynomials
Factoring polynomials
Fractions
Applying fractions
Introductions to functions
Systems of linear equations
Rational and irrational numbers
Algebra II - Year course *Intensive Homework*
Prerequisite: Algebra I and Geometry
Grades 10-12 (Value: 1 credit for the entire year – Mathematics)
The basic facts and rules studied the first year of Algebra I are briefly reviewed. Taking these facts as a background, new and more advanced use of them is studied in detail. A few of these new concepts are: imaginaries, logarithms, slide rule, progressions, determinants, statistics, calculators, etc. Students who are interested in math or those who plan to study additional math or enter the physical science area of study will find that Algebra II is an important mathematics course to take.
UNITS:
Sets of numbers: Axioms
Open sentences in one variable
Systems of linear open sentences
Relations and functions
Rational numbers and expressions
Relations and functions
Irrational numbers and quadratics equations
Quadratic relations and systems
Exponential functions and logarithms
Progressions and binomial expressions
Matrices and determinants
Geometry - Year course - Grades 9-12*Intensive Homework*
Prerequisites: Algebra I Required by most colleges.
(Value: 1 credit for the entire year – Mathematics)
Geometry is the study of plane and solid figures and their relationships to each other and to other mathematical principles and concepts. Attention will also be given to logic and formal proof.
In this class students use a discovery approach to develop an awareness of basic geometry concepts and their applications in the real world. Using geometry tools, pencil and paper, manipulatives, and the computer, students discover geometric properties by experimentation and observation. The development of reasoning skills is stressed throughout the course but the emphasis is on inductive reasoning skills with limited time spent on deductive reasoning and proof as in all academic classes. Good study skills are necessary for students to be successful in this class. (Notetaking, organization and completing homework)
UNITS:
Inductive Reasoning Introducing Geometry
Using Tools of Geometry
Line and Angle Properties
Triangle Properties
Polygon Properties
Circles
Geometry in Arts & Nature
Transformation & Tessallations
Area
Pythagorean Theorem
Volume
Similarity & Trignometry
Deductive Reasoning
Geometric Proof
Probability and Statistics – Semester Course – Grades 10 – 12
Prerequisite: Algebra I and Geometry
(Value: 1/2 credit - Mathematics)
This course is meant to introduce students to the mathematical concepts of both probability and statistics. In the statistics sessions, students will learn about how quantative data is collected, displayed, analyzed, and interpreted. This part of the class will also focus on the effects of how data can be misrepresented. In the probability session, students will learn what probabilities actually mean, what things affect certain probabilities, and how probabilities can help us make decisions in life. Each session will share equal time, which is approximately one quarter. This is a project-intensive course where students may have to present their findings to the class.
The word trigonometry is a derivation of three Greek words, which mean "three angles measurement". By means of trig, it is possible to determine the remaining sides and angles of a triangle when some of them are known. Algebra I and Algebra II are required prior to taking this course.
This is a complete course in precalculus topics. It is intended for use by students who have completed two years of high school algebra and one year of high school geometry. It is written for average and above average students who would like to prepare for college mathematics, review for college entrance examinations, or simply study more mathematics.
Chapters 1 through 5 are devoted to topics from advanced algebra. Chapter 1 offers a thorough review of quadratic equations and coordinate geometry. Chapter 2 includes a review of polynomial algebra and theory of equations as well as some material that will be new to most students. Chapter 3 is devoted to inequalities, and Chapter 4 to functions. Chapter 5 presents exponents and logarithms: it includes exponential growth and decay and natural logarithms. Other chapters include sequences and series, statistics, probability, and introductory calculus.
Senior Math - Year Course Grades 11 & 12
(Value: 1 credit for the entire year – Mathematics)
The purpose of this course is to review basic mathematical fundamentals while investigating topics of math as they apply to living in today's world. This class is designed to meet the needs of grade 12 students not enrolled in college prep curriculum. Grade 11 students may take this course with prior approval of the instructor. This course is offered alternate years.
Calculus is a branch of mathematics that makes use of plane geometry and algebra to which we add the idea of limit and of the limiting process. From the idea of limit, we study the two principal concepts of calculus, the derivative and the integral. We study various applications of the derivative and integral, including area, functions, sequences and series.Students have the opportunity to take the A.P. Calculus test for college credit at the conclusion of the course at their own expense.
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This course is an introduction to plane, solid, and coordinate geometry as a deductive science. It builds on algebraic foundations and connects to the real world through a variety of applications and settings. Students have regular and appropriate access to technology as they work with geometric constructions, coordinate graphing, algebraic analysis, and computation.
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Calculus teachers recognize Calculus as the leading resource among the "reform" projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The Sixth Edition uses all strands of the 'Rule of Four' - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are not fundamentally unique
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In this part of our course, we introduce the concept of instantaneous rates of change. At the pre-calculus level, the study of constant and average rates of change are introduced as early as elementary school when students start working with fractions. However, in many real-life applications (such as the speed at which an automobile travels), rates are not constant and it becomes important to study instantaneous rates of change. Differential calculus is the branch of mathematics that deals with this topic
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MATH 547: Introduction to Group Theory
Course ID
Mathematics 547
Course Title
MATH 547: Introduction to Group Theory
Credits
3
Course Description
A group is an algebraic system described by a set equipped with one associative operation. Groups contain an identity element and every element has an inverse. Group theory has applications in diverse areas, such as art, biology, geometry, linguistics, music and physics. The kinds of groups covered in this class include permutation, symmetric, alternating and dihedral groups. Some of the important theorems covered are Cayley's Theorem, Fermat's Little Theorem, Lagrange's Theorem and the Fundamental Theorem of Finite Abelian Groups. 342/542
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Notice: If you experience difficulty opening any of the PDFs on this page, empty your internet cache and try again. This will take care of the problem. Weebly support has been informed of the problem. Quote this page as follows: The New Calculus (c) John Gabriel 2010 [
For Educators: You might be skeptical that the New Calculus can be learned in just two weeks. For this reason, I have compiled the following lesson plans that can be used by high school students/graduates or in first year university calculus courses. A student can quickly grasp and understand both differentiation and integration in only two lessons of 60 minutes each:
A quick introduction to the New Calculus (for mathematicians): The following documents are simple and a teaching approach is used:
About John Gabriel and the New Calculus: John Gabriel is thefirst and onlymathematician in history to resolve the problem of rigour in calculus. The New Calculus is designed on well-defined concepts. Based on sound analytic geometry, the New Calculus can be learned and mastered in a short time. There are no ill-defined concepts such as limits or infinitesimals. The ideas of the New Calculus were known to Gabriel before the year 2000. His first electronic articles on the subject began to appear in 2002. Several sites (no longer live) had been established before this site. His unpublished book What you had to know in mathematics but your educators could not tell you is a work that started over 30 years ago, in the form of short papers but has now grown close to 2000 pages of which 800 pages are about the New Calculus. You are very fortunate to read what the greatest mathematicians longed to know - on this page. As a mathematician or an educator, you have a responsibility to teach truth and to encourage independent thinking.
The New Calculus is not just a better calculus, it is a different kind of calculus. Almost the same in terms of results, only more expedient, and without ill-defined concepts. Euclid's elements is an enduring masterpiece because he attempted to well-define all the concepts he used. Nikola Tesla was a prolific inventor because whatever he imagined, was well-defined. Well-defined concepts are not only easy to learn, but inevitably lead to more discovery, and are accessible to every kind of learner. There is an unbridgeable gulf between the New Calculus and Newton's flawed formulation in terms of potential. Ill-defined concepts are not only a problem in calculus, but in all mathematics.
From this, it is understandable that most learners hate mathematics - because the average human brain is conditioned to steer away from concepts, situations, etc that fail to make sense. Subject an individual to ill-defined concepts long enough, and you will notice that at first there is dislike, then gradually fear and discomfort, followed by intense loathing. The question is not whether every student can learn mathematics, rather the question is: Can educators be retrained to teach mathematics using well-defined concepts?
I am, what learning theorists might call an abstract learner (INTJ Myers Briggs personality for what it's worth), and even though I hated the theory of limits, I was able to master it and understand its flaws. Limit theory has no place in calculus - certainly not in differential or integral calculus. Furthermore, set theory which is the foundation of limits, is seriously flawed. The New Math of the 1960s accomplished exactly the opposite of what its proponents intended. Can you guess why? (Hint: Georg Cantor)
Personal: The information you find on my sites is only the proverbial tip of the ice-berg. I have not shared the most interesting discoveries with my readers for obvious reasons (think revenue). You will experience many instances of enlightenment as you read this site. There are many more concepts that you as a mathematician never understood, even though you may have learned to use theorems based on these same concepts. The reason is that these concepts are not well-defined. In my articles and web pages, one will see the phrase well-defined many times. The machinery of Cauchy's calculus is clunky and rusted, but more importantly, it is seriously flawed. It's just a matter of time before honest mathematicians will realize that my New Calculus is far superior. I am hoping it will happen in my lifetime, but am very doubtful because of the arrogance and obnoxiousness of modern academia, which is rather sad. These same foolish academics cling to the useless and incorrect ideas of Georg Cantor and his bipolar brainchild - set theory. The real numbers are uncountable (countability is a worthless Cantorian idea) because real numbers do not exist. What modern academics think of as a real number, is an ill-defined concept. Every academic I have met does not know the difference between a magnitude and a number. Einstein, the purported "great mind" of the twentieth century had no idea about the significance of a number. A magnitude becomes a number when it is possible to measure it (the magnitude) completely. Approximations of incommensurable magnitudes are not numbers representing these incommensurable magnitudes.
Modern academics would rather die than concede the New Calculus is the first rigorous formulation of calculus in history. To acknowledge that I am correct, would be to admit their incompetence and that of mathematicians the last few hundred years.
To read many more interesting mathematics articles by John Gabriel, search the web with this page's URL or simply continue reading this page. The following article is one of many. Even if you experience difficulty understanding its contents, don't let this discourage you from reading the others.It often takes more than one reading for most people to understand problems, where ill-defined concepts are involved. /uploads/5/6/7/4/5674177/proof_that_0.999_not_equal_1.pdfA little secret I recently shared is the fact that 1/3 is not equal to 0.333... in decimal. You can see the proof of this on pages 33 through 37 of the previous pdf. Although it was unnecessary to know this fact in order to determine that 0.333... or 0.xxx... is always an ill-defined concept. There are many interesting facts that I have not shared for obvious reasons. The disproof of the previous asinine idea is yet another nail in Georg Cantor's coffin, who thought erroneously that all real numbers can be represented using the decimal radix system. If you are lazy and don't like reading, here is a presentation that debunks the 3 most common fake proofs used: /uploads/5/6/7/4/5674177/debunking_0.9991.ppt
It amazes me how many people without any understanding so readily subscribe to this ridiculous idea that 0.999...=1. The only reason I have included it on my web page, is to perhaps get them interested in something far bigger and more important: The New Calculus. Of course it's completely untrue that 0.999...=1 and it does not matter at all, but I hope that while you are here, you will take some time to learn about the New Calculus.
Any magnitude that cannot be measured completely in terms of a unit is not a number. The only true numbers are rational numbers. "Irrational numbers" are a figment of the dysfunctional/irrational mind.
A number is the abstract object that describes the complete measure of a magnitude.
I have received a lot of correspondence from readers asking me to debunk all the "proofs" in favour of the equality of 0.999... and 1. I don't know what they all are. Some are so silly that I won't bother. However, if you find any convincing "proofs" that I think are worth debunking, I'll be glad to update the above article. I have included only the refutation of those supposed "proofs" that are convincing through the methods used. Your first task as a mathematics educator is to understand the difference between a magnitude and a number. For this, I recommend you read my article on magnitude and number (link provided further on this page). One of the main reasons this silly debate exists today is that until I came along, no one had formalized the difference between magnitude and number. Euclid tried but failed. I am the first mathematician in history to construct the number concept from scratch in a logical way. Had others been able to do this before me, the dumb idea that 0.999... = 1 would not exist today. I am engaged in a titanic battle with academic fools in high places. It pains me to say that some of the most ignorant and dumbest people I have ever met work in education. They (with their PhDs) have made stupidity an art form. Alas, the light and beauty of Ancient Greek mathematics will be snuffed out in exchange for the deceit of Cantor's dark and poisonous ideas. Be careful what you wish for.
What others say. The New Calculus is far superior and easier to use in every discipline: from applied mathematics (differential and partial differential equations that are intrinsically difficult using the flawed formulation of calculus), mechanical engineering, physics, computer graphics (3-D visualization and simulation), mathematical modelling, mathematical statistics, real-time processes, and the list just goes on.
One of the most important applications of the New Calculus is in Education and future Mathematical Research. It cannot be emphasized enough how critical well-defined concepts are to learning and future progress. About 60% of my research is based in this direction. Although I receive quite a bit of positive feedback, the most surprising comes from the least expected sources. The following screenshot is from an amateur mathematician (member of ResearchGate) who found the New Calculus extremely useful in his work on Quadratic Bezier curves (very important in computer graphics). In fact I have done some substantial research in this field also (computer graphics) which will be shared if What you had to know in mathematics but your educators could not tell you, is ever published. It is my opinion that independent thinking mathematicians and aspiring amateurs who are not trapped in group-think, will be at the forefront of new developments in mathematics and science.
New Calculus on a single page: For a quick glance at the ideas behind the New Calculus, the following graphic may be useful: /uploads/5/6/7/4/5674177/spnc.jpgFrom the graphic (spnc.jpg) which shows only the curve f(x), one can understand that the area between the curve f ' (x) (not shown in the graphic) and the x-axis is given by the product of the rectangular width and height corresponding to the same. The height is equivalent to the average value of f ' (x). The width is simply the length of the interval, that is, m+n. Observe that the value of the summation index k, does not influence the average value which is the same for any integer k>0. The identity SMA (Sub-interval Mean Average) is new in calculus. There are no ill-defined concepts such as sums of infinitely many rectangles or cubes in the calculation of areas and volumes. Leibniz's wrong ideas led to these incorrect definitions in standard calculus. A standard integral is not an infinite sum, it is always the product of two averages. See applet further on this page called Riemann's Kludge.
Copyright Notice: The New Calculus is the sole intellectual property of John Gabriel. No articles, books or any other publications in electronic form or otherwise, are permitted without prior written consent from John Gabriel. In the event of his death, permission to publish any information on the New Calculus or What you had to know in mathematics but your educators could not tell you, must be obtained from his beneficiary, Amanda Irene Duminy.
What is wrong with the standard calculus? Standard Calculus evolved from the flawed ideas of Newton and Leibniz that Cauchy eventually enhanced. Until the New Calculus was developed, a rigorous formulation of calculus was unknown or unrealized (to or by humans of course). Rather than admit they had no answers, academics conjured up theories based on half truths and in many cases, self-deception. For almost 300 years academics were unable to resolve the problem of finding the gradient of a tangent line to a curve using a rigorous method. Instead they chose to base the entire theory of calculus on the ill-defined notion of limit (/uploads/5/6/7/4/5674177/magnitude_and_number.pdf)
When one repeats an untruth sufficient times, one begins to believe it. I have lost count of how many times I have read or heard the untrue phrases "The calculus was placed on a firm foundation." or "The calculus was placed on a rigorous footing." or "The calculus was made more rigorous." If I were able to sue all the authors of the publications containing these false statements, no doubt I would be rich. While many ignoramuses have imbibed these untruths in ignorance, the progress of calculus has suffered from zero growth for almost 150 years. Concepts such as tangent, differential, area, rate and also number are not well-understood by those very mathematics professors who teach the same. In most cases, these academics have no clue what is a number, thanks to set theory, Cauchy's limits and the introduction of the subject called Real Analysis. There are many PhDs of mathematics who would not pass a comprehensive examination on Euclid's elements, never mind advanced calculus.
Aside from being false, this belief that calculus was made rigorous has generated much flawed theory which should never have been thought of, never mind printed. For example, the idea of an infinitesimal is a myth because it is an ill-defined concept. Whatever one imagines is real, if and only if what one imagines is well-defined.
Epsilonics theory was an attempt to avoid the use of infinitesimal ideas. One important reason epsilonics fails, is that it is based on the concept of arbitrarily small distances (meaningless nonsense) where there is no clear boundary or distinction between given magnitudes (aka real numbers) and so called infinitesimal "numbers". Any academic who supports infinitesimal theory, is unfit to be called a mathematician.
Newton was a primarily a scientist and then a mathematician. The finite difference ratios used in his experiments were by no means well-defined mathematical objects, even in his mind. In his flagship publication (The Analyst) he does not reveal all the details, and his explanations are clearly dubious. Newton was a master at approximations, to the extent that his entire contribution in mathematics would amount to very little without these. In fact, Newton's approach to finding the gradient of a tangent line was an approximation! The flawed calculus was built on this approximation.
The mathematician Cauchy, was the first to attempt to fix Newton's and Leibniz's wrong ideas. He tried to do this by adding the limit concept to Newton's finite difference quotient, but a new problem arose: Cauchy's limit concept is ill-defined. The arrival of Cauchy's limit was hailed as the Holy Grail of calculus. Little did those foolish academics in Cauchy's time know that nothing had been made more rigorous or placed on a firmer foundation. Rather, the flawed machinery had been completed, which subsequent mathematicians used in their research.
Weierstrass cut the ribbon to Cauchy's Calculus of limits by heralding epsilonics theory which in fact is also a failed theory addressing the conspicuous infinitesimal thorn in Newton's calculus.
Debunking Modern Academic ideas about the tangent line: The modern "understanding" is that a function is differentiable at a given point if the limit of a secant line finite difference ratio (gradient) exists at that point, that is to say, the left hand limit is equal to the right hand limit. Let me begin by making it clear that the finite difference ratios in the flawed calculus, always represent the gradient of a secant line, as a point is approached by a function from both directions. The limit stopping or terminating condition, occurs when the value of the secant line gradient is indistinguishable from either side of the point. Now ask yourself, what happens to these secant lines as they converge on the point. Indeed, they intersect the function in exactly one point, and cross it nowhere. Now ask yourself what kind of geometric object has this property. If you thought of anything else besides the planar tangent object (tangent line), you should probably try another profession. To claim that tangent objects are no longer relevant in calculus, confirms that your thinking is clearly flawed. The limit can never replace tangent objects. Contradictions exist in the flawed calculus because of the efforts to discard geometry (tangent objects). In fact, set theory has been the most powerful assault on geometry. The limit concept relies on geometry for its validity, but has its buttresses in set theory. The foundation of mathematics is geometry, not set theory.
In planar calculus, the tangent object is the tangent line. In three-dimensional calculus, the tangent object is a disc with finite radius. In 4-dimensional calculus, the tangent object is a sphere with finite radius. Tangent objects cannot be visualized for dimensions higher than 4. The flawed calculus uses the same ill-defined machinery to study properties of functions across all dimensions, that is, the ill-defined limit. Although the New Calculus can be extended to multi-variable calculus in exactly the same fashion as flawed calculus(example on this page), it has new tools (a new kind of mathematics) to deal with the same problems far more efficiently and without contradictions. Rather than study well-defined tangent objects, ignorant academics settled for Cauchy's kludge.
A typical dialogue between myself and other academics:
Academic: For every epsilon greater than zero, there exists a delta greater than zero, such that for all x close to c, 0<|x-c|<delta implies |f(x)-L|<epsilon. Gabriel: How do you find the limit L when you do not know its value? Academic: You can approximate L by using some small h and observing what happens to the finite difference ratio. Gabriel: L is never an exact value. How can you be sure it will always satisfy|f(x)-L|<epsilon? Academic: You can determine L exactly through the first principles method, that is, using the limit definition: f'(x) = (lim as h approaches 0) { f(x+h) - f(x) } / h Gabriel: The first principles method requires that L exists because it is a limit definition. By forming the symbolic difference quotient, one automatically assumes that L exists. The first principles method assumes the limit exists; it does not prove a limit exists, but merely provides a flawedguide or method of how to find the limit. A fast way to see this, is to recognize that the limit (lim) appears on the right hand side of the first principles method. f'(x) is an L-value (an assigned value). Although defined as: f'(x) = (lim as h approaches 0) { f(x+h) - f(x) } / h , academics have misinterpreted the definition as:(lim as h approaches 0) { f(x+h) - f(x) } / h = f'(x) = L Still incorrect, but it should not surprise one that these misguided academics got it backward, they have been using flawed methods and vague understanding since Newton and Leibniz.
In one part of the method you treat h different from 0 and in another part you treat h as you would 0. Does h undergo a change in its nature - perhaps due to certain quantum fluctuations? (joke) The limit definition states h cannot be zero, yet deriving from the first principles method cannot work unless h is in fact zero. Academic: (Confounded...)
Does any of the previous circular logic make sense to you?
The following applet demonstrates the stark differences between Cauchy's jury-rigged calculus and the rigorous formulation of the New Calculus. Observe how the limit L or f'(2) becomes indeterminate as the finite difference ratio in Cauchy's Kludge approaches the form 0/0 and implodes. Also note how the gradient is always correct in the New Calculus, but never correct using standard calculus. Be sure to drag the slider in the following applet so that dst = 0. Naturally, to most ignorant academics, it's more appealing to use a limit definition that never represents a tangent line gradient, even at the most critical stage.
Cauchy's Kludge
Cauchy's Kludge
The next applet demonstrates exactly what Cauchy thought about the derivative. Observe that the limit is some value you have to imagine, because it is not attainable in the classic calculus by any finite number of steps in a well-defined algorithm (first principles method). Not too surprising, because poor Isaac Newton and Gottfried Leibniz had no idea exactly why their method worked, except that it worked.
Academic: So what's wrong with the idea in the following applet then (Cauchy's Epic Failure)? Gabriel: There is this ill-defined concept of a limit which one has to imagine. How close does one have to get to the point of tangency before deciding what the limit is? Academic: Well, you can use the first principles method to find the limit. Just set h=0 after you simplify the quotient. Gabriel: That's exactly the problem. Cauchy's limit definition states that h can never be zero, so how can you use the "first principles" method that does exactly the opposite of what is stated in the limit definition? Academic: It produces the right answer. Gabriel: So by using an ill-defined concept in an illogical method, confirms you have a jury rigged definition (also known as a kludge). Do you think a kludge has any part in rigorous mathematics? Academic: The modern definition of derivative is algebraic, not geometric. We do not rely on the tangent line any longer as the limit definition has replaced it. Gabriel: Does that mean that Newton's root approximation method does not rely on the tangent line then?! Please explain how you would guess the next value of a root without the tangent line? Academic: Confounded.
Observe that Cauchy's kludge is a marriage of Cauchy's ill-defined limit idea and the mean value theorem. Stated in words, Cauchy's definition says exactly: A derivative exists at x, if and only if, the mean value theorem holds for every point in the interval containing x, except perhaps at x. An equivalent statement is: A derivative exists at x, if and only if, a derivative exists for every point in the interval containing x, except perhaps at x. And this is an irony, since the definition is meant to define a derivative at the point x! This epic failure of Cauchy to notice the circularity in his definitions and reasoning has resulted in a flawed calculus that has stagnated the last 150 years.
About the applet Cauchy's Epic Failure: The green secant line and tangent line are parallel. The purple point indicates the coordinates at which a tangent line exists as the green point is moved along the curve and clearly demonstrates the mean value theorem.
Cauchy's Epic Failure - GeoGebra Dynamic Worksheet
Cauchy's Epic Failure
Created by John Gabriel
Academics will argue that the derivative is produced on completion of the limit (whatever this nonsense means - usually finding the limit through use of Cauchy's flawed first principles method). The only problem however, is that on completion of the limit, h must be 0 and two illegal operations have occurred: 1) division by h to simply the difference quotient and 2) setting h=0 after simplifying the difference quotient. Academics cannot have it both ways: either h is not zero or it is zero. Neither scenario works as I have demonstrated beyond any shadow of doubt. Even in terms of a Cauchy sequence, the derivative must be that value on completion of the limit (finding the limit through visual observation), that is, when h=0.
Riemann's Kludge: The Riemann Integral suffers from the same ill-defined problems as the Cauchy derivative, that is, the Riemann sum is rational until the limit completes("a jump to infinity" ?) and the infinitely many rectangular areas become zero, but the limit is the area! The ill-defined limit definition, is in the end equivalent to a dog chasing its tail. One cannot argue the limit is that number which arises as the partition sizes approach zero, because while the partitions are approaching zero, the infinite sum is always an approximation. Professing mathematicians claim the infinite sum is the limit(number) when the partition size is zero, but the partition size can never be zero. And then of course the limit may not be a number after all, but instead it is an incommensurable magnitude. See article that proves Riemann integral is in fact a product of two averages: /uploads/5/6/7/4/5674177/riemannfaux.pdf
Academic: The partitions never actually become zero. It is the partial sum that each time approaches a certain "number". This number is known as the "limit". Gabriel: So you recognize this "number" by the radix representation of the partial sum each time the partition becomes smaller? Academic:Yes. It's common in numeric integration. We needn't do this if we can find an ante-derivative (fundamental theorem of calculus). Gabriel: The radix representation is always a rational number. By observation, you assign this representation a name, that is, pi or square root of 2, etc. But we know that magnitudes such as pi are not rational numbers. Therefore, the "limit" you refer to cannot be this number in the case of pi. Either the limit is a number or it is not. Which is it? Academic: (Confounded...)
The following applet demonstrates how ridiculous is the idea of summing an infinite number of rectangle areas. Riemann's Kludge suffers from exactly the same ill-defined problems as Cauchy's Kludge. The New Calculus definition of integral as a product of two averages is rigorous and easy to understand. Bernhard Riemann was probably a real nice guy, but some of his ideas were just plain wrong. Fix your gaze firmly on theArea and try to imagine what the limit will be, because once you get to "infinity", the sum implodes due to the partition size of zero. The limit is the "number" that describes the area, but if you don't know the limit, then you have to observe visually what happens as you approach infinity. But, you may say, "we can use the fundamental theorem of calculus", to which I respond: "What if the function does not have a closed primitive form or is described by a transcendental series?" In this case, you cannot find the exact limit, and if you are astute, you will concede the idea of summing an infinite number of rectangle areas is clearly flawed.
The flaws of Riemann's misguided thinking are summarized as follows:
1) One is informed that the sum is that "number" (unknown at this time) which is approached as n gets closer to infinity. 2) However, as n gets closer to infinity, the sum is always an approximation. 3) One must at some stage reach a stopping condition.
In order to reach a stopping condition, one must guessthe value of the sum!Can you tell why? Because one must be able to show that the difference between this number guessed (say L) and any sum (say S) close to L can be made as small as one wishes, that is, |S-L| < epsilon. Guess you say?! "Yes", I respond, "and there's the rub. In most cases, you will not be able to do any better than some approximation, because no irrational number has a finite representation of any sort". The only exceptions are well-known incommensurable magnitudes such as pi, e, square root of 2, etc that have a recognizable approximation. If L is not a rational number, then one can't even show that the difference |S-L| can be made as small as one wishes! Because one can't even begin to guess L! Unless the "irrational numbers" are already known (such as concepts defined in other ways, for example pi, e, square root 2, etc which are recognized by abbreviated radix representations), the limit, which supposedly represents the number, is an ethereal concept whose value cannot ever be known, because n never reaches infinity. An academic retort might be that one can assign each of these ethereal limits a unique name. Well, in that case, there are infinitely many such "numbers" that can be placed into a one-to-one correspondence with a unique name?! This is absurd to say the least, and also in complete contradiction to the spirit of the deified mathematician Georg Cantor, whose only claim to fame is a worthless concept that states a set is countable if it can be placed into a one-to-one correspondence with the set of natural numbers, that is, a bi-jection.Of course Cantor was a misguided fool, but that is another issue.
So once again, we have the ill-defined limit concept - similar to a dog chasing its tail. The same rot that we observed used in the definition of the derivative by Cauchy.
The integral is well-defined in the New Calculus - there is no direct computation of infinite sums, use of limits or infinitesimals. In fact, the new calculus standard integral is defined as follows in terms of well-defined averages:
Area = Average length of infinitely many vertical lines x Average length of infinitely many horizontal lines (the interval width)
An astute reader says: But your Average lengths of infinitely many lines require an infinite sum!
My response to the reader: Correct! However, my method does not compute an infinite sum directly. In the case where a primitive function exists, the average can be found because of a telescoping series (used in proof of Mean Value Theorem). In all other cases, it is always an approximation, just like Riemann's faux definition of integral. But unlike Riemann's faux definition, my definition is a well-defined product of two averages, that are understood to be approximations.
Riemann's Kludge
Riemann's Kludge
Created by John Gabriel
Instantaneous Rates: At some time in the twentieth century, an ignorant mathematics professor had a scatter-brained idea about instantaneous rates. I don't know who (Joseph La Grange from the18th Century may have been the first to get the idea) coined this term, but frankly it makes little difference, because it is easy to see that it makes no sense. Ever since the Newton/Berkeley fiasco, academics have been trying to make sense of the derivative. Any mix of the expressions average rate and instantaneous rate has a mind-altering effect on the way modern academics reach non-erudite conclusions. In fact, what they imagine to be an instantaneous rate is in fact a rate corresponding to many average rates over a given interval. Now that the New Calculus is here, and the derivative and integral are well-defined for the first time in history, there is no longer a need for redundant terms such as instantaneous rate. Instantaneous rate must be replaced by the phrase "rate at time t" or "rate at a given point".
Academic: We think about the derivative as a rate of change. Gabriel: The derivative at a given time t is a rate that is expressed as a ratio of differences (each differential is exactly a well-defined difference, not infinitesimal or any other nonsense), so why call it an instantaneous rate? Academic: We call it an instantaneous rate because it represents the ratio of two differentials. Gabriel: The derivative is not generally a rate of any kind unless a time differential is involved. Therefore it makes no sense to call a derivative an instantaneous rate. However, even if time is one of the differentials, the derivative at a given time t is a rate, so why call it an instantaneous rate? What else can it be but the rate at the time t? Academic: Confounded.
The phrase instantaneous rate is not only a redundancy, but it makes no sense even when time differentials are involved. Furthermore, to think of the derivative as the change in one differential with respect to another differential is fallacious. The derivative has nothing to do with change unless time is involved. The derivative is a ratio of finite differences. In fact, the derivatives for a given differentiable function have always existed, so that nothing is changing or has for that matter, ever changed. That the comparison of two differentials is called a rate, does not mean it is related to change. To wit, most derivatives do not contain time differentials. As the most common example, consider the thousands of functions that are everywhere differentiable and contain no time differential. The differentials are finite differences, not changes that take place in given variables. There is a significant semantic difference (excuse the pun) between change and difference.
The New Method: Although I first called my new idea the Secant Theorem (because it has been proved), it is actually also an axiom. One might equally well have called it the Secant-Tangent Axiom. Ask yourself why it is that academics have been so ignorant and incompetent all these centuries.
The idea of limit was born from the inability of mathematicians to realize what I have finally accomplished in the New Calculus - a task not completed by any one else in history.
What better method exists to find the slope of a tangent than by means of a parallel secant? The New Calculus is founded on this idea. This beautiful axiom/theorem makes it possible to develop the entire calculus without the use of limits or other ill-defined concepts such as infinitesimals. If you are astute, then you will notice that the introductory graphic on this page summarizes all calculus. Well, you might have to study a bit...
The next applet demonstrates the secant theorem. Slide the point m or c (by using slider tangentpoint) and observe how the gradient is always correct in the New Calculus. The derivative f'(c)=E(c)+Q(m,n) is also displayed. Check its values manually to satisfy yourself the secant method is true. Remember m must be less than or equal to c and n must be greater than or equal to c. m and n are the distances on either side of point c. Moving point m to the right of c will yield unpredictable results. Note that f'(c)=E(c) where E(c) is an expression for the derivative in terms of c. Q(0,0) is always equal to 0 in the secant method. Hold down CTRL; press and release + while the applet is not in focus in order to view the whole applet; then click on the applet to give it focus.
If at any time, the values of m or n are undefined, this is due to the limitations of the GeoGebra software. Observe that you can find an (m;n) distance pair for any distance m in the interval [c-m;c] regardless of whether c-m>n-c or c-m<n-c. When the software fails to work, you can determine the (m,n) pair from the auxiliary equation, that is, Q(m,n)=0. One of the powerful features of the New Calculus is its potential use in computer graphics. If the creators of Geogebra knew the New Calculus, the following applet would probably not suffer from undefined values.
Secant Theorem
Secant Theorem
For more information on why epsilon-delta theory is flawed, be sure to read: /uploads/5/6/7/4/5674177/limits.pdfThe flawed theory of epsilonics is discussed in more detail further on in this page. Cauchy's Kludge (/uploads/5/6/7/4/5674177/cauchykludge.pdf) demonstrates the errors of Cauchy's definition in about 6 pages. One of many contradictions and errors that arise from the Kludge is a theorem that states polynomial functions are generally everywhere differentiable.One can read about the fallacy regarding the differentiable cubic /uploads/5/6/7/4/5674177/cubicfallacy.pdf at the origin. The New Calculus has a rigorous method for determining inflection points: in the case of a point where a tangent line cannot be constructed, the method in the new calculus shows the existence of an inflection point by proving that an (m,n) pair other than (0,0) does not exist. For certain special functions such as the hyperbola, one uses the (0,0) pair in conjunction with the auxiliary equation in order to find a relationship between m, n and the point at which the tangent line is constructed. The flawed calculus recipe for finding inflection points fails in several cases.
For the single variable New Calculus, a tangent line at any point P is constructed by knowing the relationship between the abscissa of P and the distances m and n from the abscissa to the endpoints of a parallel secant. It is my correct opinion that tangent lines were invented by Ancient Greeks to measure smoothness of curves when continuity is given, that is, a curve is smooth if exactly one tangent line can be constructed at every point except perhaps a point of inflection. If more than one tangent line can be constructed at the same point, then the curve is no longer smooth at that point. Many modern academics do not know what is a tangent line because their definitions are ill-defined. Bad mathematics resulting from ill-defined concepts does not end with the tangent line definition. Consider that until my New Calculus, academics were completely in the dark about the nature of differentials. I now enlighten all those who are interested: /uploads/5/6/7/4/5674177/dydx_compared.pdf
Even though Cauchy's fake calculus generally "works", consider that it has given rise to more false and unsound theory (infinitesimals, limits, real analysis, etc) and obstructed the progress of mathematics in this regard.
Differentiability: The concept of differentiability is poorly defined in standard calculus, with incorrect results at points of inflection. A function is not differentiable at inflection or saddle points. This is caused by a change in concavity. If a tangent line could be constructed at a point of inflection, then there would be no change in concavity. A tangent line (of finite length) by definition cannot cross a function's path. It intersects a path in exactly one point, extends to both sides of the point and crosses it nowhere (*). If a "tangent line" does not extend to both sides of a point, then by this false definition, it is possible to have an infinite number of tangent lines at the same point provided one end of the tangent lies on the path. The file called cubicfallacy.pdf on this page explains much more. Read the following article to see what it means for a function to be differentiable at a given point in single variable calculus (not the same in new multi-variable calculus): /uploads/5/6/7/4/5674177/differentiability.pdf
A serious misconception is that one can use differentiability to draw conclusions about smoothness. Smoothness must have already been established or assumed in order for differentiation to be possible, that is, in order for any function to be differentiable over a given interval, it must first be known to be smooth over the same interval. Stating that a function is smooth over an interval if it is differentiable over the same interval, is like saying: "A function is smooth over an interval if it is smooth over the interval". The standard calculus is replete with many such circular definitions. A good example here is the work of Maurice Frechet, yet another incompetent French mathematician.
A final comment on the subject of differentiability: it seems quite odd that one should even care that a tangent line to a function exists at any given point, unless there are questions about the function behaving strangely at that point (it appears to be discontinuous or not smooth). After all, it's impossible to check every point for differentiability in a given interval using the flawed calculus. However, the New Calculus is designed to check for differentiability over an entire finite interval using only one point in that interval and a parallel secant (provided it exists)!
Confusion about the meaning of tangent line. Modern mathematicians (read as: mostly incompetent) will tell you that tangent lines are defined in terms of derivatives and then add that the motivation of the derivative definition is the tangent line. The derivative definition is based entirely on the fact that the given secant line finite difference approaches the gradient of the tangent line as discussed earlier.
(*) Webster's dictionary defines a tangent line as follows: a :meeting a curve or surface in a single point if a sufficiently small interval is considered First Known Use: 1594
One might try to argue that an endpoint of the sufficiently small interval considered, is the same as the point of tangency. This argument is quickly refuted, that is, in such a scenario, one can construct infinitely many tangents with the same endpoint, but this is obviously misguided.
To think of a function f being differentiable at a point c in standard calculus is in fact incorrect, because in order to be differentiable at c, implies that f is continuous and smooth on the interval or sub-interval containing c, that is, f is differentiable over the interval containing c, not only at c. Therefore, it is ignorant to state that if the Cauchy limit can be found, then a function is differentiable at a given point, to wit, the cubic (x^3) is not differentiable at the origin due to the fact that a finite tangent line with defined gradient cannot be constructed at the origin. In fact, it is just absurd to talk about a function being differentiable at a point. More importantly, a function f is differentiable over an interval, if and only if, one finite tangent line with defined gradient can be constructed at every point in the interval. In other words, the question to ask is: "Is the function differentiable over a given interval?" and not "Is a function differentiable at a given point?" Moreover, one assumes that a function is differentiable on a given interval and then finds the gradient of a tangent line at a point in the interval to confirm this is true or not. By calculating a derivative, one always assumes a function is differentiable.Therefore it is incorrect to think that one proves differentiability, rather one confirms or rejects that a function is differentiable over a given interval.
In the New Calculus, provided a single tangent with a defined gradient can be constructed at any point in a given interval, then it follows the function is differentiable over the finite interval.
Cauchy did not realize that in order to form the difference quotient used in his kludgy limit definition, the function must be continuous and smooth, that is, differentiable over the interval for which the secant line gradients (difference quotients) are formed. It is absurd to think that one can find tangent line gradients to functions that are not differentiable over a given interval! Yet this is exactly the implication of Cauchy's kludgy limit. The unique finite tangent lines with defined gradients are the reason a function is differentiable over a given interval. Ignorant academics did not realize this important fact for centuries until I made them aware. Calculus is about natural averages which are only possible with continuous and smooth functions.
Certain academics will shun the new calculus because they are lazy and do not like to exercise what little brains they have. A typical academic might claim the new calculus is difficult. For one who has learned the wrong methods all one's life and is resistant to change, this is true. However, I have found that students learn the New Calculus far easier than the flawed standard calculus. Their general mathematics ability also improves tenfold because the new calculus requires sound reasoning and thought processes which are rigorous.
Introduction: I decided to call my reformulation of this important branch of mathematics: The New Calculus. In fact, it is not so much a new calculus, as much as it is a sound, well-defined and easy-to-learn calculus, without the use of limit theory or real analysis, of which neither existed in Newton's period. The secant method is at the core of single variable differentiation, just as the tangent disc method is at the core of 3 dimensional differentiation (*) in the new calculus.
Tangent objects exist for higher dimensions even though these can only be visualized up to and including 4 dimensions. Calculus, in one of its aspects, is the subject that describes how to calculate attributes of tangent objects (such as gradient, normal vector, etc), to functions that are both continuous and smooth. In another aspect, it is the branch of mathematics that through the use of natural averages, describes distances, tangent objects, areas, volumes and hyper-volumes in different dimensions. Calculus developed into a very complex subject when real analysis became widely used in education.
(*) A new kind of mathematics vastly different from standard multi-variable calculus but far simpler and easier to learn. Most mathematicians like to work from a set of axioms. For a simple, no nonsense list of well-defined mathematical axioms, the following will be very useful: /uploads/5/6/7/4/5674177/mathematical_axioms.pdf
This web page is about introducing the New Calculus which will change the way mathematicians do business. Although single variable calculus is discussed, there is one example of how it can be extended to multi-variable calculus (/uploads/5/6/7/4/5674177/partial_derivatives.pdf).
The New Calculus will change the way students perceive calculus, which has nothing to do with limits, but due to Cauchy's wrong ideas, is now mistakably associated with limit theory.
If this is your first visit, you will encounter new knowledge. Whether you are a renowned mathematics professor or a high school graduate, prepare to be astounded.
The new calculus is not only the first rigorous formulation of calculus, but also the clearest exposition of calculus ever. One can also say it is the only sound formulation, given Cauchy's limit based calculus is fake.
Newton and Leibniz were both credited for inventing calculus independently of each other. Although both these individuals were fine academics, the truth is neither of them invented calculus. Furthermore, neither individual was able to formulate a sound definition of either the integral or derivative. Moreover, neither academic was able to state the mean value theorem which is the fundamental theorem of calculus. The following document explains: /uploads/5/6/7/4/5674177/mvt-indivisibles.pdf
The Real Beginning: The first three propositions mentioned in the section called Quadrature of the Parabola (The Works of Archimedes), were stated without proof. Archimedes claims these were proved in the Elements of the Conics, presumably a work by Euclid and Aristarchus, that is thought to be lost forever. Calculus began with these propositions. The geometric objects called curves and tangent lines were the source of the modern concepts such as continuity, smoothness and differentiability. These concepts are not well-defined in standard calculus but they are well-defined and easy to understand in the new calculus.
Proposition 1 (Quadrature of the parabola) states that if a straight line from a point V on a chord of a parabola is constructed parallel to the axis (or the axis itself) and meets a tangent line at some point P, then V is the midpoint of the chord. Had Archimedes thought of the tangent line concept in the same incorrect way as modern academics, the first three propositions would never have been thought of or published.
Ideas and concepts exist independently of the human mind. Those who chance to think of them are but flash moments in the history of time.
The Secant Theorem. Given a function f, that has exactly one tangent line (with defined gradient) at each point in an interval [c-m,c+n] containing some point c, the gradient of the tangent line to the function at the point c is given by:
However, we use the following notation once it is understood that c is a point, while m and n are distances on either side of c such that c-m < c < c+n. c_x means the x coordinate of the point c.
m and n are related distances corresponding to the endpoints [c-m,f(c-m)] and [c+m,f(c+m] of a parallel secant to the tangent line at c.
For any function f with a tangent line (at x=c) having gradient k, the ordinate difference of the secant endpoints is always k(m+n)because of the gradient ratio. This implies the ordinate difference is always divisible by (m+n). Provided fis continuous and smooth over any interval (c-m,c+n), there are infinitely many secant ordinate pairs f(c+n) and f(c-m), such that any secant gradient ratio [f(c+n)-f(c-m)]/(m+n) produces k.
To see how the New Calculus works dynamically, click on the slider called dst in the applet that follows. Changing the value of dst(by dragging the black dot on the slider in either direction) repositions the red line which is parallel to the blue tangent line. You will notice that there is always a relationship between m, n and c. The tangent line is drawn at the point (c,0). You can click on the function f(x) in the Objects pane, and change it to see how the secant method works with other functions. If for some reason when you change the function, nothing seems to happen, your best bet is to reload/refresh the page and try again. There are some bugs in the Geogebra applet API. Experiment by changing f(x) and moving the sliders dst and c to see the effect on the secant and tangent line. Please bear in mind that if a discrepancy occurs between the software and the algebra, it could be that the limitations of the graphing software are inadvertently misleading.
Illustration of the Secant Method in the New Calculus:
Illustration of the Secant Method in the New Calculus
Finding the relationship between m and n is not always an easy task and in certain cases, may require some ingenuity. It is always possible to determine a general derivative (explained shortly) given a primitive function that is differentiable.
In order to convince oneself the secant theorem is true, form the quotient described by f ' (c) for a given differentiable function and simplify the result. Next, one must find a relationship between m and n. This can be done by equating the sum of all the terms in m and n [denoted by Q(m,n)] to zero. Choose a suitable m or n and then find the corresponding m or n. Finally, substitute the values of m and n into the finite difference directly to obtain the gradient f '(c). Any (m,n) pair can be substituted into the finite difference quotient except (m,n)=(0,0) because the (0,0) pair does not belong to any secant.
Note that an expression such as [sin(c+n)-sin(c-m)]/(m+n) is not an expression whose terms all contain m and/or n. Thus, equating [sin(c+n)-sin(c-m)]/(m+n) to zero would yield incorrect results. However, (m+n) is a factor of every term found in the ordinate difference formed by f(c+n)-f(c-m). It can be demonstrated that m+n is a factor of every term in the ordinate difference sin(c+n)-sin(c-m). So, what do you think the relationship between m and n might be in the case of the derivative where f(x)=sin(x) ?
All differentiable functions possess the special relationship where the sum of the terms in m and n equals to 0 after cancellation (known in the new calculus as the distance pair (0;0) ) in the difference quotient as will be explained shortly. Once again, it must be remembered that although we can disregard terms in m and n, this does not matter in the case of a straight line where m and n can take any values. A straight line is the only geometric object whose gradient does not depend on secant ordinates because straight lines do not have tangents. In fact, it can be proved that the values of m and n have no effect on the secant line gradients or tangent line gradient.
For a given interval (m;n), all the parallel secants are part of a tangent space for the given (c, f(c)) which is the point of tangency. Each parallel secant has its own unique (m;n) pair. The tangent line owns the (0;0) pair. A special relationship exists between m, n and c for any continuous and smooth function. If only the pair (0;0) is possible, then a point of inflection exists.
Differences between the standard and new calculus: If f(x) = x^2 (x squared), then the general derivative is given by f ' (x) = 2x and a numeric derivative is given by f ' (c) = 2c, where c is the x coordinate of the tangent point.
The standard definition (with limit), that is, [f(x+h)-f(x)] / h as h approaches 0, never represents the tangent line gradient until cancellation has taken place (even in this case it represents only the general derivative, never the numeric derivative), that is, k(h)/(h), where h/h is replaced by 1, h is replaced by 0 in any terms still containing h, and k is the gradient.
Newton's idea of considering a denominator in a finite difference ratio that decreases to an infinitesimal value is incorrect. In fact, it is incorrect to substitute zero for h (old calculus) or m+n (new calculus) in the terms of the finite difference quotient that contain them, until after cancellation, for otherwise the ratio does not represent any meaningful gradient. In the New Calculus, one of the terms which is the gradient, does not contain m or n. The remaining terms always sum to zero. By taking the limit as per standard calculus, one is concluding that the sequence of finite differences:
converges to some real number k, which is the tangent line gradient. Although this is generally true, the sequence depends on the validity of the limit as a well-defined concept, which of course is not the case. Furthermore, k derived in this way, is always a numeric derivative, never a general derivative. In fact the value of k is always approximate, unless k is a rational number.
Bishop Berkeley was correct in being skeptical. The denominator of a finite difference ratio which describes the consequent of a gradient can never be zero. Setting h=0 in the polynomial resulting from the difference quotient (that is, after cancellation) produces correct results, although the method is jury-rigged, because as previously explained, the derivative produced is a general derivative as opposed to a numeric derivative, that is, f ' (x) for some value of x On the other hand, the new calculus definition always represents the tangent line gradient (given appropriate values of m and n for each parallel secant line in the interval considered) and does not rely on the ill-defined notion of limit - this in contrast to the standard definition which erroneously refers to a non-parallel secant line gradient until after the difference ratio is reduced by cancellation. Furthermore, it is entirely correct in the New Calculus to set m=n=0 in the polynomial resulting from the difference quotient after cancellation.
The new calculus does not suffer from absurd results such as 0/0 in the finite difference ratio, the ill-defined concept of infinitesimals or any other confusion that arises naturally from Cauchy's ill-defined derivative. One could say that standard calculus is fake because it is based on definitions that are ill-formed. If you understood the contents of the Cauchy Kludge pdf, then no doubt you will agree that Cauchy's definition is a perfect example of jury-rigging.
Simplifying Explanation of Cauchy's wrong ideas: The main problem with Cauchy's definition is that h is infinitesimal (greater than zero but less than every other magnitude?) before cancellation (*) and zero after cancellation - neither of which are permitted for a numeric derivative. As explained in the pdf called Debunking wrong ideas about the derivative /uploads/5/6/7/4/5674177/debunking_wrong_ideas_about_derivative.pdf, h cannot be infinitesimal before cancellation as it represents the horizontal component for the gradient of a non-parallel secant line and h cannot be zero after cancellation because then the horizontal component of the gradient is zero and thus the finite difference ratio is meaningless. Standard calculus performs bogus arithmetic operations twice:
1. Assumes h is not zero and performs division by h. 2. Assumes h is very small (meaningless nonsense) and discards terms in h effectively treating h as if it were 0.
Moreover, the Cauchy definition relies on the empirical approach of Newton which is an indeterminate process directed at finding a numeric derivative rather than a logical method in finding a general derivative. For example, it does not matter how small the denominator of the ratio is made, because no infinitely small denominator exists that can be the horizontal component of the tangent line gradient. This means that although f '(x) represents the general derivative (after cancellation and h=0), it never represents the derivative when x=x in Newton's empirical approach! In fact, Cauchy's definition is incorrect if it is interpreted as valid for both a numeric and general derivative. Newton was stumped by this, which I think is one of the reasons he refrained from publishing his ideas sooner. The following graphic will help clarify these ideas:
The numeric derivative presumes prior knowledge of limits. The general derivative is only possible if the red secant lines are parallel to the green tangent line. If not, then one cannot set h=0 to find the general derivative because the difference quotient is that for a non-parallel secant. By setting h=0 in Cauchy's definition after the quotient is reduced (diagram on right in previous illustration), one arrives at f '(c), where x<c<c+h, and f ' (c) is not equal to f ' (x). This is discussed in Cauchy's kludge.
In the previous illustration, the diagram on the left shows how h can never be 0. To make sense of the difference quotient, one must use parallel secants (as in the diagram on the right), in which case h can be 0, and subsequently a general derivative can be found. However, the general derivative is not the same as Cauchy's numeric derivative. In my new calculus, one arrives at the same derivative (numeric or general) every time.
Can you see how Cauchy's definition is jury-rigged? You may have to study these facts several times before your understanding becomes clear because Cauchy's error is subtle. The diagram on the right illustrates why the kludge works for general derivatives, that is, f(x+h) and f(x) in the left diagram correspond to f(c+h) and f(x) in the right diagram respectively. These facts do not affect the use of Cauchy's definition, except for pedagogical purposes and numerous incorrect theorems. The last couple of centuries have shown that students and mathematics professors never acquire a clear understanding of the derivative (where limits are not required at all).
These inconsistencies are removed in my New Calculus. As a result, differentials are well-defined in the New Calculus. dy/dx always means the same thing in the New Calculus, whereas it has a serious identity crisis in the standard calculus, that is, it can be interpreted as a limit or as a rational expression depending on context. Cauchy would have been correct if he defined the derivative as follows from the diagram on the right:
f ' (c) = [ f(c+h) - f(c-x) ] : (c+h-x)
Cauchy's first error was in his conception of the limit. His second error was using the limit in an attempt to define the derivative, where it results in his jury-rigged definition. If there were any justification for using limits in calculus, it would probably be in respect to integration, however, even in this regard, limits are not required. Using my systematic approach in finding ante-derivatives (/uploads/5/6/7/4/5674177/indefinite_integral_-_systematic_method.pdf) it may now be possible to find any ante-derivative, although not a trivial process. Leibniz's definition is really not much better than Newton's, even though it appears to be geometric and somewhat tidier. Leibniz would have defined the derivative as follows had he understood calculus as well as I :
df:dx = [ f(x+n) - f(x-m) ] : (m+n) [LD]
The following links explain more:
The differentials df and dx are exactly equal symbolically (or proportional if numeric) to [ f(x+n) - f(x-m) ] and (m+n) respectively where [LD] represents the gradient of a secant line parallel to the tangent line whose gradient is being determined. In the previous form [LD], df/dx is a symbolic fraction. df/dx becomes an exact fraction when the symbols (function placeholders and variables) are replaced with numbers. The difference between a symbolic and exact fraction is that a symbolic fraction's value is determined according to a given difference ratio whereas an exact fraction has known values. However, this difference is superficial as the fractions are used exactly the same way in algebra.
(*) Cancellation is the process of forming the separate individual quotients by considering the quotient of each term in the numerator with the denominator. The astute reader will notice that this process is assumed to be complete (according to Cauchy), before the finite difference is reduced later (through cancellation) in order to find the general derivative. Therefore, Cauchy's definition is not only fake, but is also fraudulent in terms of the simple properties of arithmetic.
The following file called newcalculus.pdf contains a few general examples. Divisibility_identities.pdf contains a proof that supports the ideas in newcalculus.pdf.
Back to the Beginning. If you read and understood the previous files containing some general and actual examples, then I have no doubt you will want to know more. What happened between the time of Archimedes and Newton? It is a well-known fact that the foundations of integral calculus were laid by Archimedes. He used the method of exhaustion to approximate incommensurable magnitudes such as Pi and to calculate irregular areas. The mathematical objects that Archimedes used were the rational numbers.
Over 2,000 years later, these are still the same objects we use today. There is some new terminology (real number) and also some ill-defined concepts (limit, complex number). 'Real number' is terminology used to describe any magnitude that is incommensurable in addition to well-defined magnitudes that are known as rational numbers. Contrary to the opinion of most academics, a real number has yet to be well-defined. In fact, most academics don't know the difference between number and magnitude. Following article explains: /uploads/5/6/7/4/5674177/magnitude_and_number.pdf
A mathematician is like an artist: the objects arising from concepts in a mathematician's mind, are only as appealing as they are well-defined. Zero enters the number club. Many centuries passed after rational numbers had been introduced. Eastern mathematicians were researching the concept of zero magnitude that was initially rejected by the Greeks. At first, zero failed almost every requirement needed to qualify as a number. The main requirements were: (i) the mathematical object must be a magnitude (ii) the object must be measurable by the magnitude of an object different to itself, presumably the larger measured by the smaller as Euclid states in his masterpiece, The Elements.
In order for an object to qualify as a magnitude, it was considered imperative that such an object could be instantiated, if not physically, then at the very least it must be well-defined in one's imagination, which implies its physical instantiation is irrelevant. If what one imagines is well-defined, then that which is being imagined is as good as real!
Zero was not immediately accepted, for it could not even satisfy the first requirement, that is, it is not a magnitude. Although the first requirement was eventually waived, zero still failed to satisfy the second requirement, for no other object or magnitude can measure zero, except zero itself.
Zero manifested great potential in serving as a placeholder in radix systems and also in denoting equality between arithmetical operations, as opposed to the unit which denotes equality between numbers. For example, comparing p with p, that is, taking the difference (p - p), results in equality where p is some number, while a - b = c/d results in equality between the expressions a - b and c/d, that is, (a - b) - c/d = 0. It was eventually realized that granting zero membership had certain potential benefits. Provided there was a way to resolve this conflict, zero could be quite useful as a number.
What are the irrational numbers? Given any integers c and d and some real number k, one can form averages k/c and 1/d. If no c and d exist such that k/c=1/d, then k is said to be irrational. In other words, no proper fraction p/q exists (where p and q are integers), such that p/q equals the fractional part of k, given that k is irrational. This definition states that k is irrational if it cannot be written as c/d. The definition states what is not a rational number, but says nothing about what constitutes an irrational number. Consider the following analogy:
Humans are not human because they don't have tails. They are human because they think and reason. Chimpanzees do not have tails, yet they are not human. A quality or trait or characteristic, must indicate a feature of the object or idea being defined.
Would it have made any sense to say, Primates without tails are human ? Obviously not. The analogy here is that primates are magnitudes where humans are rational and chimpanzees are irrational.
A property of being well-defined is evident by defining a concept through an attribute it possesses, not by an attribute it lacks. This property is one of the core characteristics of an object that is well-defined. To state that irrational numbers are numbers that are not rational, does not say much, except that such magnitudes cannot be described by any known rational number. So, exactly what is an irrational number?
Unlike a rational number, an irrational number is an incommensurable magnitude that can neither be defined, nor represented in terms of comparison. It is possible only to represent incommensurable magnitudes through averages that are mere approximations. However, these same approximations never define incommensurable magnitudes which came to be known over time as irrational numbers.
In fact, irrational 'numbers' are not numbers; rather these are incommensurable magnitudes, which are now represented and used as approximations through sums of averages.
What are real numbers? Real numbers came into existence as a result of irrational "numbers". Before incommensurable magnitudes (aka irrational numbers) were discovered by the ancient Greeks, all magnitudes were considered to be rational numbers, that is, all magnitudes were thought to be measurable. Therefore, incommensurable magnitudes, were simply renamed irrational numbers and lumped together with the rational numbers to form the new real numbers.
Given these facts, it is immediately evident that an irrational number is ill-defined. Now since real numbers are defined to be the union of a set which includes the set of rational numbers and the set of irrational numbers, it follows the real numbers also, are not well-defined.
It would thus be in order to say that when a real number is not rational, the same number represents some incommensurable magnitude, for example Pi, the square root of two or any other incommensurable magnitude.
Academics after the ancient Greeks (with the exception of John Gabriel) have failed to realize that a magnitude becomes a number once it is completely measurable.
The Limit. It did not take long for mathematicians to become dissatisfied with their inability to well-define incommensurable magnitudes. Rather than concede these incommensurable magnitudes cannot be represented by numbers, those misguided 'mathematicians' devised a new concept called the limit which they would use to redefine, not only all real numbers, but also attempt to provide rigour to the many problematic definitions in calculus. They failed, and what is taught in today's lecture rooms is the nonsense created by them. Limits are not required in calculus or any branch of mathematics. If set theory (not all set theory is unsound) can be called mathematics, perhaps this is the only place where limits might be studied. Indeed, the arrival of the theory of limits is a major setback in the progress of mathematics. The theory is flawed for many reasons, but the most important reason is that limit theory was developed from the theory of sets, which is fundamentally in error.
The popular notion regarding the concept of limit being at the core of calculus is entirely false. In fact, as stated earlier, the inventors of calculus knew nothing about limits. The limit itself is poorly defined and takes on several different meanings, depending on context. For example, a limit can denote some value of an expression through cancellation of certain factors in the expression (*). A limit can also denote the upper bound of an infinite sum as in the case of a convergent sequence (also known as Cauchy sequence).
The most popular use of limits is for determination of gradients where they are not required at all. A gradient of a curve at a point is the gradient of the tangent line at that point, provided the tangent line exists and has a defined slope. The only geometric object that has a gradient or slope is a straight line. By gradient of a curve at a point, the implied meaning is the gradient of the tangent line to a curve at that point.
(*) This notion is extended to expressions where very small magnitudes (ill-defined nonsense) are discarded and the result is called a limit. How did we get the idea of limit? The great mathematician Archimedes was the first to formalize the ideas that eventually led to the unfortunate adoption of the limit by modern academics, and consequently also the theory that arose from this idea. The Archimedean Property states:
If x is any magnitude, then there exists a well defined natural number N such that x < N.
Note that I use the word magnitude - because a real number is not well-defined. In fact, Real Analysis, which was created from the nonsense of Georg Cantor's misguided ideas, defines real numbers using the limit concept (Dedekind cuts or Cauchy sequences). Arbitrarily small distances between rational numbers are used to demonstrate convergence which leads to the notion of a limit. That which is being defined is used in its own definition! Refer to my article on magnitude and number (section called Debunking Real Analysis Myths).
The Archimedean property is misunderstood by most professors of mathematics. The property was not intended to be used as a prelude for the concept of a least upper bound of a set which is bounded above. Its purpose was to establish a method for measuring incommensurable magnitudes by means of approximation. For example, Archimedes knew that his method of exhaustion would never result in a value greater than some rational number that is larger than Pi. In fact, pi is the reason Archimedes arrived at the Archimedean property. Archimedes knew after he discovered the property, that the perimeter of a polygon inscribed in a circle, would always be some rational number less than pi and that the perimeter of a polygon circumscribing the same circle would always be some rational number greater than pi. This is the sole reason for the discovery of the Archimedean property revealed for the first time in history on this page.
The property is stated more accurately as follows:
If x is any magnitude, then there exists a commensurable magnitude N such that x<N.
What every academic (except John Gabriel) failed to understand is that the following is also true:
If x is any magnitude, then there exists a commensurable magnitude N such that x>N.
These last definitions use commensurable magnitude because natural numbers are an abstraction derived from ratios of commensurable magnitudes - a fact that every professor of mathematics does not know, unless such a professor learned it from me. What does the Archimedean property mean? The property simply states that given any commensurable or incommensurable magnitude, a commensurable magnitude exists which is greater or smaller. I suppose one can write the revised Archimedean property as follows:
If x is any magnitude, then there exist commensurable magnitudes M and N such that M > x > N. The Archimedean property is best illustrated geometrically through the triangle inequality which states that the sum of any two sides of a triangle are greater than the remaining side. One can therefore let the magnitude be a side of the triangle and construct the other two sides such that they are commensurable, and thus the property is proved.
Note that Archimedes knew nothing about infinitesimal/s. In fact, Archimedes rejected the idea of real number as "understood" by ignorant modern academics. He knew that irrational numbers do not exist and hence neither do real numbers exist. Archimedes understood that pi is an incommensurable magnitude. It could not be a number because it is impossible to completely measure the magnitude known as pi.
Anal retentive academics were not satisfied with Archimedes' ideas. They preferred to obfuscate matters by introducing non-issues that are unrelated to Archimedes' original ideas. What started out as simple and elegant ideas morphed into:
(i) a least upper bound concept for a given set called a limit. (ii) the name given to a result of simplifying a given expression through cancellation of its factors, is called a limit. (iii) zero which assumes the role of a limit when the denominator of a given fraction is assumed to increase or decrease without bound. (iv) the name given to a result of simplifying a given expression by disregarding terms whose values are decided by an indeterminately large denominator. This is an extension of (iii). The categorization of these various aspects based on this ill-defined concept of limit evolved into the ubiquitous calculus limit as it is known today.
Epsilonics (Epsilon-Delta) proofs. Unfortunately, an entire semester is usually wasted learning epsilonic proofs. The fact that many students drop out of math courses because they think they do not have what it takes, is good reason to be alarmed. To those students who are contemplating dropping out, my advice to you is to simply tough it out. Your difficulty in understanding is mainly due to the fact this theory is ill-defined. And then you have lecturers who are absolutely incompetent, which compounds the problem. If only they understood some mathematics... To help you understand what the non-noteworthy fuss is about, I have compiled a document called limits.pdf (link follows). Use this document in conjunction with the GeoGebra applet to understand the meaning. Whatever you do, don't quit your math studies because of this unimportant topic in Real Analysis. I expect that when my New Calculus eventually replaces standard calculus, this nonsense will be a thing of the past.
Stating a theory using numbers and symbols does not make it more rigorous. In most cases, unless the theory can be properly worded and more importantly well-defined, it usually ends up containing many flaws. Prior to learning standard calculus, students are required to spend much fruitless time trying to prove limits exist, when all they are doing is stating a limit exists using epsilonics. Finding a delta and epsilon, is analogous to saying that a triangle is a triangle because it has three angles. In fact it is insufficient to find a delta and epsilon - one has to find a relation between the two. Again, it is insufficient because of the reasons explained in limits.pdf. Whichever way one chooses to look at this epsilon-delta theory, it is ill-defined as the following article explains: /uploads/5/6/7/4/5674177/limits.pdf
The following applet calculates the exact epsilon-delta region for a given planar function f(x), which you can change by clicking on f(x) in the Free Objects region and entering your own function. The anchor button (rightmost button on toolbar) allows you to move the graph (move graphics view mode), to re-scale the axes and to zoom in and out. To re-scale an axis, click on any part of it whilst in move graphics view mode and whilst holding your mouse button down, drag in either direction (north-south or east-west) until you are satisfied and then release the mouse button. You may have to move the graph back into view (using anchor button) after re-scaling the axes. To move the sliders you must be in pointer mode (leftmost button on toolbar). Move the sliders called delta and c to observe the effect on the limiting region.
In some cases it might appear as if there is no graph, but what this means is that you have to re-size and re-centre the graph. Re-scaling the axes helps to see more granular details when the limiting region is very small.
Purpose of this applet: To demonstrate the uselessness of epsilonics in limit theory.
Calculus without the use of limits. Calculus is possible because of the properties of natural averages. Calculus is about natural averages, not limits. The New Calculus does not use limits. The following pdf provides a proof of the average value theorem and the fundamental theorem of calculus without the use of limits.
Newton and Leibniz would have loved to know this information. Both these academics were challenged to provide rigorous definitions of their calculus. Both failed.
Newton's greatest mathematical accomplishment and supreme work was his discovery of the finite difference interpolation polynomial. In terms of mathematics, all of Newton's remaining works are in my opinion, insignificant. Leibniz on the other hand, almost succeeded in redefining his definite integral correctly. He was on the right track by researching moments, but was unable to complete his task. Leibniz may not have been too surprised to learn from my redefinition of the integral, that all integrals are path or line integrals.
Most modern academics fail to understand that calculus only applies where natural averages are present, that is, continuous and smooth functions. For example, they have been known to apply calculus to conditional functions (aka piece-wise functions) where they remove discontinuities inadvertently or purposely, and then try to deduce conclusions about continuity and differentiability. Newton and Leibniz knew nothing about conditional functions.
To help you learn all about Newton's interpolation polynomial (and much more...), I have compiled the following publications:
How we got calculus is a tour de force of Newton's most important work which I think is his interpolation polynomial. Only standard calculus is used in How we got calculus. Taylor's theorem can be derived from Newton's interpolation polynomial by using only the new calculus. Start with formula [T1] on page 15 of How we got calculus. Observe that my new calculus version of the Taylor polynomial is an equality (Gabriel polynomial), unlike Taylor's polynomial which is always an approximation. The main idea behind Taylor's theorem is ease of calculation even though convergence is a disadvantage in terms of acquiring sufficient accuracy.
Newton's methods were based largely on approximation because he lacked certain knowledge, that is, he probably did not know of the mean value theorem and certainly did not know of my New Calculus. The mean value theorem is hardly understood by most mathematics professors. There are many GeoGebra applets supposedly demonstrating the workings of the mean value theorem, but all of them are superficial and miss the most important aspect which is captured in the following GeoGebra applet: /uploads/5/6/7/4/5674177/mvtvisual.ggb
Roger Cotes is a mathematician who is not well-known. Yet the impact of Cotes' work has been significant. Without it there would have been very little progress in numeric integration and therefore the solving of differential equations. The following document shows how Cotes arrived at the general formula: /uploads/5/6/7/4/5674177/cotesintegration.pdf
A proof of L'Hopital's Rule using the New Calculus. I am often asked how I would explain L'Hopital's rule without the use of limits. The following file contains an elegant proof that is less than one page. /uploads/5/6/7/4/5674177/lhopital.pdf
Finally, a rigorous calculus without the use of limits. Almost 330 years later, I have presented a rigorous calculus and redefined not only the derivative, but also the definite integral without the use of limits. Newton, Leibniz and Cauchy were mistaken, but their wrong ideas have been corrected in the new calculus. Even differentials are now well-defined in the new calculus. PDE experts normally spend their entire lives mastering DEs. The New Calculus has already changed this (personal research) and when it is adopted, mathematicians will no longer require a lifetime to become an expert in a topic as complex as PDEs (partial differential equations).
Conclusion: I trust your visit to my site has been informative and also entertaining. These few ideas and secrets I have shared with you, are only a fraction of what is contained in my new calculus. One of the secrets I recently shared is the existence of a cumulative probability function for the normal density function. It might have been impossible or very difficult without the methods of the New Calculus.
Future of the new calculus: 1. The primary purpose of the new calculus is for academics to adopt a sound and rigorous branch of mathematics that one can learn and master in a short time. Standard calculus has proved to be difficult and hard to master in any reasonable period of time, one of the key problems being its use of the limit concept and real analysis.
2. Real calculus reform: there are no ambiguities, paradoxes or contradictions. Concepts are well-defined unlike the standard calculus which is fake.
3. The following key fields have been identified as important research areas in the new calculus:
Solution of differential and partial differential equations, including the use of numeric integration techniques. Solution of area, volume and hyper-volume problems using well-defined concepts such as natural averages. Spatial representation of curves in n-dimensions (useful in computer graphics). Accurate curve fitting. Systematic indefinite integration and differentiation. New regression analysis (statistics). Application to abstract algebra and discrete mathematics. And much more...
4. Due to decreased complexity in the new calculus, research and understanding is facilitated and thereby enhanced.
I have done substantial research in these aforementioned areas. With regards to numeric differentiation and integration the New Calculus has significant advantages over the n-Point central difference formulas and well-known quadrature/cubature algorithms. This information is currently withheld pending the publication of What you had to know... I am certain there are many more benefits to be realized by adopting the new calculus which the future is certain to reveal.
(C) John Gabriel, The New Calculus, 2010 All Rights Reserved
Fools in academia:
These foolish academics need to be exposed, not because they are jealous and arrogant (they generally are), but rather because they are ignorant beyond belief. It is a tragedy that today's most influential academics censor and denigrate those, whose ideas they neither understand nor like. While ignoramuses are abundant throughout the earth, it is a crying shame that modern academia is the new Catholic church of knowledge. As the Catholics suppressed those who in the middle ages differed from their ideas, modern academia has established a biased knowledge repository in the form of academic journals. Modern academia have through the medium of journals formed an elite clique which one might call the academic bourgeoisie. If new knowledge is not printed in one of the recognized journals, then sheepish learners are trained to shun it, just as Catholics of the middle ages were warned of committing blasphemy. In the end, everyone is a loser.
Georg Cantor: Believed he was onto something with regards to countability of sets, in particular infinite sets. Cantor foolishly assumed that the set of real numbers is uncountable for all the wrong reasons, when in fact the set is ill-defined for two reasons: it is infinite and its members are not well-defined. If the real numbers can be represented in base 10 (as Cantor assumed), then the set of real numbers is indeed countable because any radix system only represents rational numbers. In fact, the set of real "numbers" is uncountable because real numbers are not well-defined. Indeed, how can one count anything if one does not know what it is? Preposterous, of course! See article on magnitude and number at the top of this page.
Not too long ago, I had some correspondence with a theoretical physicist (who by his own admission stated he is not a good mathematician) who claimed that it is not possible to formulate (construct) the numbers in only a few pages. Without even reading my formulation (see magnitude and number article and also power-point document called construction of numbers), he dismissed it as spurious. To show the world how ignorant this academic is, I have demonstrated (further down) on this web page the development of the number concept up to rational numbers in just a few paragraphs.
Guiseppe Peano/Abraham Fraenkel/Ernst Zermelo: These "mathematicians" formulated the ZF axioms which are nothing more than a juvenile attempt to form what they perceived as the foundations of arithmetic. Built on Cantor's dumb ideas, ZF is an attempt to define numbers without an understanding of ratio or measure.
David Hilbert: An ardent follower of Cantor whose efforts placed Cantorian ideas at the forefront of mathematics - a devastating action that set mathematics off course the last 120 years. Set theory is a failed attempt at understanding numbers in terms of containment rather than measure. Containment disregards the aspect of comparison, whereas measure is defined by it. The New Mathematics (of set theory) espoused these wrong ideas and till this day mathematicians never fully grasp the concept of ratio. In fact, most mathematics professors do not know the difference between a magnitude and number. I had a foolish professor write and tell me that the number of elements in each set can be compared and thus there is measure in set theory! What the nitwit failed to realize, is that to find the number of elements, one must assume prior knowledge of measure, that is, "number".
Kurt Godel: The father of the completeness/incompleteness theorems in logic. A little known fact is that Godel's own theorems disprove themselves. Only academic ignoramuses can be trusted to miss such a simple fact as this.
Bertrand Russell: Discovered paradoxes in set theory (unsurprising because set theory is ill-defined) and prepared the stage for flawed modern logic theory. Russell was an overrated logician whose debating skills earned him a place in the history of mathematics. He was a notorious smoker who never failed to appear in any photograph without a tobacco pipe in mouth or hand. Russell's Principia Mathematica is less worth than the paper used to print it.
Abraham Robinson: A twentieth century American Jewish mathematician,who designed non-standard analysis on Cauchy's kludgy idea from classic analysis, that is, the ill-defined concept of infinitesimal. The infinitesimals according to Robinson are a subset of the interval (0,1) with no least upper bound. Ignorant academics claim that an infinitesimal is greater than 0 but less than every positive number. How they arrive at the plural is absurd, that is, there is no way of comparing infinitesimals with each other.
It is impossible to tell where infinitesimals end and the "real" numbers begin since the infinitesimal set has no least upper bound. Not one instance of an infinitesimal number can be demonstrated either in theory or practice. Rather an attempt is made to draw conclusions about more theorems using the same ill-defined concepts, unsurprisingly often with incorrect results. It is impossible to compare infinitesimals (measure them). Ironically, Robinson based his theory on the assumption that the real numbers are well-defined (and used the transfer principle to validate his non-standard theory) - I have proved this to be false, that is, the real numbers are not well-defined. In fact, real numbers do not exist. See my article on magnitude and number.
Robinson's useless theory survives because of the Jewish influence in academia. Wikipedia's Jewish sysops and admins tried to give this unsound theory of Robinson's more credibility by claiming Archimedes used infinitesimals, until they finally understood that Archimedes had no idea about any such nonsense, nor is his method of exhaustion in any way related to Robinson's absurd ideas.
Stephen Hawking: The obscurities of his theories are tantalizing to the community because they sound exotic and alluring. It is my opinion that most theoretical physicists are duller than dish water. In 2012, Stephen Hawking was awarded $3 million dollars for his useless theory on black holes. What Hawking knows about calculus is quite precarious. When Hawking creates theory using a calculus that's questionable and non-established "facts" about the universe, what you get is sheer speculation. Just one more example of ignorant academics awarding each other accolades and prizes for theory that's not worth the ink used to publish it. If future historians are honest (somehow doubtful), Hawking will be remembered as a speculating ignoramus. This type of news almost makes me want to take down this site and all others in which I share my knowledge of the New Calculus. While Hawking has been awarded millions, cowardly academics who have acknowledged my work privately, remain silent in public. My horoscope has more chances of being true than a theoretical physicist's theories ever being proved sound - at any rate, such theories will not be confirmed or discarded for centuries. Perhaps I have been too harsh in my criticism of Hawking, but frankly I think he knows his theories are bogus. He does not think it is his problem that so many ignorant academics and people consider his theory to be doctrine. After all, it puts more money in his pocket and earns him many awards.
Thanks to Einstein's nonsensical theories, theoretical physics is now a lucrative occupation, especially if you are an academic suffering from some kind of handicap. For those of you that hate mathematics, your next best bet might be a career in theoretical physics. If "physicists" can tell so much from ancient light sources and a flawed calculus, surely there must be some truth in Astrology also. AMS (American Mathematical Society) and MAA (Mathematical Association of America): Both societies will gladly publish anti-mathematical nonsense on ill-defined calculus limits and non-existent infinitesimals. Unfortunately most sheepish journals follow their lead. That these societies are run by misguided and short-sighted academics, is beyond question.
ACTA (The Royal Swedish Academy of Sciences): A journal which does not bother refereeing papers. The editors glance at new submissions while drinking tea and base their "expert" opinions on what "appears or seems to be".
I could spend my entire life refuting all them or continue working on my new ideas in mathematics. I choose to sound the warning bells and let those who ignore pay the price.
To contact me: john underscore gabriel at Ya hooooo dot com
Laughs and Criticism: The following section contains criticism of articles I sometimes read when I am bored. There is so much nonsense constantly being published by mathematics professors and other academics. The same nonsense is read and digested by naive students worldwide. The results are undesirable.
The Cantor Debate with Crank Chu Carroll. A few years ago I wrote a Knol disproving Cantor's diagonal argument. Firstly, let me say that the real numbers are uncountable because they don't exist, that is, they are not well-defined. Secondly, I wanted to prove that if the real numbers are represented in decimal, I could list all of them in a tree using a sound algorithm and locate the start node of any real number. In fact, this is what Cantor was trying to do. Cantor knew nothing about a mapping or inverse pairing function - these came much later and were named after him. It is possible to find a mapping and an inverse pairing function for the rational numbers because they are well-defined. My primary goal was to disprove the false Diagonal Argument that it was not possible to list all the real numbers, if they can be represented as decimals. Also note that when Cantor came up with his idea, a bijection was not known. The first known use of bijection was in 1963 (according to Webster), not too long after WWII (and the Holocaust). Of course this does not mean it was unknown under another name or the idea was unknown, but it is highly unlikely Cantor knew anything about bijections. The bijection came much later, as did the pairing and inverse functions for rational numbers. Were Jewish mathematicians scrambling to defend Cantor's wrong ideas?
Crank Chu Carroll was immediately hung up on many things at the start of the debate. He began by trying to differentiate between representation and enumeration. I never got round to explaining my enumeration (part of the mapping algorithm) because Chu Carroll knew he lost the argument and continued to insult me. Aside from this objection, Carroll would just not accept that 1/3 (or 0.333....) was present in my tree. He was comfortable that 1/3 = 0.333... (with endless 3s following) but would not accept that I could do the same thing in my decimal tree traversal. He became obsessed and infuriated that I was able to do this in my tree traversals. For someone who has a PhD in computer science, Carroll knows very little about traversals. Any astute mathematician will see that this is a contradiction, for if one cannot represent 0.333... in my tree, then 0.333... cannot represent 1/3 in decimal. Last I looked at the above link, it has been heavily edited: the comments where Chu Carroll raised the subject of his Jewish ethnicity are completely removed. The comments where I corrected his wrong ideas regarding tree traversals are also removed. Many of the comments are either modified or deleted to suit Carroll's agenda.
One of the funny comments Carroll made was "No, you can't take 0.333... to infinity. You just can't!". And in the same breath, all the idiots who were agreeing with him in the forum, would say that 0.333... is equal to 1/3, which is only possible if the limit is considered. Well, if ignorant mathematicians can use the ill-defined limit in their mathematics, why can't I also use it in my mapping algorithm? Answer: It refutes the dumb ideas of Cantor regarding countable infinite sets.
By all means, read what others say about me, but remember to form your own opinion after you read what I have to say about myself. I have found that one can have all the knowledge in the universe but is unable to form an opinion. Being able to remember many facts is not a sign of intelligence but that is a different topic. Some of the false criticism leveled against me by the "erudite" mathematics community: - lunatic - psychotic - crank - kook - narcissist - sufferer of BPD (border-line personality disorder) - schizophrenic - nutcase - exhibitor of the Dunning Krueger effect - Your choice of mental disorder/syndrome/dysfunctional effect here...
I have to say that inside my mind there is an immeasurable hatred for these scoundrels who have leveled baseless accusations against me and in the process destroyed or stymied my chances of success. I hate them with a perfect hatred and sincerely hope that every worthless dog who has leveled these accusations against me, suffers a terrible life and an even worse death. They have no excuse and ignorance is not an excuse. In fact, words cannot even begin to describe the loathing I feel for these academics - many of whom are pedophiles and perverts. Perhaps I am not such a nice person, because I cannot find it in me to forgive and forget. Why should I... Hatred is not illogical. It is every bit as logical as love or kindness or gentleness or any other human emotion. Anyone who claims not to hate is a liar. Many of my accusers overlook the fact that I am not good at expressing myself in writing. Not everyone is a natural born writer or speaker. I find it extremely difficult to write especially seeing that I loathe writing (*). There are many other factors that influence my writing, including foreign language and culture differences. I speak/read/write several languages and have lived in many parts of the world. I often think about how I will be understood or perceived. Many times I am horrified at what I once wrote. I can't change it soon enough, but by the time the realization has sunk in, it may already have been read and the damage done. Unlike my accusers, I am not afraid to admit I might have been wrong or to apologize. In my opinion, this makes me a better human than they all. (*) If you have visited this page, you will know that it has changed many times. Even as I write this, I am not fooled about not having to change the page contents at some future time. I am especially amused when academic ignoramuses attribute the title crank to me. If I am a crank, what are they? Anyone who disagrees with what they think are "well-established" views, is by their definition, a crank. In the United States, it is permissible (legal) to call someone a crank. For example, take a look at Mark Chu Carroll: an imbecile, who knows nothing about anything, but can mouth off his stupidity on his blogs without any fear of being sued. This vile reptile (Chu Carroll) is protected by a ruling passed by Chief Judge Posner, in the following court case:
One might say, "Why don't you wait a while before you publish?". My response is that I will probably end up waiting indefinitely because I always find something I don't like about my writing. My confidence is often interpreted as arrogance. I understand mathematics better than anyone I have ever known or any of the mathematical authors I have studied (including Newton, Leibniz and Cauchy).
On occasion I have shared some personal information. Probably not a wise thing to do. The reason is that I want my readers to know a little more of me, aside from the mathematics. Sharing information was also an attempt to be more approachable, but evidently not too successful. Oh well, at least I tried, which is more than what many others might do. Often I think that these last few paragraphs (under Laughs and Criticism) do not belong on this web page. I ask myself many times what to include and omit. It would be nice if there were a science language which lacked inference, so that when communicating in this language, one would not have to worry about hurting or offending others, even though the fault usually lies with those who take offence. Unfortunately, there is no such language. If it were possible to describe everything in terms of mathematical symbols, then mathematics would be a language in its own right. However, it is common knowledge that mathematics is not a language.
One of the reasons I have created this page is for the sake of young learners. I hope to make it easier for them to learn calculus. I also encourage students to study the old flawed methods of calculus. Graduates can also learn much from studying the New Calculus - it can help to consolidate and dispel some of the wrong concepts they are taught at university.
So, try to focus on the facts rather than the inferences, which might pop out at you while reading this web page. If you try to psychoanalyze me from my writing, you will be completely misled.
Educating the uneducated: The following exercises are appropriate for professors of mathematics and mathematics teachers because most of them do not know what is a number.
Study the following publications: /uploads/5/6/7/4/5674177/magnitude_and_number.pdf 1. Explain why a real number is ill-defined in standard calculus by considering that the decimal representation (or any radix representation in base 10) of an "irrational" number is always rational. Observe that a limit may itself be a number or not, then argue that since "real" numbers that include "irrational" numbers are defined in terms of limits (equivalent Cauchy sequences), it follows that "real" numbers also are not well-defined. Submit solution in not more than 3 pages. The convergent sequence [pi] = (3; 3.1; 3.14; .... ) is bounded above (*) by the magnitude known as pi. Yet pi is NOT a number (pi is a ratio of two magnitudes). Therefore it is incorrect to state that the sequence [pi] has a limit pi, because the ill-defined concept of "limit" has a dual nature: sometimes it is a number and sometimes an incommensurable magnitude. In any event, neither pi nor any other incommensurable number (known as irrational numbers to modern academics) can be well-defined using Cauchy sequences.
(*) While the concept of being bounded above by a rational number makes perfect sense, it is ridiculous to think of an incommensurable magnitude being an upper bound because its complete dimensions are unknown, that is, it is an ill-defined concept.
2. Prove that radix systems can only be used to represent rational numbers.
However, qualitative measurement can be inaccurate. For example, if a mathematician were able to discern that two magnitudes are not equal, it would still be impossible to determine how much larger or smaller the one is from the other, that is, a difference is not well-defined.
To define difference, the brilliant Ancient Greeks began by formalizing the process of comparison, that is, they formed ratios of magnitudes. For example: magnitude (object A) : magnitude (object B) which means literally: The magnitude of A compared with the magnitude of B qualitatively, that is, one of two outcomes only: magnitude(A) = magnitude(B) or magnitude(A) not equal to magnitude(B). Note that the magnitudes are unknown till this stage. They are only being compared qualitatively (by visual observation).
The incredible breakthrough occurred when they discovered the abstraction of a unit, from which the natural numbers were born. A unit is defined very simply as the comparison of any magnitude with itself or another equal magnitude, that is,
magnitude (X) : magnitude (X) = UNIT (**)
(**) This great accomplishment led to quantitative measurement. Now, it was possible to tell the difference between two magnitudes being compared (provided both are measurable in whole units), and a known symbol could be associated (having a known magnitude value) with both magnitude(A) and magnitude(B), that is, both magnitudes are now comparable quantitatively as natural numbers. Note that the magnitudes are known as natural numbers at this stage. They are being compared quantitatively (by finding the difference in terms of a natural number). Recap: We started with an unknown magnitude (X) and arrived at a quantitative measurement of X in terms of natural numbers. A natural number is a ratio X : N where X is a ratio of measurable magnitudes and N is a unit. Now, this great knowledge was insufficient where incommensurable magnitudes (pi, sqrt(2), e, etc) are concerned. That is, incommensurable magnitudes can be represented only by approximations through use of fractions (or radix systems).
I am not unaware of how dull mathematics professors and teachers can be, so I suggest you find a quiet room and study these things long and diligently.
These few paragraphs are meant only to whet your appetite. For the full story of the intricate thought processes of the Ancient Greeks and much more on the development and history of numbers, calculus and mathematics in general, you will have to wait for the publication of the greatest unpublished work in mathematics: What you had to know in mathematics, but your educators could not tell you.
This great work is the correction and completion of Euclid's Elements, The Works of Archimedes and Apollonius' Conics. It incorporates the New Calculus. Contained therein is new knowledge of averages, natural averages, area, volume and hyper volume, which the Ancient Greeks understood but were unable to formalize.
Reader FAQs:
May I write articles about the New Calculus? The short answer to this question is: NO. The current circumstances are such that the academic community as a whole, needs to publicly acknowledge the New Calculus is the first rigorous formulation. Until this happens, I cannot grant permission to anyone wishing to write about the New Calculus.
May I teach the New Calculus in class? Yes, on one condition: You must use the correct attribution: "The New Calculus was discovered by John Gabriel in 2002 and published online in 2010." It would also not hurt to refer your students to my websites.
Can I write a review? Unfortunately, I can't stop you from writing a review. However, be warned, if you misrepresent information, you may be sued. Truth is, you probably won't be sued because I don't have money to hire a lawyer, but that could change at any time, and you could be in trouble if the statute of limitations has not run out.
Above photograph has been edited to prevent positive identification using facial recognition software.
Well, if you read this far, you are entitled to know a little more about the greatest mathematician ever:
I once had an acquaintance who would tell me that one should never call oneself the greatest, but let others say this of him. What this acquaintance with his limited intelligence (IQ of 139 according to him) failed to realize, is that the opinion of others is completely worthless to those who are the greatest. Needless to say, he is/was a bumbling religious fool. Thus, if others call me the greatest, I would be very worried, because the world is full of idiots.
Throughout my life I have always kept an open mind. While I do not claim to know everything, I do claim to know what I know, better than anyone else. Of course I have doubted myself. Not once, but many times. Over and over again, I have questioned my logic and conclusions. In fact, I never cease to question my conclusions. Knowledge of any subject is like a model. My approach to knowledge, is that if a flaw is found, the entire model, or at the very least those flawed parts of it, must be revised, discarded or replaced completely. Never have I closed the door on any knowledge and by so doing, considered it beyond further investigation. I openly acknowledge my imperfections and admit I could be wrong. What I find most irksome, are ignorant academics (and they are in no short supply!) who will argue a subject they do not and have not ever understood.
My life's journey thus far has been very "interesting". I am certain my days are numbered (as are everyone else's) and whether I am recognized or not, will mean nothing once I am gone. True knowledge is always discovered, never invented. There is never anything new - neither ideas, nor inventions. Future inventions and ideas have always existed - we just cannot think of them all, because we exist a finite time in our reality. What one imagines is as good as real, if and only if it is well-defined.
So what where you hoping to realize by reading this far? I do not have all the answers - neither to the meaning of life(if there is any), nor to mathematics. The New Calculus is only "new" because I was the first known to discover it in the history of man. However, the ideas and knowledge of the New Calculus have always existed, just as all other knowledge has always existed. In this sense, attributing greatness really means nothing more than acknowledging the individual who first thought of useful knowledge in a given time frame of history.
Ethnic roots: Cyprus and Lesvos are the islands of my recent ancestors. Before them, your guess is as good as any.
Other passionate views: I am a supporter of assisted suicide. It is my hope that assisted suicide will be legalized in the United States, as it has been in Switzerland and other more advanced civilizations. In my opinion, terminal illness or lack thereof, should not be a consideration if an individual chooses to end his life. Even healthy individuals should have access to assisted suicide. It's too bad one does not have a choice regarding entry into this earthly reality, but it's very sad and unfortunate that one cannot choose a time to exit peacefully. The most powerful mind-control drug (religion) is protected by the constitution of the United States, but the most basic right of choice to die is illegal in most states. An irony?
Hall of Infamy: Slander and Disrepute Infamers: The following slanderers/murderers have at one or other time attributed slander and disrepute to my name.
All it takes to destroy a man's reputation is a few slanderers. Once a reputation is destroyed, it's not possible to accomplish success orcontinue earning a living.
This day, 16th February 2013, I swear that I will take the most prized knowledge and ideas, which I have not shared with the world to the grave with me. I hope that my accusers and slanderers suffer a long and terrible life, followed by a slow, agonizing death. Thanks to the efforts of slanderers, I have become a pariah amongst the mathematics community. I have been compared to the likes of Gene Ray of Time Cube - a delusional individual, whom I knew nothing about until recently (Feb 2013). Such comparisons are irresponsible and destructive, and amount to slander and disrepute.
How to commit murder and not be prosecuted: In the United States, it is legal to call someone a crank. You have the right by law to call anyone you like or don't like, a crank. It does not matter if you are correct or not. It does not matter if you know your subject well enough or not. It does not even matter if you are educated. In fact, you do not even have to prove that the person you are defaming is a crank! You may call anyone you like a crank.
Disrepute and slander are the most effective tools of murder. One is never charged and never pays for one's heinous crimes. There are no court appearances, no jail time, no execution. It is a fact that once a reputation is destroyed or severely blemished, then it is effectively the same as murder. The reasons are not hard to see. In our modern age, people frequently are attracted to the most negative reports of others they hear about. Often individuals accept such reports to be true at face value. With the advanced media of the internet and television, it takes only a few seconds to destroy a good man's reputation. Given the varying degrees of intelligence, many people will often not bother verifying negative reports. For them, where there's smoke, there must be fire.
Whatever you do, don't call anyone a crank. Even if a person is wrong, no one has any right to call any person a crank. We are not all equal in terms of intelligence. History has shown over and over again how the greatest minds were often perceived to be cranks. If you have nothing good to say, just don't say anything. I am the first to admit, that even I have on many occasions used profanity out of anger. Just as profanity is damaging and wrong, so is slander and disrepute even more deadly than profanity. Hardly anyone takes profanity seriously, but slander and disrepute are cankerous. Once the seed of doubt has been sown, it grows almost like an avalanche. An avalanche is unstoppable in nature. Slander and disrepute are also unstoppable, but unlike an avalanche, slander and disrepute are preventable.
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Adobe PageMaker Pro for print, press, and electronic distributionThe element calculates determinants, linear equation systems and generates matrices. It provides additional basic functionality like faculty, subdeterminant and matrix reduction calculations. Further more an event is implemented to support a progress bar for time intensive operations
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DreamCalc Scientific Edition is the smarter alternative to a hand-held ScientificCalculator for your PC or laptop! You'll get the intuitive feel and productivity of using a professional hand-held, but one...
ScienCalc is a convenient and powerful ScientificCalculator. ScienCalc calculates mathematical expression. It supports the common arithmetic operations (+, -, *, /) and parentheses. The program contains...
ScientificCalculator Decimal for scientists, engineers, teachers, and students. Symbols can be entered by clicking buttons and/or using keyboard and num pad. Calculation history can be stored into text fileFull featured FreeCalculator providing all the essential features of a Calculator and more. The software handles all basic and advanced features required by most users and includes several advanced features...
ScientificCalculator is a useful program that let you calculate basic math, advanced math and trigonometric functions. It has all of advanced math functions like log, square root, trigonometric functions...
A+ Calc is a ScientificCalculator with a multi-line edit display that shows all calculations and allows editing and recalculating equations. A+ Calc includes hexadecimal, statistics, and unit conversioneCalc is an easy to use ScientificCalculator with many advanced features including unit conversion, complex number math, equation solving, and even support for both algebraic and RPN operating modes. The
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WOW! I had no idea how easy this really was going to be. I just plug in my math problems and learn how to solve them. Algebra Buster is worth every cent! Billy Hafren, TX Dan Mathers, MI
I was searching for months for a piece of software that would help me improve my Algebra skills. My solution was the Algebra Buster, and now I can tell you how it feels to be an A student12-20:
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1) What is the difference between evaluation and simplification of an algebraic expression?
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Fort Mcdowell StatisticsWhen students can grasp the concept first, they are more likely to grasp more complex concepts later (math builds). At the least, they develop critical thinking skills grow more comfortable visualizing abstract relationships. I took college calculus and might only need to briefly brush-up on so... of learning.
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Find a SoutheasternMajor topics studied include: probability, combinatorics, set theory and graph theory. Set theory is the study of sets, both infinite and finite. Some basic operations of set theory include the union and intersection of sets
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Getting Started with Interactive Geometry Software
A series of books each covering the use of a different piece of Interactive Geometry Software. Learn how to use the software and create interactive files using these excellent A5 books.
Includes
• Detailed worked examples with step by step instructions for using key techniques
• Open 'try this' activities invite you to apply techniques and explore ideas for yourself.
• Details onhow to create Objects based on points, lines and circles.
• Includes introduction to measurement, transformation and the exploration of loci.
Suitable for pupil, teachers and parents of pupils from upper primary to sixth form.
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Endorsed by Edexcel, this book is ideal for re-sits, returners to study. Based on Edexcel's official 1-year linear scheme of work and targeted on the Foundation tier, it offers plenty of support for C/D borderline.
Features
The only dedicated resource for students who want to get a grade C in Edexcel GCSE maths in just one year
Spread-based format means that students can take ownership of the pace of their learning: left page + right page = 1 lesson!
Based on Edexcel's official 1-year schemes of work, with all the learning condensed into two
terms
Powerpoint worked examples and exam question solutions on CD-ROM, so students can be even more fully prepared for their assessments
Focus on more mature students (not just 16-17 year olds), with practical everyday contexts to satisfy the functional elements of maths
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Technical Computing Magazine - Issue 41
'Mammath' Potential
Elizabeth Northrop,
Maple T.A. Product Manager
With inbuilt maths intelligence and an understanding of the theories that form the foundation of learning for multiple disciplines, the Maple T.A. online assessment solution has the potential to improve understanding and enrich education across the curriculum.
Maths is a true universal language - whether you're in London or Lisbon, Milan or Melbourne,
Pi is still 3.14159 no matter where you are. As well as connecting continents, the principles and foundations of maths also underpin all kinds of disciplines and developments in modern technology. A solid foundation in maths is essential for success in engineering, chemistry, biology, medicine, economics, finance, business, computer science - the list goes on!
And when we say 'assess student responses', we're not talking a blanket right or wrong. The latest release, Maple T.A. 8 supports adaptive questions, partial grading and intelligent evaluation of responses. So if a student is having trouble, the question can be adapted to provide more information or broken down to walk the student through the problem step by step. Alternatively, the student can try a simpler version of the question before retrying the original problem or try the same question again for reduced credit - whatever approach the teacher feels is appropriate. Flexible partial grading lets teachers control how generous or rigorous the grading will be, with the option of giving partial credit for responses that are not completely correct. With Maple T.A. course leaders can ask free-response questions, including questions that have more than one correct answer. Maple T.A. will evaluate responses intelligently, just like a human examiner.
Of course, questions and content don't just materialise out of thin air, and for time-pressed teachers, generating assessments and exercises can often form a stumbling block to adopting testing solutions, no matter how easy to use the authoring tools are! Thankfully, Maple T.A.'s developers have already thought of this and they've made thousands of questions, many of which have been field tested in actual courses, freely available for users to employ and customise for their own classes. Adept are also in the first stages of developing a UK Maple T.A. user group to provide a supportive forum for sharing knowledge and experiences of using Maple T.A., as well as collaborating on content and question creation.
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Video Summary: This learning video presents an introduction to graph theory through two fun, puzzle-like problems: "The Seven Bridges of Königsberg" and "The Chinese Postman Problem". Any high school student in a college-preparatory math class should be able to participate in this lesson. Materials needed include: pen and paper for the students; if possible, printed-out copies of the graphs and image that are used in the module; and a blackboard or equivalent. During this video lesson, students will learn graph theory by finding a route through a city/town/village without crossing the same path twice. They will also learn to determine the length of the shortest route that covers all the roads in a city/town/village. To achieve these two learning objectives, they will use nodes and arcs to create a graph and represent a real problem. This video lesson cannot be completed in one usual class period of approximately 55 minutes. It is suggested that the lesson be presented over two class sessions
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Secondary Curricula
Student Textbook Set
When a school implements Carnegie Learning textbooks, each student receives a consumable textbook set that contains the following books.
Student Textbook
The Student Text is a consumable textbook designed for students to take notes and work problems directly in each lesson. Each lesson contains objectives, key terms, and problems that help the students to discover and master mathematical concepts.
Student Assignments and Skills Practice
The Student Assignments book contains one assignment per lesson and skills practice activities. It is designed to move with the student from classroom to home to lab time so that students can repeatedly practice the skills taught in the lesson.
Homework Helper*
The Homework Helper book is designed to help parents and care givers be more informed about the concepts being covered in the student's math course. Students are encouraged to keep the Homework Helper at home. It contains one activity per lesson including examples of the skills taught in the lesson and several practice problems. Answers to the practice problems are provided in the back of the Homework Helper book.
*Homework Helper included in Bridge to Algebra and Algebra I curricula.
What Makes Carnegie Learning Student Texts Engaging?
Learning By Doing Principles
Carnegie Learning believes that students develop math understanding and skills by taking an active role and responsibility for their own learning. With Carnegie Learning textbooks students become engaged in solving contextual math problems that strengthen their conceptual understanding of math topics. Rather than encouraging students to memorize procedures, we provide them opportunities to think and work together in small groups.
Real-World Context
Students work with their peers to solve real-world problem situations like using percents for leaving a tip in a restaurant or using a graph of an equation to determine the number of days it will take to build miles of highway. They become more engaged in learning mathematics when they see how it plays a significant role in everyday life.
Mathematical Discourse
Throughout the student text icons prompt different forms of student communication. These icons may instruct students to work independently, work with groups, or share ideas with the class. Encouraging mathematical discourse provides opportunities for students to explain their thoughts and processes for solving math problems.
Carnegie Learning Textbooks
Documents & Brochures
2012 Program Guide (Middle & High School)Explore our Middle School and High School Math Series featuring our innovative, research-based software and textbooks for students in grades 6-12, and Professional Development for educators of Grades K-12.
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Number Theory 1st Edition
0387498931
9780387498935 or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject. «Show less... Show more»
Rent Number Theory 1st Edition today, or search our site for other Cohen
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Intermediate Algebra, 9th + Student Solutions Manual
ISBN10: 1-111-49672-2
ISBN13: 978-1-111-49672-2
AUTHORS: Kaufmann/Schwitters
Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; practice the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundamental problem solving skills necessary for future mathematics courses. Algebraic ideas are developed in a logical sequence, and in an easy-to-read manner, without excessive vocabulary and formalism.
The open and uncluttered design helps keep students focused on the concepts while minimizing distractions. Problems and examples reference a broad range of topics, as well as career areas such as electronics, mechanics, and health, showing students that mathematics is part of everyday life.
The text's resource package—anchored by Enhanced WebAssign, an online homework management tool—saves instructors time while also providing additional help and skill-building practice for students outside of class.
Additional versions of this text's ISBN numbers
Purchase Options
List$335
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Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. (Source: Wikipedia)
To find the derivative of inverse function of f(x) = 2x-1, first we need to find the inverse function of f(x), which is, f-1(x)=(x+1)/2. In the next step we find the derivative of this inverse function. [f-1(x)]'= d/dx[x/2 +1/2]=1/2 [applying the derivative rule] . So, the derivative of inverse function of f(x)=2x-1 is 1/2
Volume of a three dimensional shape is defined as the amount of space it occupies. For a given density of a material, it is also related to its heaviness. Cylinder is a three dimensional shape which has a uniform cross section of circle along its vertical height.
The OMICS Group is keen on serving the scientific community for upliftment and progress of overall scientific activities in various parts of the globe. According to experts, the scientific progress is all the more smooth through interaction of professors and delegates on a single platform, and OMICS works on the same lines to organize events for sharing of knowledge and building industry-academic collaborations.
The matrices multiplication need two input matrices minimum and it produce the result in one matrix. If we have to multiply two 3 x 3 matrices we have totally 18 elements(each 3 x 3 matrix have 9 elements, three elements in each row) to multiply but we have only 9 elements in the resulted matrix. In this article how this process done with the calculator and its operations.
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Welcome to the Algebra portion of the site! On this and the following pages,
we'll try to clear up some common problems people have with algebra (a subject
that has been stumping everyone from seventh graders to college students throughout
the ages). Everything from the basics of solving equations to exponents, and from
graphing to word problems (which people seem to absolutely love) will be covered.
After each section,
there is an optional (though highly recommended) quiz that you can take
to see if you've fully mastered the concepts. Don't forget to visit the
message board
and the formula database.
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Survey of Mathematics with Applications, A (9th Edition)
9780321759665
ISBN:
0321759664
Edition: 9 Pub Date: 2012 Publisher: Addison Wesley
Summary: This textbook serves as a broad introduction to students who are looking for an overview of mathematics. It is designed in such a way that students will actually find the text accessible and be able to easily understand and most importantly enjoy the subject matter. Students will learn what purpose math has in our lives and how it affects how we live and how we relate to it. It is not heavy on pure math; its purpose ...is as an overview of mathematics that will enlighten students without an intense background in math. If you want to obtain this and other cheap math textbooks we have many available to buy or rent in great condition online ***Warning***Text Only. Still in Shrink Wrap Annotated Instructor's Copy, 9th edition but No Supplementary Materials otherwise same as student with help added tips,and answers.Shipping from California.[less]
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Mathematics
Why study Mathematics?
An essential element of mathematical learning is the development of mathematical knowledge in a way which encourages confidence and provides satisfaction and enjoyment. It is expected that students will gain an appreciation of the use of mathematical skills within other subjects as well as an understanding of problem solving in the real world.
Which specification is followed?
Girls in Lower School follow the National Framework for Mathematics as laid down by the National Curriculum and the National Numeracy Strategy. From Lower 4 onwards pupils are taught in sets according to attainment.
In 2009, Bradford Girls' Grammar School embarked on the iGCSE course following the Edexcel specification. Candidates are encouraged to develop a feel for numbers; to recognise patterns and relationships; to generalise results; and to use the language of mathematics to communicate their ideas effectively and efficiently. All candidates will be studying a course leading to the higher tier examinations, but individual students may be entered at Foundation Level if appropriate. The iGCSE does not have a coursework element. Marks are obtained from sitting two examination papers at the end of Upper 5. The iGCSE is an excellent preparation for students intending to study mathematics at A-level.
Mathematics and Further Mathematics are offered at both AS and A level following the Edexcel specifications. In recent years the role of Further Mathematics has changed. It is seen as enriching and deepening the curriculum: truly further maths rather than just harder maths – it provides able students with a course which stimulates them mathematically and prepares them for a wide range of higher education options.
At A level students study three main areas of mathematics:
Pure (or Core) Mathematics which develops and extends topics already met at GCSE including algebra, trigonometry and graphs, it also introduce new topics such as calculus.
Statistics which includes the presentation, analysis and interpretation of data and the study of probability.
Mechanics which involves the study of the motion of objects and how they respond to forces acting on them.
Further Mathematics will cover Decision Mathematics in addition to the three areas mentioned above.
Workshops, conferences and visits
Mathematics workshops are offered during lunchtimes, providing help and support to students in all years.
Girls throughout Bradford Girls' Grammar School participate in the Mathematical challenges, including the Junior and Senior Team Challenges, run by the UK Mathematics Trust.
Degree and career choices
There are many opportunities to study Mathematics in the Sixth Form at Bradford Girls' Grammar and it is an excellent support subject for any combination of 'A' levels. It is not an easy subject to study at this level and its academic rigour means that it is highly valued by universities for entry into most degree programme,s particularly the sciences, geography, economics, psychology, medicine and engineering.
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MATH40237 Fundamental Mathematics for University
Course details
Fundamental Mathematics for University is designed to provide students with foundation concepts, rules and methods of elementary mathematics. The main aim of this course is to provide the fundamentals of mathematics, which are necessary to develop a unified body of knowledge. Topics covered in the course include operations, percentages, introductory algebra, simple equation solving, exponents, linear equations, introductory statistics, and units and conversions
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Math software for students studying precalculus. Can be interesting for teachers teaching precalculus. Math Center Level 1 consists of Graphing calculator 2D, Advanced Calculator, Simple Calculator, Simple Calculator, Simple Rational Calculator, and Simple Integer Calculator called from the Control Panel. Simple calculator is a general purpose calculator which combines use simplicity and calculation power. It handles simple arithmetic operations and complex formulas. Advanced Calculator is a step farther in complexity comparing to the Simple Calculator. The Advanced Calculator has two editing windows. One is for editing x, and the second is for editing f(x). In the x window you can enter any number or formula which contains numbers. In the f(x) window you can enter formulas containing numbers and formulas containing x. First, x will be calculated. Then the result for x will be substituted into the formula for f(x). The presence of two editing windows demands switching between windows. You can do it by clicking buttons "go to x" and "go to f(x)", or by clicking inside the window. If you forget to enter x, then the x=1 will be assumed. If you forget to enter f(x), then f(x)=x will be assumed. Advanced Calculator works in scientific mode. All numbers in internal calculations are treated in scientific format. Graphing Calculator 2D has two panels. The Left Panel has the Magnifying Square represented by Small Square with gray border on the Left Panel. It is 16 times smaller than the Left Panel. The Right Panel shows content of the Magnifying Square magnified 16 times. You can press button "zoom +". Then the Left and Right Panels will be zoomed twice each. Maximum zoom is 8 (tree clicks of "zoom +"). Clicking button "C" (for Center) on Zoom returns picture to starting position with no zoom and Magnifying Square at the center of Left Panel.
Arcade Math Blocks for Mac OS - Arcade Arithmetic Game.Arcade Arithmetic Game. Set-up and solve equations while searching for treasures and avoiding bad guys. There are many options for math and arcade difficulty. This math game puts you in charge of seeing how numbers relate to...
FASTT Math - FASTT Math ensures that all students, regardless of their fluency level, build the long-lasting fluency they will need to tackle higher-order math.FASTT Math ensures that all students, regardless of their fluency level, build the long-lasting...
Algebrator - Algebrator is one of the most powerful software programs for math education ever developed.Algebrator is one of the most powerful software programs for math education ever developed. It will tackle the most frustrating math problems you throw at...
ScientificCalculatorDecimal - Scientific Calculator Decimal is programmed in C# and is similar to Scientific Calculator from Math Center Level 2 except that all calculations are done in decimal data type instead of double.Scientific Calculator Decimal is programmed in C# and...
Math Stars Plus - Math Stars Plus is an educational application which includes a series of games that will help kids improve their Math skills.Math Stars Plus is an educational application which includes a series of games that will help kids improve their Math
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Math
Math Assistive Techologies are typically used by individuals with dyscalculia, dyslexia and dysgraphia learning disabilities but can benefit most any user in accomplishing educationally or occupationally in relevant tasks involving Pre-Algebra, Algebra, Geometry, Trigonometry, Chemistry, Calculus.
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Features
Provides information in an accessible, easy to comprehend, self-study format
Facilitates the reader's understanding of the material and mastery of basic math
Presents information in a style that has been thoroughly tested and has proven successful
Summary
Accurately calculating medication dosages is a critical element in pharmaceutical care that directly affects optimal patient outcomes. Unfortunately, medication dosage errors happen in pharmacies, in hospitals, or even at home or in homecare settings everyday. In extreme cases, even minor dosage errors can have dire consequences. Careful calculations are essential to providing optimal medical and pharmaceutical care.
Essential Math and Calculations for Pharmacy Technicians fills the need for a basic reference that students and professionals can use to help them understand and perform accurate calculations. Organized in a natural progression from the basic to the complex, the book includes:
Roman and Arabic Numerals
Fractions and decimals
Ratios, proportions, and percentages
Systems of measurement including household conversions
Interpretation of medication orders
Isotonicity, pH, buffers, and reconstitutions
Intravenous flow rates
Insulin and Heparin products
Pediatric dosage
Business math
Packed with numerous solved examples and practice problems, the book presents the math in a step-by-step style that allows readers to quickly grasp concepts. The authors explain the fundamentals simply and clearly and include ample practice problems that help readers become proficient. The focus on critical thinking, real-life problem scenarios, and the self-test format make Essential Math and Calculations for Pharmacy Technicians an indispensable learning tool.
Table of Contents
Working with Roman and Arabic Numerals Using Fractions and Decimals in Pharmacy Math Using Ratios, Proportions and Percentages in Dosage Calculations Applying Systems of Measurements Interpreting Medication Orders Identifying Prescription Errors and Omissions Working with Liquid Dosage Forms Working with Solid Dosage Forms Adjusting Isotonicity Working with Buffer and Ionization Values Dealing with Reconstitutions Determining Milliequivalent Strengths Calculating Caloric Values Determining IV Flow Rates Working with Insulin and Heparin Products Appendices: A: Working with Temperature Conversions B: Working with Capsule Dosage Forms C: Dealing with Pediatric Dosages D: Understanding Essential Business Math
Editorial Reviews
"Calculations are explained in great detail in a logical fashion. The formulas are helpful and explained in several ways. The information is presented so that verification for accuracy is easily done. … Information provided in this text is more complete and detailed than in other math textbooks. I recommend this volume as a textbook or as a supplemental course book. Students could also benefit from Essential Math as a reference book. It should be a very valuable addition to their pharmacy technician library." - Ray Vellenga, Journal of Pharmacy Technology
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Pre-Calculus for Dummies
9780470169841
ISBN:
0470169842
Pub Date: 2008 Publisher: Wiley, John & Sons, Incorporated
Summary: Brush up on algebra and trig concepts and get a glimpse of calculusUnderstand the principles and problems of pre-calculusGetting ready for calculus, but feel confused? Have no fear! This unintimidating, hands-on guide walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. You'll understand the concepts -... not just the number crunching - and see how to perform all tasks, from graphing to tackling proofs.Apply the major theorems and formulasGraph trig functions like a proFind trig values on the unit circleTackle analytic geometryIdentify function limits and continuity[read more]
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COURSE SYLLABUS
Course Number: MATH1000
Course Title: COLLEGE ALGEBRA
Credit Hours: 3
Prerequisites: High school geometry and second year high school algebra.
Corequisite:
Objectives:
To develop student mathematical and analytic skill with particular emphasis on establishing student skills at algebraic manipulation. To give students the background and preparation necessary for core mathematics. Does not satisfy the core requirement in mathematics.
Schedule and Outline of Course Content:
Fundamental concepts of algebra ( 3 weeks).
Equations and inequalities ( 3 weeks).
Functions and graphs ( 5 weeks).
Polynomial and rational functions ( 4 weeks).
Possible textbooks:
Algebra and Trigonometry, 9th edition by Swokowski and Cole, 1997, Brooks/Cole.
Sample Grading and Evaluation Procedures:
Students will be expected to have prepared the daily homework assignments. Homework will occasionally be collected. This will be part of the participation grade.
Grade Calculation
Participation grade
(includes: blackboard presentation and classwork, attendance, homework): 10%
Quizzes (quizzes are approximately 10-minutes long and may be announced or unannounced): 10%
Hour Tests (four tests at 12 % each): 48%
Final Exam: 32%
Tentative Test Schedule
Hour tests are given approximately every three weeks and will be announced a week ahead of time. Quizzes may or may not be announced; at least four quizzes will be given in the course of the semester. Friday is typically a good day for quizzes
The course will be used in conjunction with the mathematics placement test and is designed to bring students to the level of preparation which will enable them to take core mathematics courses.
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Books on Mathematics > Algebra > Linear
8 new & used from sellers starting at 1,721 In Stock.Ships Free to India in
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Renowned professor and author Gilbert Strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. While the mathematics is there, the effort is not all concentrated on proofs. Strang's emphasis is on understanding. He explains concepts, rather than...... more
4 new & used from sellers starting at 3,405 In Stock.Ships Free to India in
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This clear, concise and highly readable text is designed for a first course in linear algebra and is intended for undergraduate courses in mathematics. It focusses throughout on geometric explanations to make the student perceive that linear algebra is nothing but analytic geometry of n dimensions. From the very start,... more
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An effort has been made to present the various topics in the theory of graphs in a logical order, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results. In addition, there are three appendices which provide diagrams of graphs, directed graphs,... more
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In this international version of the first edition, Principles of Signal Processing and Linear Systems, the author emphasizes the physical appreciation of concepts rather than the mere mathematical manipulation of symbols Avoiding the tendency to treat engineering as a branch of applied mathematics, the text uses mathematics not so much... more
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This text offers a comprehensive and coherent introduction to the fundamentals of graph theory. Written in a reader-friendly style and with features that enhance students- comprehension, the book focuses on the structure of graphs and techniques used to analyze problems. Greatly expanded and reorganized, this edition is integrated with key... more
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About the Book : The present book is intended for the advanced level undergraduate, and postgraduate students, in mathematics and other disciplines, who need a comprehensive knowledge of linear algebra. It can also be a reference source for teachers, looking for detailed proofs of results, given in elementary books, without... more
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About the Book : The second edition of the authors acclaimed textbook covers the major topics of computational linear algebra, including solution of a system of linear equations, least-squares solutions of linear systems, and computation of eigenvalues, eigenvectors, and singular value problems. The important features of the original edition have... more
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From the reviews: "The book is well written. We find here many examples. Each chapter is followed by exercises, and at the end of the book there are outline solutions to some of them. [...] I especially appreciated the lively style of the book; [ ] one is quickly able... more
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Breadth of scope is unique Author is a widely-known and successful textbook author Unlike many recent textbooks on chaotic systems that have superficial treatment, this book provides explanations of the deep underlying mathematical ideas No technical proofs, but an introduction to the whole field that is based on the specific...
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What can we learn from fish in a pond? How do social networks connect the world? How can artificial intelligences learn? Why would life be different in a mirror universe?. Mathematics is everywhere, whether we are aware of it or not. Exploring the subject through 35 of its often odd and unexpected applications, this book provides an insight into the... more...
A is for Algebra-and that's the grade you'll pull when you use Bob Miller's simple guide to the math course every college-bound kid must take. With eight books and more than 30 years of hard-core classroom experience, Bob Miller is the frustrated student's best friend. He breaks down the complexities of every problem into easy-to-understand pieces... more...
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Wikipedia in English (1)
If you want top grades and a thorough understanding of abstract algebra this powerful study tool is the best tutor you can have! It takes you step-by-step through the subject and gives you sample problems with fully worked solutions, including proofs of all important theorems. You also get additional practice problems to solve on your own, working at your own speed. In addition, this superb study guide gives you chapters on sets, integers, groups, polynomials, and vector spaces. Students' favorite, with more than 30 million copies sold, Schaum's study guides are the best value for your student dollar--clear, complete, and low-cost.
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Helping Undergraduates Learn to Read Mathematics
Although most students "learn to read" during their first
year of primary school, or even before, reading is a skill which continues
to develop through primary, secondary and post-secondary school, as the
reading material becomes more sophisticated and as the expectations for
level of understanding increase. However, most of the time spent deliberately
helping students learn to read focuses on literary and historical texts.
Mathematical reading (and for that matter, mathematical writing) is rarely
expected, much less considered to be an important skill, or one which can
be increased by practice and training.
Even as an undergraduate mathematics major, I viewed mathematical reading
as a supplementary way of learning--inferior to learning by lecture or
discussion, but necessary as a way of "filling in the gaps."
Not until graduate school was I responsible for reading new material at
a high level of comprehension. And, as I began to study primarily written
mathematics (texts and articles) rather than spoken mathematics (lectures),
I discovered that the activities and habits needed to learn from written
mathematics are quite different from those involved in learning from a
mathematics lecture or from those used in reading other types of text.
As I consciously considered how to read mathematics more effectively and
to develop good reading habits, I observed in my undergraduate students
an uneasiness and lack of proficiency in reading mathematics.
In response to this situation, I wrote for my students (mostly math
majors in Introductory Abstract Algebra at the University of Chicago) two
handouts, one on reading theorems and the other
on reading definitions. These describe some
of the mental activities which help me to read mathematics more effectively.
I also gave a more specific written assignment, applying some of these
questions to a particular section of assigned reading. My hope was that,
as they were forced to actively engage in reading, they would discover
that reading mathematics could be a profitable pursuit, and that that they
would develop habits which they would continue to use.
More than one such written exercise is needed to significantly affect
the way that students view reading. While the students seemed to understand
the types of questions that are helpful, they needed some practice in carrying
these out, and even more practice using these activities in the absence
of a written assignment.
A Few Mathematical Study Skills... Reading Theorems
In almost any advanced mathematics text, theorems, their proofs, and
motivation for them make up a significant portion of the text. The question
then arises, how does one read and understand a theorem properly? What
is important to know and remember about a theorem?
A few questions to consider are:
What kind of theorem is this? Some possibilties are:
A classification of some type of object (e.g., the classification of
finitely generated abelian groups)
An equivalence of definitions (e.g., a subgroup is normal if, equivalently,
it is the kernel of a group homomorphism or its left and right cosets coincide)
An implication between definitions (e.g., any PID is a UFD)
A proof of when a technique is justified (e.g., the Euclidean algorithm
may be used when we are in a Euclidean domain)
Can you think of others?
What's the content of this theorem? E.g., are there some cases in which
it is trivial, or in which we've already proven it?
Why are each of the hypotheses needed? Can you find a counterexample
to the theorem in the absence of each of the hypotheses? Are any of the
hypotheses unneccesary? Is there a simpler proof if we add extra hypotheses?
How does this theorem relate to other theorems? Does it strengthen
a theorem we've already proven? Is it an important step in the proof of
some other theorem? Is it surprising?
What's the motivation for this theorem? What question does it answer?
We might ask more questions about the proof of theorem. Note that, in
some ways, the easiest way to read a proof is to check that each step follows
from the previous ones. This is a bit like following a game of chess by
checking to see that each move was legal, or like running the spell checker
on an essay. It's important, and necessary, but it's not really the point.
It's tempting to read only in this step-by-step manner, and never put together
what actually happened. The problem with this is that you are unlikely
to remember anything about how to prove the theorem. Once you've read a
theorem and its proof, you can go back and ask some questions to help synthesize
your understanding. For example:
Can you write a brief outline (maybe 1/10 as long as the theorem) giving
the logic of the argument -- proof by contradiction, induction on n, etc.?
(This is KEY.)
What mathematical raw materials are used in the proof? (Do we need
a lemma? Do we need a new definition? A powerful theorem? and do you recall
how to prove it? Is the full generality of that theorem needed, or just
a weak version?)
A Few Mathematical Study Skills... Reading Definitions
Nearly everyone knows (or think they know) how to read a novel, but
reading a mathematics book is quite a different thing. To begin with, there
are all these definitions! And it's not always clear why one would care
to know about these things being defined. So what should you do when you
read a definition?
Ask yourself (or the book) a few questions:
What kind of creature does the definition apply to? integers? matrices?
sets? functions? some pair of these together?
How do we check to see if it's satisfied? (How would we prove that
something satisfied it?)
Are there necessary or sufficient conditions for it? That is, is there
some set of objects which I already understand which is a subset or a superset
of this set?
Does anything satisfy this definition? Is there a whole class
of things which I know satisfy this definition?
Does anything not satisfy this definition? For example?
What special properties do these objects have, that would motivate
us to make this definition?
Is there a nice classification of these things?
Let's apply this to an example, abelian groups:
What kind of creature does it apply to? Well, to groups... in particular,
to a set together with a binary operation.
How do we check to see if it's satisfied? The startling thing is that
we have to compare every single pair of elements! This would be a big job,
so:
Are there necessary or sufficient conditions for it? Well, it's sufficient
that the group be cyclic, as we saw in the homework. Do you know of any
neccesary conditions?
Does anything satisfy this definition? Well, yes... the group of rational
numbers under addition, for example. We have a whole class of things which
satisfy the definition, too -- cyclic groups.
Does anything not satisfy this definition? Yes, matrix groups come
to mind first. There are finite non-abelian groups, but this is
harder to see... do you know of one yet?
What special properties do these objects have, that would motivate
us to make this definition? Some of these properties are obvious, others
are things which we had to prove. One example: If H and K are subgroups
of an abelian group, then HK is also a subgroup.
Is there a nice classification of these things? Why, yes, at least
for a large subcategory of them. We'll get to it later... it says, basically,
that a finite abelian group is always built in a simple way from cyclic
groups (Zn's).
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Discrete Mathematics Demystified - 08 edition
If you're interested in learning the fundamentals of discrete mathematics but can't seem to get your brain to function, then here's your solution. Add this easy-to-follow guide to the equation and calculate how quickly you learn the essential concepts.
Written by award-winning math professor Steven Krantz, Discrete Mathematics Demystified explains this challenging topic in an eff...show moreective and enlightening way. You will learn about logic, proofs, functions, matrices, sequences, series, and much more. Concise explanations, real-world examples, and worked equations make it easy to understand the material, and end-of-chapter exercises and a final exam help reinforce learning.
This fast and easy guide offers:
* Numerous figures to illustrate key concepts * Sample problems with worked solutions * Coverage of set theory, graph theory, and number theory * Chapters on cryptography and Boolean algebra * A time-saving approach to performing better on an exam or at work
Simple enough for a beginner, but challenging enough for an advanced student, Discrete Mathematics Demystified is your integral tool for mastering this complex subject
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Homework is worth 3 points and is a completion grade. A grade less than 3 indicates that the student did not complete/turn in the assignment or that they did not show their work. I spot check a few problems for the students so that they can see how they did. "Did We Get It (DWGI)?" questions are given daily. This is graded and scored for accuracy.
If a student is absent, DWGI will be marked as a 0 and no count, which indicates they did not do the questions but that it is not counted against their grade. Homework is marked a 0 until the missing assignment is turned in and it does count against the grade. If a student has an extended absence, they may want to meet with me to discuss a plan for making up missed work. An asterisk (*) in the gradebook indicates that I have not entered a grade for that assignment, not that it is incomplete.
I typically add the homework to the webpage before class. If there are changes made to the assignment during class, I will change the website as soon as possible. If you receive an assignment in class that is different from the webpage, do the assignment from class.
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This is alarming as around 60 of the students could not define the
function of ``y sin x . One of the supposed reasons for this is that
the students only wanted to find or choose the answers to the questions
given to them because they wanted to get high scores in college
examinations instead of understanding the basic principles and concepts
of mathematics . Moreover , the 21 figure only proves that some students
are also confused of the concepts of trigonometric...
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The credit crisis that started in 2007, with the collapse of well-established financial institutions and the bankruptcy of many public corporations, has clearly shown the importance for any company entering the derivative business of modelling, pricing, and hedging its counterparty credit exposure. Building an accurate representation of firm-wide credit... more...
This book covers the necessary topics for learning the basic properties of complex manifolds, offering an easy, friendly, and accessible introduction into the subject while aptly guiding the reader to topics of current research and to more advanced publications. The first half of the book provides an introduction to complex differential geometry and... more...
Algorithmic composition - composing by means of formalizable methods - has a century old tradition not only in occidental music history. This book provides an overview of prominent procedures of algorithmic composition in a pragmatic way. It is suitable for the musicians and the researchers. more...
Accessible Mathematics is Steven Leinwand?s latest important book for math teachers. He focuses on the crucial issue of classroom instruction. He scours the research and visits highly effective classrooms for practical examples of small adjustments to teaching that lead to deeper student learning in math. Some of his 10 classroom-tested teaching shiftsThis book is devoted to Quisped, Roberts, and Thompson (QRT) maps, considered as automorphisms of rational elliptic surfaces. The theory of QRT maps arose from problems in mathematical physics, involving difference equations. The application of QRT maps to these and other problems in the literature, including Poncelet mapping and the elliptic billiard,... more...
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Coloring Maps and Related Problems
This is one of a series of interactive tutorials introducing
the basic concepts of graph theory. This six page tutorial introduces coloring
problems as well as one of the most famous theorems in mathematics: The Four Color
Theorem.
Most of the pages of these tutorials require that you pass a quiz before continuing
to the next page, while others ask for a written comment. To keep track
of your progress we ask that you first register for this course by selecting
the [REGISTER] button below (press [help] for more information). After
you are registered, you will be able to start this tutorial, moving back and
forth in it using the buttons on the bottom of each page. If you
are already registered, you may continue where you left off by again pressing
the [REGISTER] button (and then re-entering your name and password).
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Discrete Mathematics Study Materials
Discrete Mathematics
Lectures Ppt
Click on the blue colored links to download the lectures.
Course Description
This course covered the mathematical topics most directly related to computer science. Topics included: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. The analysis of algorithms requires the ability to count the number of operations in an algorithm. Recursive algorithms in particular depend on the solution to a recurrence equation, and a proof of correctness by mathematical induction. The design of a digital circuit requires the knowledge of Boolean algebra. Software engineering uses sets, graphs, trees and other data structures. Number theory is at the heart of secure messaging systems and cryptography. Logic is used in AI research in theorem proving and in database query systems. Proofs by induction and the more general notions of mathematical proof are ubiquitous in theory of computation, compiler design and formal grammars. Probabilistic notions crop up in architectural trade-offs in hardware design.
Lecture 1:What kinds of problems are solved in discrete math?What are proofs? Examples of proofs by contradiction, and proofs by induction:Triangle numbers, irrational numbers, and prime numbers. (3.1-3.2)
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Geometry Kit with Solutions Manual
Presented in the familiar Saxon approach of incremental development and continual review, topics are continually kept fresh in students' minds. Covering triangle congruence, postulates and theorems, surface area and volume, two-column proofs, vector addition, and slopes and equations of lines, Saxon features all the topics covered in a standard high school geometry course. Two-tone illustrations help students really "see" the geometric concepts, while sidebars provide additional notes, hints, and topics to think about. Parents will be able to easily help their students with the solutions manual, which includes step-by-step solutions to each problem in the student book; and quickly assess performance with the test book (test answers included). Tests are designed to be administered after every five lessons after the first ten.
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This interactive course covers the basics of geometry for high-school and senior high-school students. From the basics of what are points, and lines, this course goes all the day to explain dynamically and interactively what are similarities and dilatations.
This Interactive Learning App was developed by experienced teachers who have fine-tuned how to visually teach to students these fun concepts. An emphasis is put on interactive and dynamic learning.
Like all of our courses, it is meant to be fun and engaging and to allow the learner to progress by being challenged throughout!
The most complete interactive Trigonometry course on the web (or everywhere).
This course provides really solid training in the topic of high-school Trigonometry. What is viewed as a dry topic, is explained in an engaging and interactive way for students to learn and absorb what should be an important topic to learn, for students vying for scientific fields.
This topic starts from the basics of 'trig', to them explain the basics of the different important laws, as well as 'trig' functions and identities.
No other online Trigonometry course we know of does it so easily and gets the students learning, better and faster.
Probability is one of those fun topics that many students have a hard time with. This topic is more important that many think, as it teaches systematic and sequenced thinking.
In this fun and interactive course, the student will go on a journey to master high-school probability.
From understanding the basics of what outcomes are to computing basic linear probability, to then end with Compound random experiments.
Our collective teaching experience of over 100 years has helped us bring it down to basics, to help the student think through the process in an organized way. This approach is evident in all the courses on this site or on the mobile app stores.
Statistics is normally a very dry topic, that is meant as a 'toolbox' for students to almost memorize. At the right time, the right 'swiss army knife' is brought out to solve the problem at hand.
This course deals with the topic in a different way! The student progresses through the learning at a steady pace, while 'absorbing' the notions through strong visuals and interactions.
For example, one of the hardest topics to learn in stats is correlation. In this course, it will be shown through interactive learning where the student will understand it naturally by watching the examples and going through the directed exercises.
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Casio Takes New Approach to Graphing Calculator for Students
Casio's Prizm fx-CG10 plots graphs over full-color images to help students visualize concepts.
Casio Education has introduced the Prizm fx-CG10, a new concept in educational graphing calculators that aims to impart mathematical concepts in addition to providing standard graphing functions. Using a new tool known as Picture Plot, the Prizm enables users to plot graphs over full-color photographic images, such as an Egyptian pyramid or the jets of an outdoor fountain, as way of relating complex mathematical functions to real-world concepts such as design and engineering.
Casio also offers teachers online training using streaming video and downloadable supplemental activities, as well as a loaner program, which enables interested educators to try the Prizm for 30 days. An application for the program is
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The third edition of Languages and Machines: An Introduction to the Theory of Computer Science provides readers with a mathematically sound presentation of the theory of computer science at a level suitable for junior and senior level computer science majors. The theoretical concepts a...
Mathematics on the Internet: A Resource for K-12 Teachers, third edition, helps teacher educators, college students preparing to become mathematics teachers, and teachers in elementary, middle, and high schools to become better acquainted with some of the resource materials and information available...
For non-majors introductory energy courses in the departments of Physics, Geology, and Geography. The most complete book of its kind on the market, this text focuses on energy needs, trends, and long-term prospects and resource supplies. It addresses all the various issues involved w...
Far more "user friendly" than the vast majority of similar books, this text is truly written with the "beginning" reader in mind. The pace is tight, the style is light, and the text emphasizes theorem proving throughout. The authors emphasize "Active Reading," a s...
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Skills Improvement Service (LSIS), this case study tackles the theme of progression through STEM subjects. By Newham Adult Learning Service, it describes an action research project that aimed to capture progression and learner voice data from a sample of learners who had recently completed a level two mathematicsA Mathematics Matters case study which looks at how advances in statistics allow us to analyse risks and consequences and so make informed decisions. Risks are an unavoidable part of modern life, but mathematicians and statisticians have developed a variety of methods to help mitigate its effects. These techniques enable hospitals, mathematicians work with epidemiologists to understand the spread of infections and mitigate their effects. Epidemics can threaten the can help industry to manage their use of fluids. Many industrial processes involve the complex movement of fluids, but predicting fluid behaviour can often be difficult. Mathematical models of fluid flow can help to improve manufacturing efficiency and reduce costs, looks at how Formula One teams use mathematical methods such as fluid mechanics and Navier-Stokes equations to improve performance. Every second counts in the fast-paced world of Formula One, so race teams use advanced mathematics to squeeze the best performance out of their cars. Computational…
This Mathematics Matters case study describes how mathematicians help to boost efficiency in the energy industry by mapping buried oil reserves. As oil supplies become harder and more expensive to reach, it's essential that we maximise the yield from available reservoirs in any way possible. Mathematicians are contributing…
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When 'studying calculus, you should have a good understanding of the following tables of formulas so you can efficiently and correctly solve calculus problems. An introduction to the basic concepts of calculus. The derivative ... The derivative, then takes a type of formula and turns it into another simiilar type of formula.
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Ms. Hershey
NewsFlash
Pre-Calculus
2011-2012
Instructor: S-A Hershey
Course Description: The purpose of this course is to prepare students in skills necessary for success in a first course in Calculus. A firm foundation in Algebra is needed, thus a large emphasis is placed on the students' proficiency in Algebra. The concepts of Trigonometry and Analytic Geometry are also included in this course. Examples, exercises and activities provide a real-life context to help students grasp mathematical concepts. Technology is utilized throughout the course.
Course Outline: Class begins with a brief discussion concerning the homework assignment. The lesson will consist of notes with examples, teacher lecture, and class discussion.
Core Performance Standards: Upon completion of the course, students should be able to:
Required Materials: Students will be required to bring to class daily:
1)Textbook
2)Notebook (preferably a three ring binder) devoted solely to this course
3)Pencil (Homework done in pen will result in a grade of 0.)
4)A graphing calculator (TI-83 or TI-84 is recommended)
Student Evaluation:Grades will be based on the Duval County Grading Scale as follows:
90-100: A80-89: B70-79: C60-69: DBelow 59: F
Grades will be broken into three categories: (1) Tests, (2) Quizzes, (3) Homework
All assignments are given a point value. Final averages will be determined by dividing points earned by points possible. For example, if you have earned 315points out of a possible 395 points, your grade would be , which is equal to 0.797, or 79.7% or C.Averages will not be "rounded up".
Tests: Tests will be administered periodically throughout each grading period to see which concepts have been mastered. Each test may contain problems from previously tested material. No test will be dropped at the end of a grading period.Tests will be done in ink.Appropriate work must be shown.
Quizzes: Quizzes may also be given throughout a grading period. Point values will vary depending on the length and content.Appropriate work must be shown.
Homework: Homework is assigned daily. Doing assignments regularly is essential to being successful in this course. Homework is due at the beginning of the period. Late work will not be accepted.A homework paper is worth 5 points. (On rare occasions, the paper could be worth 10 points.)
Points are assigned based on the following (all work must be shown):
at least 90% complete:5 points
at least 80% complete:4 points
at least 70% complete:3 points
at least 60% complete:2 points
at least 50% complete:1 points
less than 50% complete:0 points
If you are absent the day a homework assignment is due, you must turn it in the next class period you are present. If you are absent the day homework is assigned, it is the student's responsibility to get the make up from my website and turn it in within one day.
If a student is absent from theirThis includes field trips.Students may come to Room 211 before school, between classes, or before they leave campus to turn in the work.They may also ask the office staff in the main office to place it in my school mailbox.
Heading for Homework(One point will be deducted for incorrect headings)It will be written on the side whiteboard.
Write in the upper right-hand corner the following:Last name, first name
Date the assignment was assigned
Period
what work is being turned in [e.g, Pg 254 (2-40even)
Class Participation: This involves more than just showing up for class. I expect you to be here physically and mentally. Participation includes coming to class regularly and on time, completing assigned work, using your time wisely, being respectful to others, and answering when called upon. It is important to pay attention in class and ask question as soon as you become confused. Understand, I cannot stop teaching the concepts of pre-calculus andteach a lesson on algebra 1 and 2 concepts (That may require coming to a help session.), but I can quickly re-fresh your memory on a concept if you get confused during the lesson.
Attendance/Tardies: Being in class is essential to learning. I expect you to be in your seat when the bell rings, with the homework out, the textbook open and ready to begin class.You miss important information if you are not in class.You will be held accountable for all standards.
In order to earn a passing quarter grade in a course, any student who misses more than four classes, not including school-related absences*, will be required to pass a comprehensive quarter exam or project, in addition to having an overall passing grade for the quarter. A student must be in class for at least half the period to be considered present.
*It is the responsibility of sponsors supervising field trips and other school-related activities to provide a list of students who missed class for these reasons to all faculty and the attendance clerk.
Make-Ups: If you are absent the day a test or quiz is given you must do a make-up test/quiz to receive credit.
**Tests are to be made up before school.You have until 8:15 to take the test, so the earlier you arrive to school, the more time you will have.I arrive to school by 6:45AM.
**Quizzes are to be made up before school.They must be made up the day a student returns to school, or make an appointment with Ms. Hershey for another day.
**Failure to make up tests and quizzes within the timeframe will result in a grade of "0".
If a student is absent from his/herStudents may come to Room 211 before school, between classes, or before they leave campus.This includes field trips.
Conferences: Parents/guardians can call the guidance office at 693-7583 to schedule a conference. If you would like to reach me by phone, call the main office (693-7583) then dial extension 158 and leave me a message. Also, you can e-mail at hersheys@duvalschools.org .
Safety Net:Help outside of the school day is available from me on Mondays from 7:30am to 8:10am .
On Tuesday-Friday, if you have a quick question, I am usually in Room 211 to answer your questions.If you need a more in-depth explanation on a concept not mastered, I can help you on Monday mornings.
Student Integrity and Decorum: It is the responsibility of the student to be an active participant in the learning process and to uphold the academic integrity policy. All work turned in by a student should be his/her own. It is also expected that students behave according to and adhere to the Code of Student Conduct. Giving or receiving help on tests or quizzes or copying homework is cheating. Calculators are not to be shared during tests. Cheating will result in an automatic zero for the assignment assessment as well as a referral to the appropriate administrator.
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11th Grade: Logarithmic Equations Word Problems
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Eleventh Grade: Logarithmic Equations Practice Questions
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I hope it will teach me the fundamentals for 11th grade and 12th grade Math. I am helping a student with her 11th grade math and will be using this website as a refresher. Because i am in 9th grade n im supposed 2 be in the 11th grade and i was told this could catch me up. I teach resource math 6th grade-11th grade and am always looking for sources. 11th grade math is hard, need examples of solved problems. As a math coach I am always trying to improve upon my own skills before implementing ideas in staff development. I also have an 11th grader who needs my help.
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I am a math-major bachelor student. And I want to get some advice about the approach I'm trying now for learning maths, not for efficiency, but for depth and fully-mastered.
Firstly, I want to know how does it look like when the real mathematicians were learning a subject. Because recent a few months when I'm reading the Calculus and Linear Algebra textbooks, I frequently feel sort of unsafe or uncomfortable by just read, try to understand what the author saying and finish the exercises. Then, I tried to do it in another way, close the book, and try to establish all the staffs on the blank papers. During this procedure, there're really many obstacles and difficulties, when I'm confused with a problem for more than ,say, half-hour then refer to book to find ideas. But after all these, when you see you yourself establish not entire but still a small theory-building on papers, then we call it comfortable and safe feelings coming out.
Secondly, the approach used above, I feel it improves some creative feeling or say to be easier to connect different concepts/theorems, better than the classical approach I used before as most classmates do, which is only reading books and doing exercises after each chapter. But apparently, I don't mean doing exercises is not important, it's very important I think. Just like playing music instruments, eg. for me when playing flute, you can't enjoy or even play an advanced music if you don't have extremely strong skills for fundamental fingering 'rule' , same as maths. But the point here is , I don't like or say adjusted with the classical learning approach. Same example with playing flute, in an classical way when a student learns to play flute, just as in ordinary music training school, they learn basic music knowledge, and train basic fingering for a very long time, and from like 'do do do re re re mi mi mi' to other harder permutations, to simple rhythm, to easy music, to... yeah, it's systematic and maybe efficient, but I really don't like this way, when I tried to learn flute in a very first time, I used this classical way but just for a few days, it bored me a lot and cannot even make it sound. But after a few months, I tried my own way, skip all the stuffs, just find a fingering table and a real my favorite music notation, and just try to play directly, at first place I tried every sound(symbol) one by one slowly. But it speed-up dramatically, I remember at that time only after 2-3 hours, I can just play that music, even though it's still slow when try a new music. And by this approach, more than a dozen of songs could be played by flute. So, I think this approach is somehow similar to when a baby try to learn language or something, the babies, they don't have systematic taught, but just be around or say inside the natural, chaotic, complex world directly. Therefore, I'm thinking whether it's also could be perfect way in maths. Because even though I tried this approach for 3 maths courses and I see the similar good results(especially when talking in details with classmates during discussion-session, it's fluent and comfortable to explain materials learned by this way.) , but I still want to know how some other people think whether this approach would be great to improve creative or imaginative ability in the long run, especially ones who are doing research in maths.
Thirdly, connected with the approach above, when dealing with the detail material like definition/theorems, how the real mathematician do that. I mean, when I encounter with a definition or theorem, I always try to 'see' a kind of 'image' in my mind, sometimes corresponding geometric viewing, sometimes algebraic expression, sometimes like 'invisible image', in order to really understand what it really means. For first example, when learning with mean value theorem, first is more geometric, if you get two pairs $(a, f(a)), (b, f(b))$ one could easily calculate the slope. And 'in between', if changing-rate has positive increment, then in order to reach the end point finally, there must be somewhere has negative increment to balance, since it's continuous from positive(negative) to negative(positive), it must cross the slope somewhere, and in this way, it's obvious to see it's a generalization of rolle's theorem which is only special case when $f(a)=f(b)$. But when proof, it's not geometric anymore, I could only imagine the algebraic expression in my mind and try it step by step. And especially, I don't know why, 'imagining' proof or theorem or definition in mind is easier to get 'big-picture' feeling than only writing it down on papers. For second example, when it comes to say Schwarz-inequality, especially proof, I could only 'see' algebraic reasoning, in one way, to get some expression contains both $AB$ and $\|A\|, \|B\|$, then we think a right triangle $AOB$, there could be projection for $A$ to $B$, combined with orthogonal property for dot product $(A-tB)B=0$ to get $t=\frac{AB}{BB}$, then we could construct $A=A-tB+tB$ through Pythagoras theorem we could prove the Schwarz-inequality.
1 Answer
Indeed, this is a very long question. Maybe you get more to the point if you expect answers. Anyhow. Let me point out some things.
In mathematics you should probably have some interaction with a more experienced person, i.e., teaching assistant, professor, etc. just to be sure you are on the right track. For some people it is difficult to find the right balance between formalism and intuition or just to understand what a proof in mathematical practice really is. So, talk to people (ideally people more experienced than you) and have your exercises corrected etc.
Something that worried me slightly is that you said that if you can't solve a problem after half an hour or so, you turn to books and other sources to get some ideas. I find it pretty normal to think about problems for a couple of days, or even months and years (weeks, months, and years are obviously inappropriate for exercises, but not for research problems).
Work on a problem, and if you don't get anything, let it sit for a bit and try again later.
Finally, you usually develop the best understanding of something if you try to develop the theory yourself. I.e., you read some theorem and think about its proof for some time. You will get some understanding of what is really going on. Then you look at the proof, and usually you only have to see some important ideas and you will be able to fill in the details yourself.
This way you only have to remember the crucial ideas, but not every detail.
Also, if you learn something new that builds on things that you learned before,
go back to the old stuff, in your memory, and try to reconstruct why these things worked the way they did. In this way you will learn about the connections
of various results and you get a complete picture.
I have to say though, that in my own research I sometimes look up what I need
and try to understand only the bit that I really need without going through the theory that surrounds it. I simply don't have the time and energy to learn everything that somehow touches my own research. If I notice that some concepts turn out to be important for me, I spend more time on learning this.
I often I just go into the details of a specific proof that seems relevant
and use other thinks as black boxes before I really need them.
This saves some time and energy, which is also important at some point.
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Course Description:Sentential and quantifier logic, axiomatic systems, and set theory. Emphasis is on the development of mathematical proofs. Pre-requisite: grade of C or higher in Math 221.
Text: An Introduction to Abstract Mathematics, by Robert Bond and William Keane, Brooks/Cole, 1999.
Course Goals:
1.The over-arching goal in this course is to learn to read and write mathematical proofs.
2.A second major goal is to be able to communicate mathematical concepts and mathematical proofs.
3.While the course is more about process than content, it is still a goal of this course to consider content which is important to further study of mathematics: logic, set theory, relations and functions, mathematical induction, number theory, and cardinality.
Course Objectives: These are basically the topics we will be covering; throughout, the emphasis will be on reading, writing, and communicating proofs.
This course is a very significant step for every mathematics major, and probably more than any other single course, certainly up to this point in your course work, lets you see what it is like to be a mathematician and whether you can find fulfillment as, or have the ability to be, a mathematician. I certainly have vivid memories of my equivalent course.
All mathematics educators believe in the adage, "Mathematics is not a spectator sport!" In this course, more than in most, however, your contributions will MAKE the course. I intend to allow a healthy fraction of our class time to consist of student presentations of attempted proofs. In addition to learning to read or listen to proofs, with comprehension, as well as learning to problem solve and write proofs, it is important to acquire a disinterested objectivity - like a scientist, we must strive to view a proof in its own terms and not take corrections personally. The "craft" of mathematics is in one sense very personal but at the same time your work must be held to the highest standard of objectivity.
I have chosen this textbook largely for its philosophy; it is designed to be used in a student-oriented class. I will make assignments virtually on a daily basis and it is very important that you work on these as best you can. I hope that at least once a week, on average, each of will get a chance to present some of your work. It is important that we not miss classes; this will be almost a "seminar" course - the size of the group and the philosophy ll allow us to do that.
Because it is important that we learn to WRITE proofs which are correct and clear, as well as to present them orally, I will collect most of the daily assignments and grade them. There will also be two in-class, written, exams - one at the mid-term and one during the final exam time slot; I do believe in seeing what students can do on their own and "under pressure". At the same time, there will be frequent take-home assignments, both daily homework and other problem sets.
Grading: I will use the traditional 90-80-70-60 scale as a framework for assigning grades. I do realize the artificiality of such a scale - but I usually try to make the assignments reasonable enough so that students can earn an appropriate grade. It is harder in this course in particular to make things seem "objective", since writing proofs is something like writing essays in an English course - there is "correct" and "complete", but there is also "elegant" and "insightful".I will try to keep you informed along the way about your progress in the course.
Portfolios: Because this course is all about developing skills, albeit subtle and substantial skills, it is especially important to monitor your development. In fact, we are trying to compile a portfolio of the work of each mathematics major as they work their way through the courses in the major. I would like to ask you to turn in a collection of your best efforts during the course, probably five problems, which can be added to your portfolio. These portfolios should not only give you a chance to pay attention to your progress, but they also will give the department a picture of how well our students are learning what we want them to learn.
Journals: I realize that sometimes "journaling" is overdone in this era of Journal-as-a-verb, but I do think it is important that each of you have a vehicle for thinking about your performance in the course and for letting me know how things are going. Consequently, I'd like to ask you each to turn in a journal entry, at least a few paragraphs, every two weeks - I don't want to make the task too onerous but I also want to stay on top of things. So let's aim for Mondays: Jan 29, Feb 12, Feb 26, Mar 19, Apr 2, Apr 17, Apr 30. I'll award 5 points for each of these
The Schedule: For many courses I teach, I construct a fairly detailed schedule, with assignments and material to be covered each day throughout the semester. For several reasons, however, this time we will be playing it a bit more by ear. It is difficult for me to know just how much text material we can comfortably cover (I do hope to at least cover 5 chapters, perhaps 6). The emphasis I want to give to looking at your work also requires a certain flexibility in schedule - and it is difficult to say just what effect this will have. I do think it is much more important that you LEARN the material than that we "cover" it.
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Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.
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Challenges for Teaching Electricity and Electronics, Version 4.0g
Source: ETCAI Products
Challenges for Electronics is a suite of seven educational programs for electricity, electronics and math. The titles of the seven programs are Basic Circuits Challenge, DC Circuits Challenge, AC Circuits Challenge, Trigonometry Challenge, Digital Challenge, Solid State Challenge and Power Supply Challenge. Each program contains several interactive activities. Activities for both circuit analysis and troubleshooting are included. The programs grade and correct all student work immediately. Student scores can be stored on diskettes or printed. The material is suitable for use as a supplement to classroom or tutorial instruction. The programs can also be used as a refresher course for employees who have already had basic electrical or electronics training. The programs run for ten executions of any time length. After ten executions, the programs become limited capability demos
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Maths and Numeracy
Mathematics is a subject that all students need. In addition to the acquisition of vital numeracy skills needed for the future, it is a subject that should be enjoyed. In developing the application of maths, necessary problem and logic skills are also developed. All students will find one area of mathematics that they enjoy. It is our goal to ensure that this enthusiasm and confidence in that area is transferred into all aspects of mathematics.
It is key that all students develop logical reasoning and problem solving skills. The Year 7 and 8 scheme of work builds on these skills and includes regular project and themed approaches to study ranging from designing a house and budgeting to making a spectacular sports car. In addition to a more engaging approach, we focus on independency and developing students to assess their own work and build a sound knowledge to guide them to further progress.
At KS4 we offer a personalised approach depending on a students' individual mathematical ability:
GCSE Mathematics (linear)
GCSE Mathematics (modular)
Our absolute aim is to ensure that every student has achieved his/her potential by the end of Year 10, with GCSE studies starting in Year 9.
At KS5, we offer both A Level Maths and AS Further Maths, covering the modules of Core, Mechanics, Statistics, Decision and Pure Maths - subjects that are the key areas needed for any students wishing to go further with their mathematical studies at university.
Please note: Sixth Form students have to re-take GCSE Maths if they have not achieved a C grade
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Temporarily out of stock
Elementary Statistics : A Step-by-Step Approach
Allan Bluman explains the basics of statistics in an intuitive and non-theoretical way, using worked examples and step-by-step instructions. This edition places more emphasis on conceptual understanding and Excel, MINITAB and other computing technologies.
Allan G. Bluman is Professor of Mathematics at Community College of Allegheny County, near Pittsburgh. For the McKeesport and New Kensington Campuses of Pennsylvania State University, he has taught teacher-certification and graduate education statistics courses. Prior to his college teaching, he taught mathematics at a junior high school. Professor Bluman received his B.S. from California State College in California, Penn.; his M.Ed. from the University of Pittsburgh; and, in 1971, his Ed.D., also from the University of Pittsburgh. His major field of study was mathematics education. In addition to Elementary Statistics: A Step by Step Approach, Third Edition, and Elementary Statistics: A Brief Version, the author has published several professional articles and the Modern Math Fun Book (Cuisenaire Publishing Company). He has spoken and presided at national and local mathematics conferences and has served as newsletter editor for the Pennsylvania State Mathematics Association of Two-Year Colleges. He is a member of the American Statistical Association, the National Council of Teachers of Mathematics, and the Mathematics Council of Western Pennsylvania. Al Bluman is married and has two children. His hobbies include writing, bicycling, and swimming.
List price:
Edition:
5th 2004
Publisher:
McGraw-Hill Higher Education
Binding:
CD-ROM
Pages:
N/A
Size:
5.50" wide x 7.40" long x 0.05
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Algebra ½ covers all topics normally taught in prealgebra, as well as additional topics from geometry and discrete mathematics (used in engineering and computer sciences). With Algebra ½ , students can deepen their understanding of prealgebraic topics. Algebra ½ includes: instruction and enrichment on such topics as compressions, approximating roots, polynomials, advanced graphing, basic trigonometry, and more. [via]
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Mathematics All Around Plus MyMathLab Student Access Kit
Average rating
4 out of 5
Based on 5 Ratings and 5 Reviews
Book Description st... More students understand the math, not just get the correct answers on the test. Useful features throughout the book enable students to become comfortable with thinking about numbers and interpreting the numerical world around them.
Problem Solving: Strategies and Principles; Set Theory: Using Mathematics to Classify Objects; Logic: The Study of What's True or False or Somewhere in Between; Graph Theory (Networks): The Mathematics of Relationships; Numeration Systems: Does It Matter How We Name Numbers?; Number Theory and the Real Number System: Understanding the Numbers All Around Us; Algebraic Models: How Do We Approximate Reality?; Modeling with Systems of Linear Equations and Inequalities: What's the Best Way to Do It?; Consumer Mathematics: The Mathematics of Everyday Life; Geometry: Ancient and Modern Mathematics Embrace; Apportionment: How Do We Measure Fairness?; Voting: Using Mathematics to Make Choices; Counting: Just How Many Are There?; Probability: What Are the Chances?; Descriptive Statistics: What a Data Set Tells Us
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Calculus for the 21st Century
Philosophy of the Course
Traditionally, calculus has been presented from an analytical point of view often devoid of meaningful applications. Calculators and computers, with their powerful numeric, graphic, and symbolic tools, provide new opportunities for taking a multiple representation approach to the study of calculus. In particular, greater use of visualization, approximation, and prediction can be made in calculus instruction.
A modern calculus course should foster in students an appreciation and skill that allows them to apply their mathematical knowledge in a variety of practical situations. Ideally, successful completion of the course would provide students with the ability to pick up a newspaper and recognize the calculus that surrounds them. We hope students will apply their knowledge to situations that they face in their everyday lives.
Calculus students should look back on their learning experience with favor and in such a way that they desire to continue their pursuit of mathematics. Through technology, students may now take an active role in their learning. We are now able to create an environment which is rich with technology, nurtures curiosity, and promotes action. Mathematics is an experimental science and should be treated as such. Therefore, employing lesson plans that include laboratory activities, discovery exercises, individual projects, applied problems, writing exercises, and open-ended questions should be an integral part of the course.
A Sampling of the Course
It is not our purpose to detail a first year calculus course. However, the following illustrates an approach to calculus that utilizes hands-on experience and technology. This approach makes learning functions, limits, continuity, derivatives, integrals, approximation, and their applications a more enriching experience for the students and teachers.
LIMITS AND CONTINUITY
Limits are critical to the study of calculus. While the development of a rigorous definition is necessary, formal proofs may be de-emphasized in a first course. Limits should be approached numerically, graphically, and analytically. Graphing calculators are wonderful tools to help develop a clear-sighted concept of limits. An intuitive understanding of the e and d definition can be explored using the idea of local linearity.
Looking at problems like
(1 + )x
both numerically and graphically greatly enhances understanding. Introducing L'Hopital's Rule early in the course is desirable. Students need to examine, graphically and analytically, the relationship between left and right hand limits, continuity, and local linearity.
DERIVATIVES AND THEIR APPLICATIONS
Because calculus is the study of change, the derivative and anti-derivative continue to be the focal point of this study. The definition of derivative and the relationship between differentiation and continuity must be emphasized as well as important theorems like The Mean Value Theorem. The derivatives of polynomial, rational, trigonometric, exponential, logarithmic and piece-wise functions must be studied. In addition, students need to have a working knowledge of implicit differentiation, logarithmic differentiation, the chain rule, product rule and quotient rule. Numerical estimates of the derivative should also be emphasized.
Applications of the derivative should include related rates, maximum/minimum, and motion problems. Questions in these areas should be realistic and focus on applications. The derivatives and their relationship to slopes, concavity, and the linearization of a curve continue to be important components of calculus. Newton's Method and similar iterative techniques should be part of the curriculum.
INTEGRALS AND THEIR APPLICATIONS
Relating motion and the anti-derivative to area and the Fundamental Theorem of Integral Calculus is a primary goal. Given a rate of change, a student should be able to construct the function. The integral as the infinite sum should be explored by several methods including rectangles, mid-point, trapezoid, and Simpson's Rule. With technology, these methods can be explored without tedious computations.
DIFFERENTIAL EQUATIONS
Differential equations are a common theme throughout a first year calculus course. They provide a wonderful opportunity for students to model real life situations. Numerical methods to solve differential equations, along with associated error analysis, help the student to understand the real world of applied mathematics. It is not necessary to solve differential equations solely by analytical methods when other approaches are just as enriching.
SEQUENCES AND SERIES
Students should have a thorough understanding of geometric series and the concept of estimating functions with infinite series. Graphical relationship, the ratio test, the comparison test, intervals of convergence, and error analysis should be addressed, especially with new technology.
Course Books and Resources
Several major calculus reform projects are currently in progress throughout the country. In selecting a calculus textbook, much consideration should be given to both the use of technology and the curriculum content. Major projects have been undertaken at Duke, Harvard, NCSSM, Ohio State, Oregon State, Smith, and St. Olaf's College among others. Associated with many of these are training institutes which are funded by the NSF. Other institutes and inservice opportunities exist. These programs are exciting opportunities for teachers to learn how to incorporate technology into the teaching of calculus and select curriculum materials based on current thinking in the field. Also, close attention should be paid to periodicals, newsletters and journals as a means for staying current with new ideas, technology, and trends in calculus reform.
Focus on the Future
The power of current and future technology can no longer be ignored in classroom instruction. We are faced with technology that is changing at an exponential pace. Teachers must look at how and what they teach in this environment of change. The availability of technology has caused changes in the curriculum. Some content will receive less emphasis, some content will receive more emphasis, and solutions to problems that were previously inaccessible are now possible. Open-ended problems and mathematical modeling need to be an integral part of the calculus curriculum. Writing and group activities are important to constructing and applying knowledge. Teachers must assume the role of a life-long learner and must convey this role to their students. Modes of assessment, including the Advanced Placement examination will, of necessity, change to reflect the use of technology in instruction.
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to see why it comes out as it does. If they see that this is all perfectly logical, and they know how to do it, the day is a big success—and in fact, this sort of justifies the whole unit on matrices.
As a final point, mention what happens if the equations were unsolvable: matrix A will have a 0 determinant, and will therefore have no inverse, so the equation won't work. (You get an error on the calculator.)
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SUPPLIES: Texas Instruments TI-83 Graphing
Calculator (note: If you are purchasing a calculator for this class, you are
required to purchase the TI-83. If you already have a graphing calculator,
consult your instructor about its acceptability)
EXPECTED STUDENT COMPETENCIES TO BE
ACQUIRED: The
successful student, at the end of the course, will be able to produce
well-written correct solutions for problems similar to those assigned for
homework in this course.
COURSE OBJECTIVE: To solve, both graphically and by
calculation, mathematical problems that involve:equations and inequalities, graphs,
functions, and inverse functions, polynomial, logarithmic, exponential, and
trigonometric expressions.
ASSIGNMENTS: Homework will be assigned daily and may
occasionally be collected as a check on how you are keeping up. Although most
of the homework assignments will not be collected, that doesn't mean you don't
have to do it! A major part of learning mathematics involves DOING mathematics! Also, homework is useful
in preparing for the type of questions, which may appear on quizzes or
exams.Many homework problems will be
given on quizzes and some on tests.
Evaluations:There will be given three tests and one final exam during the
semester.There will also be given
quizzes once a week approximately.It is
important that you work all of the assigned homework problems to practice for
quizzes, tests and the final exam. Also rework at home examples done in class.Some of these examples or homework problems
might given on a quiz, a test or the final exam.
Here are tentative dates for
the tests:
Test 1
Week of September
25-29
Test 2
Week ofOctober 23-27
Test 3
Week of Nov.
27-Dec. 1
Final Exam: The
final exam will be given on Wednesday December
13 at 11:00 AM.The final exam is comprehensive.
GRADING
The weights
of the various components of your grade in determining your final course grade
are shown below, along with the grade scale for the course.
WEIGHTS:
GRADE SCALE
1. Three Tests : 300 points) (100 each)
90-100
A
70-75
C
2. Quizzes, homework 150 pts
86-89
B+
66-69
D+
3. Cumulative Final Exam 150 pts
80-85
B
60-65
D
76-79
C+
0-59
F
NOTES:
Two quiz/homework
grades will be dropped to determine your final quiz/homework average.There will be no makeup quizzes.There will be no makeup tests, except under
special (documented) circumstances.In
the case you cannot take an exam at the scheduled time, contact the instructor
as soon as possible after (or before) the test, to arrange a make up.Exams not made up within one week of the
scheduled exam date will be recorded 0.
SPECIAL NOTES: If you have a physical, psychological,
and/or learning disability which might affect your performance in this class,
please contact the Office of Disability Services, 126A B&E, (803) 641-3609, and/or see
me, as soon as possible. The Disability Services Office will determine
appropriate accommodations based on medical documentation.
ATTENDANCE POLICY: I may occasionally take attendance. It
is highly recommended that the student not miss any class.However, the Attendance Policy established by
the Department
of
Mathematical Sciences states that the maximum number of unexcused absences
allowed in this class before a penalty is imposed is four for a regular
semester.
ACADEMIC CODE OF HONESTY: Please read and review the Academic
Code of Conduct relating to Academic Honesty located in the Student Handbook.
If you are found to be in violation of this Code of Honesty, a grade of F(0)
will be given for the work. Additionally, a grade of F may be assigned for the
course and/or further sanctions may be pursued.
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53
ESL/ELL
Additionally, the course includes additional practice activities (such as cloze activities), as well as pre-topic vocabulary lists, that introduce key vocabulary in English and in SpanishA calculator with the normal functions. A scientific calculator with exponent and square root keys would be most helpful; if students do not have one, they should check the accessories on their computerThe purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics in a year long algebra course. Topics included are real numbers, simplifying real number expressions with and without variables, solving linear equations and inequalities, solving quadratic equations, graphing linear and quadratic equations, polynomials, factoring, linear patterns, linear systems of equality and inequality, simple matrices, sequences, and radicals. Assessments within the course include multiple-choice, shortanswer, or extended response questions. Also included in this course are self-check quizzes, audio tutorials, and interactive games• Read, write, evaluate, and understand the properties of mathematical expressions including real numbers, radicals, and polynomials
• Add, subtract, multiply, and divide radical expressions, polynomials, and polynomial expressions
• Read, write, solve, and graph linear and quadratic equations and inequalities
• Students will solve absolute value equations and inequalities
• Work effectively with ratios and direct and inverse variation
• Solve systems of linear equations and inequalities
• Work with arithmetic sequences and linear patterns
• Understand basic statistics including measures of central tendencies and box plots
• Understand different
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Mathematics
Apply the central concepts of Mathematics and Logic to any career path.
If you like solving puzzles and figuring things out, then our mathematics major may interest you. Applications of mathematics are everywhere and a strong background in mathematics can help you in many different careers.
Ashland's Mathematics program is taught in small classes that allow you to explore and learn the language of numbers in-depth.
You will have many opportunities to participate actively in class and to get to know your professors well. Your professors have earned three awards for excellence in teaching, a testimony to their exceptional classroom skills and the strong mentoring relationships they develop with students.
What You'll Love About the Mathematics Program:
Unlike Mathematics programs in large universities where you're no more than a face in a crowd, at Ashland you get to know each of your professors well and benefit from their knowledge of Mathematics.
All classes are taught exclusively by our Mathematics professors and not by graduate students or teaching assistants.
You have many opportunities to participate in mathematics challenges and contests such as programming competitions and activities made available through the Ohio conference of the Mathematical Association of America (MAA).
The department actively participates in the national Mathematical Association of America (MAA) as well as in the Ohio section of the organization, attending conferences and presenting papers and workshops.
As a student you will have opportunities to make presentations at the Ohio section of MAA and at national meetings.
You have full access to the extensive computer resources within the department's technology center including a variety of hardware running Linux, Solaris, MacOS and Windows operating systems. The resources also include computer algebra systems, statistical and geometric software and other applications that will facilitate your learning.
Reach Your Career Goals
In addition to pursuing graduate school, our graduates are well prepared to begin careers as:
Actuaries in insurance companies
Operations and research analysts
Quality control engineers
Mathematics consultants
In addition, the analytical and logical abilities developed through the program equip you to pursue further study in areas such as business, law or medicine.
Outstanding Educators
Faculty members are very active in the field of mathematics often attending regional and national meetings with student groups.
Professors are excellent classroom educators who mentor students both in and out of the classroom through problem solving challenges and in-depth discussions of math topics.
Organizations for Mathematics Majors
Academic Department Info
Explore the language of numbers in small class settings.
Mathematics is the doorway to science and technology. Computer Science is the study of algorithmic processes to ultimately...read moreMeet our Faculty...
Also Solved by Paul S. Bruckman: An Interview by Dr. Thomas P. Dence, Professor of Mathematics Read More
Career Outlook for Mathematics Majors
Mathematics training often leads to careers in actuarial science, statistics, engineering and physics. Those with a bachelor's degree and proper licensure often become teachers of mathematics. Many mathematicians work for the federal and state governments including the Department of Defense. Private sector employers include research and development companies, technical consulting firms and insurance companies. Experts anticipate average job growth of about 10 percent between 2006 and 2016.1 Learn More about careers for mathematics majors!
What Students Say About Ashland
"Ashland University has played a large role in developing my character, providing me with the opportunities, resources, and experiences I needed to become a scholar, a leader and servant to others." -- Rachel Cordy from Elyria, Ohio, involved in Alpha Phi sorority, Judicial Board, Math Club, Orientation Team and Kappa Delta Pi.
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Course provides an additional hours of support to developmental courses that require it. Content includes classroom activities which vary depending upon instructor's methods of addressing student need and specific mathematics course.
IV. Learning Objectives
Students will learn techniques and/or applications related to the specific topic of the tandemMethods of presentation can include lectures, class discussions, and individual and group assignments. Calculators / computers will be used when appropriate. Course may be taught as face-to-face, media-based, hybrid or online course.
VIII. Course Practices Required
(To be completed by instructor)
Reading of the text and/or handouts is required as a reference to the materials and the techniques under study.
Quizzes, examinations, final examination, individual and group
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Title: Patterns and Functions Description: Introduces various types of numeric and geometric patterns. Also includes lessons on function tables with one and two operations. Source: Teacher's Helper: Intermediate. April/May 2010. p.15-25
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Sequences with the TI-83 is a "tutorial (that) shows how to create tables and graphs for recursively defined functions and their corresponding explicit functions." Created through the University of Illinois Math teacher link.
Written by Tacoma Community College math faculty member Scott MacDonald, this tutorial is designed to help TCC students learn to operate their TI-83. Wait for file to load. Then select magnifying glass to enlarge.
This website is designed to develop graphing calculator skills for students who use Ti-83 or Ti-84 graphing calculators. This site is being developed by a permanently certified New York State mathematics teacher and uses functions from the calculator that he has found to be useful in helping his students check their problems.
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Next: Polar Coordinates
Previous: Real-World Triangle Problem Solving
Chapter 6: Polar Equations and Complex Numbers.
Chapter Outline
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Chapter Summary
Description
In this chapter, students will plot points in a polar coordinate system, graph and recognize limaçons and cardiods, and work with real-world applications involving polar coordinates and polar equations.
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Set theory permeates much of contemporary mathematical thought. This text for undergraduates offers a natural introduction, developing the subject through observations of the physical world. Its progressive development leads from concrete finite sets to cardinal numbers, infinite cardinals, and ordin... read more
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Axiomatic Set Theory by Paul Bernays A historical introduction by A. A. Fraenkel to the original Zermelo-Fraenkel form of set-theoretic axiomatics, plus Paul Bernays' independent presentation of a formal system of axiomatic set theory.
An Introduction to Algebraic Structures by Joseph Landin This self-contained text covers sets and numbers, elements of set theory, real numbers, the theory of groups, group isomorphism and homomorphism, theory of rings, and polynomial rings. 1969Sets, Sequences and Mappings: The Basic Concepts of Analysis by Kenneth Anderson, Dick Wick Hall This text bridges the gap between beginning and advanced calculus. It offers a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces. 1963 edition.
Lattice Theory: First Concepts and Distributive Lattices by George Grätzer This outstanding text is written in clear language and enhanced with many exercises, diagrams, and proofs. It discusses historical developments and future directions and provides an extensive bibliography and references. 1971 edition.
Product Description:
Set theory permeates much of contemporary mathematical thought. This text for undergraduates offers a natural introduction, developing the subject through observations of the physical world. Its progressive development leads from concrete finite sets to cardinal numbers, infinite cardinals, and ordinals. Although set theory begins in the intuitive and the concrete, it ascends to a very high degree of abstraction. All that is necessary to its grasp, declares author Joseph Breuer, is patience. Breuer illustrates the grounding of finite sets in arithmetic, permutations, and combinations, which provides the terminology and symbolism for further study. Discussions of general theory lead to a study of ordered sets, concluding with a look at the paradoxes of set theory and the nature of formalism and intuitionalism. Answers to exercises incorporated throughout the text appear at the end, along with an appendix featuring glossaries and other helpful information
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Summary:
In a linear program, we are given constraints of the form a_i1*x_1 +
a_i2*x_2 + ... + a_in*x_n <= b_i, and we are asked to find the x_j that
maximize an objective c_1*x_1 + c_2*x_2 + ... + c_n*x_n, subject to the
constraints. In an integer (linear) program, the x_j must be integers. In
a mixed integer (linear) program, only some of the x_j must be
integers.
Surprisingly many optimization problems can be modeled as linear or integer
programs. In this course, we will study how to model problems as linear or
integer programs, and study basic methods (algorithms) for solving
them.
Relevance to computer science theory and AI:
In spite of the strong algorithmic component of linear and integer
programming, for historical reasons, much of the development of the
techniques for these problems has taken place outside the computer science
community. Because of this, computer scientists in general are perhaps
less aware of them than they should be. Computer science theorists tend to
be familiar at least with the techniques that allow for proving
polynomial-time solvability, but are perhaps less interested in the
(worst-case exponential-time) techniques for solving integer programs to
optimality. Among AI researchers, the familiarity is perhaps even lower
(with the exception of some subareas), even though they are generally not
averse to algorithms that require exponential time in the worst case if
they "usually" perform well. Indeed, many of the algorithms for integer
programming are very similar to AI search algorithms.
Nevertheless, computer scientists (both in theory and AI) are increasingly
looking at problems where these methods can be fruitfully applied. For
example, the use of probabilities is becoming more common, which are
continuous quantities that are naturally expressed in linear and integer
programs. Moreover, both in theory and AI, there is increasing interest in
problems from economics and game theory, which often lend themselves
especially well to formulation as a linear or integer program.
You may discuss assignments with at most one other person. Each person must
do her or his own writeup. Moreover, you may not simply write down a
solution and give it to the other person. You may present things to each
other on (say) a whiteboard, but the other person should not simply be
copying things over. You should acknowledge everyone you worked with, as
well as all other sources, on the writeup. Assignments are always due
immediately at the beginning of class.
Projects
As you can see from the grading, the project is an important part of this
course. You may work on it alone or in a team (please check with Vince
first if you want to have a team of more than 3 people). I imagine that
most projects will consist of using linear/integer programming in some
application domain, perhaps in your own research area. Because of this, we
will start discussing some example applications very early in the course,
so that you can start thinking about how you might apply these techniques
to something that you care about. We will have some checkpoints
(e.g. project proposal) during the course, but feel free to discuss
possible project ideas with Vince earlier.
The goal of the project is to try to do something novel, rather than merely
a survey of existing work. Projects may be theoretical, experimental
(based on simulations), experimental (based on real-world data), a useful
software artifact, or any combination thereof. The only real constraint is
that it has something to do with linear/integer programming. Talk to Vince
if you are not sure about whether something is an appropriate project. The
final product is a writeup (in the form of a research paper) and a
class presentation (all team members must participate in the
presentation). Some projects may well lead to publishable papers (perhaps
with some additional work).
In your project proposal, you should explain the topic of your
project, what types of results you hope to obtain, and what some of the
technical issues are that you will need to address. If necessary, Vince can
help with finding topics. Something related to your own research is
definitely OK as long as it also has something to do with
linear/integer programming.
An intermediate project progress report is also required. This
report should explain what results you have obtained already, what (if any)
difficulties you encountered, and what you plan to do to complete your
project. Ideally, at this point, you should already have some good results,
so that you can spend the rest of your time on answering questions
generated by your results, as well as preparing your writeup and
presentation.
Dates:
Project proposal (1+ pages) due before class March 6.
Project progress report (~2 pages) due April 4.
Final project writeup due April 16 under my door.
Schedule
This is the first time I am teaching this course so we will be flexible
with the schedule. One set of lecture notes will not necessarily take one
lecture to finish.
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The Mathematics Department offers courses from the intervention level through Advanced Placement levels. All courses cover the state standards relevant to that course and many cover additional enrichment topics, especially academy and pathway sections. Students also have opportunities to participate in contests throughout the school year and the Math-Science Club meets weekly during lunch.
Math Department Course Offerings
Algebra 1-2
This course is required for graduation and for admission to the University of California or California State University systems. College eligibility requires a grade of C or better.
Students complete one year of college preparatory Algebra in this two semester sequence. Incoming ninth graders are placed in this class based on scores on the placement exam and record from middle school.
This course is required for graduation and for admission to the University of California or California State University systems. College eligibility requires a grade of C or better.
Students take this class when they have successfully completed Algebra 1 and 2. The course covers all of the California State Standards for Geometry. Topics include proofs, congruence, similarity, properties of parallel lines cut by a transversal, polygons, circles, perimeter, area, volume, three dimensional figures, special triangles and polygons such as squares, Pythagorean theorem, basic constructions, coordinate geometry, basic right triangle trigonometry, and transformations. ALHS uses the Michael Serra textbook Discovering Geometry.
Advanced Algebra 1 and 2
This course is required for graduation for the class of 2014 and beyond. It is required for admission to the University of California or California State University systems. College eligibility requires a grade of C or better.
Topics included in this course cover all of the state standards. The topics include solving equations and inequalities using absolute value, solving systems of equations in a variety of methods, performing operations on polynomials, complex numbers,, rational expressions, graphs of quadratic functions, logarithms, exponential functions, conic sections, combinations and permutations, probability involving combinations and permutations, binomial theorem, arithmetic and geometric series, inverse functions, and composition of functions.
Statistics and Probability 1 and 2
This is an introductory high school level course to statistics and probability. Students taking this course should have completed at least through Geometry and Advanced Algebra is highly recommended since topics covered in that course are often used in this course.
This course covers all of the standards listed in the California Framework, but covers many other topics in addition to those listed in the Framework. The course tends to have a hands-on approach so that students can generate their own data in many situations. Topics include linear regression, analysis of one variable data, introductory inference, and probability. Teachers often use a variety of sources for the class, including websites for additional enrichment.
Pre-Calculus 1 and 2
Prerequisites for this course are Algebra, Geometry, and Advanced Algebra. Success in this course is very dependent on students having a thorough knowledge of second year algebra and geometry since both are used and built on throughout the course. Students must have teacher recommendation to take this course and scores on CST math subjects are considered. A diagnostic test is given to determine if placement in this course is appropriate.
This course covers all of the traditional trigonometry topics as well as many analytic geometry topics. The standards listed under Trigonometry and Mathematical Analysis in the California Mathematics Framework are addressed in this course as well as almost all the topics in the Linear Algebra listing in the Framework.
Advanced Placement Statistics 1 and 2
This course is the College Board Advanced Placement Statistics course and includes all of the topics listed on the College Board website. The course has four main components: exploratory analysis, probability, experimental design, and inference. Students enrolling in this course must have fairly high reading and writing ability and must have successfully completed through Advanced Algebra (An A or B in that course is recommended). Teacher recommendation and parent approval is required to be eligible to take this course.
Advanced Placement Calculus 1 and 2
Students must complete Precalculus successfully to enroll in this course. Students must also pass a placement test and have teacher recommendation to enroll. This course is very similar to the UC Berkeley course and assumes students have had some background in Analytic Geometry in addition to Trigonometry before entering this course.
The course covers all of the topics in the College Board syllabus for this course as well as a few additional topics. Students taking this course must take the Advanced Placement exam in May of the year they take the course. Students who pass the Advanced Placement exam are then eligible for college credit at many universities.
How to Contact Faculty & Staff
To the left of Faculty/Staff names click on the icon to send a message to that person through Loop Mail.
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Mathematical models
Mathematical Models
Introduction
Tom and Jane are friends. Tom picked up Jane's Physics test paper, but will not tell Jane what her marks are. He knows that Jane hates maths so he decided to tease her. Tom says: "I have 2 marks more than you do and the sum of both our marks is equal to 14. How much did we get?"
Let's help Jane find out what her marks are. We have two unknowns, Tom's mark (which we shall call tt) and Jane's mark (which we shall call jj). Tom has 2 more marks than Jane. Therefore,
t=j+2t=j+2
(1)
Also, both marks add up to 14. Therefore,
t+j=14t+j=14
(2)
The two equations make up a set of linear (because the highest power is one) simultaneous equations, which we know how to solve! Substitute for tt in the second equation to get:
This problem is an example of a simple mathematical model. We took a problem and we were able to write a set of equations that represented the problem mathematically. The solution of the equations then gave the solution to the problem.
Problem Solving Strategy
The purpose of this section is to teach you the skills that you need to be able to take a problem and formulate it mathematically in order to solve it. The general steps to follow are:
Read ALL of it !
Find out what is requested.
Use a variable(s) to denote the unknown quantity/quantities that has/have been requested e.g., xx.
Rewrite the information given in terms of the variable(s). That is, translate the words into algebraic expressions.
Set up an equation or set of equations (i.e. a mathematical sentence or model) to solve the required variable.
Solve the equation algebraically to find the result.
Application of Mathematical Modelling
Exercise 1: Mathematical Modelling: Two variables
Three rulers and two pens have a total cost of R 21,00. One ruler and one pen have a total cost of R 8,00. How much does a ruler costs on its own and how much does a pen cost on its own?
Solution
Step 1. Translate the problem using variables :
Let the cost of one ruler be xx rand and the cost of one pen be yy rand.
George owns a bakery that specialises in wedding cakes. For each wedding cake, it costs George R150 for ingredients, R50 for overhead, and R5 for advertising. George's wedding cakes cost R400 each. As a percentage of George's costs, how much profit does he make for each cake sold?
Summary
Linear equations
A linear equation is an equation where the power of the variable (letter, e.g. x) is 1(one).
Has at most one solution
Quadratic equations
A quadratic equation is an equation where the power of the variable is at most 2.
Has at most two solutions
Exponential equations
Exponential equations generally have the unknown variable as the power.
ka^(x+p) = m
Equality for Exponential Functions
If a is a positive number such that a > 0, then:
a^x = a^y
if and only if:
x=y
Linear inequalities
A linear inequality is similar to a linear equation and has the power of the variable equal to 1.
When you divide or multiply both sides of an inequality by any number with a minus sign, the direction of the inequality changes.
Solve as for linear equations
Linear simultaneous equations
When two unknown variables need to be solved for, two equations are required and these equations are known as simultaneous equations.
Graphical or algebraic solutions
Graphical solution: Draw the graph of each equation and the solution is the co-ordinates of intersection
Algebraic solution: Solve equation one, for variable one and then substitute it into equation two.
Mathematical models
Take a problem, write equations that represent it, solve the equations and that solves
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Other Resources
Mathematics
Mathematics
Mathematics
The goal of the Mathematics Department is to prepare students for college and to be successful in their careers and everyday life using mathematics in rigorous and challenging ways. We offer the following courses on four levels: Honors (H), Advanced Placement (AP), International Baccalaureate (IB), and Mentally Gifted (MG).
ALGEBRA I (H, MG): 9th grade.
Students will study Real numbers and their properties: algebraic expressions; equations and inequalities in one variable; polynomials including operations and factoring; operations with rational expressions; linear equations and functions; operations with irrational numbers; quadratic equations; systems of equations; and problem solving.
An extension of Algebra II and trigonometry; analytic geometry; functions and their inverses; graphing; logarithmic and exponential functions; complex numbers; radian and degree measure of angles; polar and rectangular coordinates; six trigonometric functions and their inverses; graphs; identities; trigonometric equations; formulas for the sum and difference of two angles, for double angles; law of sines and cosines; operations of vectors and complex numbers. Prerequisites: Algebra I, II and Geometry, grade of B or higher in Algebra II.
CALCULUS (H): 11th or 12th grade.
Topics include properties of functions; limits; the derivative and applications; anti-derivatives, integrals, the definite integral and applications. Prerequisite: 85 or higher grade in Precalculus, permission of the Department Head.
AP CALCULUS AB: 11th or 12th grade.
This course will follow the prescribed advanced placement college curriculum level AB, including: functions, graphs, limits, derivatives, applications of integrals, Fundamental Theorem of Calculus, anti differentiation, polynomial approximations and series. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions. Prerequisites: Algebra 1, 2, Geometry, Precalculus.
IB MATHEMATICS SL: 11th and 12th grades (a 2 year course).
Mathematics SL course is a two-year course designed for students needing a strong background in mathematics as they prepare for entrance into college and continued studies in the sciences, business, or engineering. The curriculum will fully integrate the treatment of Algebra, Functions and equations, Trigonometry, Matrix Algebra, Statistics and Probability, Vectors, and elements of basic first year Calculus. The curriculum will include mathematical modeling of concepts such as data collection, prediction, and simulation. Elements of Calculus will be covered from Limits through Differential and the beginnings of Integral calculus. A portfolio is required. Prerequisites: Algebra 1, 2, and Geometry.
IB MATH STUDIES SL: 12th grade.
The course aims to cover a one-year (150 hours) mathematics curriculum in Group 5 at the standard level. Also, this course aims to enable candidates to experience international mathematical topics, enjoyment and appreciation for various dimensions of mathematics culturally, aesthetically, historically, creatively, generally, technically, and scholarly. Students completing this course will be equipped with fundamental skills and a knowledge of basic processes which can be applied to multiple disciplines, general real world situations, and internationally. In addition, students will complete a project comprised of an investigation of a self-selected topic using mathematical skills learned in the course. Prerequisites: Algebra 1, 2, and Geometry.
Explore math through daily life situations. For example, put your decision-making skills to the test by deciding whether buying or leasing a new car is right for you, and predict how much money you can save for your retirement by using an interest calculator.
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This handbook is designed as a work of reference, and provides a convenient source of basic definitions and formulae for use throughtout the three blocks. In addition to this it also contains a quick reference guide to Maple commands; and summaries of the main concepts, definitions and techniques of each of the units.
Purchase Mystery Math: A First Book of Algebra by David A Adler,Edward Miller and Read this Book on Kobo's Free Apps. Explore Kobo's Vast Collection of eBooks - Over 3 Million Titles, Including 2 Million Free Books!
Develops algebraic ideas in a logical sequence, and in an easy-to-read manner, without excessive vocabulary and formalism. This text reinforces the following common thread: learn a skill; practice the skill to help solve equations; and, then apply what you have learned to solve application problems.
Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The first half of the book deals with classification of the finite dimensional simple Lie algebras and...
A Collection of Problems in Algebra with Solutions, Book, 2 Matrices and Vector Spaces
The Title "A Collection of Problems in Algebra with Solutions, Book, 2 Matrices and Vector Spaces" is written by E.F. Robertson. This book was published in the year 1984. The ISBN number 0521272866|9780521272865 is assigned to the Paperback version of this title. This book has total of pp. 112 (Pages). The publisher of this title is Cambridge University Press . We have about 80746 other great books from this publisher. A Collection of Problems in Algebra with Solutions, Book, 2 Matrices and Vector...
Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way of a series of 164 problems, each with hints and, at the back of the book, full solutions. This book...
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This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.
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In this book the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will also want this book for their library since...
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They form a part of the field of algebraic logic, and have applications in proof theory, modal logic, and computer science. This research text uses combinatorial games to study the fundamental notion of representations of relation algebras. Games allow an intuitive and appealing approach to the subject, and permit substantial advances to be made. The book contains many new results and proofs not published elsewhere. It should be invaluable to graduate students and researchers interested in relation...
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The mathematics is presented in contexts relevant to students' studies, linking maths to other areas of the curriculum with a strong focus on integrating ICT. This work includes advice on developing the assessed portfolio and practice exam questions at the end of each chapter. Full colour comprehensions provide practice for the Applying Maths exam.
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Can you be a bit more precise about history of exponents in algebra ? I may perhaps be able to help if I knew some more . A proper software can help you solve your problem instead of paying for a algebra tutor. I have tried many math program and guarantee that Algebra Buster is the best program that I have come across . This Algebra Buster will solve any math problem that you enter and it also make clear every step of the – you can exactly reproduce as your homework assignment. However, this Algebra Buster should also help you to learn math rather than only use it to copy answers.
I couldn't agree more with what has just been said. Algebra Buster has always come to my rescue, be it an assignment or be it my preparation for the midterm exams, Algebra Buster has always helped me do well in algebra. It really helped me on topics like equation properties, like denominators and interval notation. I would highly recommend this software.
Algebra Buster is a great software and is surely worth a try. You will find lot of interesting stuff there. I use it as reference software for my math problems and can swear that it has made learning math much more fun.
I guess you can find what you need at this From what I understand Algebra Buster comes at a cost but it has 100% money back guarantee. That's how I got it. I would advise you to try it out. Don't think you will want to get a refund.
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College Algebra Concepts and Models
9780618492817
ISBN:
061849281x
Pub Date: 2005 Publisher: Houghton Mifflin College Div
Summary: "College Algebra: Concepts and Models" provides a solid understanding of algebra, using modeling techniques and real-world data applications. The text is effective for students who will continue on in mathematics, as well as for those who will end their mathematics education with college algebra. Instructors may also take advantage of optional discovery and exploration activities that use technology and are integrate...d throughout the text.A brief version of this text, "College Algebra: A Concise Course," provides a shorter version of the text without the introductory review."Make a Decision" features thread through each chapter beginning with the Chapter Opener application, followed by examples and exercises, and ending with the end-of-chapter project. Students are asked to choose which answer fits within the context of a problem, to interpret answers in the context of a problem, to choose an appropriate model for a data set, or to decide whether a current model will continue to be accurate in future years."Chapter Projects" extend applications designed to enhance students understanding of mathematical concepts. Real data is previewed at the beginning of the chapter and then analyzed in detail in the Project at the end of the chapter. Here the student is guided through a set of multi-part exercises using modeling, graphing, and critical thinking skills to analyze the data.Questions involving skills, modeling, writing, critical thinking, problem-solving, applications, and real data sets are included throughout the text. Exercises are presented in a variety of question formats, including free response, true/false, and fill-in the blank.""In the News"" Articles from current mediasources (magazines, newspapers, web sites, etc.) are found in every chapter. Students answer questions that connect the article and the algebra learned in that section. This feature allows students to see the relevancy of what they are learning, and the importance of everyday mathematics."Discussing the Concept" activities end most sections and encourage students to think, reason, and write about algebra. These exercises help synthesize the concepts and methods presented in the section. Instructors can use these problems for individual student work, for collaborative work or for class discussion. In many sections, problems in the exercise sets have been marked with a special icon in the instructor's edition as alternative discussion/collaborative problem."Discovery" activities provide opportunities for the exploration of selected mathematical concepts. Students are encouraged to use techniques such as visualization and modeling to develop their intuitive understanding of theoretical concepts."Eduspace" Houghton Mifflin's online learning tool powered by Blackboard, is a customizable, powerful and interactive platform that provides instructors with text-specific online courses and content and cover are intact. Dust jacket is torn or missing. The book has moderate to heavy wear. Covers have wear; Edges are yellowed and/or dirty; The i [more]
A readable copy. All pages and cover are intact. Dust jacket is torn or missing. The book has moderate to heavy wear. Covers have wear; Edges are yellowed and/or dirty; The inside of the front cover has quite a few smiley faces drawn on it; The corners are worn and bent
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Modular Course for Strands 3 & 4 (Number & Algebra)
(10 Feb 2012)
The Project Maths Development Team, is pleased to offer this modular course on Strands 3 & 4: Number & Algebra. The course will focus on new elements of the syllabus, methodology and provide student centered activities. The course will cover topics from Junior Certificate and Leaving Certificate syllabus.
Modules (4 x 2.5 hours each)
1. Linear Patterns: Arithmetic Sequences & Series
2. Exploring properties of quadratic, cubic and exponential graphs.
3. Further exploration of patterns using Arithmetic & Geometric Series
View Course Notes
Using ICT in Strands 3, 4 & 5
(29th September 2011)
The Project Maths Development Team, in conjunction with the NCTE, is running a series of workshops to support the integration of ICT into the Teaching and Learning of mathematics for strands 3, 4 & 5 of the Project Maths Syllabus. These modules focus on the use of software to teach geometry, trigonometry and calculus.
The course consists of 3 modules of 2.5 hours each. All participants will receive a certificate of completion.
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Fearful Symmetry: Is God a Geometer? by Ian Stewart, Martin Golubitsky From the shapes of clouds to dewdrops on a spider's web, this accessible book employs the mathematical concepts of symmetry to portray fascinating facets of the physical and biological world. More than 120 illustrationsAlgebraic Geometry by Solomon Lefschetz An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
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Famous Problems of Geometry and How to Solve Them by Benjamin Bold Delve into the development of modern mathematics and match wits with Euclid, Newton, Descartes, and others. Each chapter explores an individual type of challenge, with commentary and practice problems. SolutionsFrom Geometry to Topology by H. Graham Flegg Introductory text for first-year math students uses intuitive approach, bridges the gap from familiar concepts of geometry to topology. Exercises and Problems. Includes 101 black-and-white illustrations. 1974 edition.
Fundamental Concepts of Geometry by Bruce E. Meserve Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.
Geometry: A Comprehensive Course by Dan Pedoe Introduction to vector algebra in the plane; circles and coaxial systems; mappings of the Euclidean plane; similitudes, isometries, Moebius transformations, much more. Includes over 500 exercisesA Modern View of Geometry by Leonard M. Blumenthal Elegant exposition of the postulation geometry of planes, including coordination of affine and projective planes. Historical background, set theory, propositional calculus, affine planes with Desargues and Pappus properties, much more. Includes 56 figuresProduct Description:
spaces, projective space, much more. Advanced undergraduate level. Problems. Bibliography. 1968
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for Physicists
This best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital ...Show synopsisThis best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital addition to the bookshelf of any serious student of physics or research professional in the field. The authors have put considerable effort into revamping this new edition. * Updates the leading graduate-level text in mathematical physics * Provides comprehensive coverage of the mathematics necessary for advanced study in physics and engineering * Focuses on problem-solving skills and offers a vast array of exercises * Clearly illustrates and proves mathematical relations New in the Sixth Edition: * Updated content throughout, based on users' feedback * More advanced sections, including differential forms and the elegant forms of Maxwell's equations * A new chapter on probability and statistics * More elementary sections have been deleted Mathematical Methods for Physicists
The book covers a very large range of mathematical issues. Some topics are well developed, like the ones covering complex analysis, while others, like the group theory, are much concise (in my opinion). In general, the book offers a good introduction to several topics, not only for the physicists
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Workshop: Solving Mathematical Problems with MuPAD
Workshop leader: Bernhard Stöger (University of Linz, Austria)
What is a Computer Algebra System?
A computer algebra system is a program that carries out mathematical calculations for you. But, unlike a calculator program such as Windows Calc or Excel, computer algebra systems are able to calculate not only with numbers, but also with variables. This capability lets you simplify complex mathematical expressions, but also solve equations in which variable symbols occur.
What is MuPAD?
MuPAD is a powerful computer algebra system developed at the University of Paderborn, Germany. It is quite well accessible and runs both on the Windows and on the Linux platform.
What will I learn at this workshop?
To organize my calculations in so-called notebooks, and to navigate within them
To use the simple and intuitive syntax of MuPAD for reading and writing my mathematical expressions – both numbers and variables
To do simple algebraic tasks like simplification of an expression or solution of linear and quadratic equations
To carry out basic vector calculations such as the formation of scalar and vector products
To do basic operations of differential and integral calculus
To work with matrices, determinants, and systems of linear equations
Finally, to browse through the extensive MuPAD documentation in order to find the mathematical functions that are most useful for myself
Okay, I forgot one particularly funny thing: You will be able to decompose large integers into their prime factors, and you'll discover some very large prime numbers.
Why shouldn't I miss that workshop?
Because it will, in any event, raise your power of solving mathematical problems, be it in algebra, calculus, or in numerical Mathematics.
For which level of study and which kind of disability is the workshop suited?
Each level – after a general introduction that is relevant for everyone, I shall provide tasks for every specific level, from middle secondary education until University. Bacause of the easy-to-use interface of MuPAD, the program, and thus the workshop, is manageable both for sighted, particlly sighted, and blind students.
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books.google.com - The... graph theory
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Bringing a new vitality to college mathematics
Algebra is so basic that all students need to have a good grasp, and adults without this capability will be limited in their economic and social choices. Algebra is so esoteric compared to daily life and work that only those in STEM careers need to bother learning it. Both of these statements can be true; the apparent inconsistency is based on what is meant by the word 'algebra'.
As a mathematician, I view algebra as a language system used to describe and manipulate features (whether known or not) of the physical world based on arithmetic operators . Basic literacy in this language system is essential in both academia and 'real life'; translations into and out of algebra are the most basic literacies, followed by different representations (symbolic, numeric, graphic).
Unfortunately, the algebra of mass education tends to focus on procedures and complexity of limited value to anybody combined with a focus on solving algebraic puzzles, as if completion of a crossword puzzle is a basic skill for a language. True to our current binary approach, people who agree with a literacy approach will invest great effort to avoid all procedures, complexities, and puzzles. The truth is that we undertake these problems ourselves just for fun, and this is one element in our transition to being mathematicians; how are we to capture the attention of potential STEM students if we avoid the fun stuff?
As a language system, basic literacy in algebra means that a person can read the meaning of statements; transformations to simpler forms is based on that meaning. I failed to help my students in the algebra class this summer … I know that because students could distribute correctly in a product but failed routinely with a quotient. [In case you are wondering, the problems involve a 3rd degree binomial and a 1st degree monomial; in the division case, many students 'combined' the unlike terms in the binomial instead of distributing. This is a basic literacy error; very upsetting!]
Hey, I know … nobody needs to distribute algebraic expressions on their 'job' (except us!). That type of reason is enough for me to conclude that we do not need to cover additional of rational expressions (prior to college algebra/pre-calculus); that process is complex, and is based on a higher level of understanding of the language. Distributing is a first-order application of algebraic literacy; avoiding that topic means that we present an incomplete picture of algebra as a language. A pre-college mathematical experience needs to provide sound mathematical literacy — including algebra.
Everybody needs algebraic literacy, as part of basic mathematical literacy. We can design courses that provide the needed mathematical literacy as a single experience — no need for a numeric literacy course ('arithmetic') and an algebra course and a geometry course; all of that, plus some statistical literacy, can be combined into one course. This is the approach of the New Life model, and is imbedded within the Quantway & Statway (Carnegie), and in the New Pathways (Dana Center). I encouage us all to include some transformations ('simplifications') in the algebraic language.
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Most everybody does NOT need "algebraic literacy".
However, when appropriately design, an algebra course can be an area where people can learn when to stop and consider, how to conjecture and how to act on said conjectures, how not to be taken in by appearances, when to insist on a case being made, etc. All of which everybody needs.
The specific contents of such a course are relatively unimportant. But the contents' architecture is paramount for the students to learn focussing on the "connective tissues".
Regards
–schremmer
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Wolfram Alpha is a website that is very very cool. you can almost type in anything and it will give you an answer right away, for example lets say you type in a really hard math problem it will give you the answer and explain to you how to figure it and there are different charts and things like that with your answer.
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Algebra: Gateway to a Technological Future
In early November of 2006, the Mathematical Association of America, in
a project funded by the National Science Foundation, brought together
representatives from the mathematics and mathematics education
communities across the entire K-16 spectrum to survey what has been
learned about the teaching of algebra and to identify common principles
that can serve as models for improvement. The approximately 50
participants were divided into five groups, corresponding to five
different levels of algebra instruction. The groups were (1)
Early Algebra, (2) Introductory Algebra, (3) Intermediate Algebra, (4)
College Algebra, and (5) Algebra for Prospective Teachers. Each
group reviewed research on what was known about the teaching of algebra
on that level and made suggestions for future directions that would
improve both the knowledge base and the actual teaching and learning of
algebra.
The following is a summary of the findings and recommendations of each
of
the five groups.
Algebra: Gateway to a Technological
Future was funded by the National Science Foundation Division of
Elementary, Secondary and Informal Education Grant ESI-0636175.
Early Algebra
It is now widely understood that preparing elementary students for the
increasingly complex mathematics of this century requires an approach
different from the traditional methods of teaching arithmetic in the
early grades, specifically, an approach that cultivates habits of mind
that attend to the deeper, underlying structure of mathematics and that
embeds this way of thinking longitudinally in students' school
experiences, beginning with the elementary grades. This approach
to elementary grades mathematics has come to be known as early
algebra. There is general agreement that early algebra comprises
two central features: (1) generalizing, or identifying, expressing and
justifying mathematical structure, properties, and relationships and
(2) reasoning and actions based on the forms of generalizations.
When early algebra is treated as an organizing principle of elementary
grades mathematics, the potential payoffs are tremendous: (1) It
addresses the five competencies needed for children's mathematical
proficiency: conceptual understanding, procedural fluency,
strategic competence, adaptive reasoning, and productive disposition.
(2) It creates children who understand more advanced mathematics in
preparation for concepts taught in secondary grades. (3) It
democratizes access to mathematical ideas so that more students
understand more mathematics and, thus, have increased opportunity for
lifelong success.
There is already much research telling us what mathematics young
children are capable of learning in the early grades. Yet there
is still much to learn. Thus, we recommend three critical areas
for future research in early algebra:
The development of "Early Algebra Schools"
These are
schools that integrate a connected approach to early algebra across all
grades K-5 and provide all teachers with the essential forms of
professional development for implementing early algebra. The
systemic change implied by these schools should involve not only
elementary teachers, but also middle school teachers, principals,
administrators, education officials, math coaches, parents, and even
university personnel.
Developing a coherent, connected early algebra content.
Although much is known about children's learning of basic ideas at
certain grade levels, there is a need to develop a more coherent
picture of early algebra throughout grades K-5 and connect this with
what follows in the higher grades.
Understanding the pervasive nature of children's algebraic
thinking. Research has provided us with "existence
proofs" of the
kinds of algebraic thinking of which children are capable. But we
still need an understanding of how pervasive this knowledge can become.
Introductory Algebra
One central problem in the teaching of introductory algebra is that
there are too many topics. Thus, the attempt to cover them all
impedes student learning of core concepts in depth. This problem
is exacerbated by the lack of logical connections between core concepts
and procedures. In addition, it is apparent that the transition
from using numbers to using symbols is much more difficult for many
students than has been assumed.
Basing our recommendations on these major problems and several others
as well, we suggest six major research directions for the future.
Identify core concepts and procedures that should form the
content of introductory algebra. Through a number of conferences,
NSF should build a consensus on the content of the core of
algebra. Organizing introductory algebra around a core should
make the study more coherent. Of course, teachers themselves must
recognize the centrality of the important ideas, and must keep asking
students "why" so that the students become aware that
most ideas arise
in a number of contexts.
Investigate the transition to symbolization and how teachers can
effectively facilitate it. Relatively little work has been done
investigating what occurs in the transition from work with numbers to
work with symbols. But this understanding is important in
designing ways that teachers could effectively facilitate students'
transitions. In addition, we need to know whether the process of
transition to symbolization differs in adults and in children.
Investigate models that promote learning for students with
different needs, preparation, and backgrounds in the same
classroom. New pedagogical methods including community building,
group work, and inquiry learning can help all students, but we do not
know the best balance between such methods and more traditional ones
such as direct instruction or individual work. Projects should
study the use of these methods in different ways and pay particular
attention to their use in diverse classrooms.
Prepare and sustain teachers in implementing good instructional
practices and curricular materials. There is a dearth of
curricular materials for professional development of new and practicing
teachers, especially materials which enable teachers to transfer their
own learning into new teaching practices. Investigation of the
particular content that would best support algebra learning is
particularly needed.
Identify systemic changes needed to support teacher growth.
Teachers need more structured time during the school day for
collaboration and growth. For school districts to fund such
expensive time, they need strong evidence that it will pay off and that
there are no cheaper alternatives. We therefore need to
investigate a variety of models that try alternative approaches to
providing such structured time and document the changes they produce.
Determine what use of technology is appropriate in the
introductory algebra classroom. Research has show that graphing
calculators can enhance learning and computers can provide useful
practice. We need to compile evidence of what actually happens
when these are used, including what students learn with calculators
that they do not learn without them and what they fail to learn when
they use calculators that they learn without their use.
Intermediate Algebra
Intermediate algebra is generally designed as part of the
college-intending tract, and thus includes topics thought necessary for
students' later success in freshman college courses, primarily
calculus. Since the content of this course varies widely, our
focus was on describing the mathematical ways of thinking that under
gird algebra. We also propose an action plan to identify and
implement the best approach to preparing students for further study of
college mathematics
Identify mathematical ways of thinking that are central to
algebraic reasoning. Weak coherence of algebra curricula results
in part from the absence of a central core of the subject. We
therefore propose a research and development program involving
collaboration between education researchers, mathematicians, and
teachers that will focus on mathematical ways of thinking as a way of
providing a common set of principles to guide the development of
curricula. Included among "mathematical ways of
thinking" are the
following:
The habit of finding algebraic representations even in a
problem that does not necessarily look mathematical.
Anticipating the results of a calculation without doing it.
Abstracting regularity from repeated calculation.
Connections between representations, such as tables, graphs,
formulas.
Build capacity to focus algebra instruction on mathematical ways
of thinking. We envision an evolutionary approach seeded by
smart, strategic moves that, after an initial phase of testing and
refinement, become self-disseminating and self-replicating. Some
of these moves could be
Equipping instructors with a framework for putting to new uses
the materials they have, especially in supporting mathematical ways of
thinking.
Establishing collaborative communities of mathematicians,
mathematics educators, and teachers that share a focus on student
learning of significant mathematical ideas.
Finding ways of opening discussions about pedagogical
techniques among two and four year college faculty
Encouraging all college faculty to understand that their course
is a potential teacher preparation course
Approaching teacher professional development programs with a
focus on deepening mathematical content knowledge, encouraging
mathematical ways of thinking, and examining why students are not
learning.
Advise policy makers on barriers and avenues to successful
implementation of sound instructional practices. Because
capacity-building is wasted if new ways of teaching run into
institutional or systemic barriers, we need to do determine those
institutional structures and policies that prevent the flow of
innovation as well as those that channel it in the right
direction. Thus mathematical researchers and practitioners must
collaborate with policy makers, administrators, parents, textbook and
testing companies, and the wider business community, to learn from each
other their respective concerns, to create a sense of shared
responsibility, and to encourage creative informed decision making.
College Algebra
Extensive studies over the past several years by the MAA and AMATYC
have shown that, to a large extent, the College Algebra courses taken
annually by some 700,000 college students are not successful.
Thus, the curriculum committees of both AMATYC and MAA have called for
replacing the current college algebra course with one in which students
address problems represented as real world situations by creating and
interpreting mathematical models. We therefore recommend several
programs designed to change the nature of college algebra courses
throughout the country and greatly improve the student success rate.
A large scale program to enable institutions to refocus college
algebra. This would help a large number of institutions implement
the new guidelines, beginning with faculty development.
Research on impact of refocused college algebra on student
learning. In connection with large scale implementation, there
needs to be a few extended longitudinal studies of student learning in
the refocused college algebra courses.
Electronic library of exemplary college algebra resources.
This would provide classroom activities, extended projects, and videos
of lessons that would help instructors implement new ideas for student
learning.
Establishment of national resource database on college
algebra. This resource would include information on funded
projects, textbooks, research articles, etc. that could help widely
disseminate positive results from exemplary algebra programs.
Algebra for Prospective Teachers
To improve the content and pedagogical knowledge of algebra teachers in
middle and high school, and thus to improve the achievement of their
students, we need answers to some basic research questions.
What is the role of teachers' algebraic knowledge for teaching as
it shapes their instructional practice? This needs to come from
observational studies of algebra teachers in action in a wide variety
of settings. Among other questions, we need to understand how
teachers respond to students' algebraic thinking as it occurs.
How does the content and design of the abstract algebra course
typically taken by future teachers of algebra affect their later
teaching of school algebra? New types of abstract algebra courses
for teachers have been developed in recent years, but there is a need
for solid research studies on their effect.
How does professional development in algebra content and pedagogy
affect teachers' classroom practices? This research must
begin with a careful analysis of the important algebraic concepts that
should inform teachers' understanding of the mathematics they are
teaching. Furthermore, we need to learn how teachers'
understanding of algebra and its teaching develops from the use of
different kinds of instructional materials.
Given that answers to all of the above questions require collaborative
efforts among mathematicians, mathematics educators, and classroom
teachers, we also need to identify strategies that successfully nurture
such collaboration.
Contact Information
Lead editor for the project report: Victor
Katz, University of the District of Columbia (emeritus), vkatz@udc.edu
Project Director at MAA:
Michael Pearson, Director of Programs and Services, pearson@maa.org
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MATH 1240 Applied Mathematics
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This course covers the development and application of the mathematical skills needed to solve problems related to industrial occupations. Topics include applications of arithmetic skills, measurement, and elementary algebra, geometry, and trigonometry. NOTE: MATH 1220 and MATH 1240 do not require MATH 0930, 0931, or 0960 as a prerequisite; however, MATH 0910 skills are necessary. MATH 1220 and MATH 1240 satisfy the math requirements in certain programs only. Check to see what the program advises to fulfill the general education math requirement. In most cases, these courses do not transfer to other institutions as math credit.
Prerequisites
(1) Within two years prior to beginning the course, either successful completion of MATH 0910 with a grade of P, or MCC placement test
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Brand New!
Who doesn't need help with math? Practically everybody… And we have the answer! Basic Shop Math is designed to assist students who need extra help or simply need to refresh their basic math skills. A practical, crystal-clear, Auto Shop centered math course that makes math understandable!
Most instructors start the school year with their first priority, shop safety. Right after that, math is often the next topic.
Basic Shop Math is designed to assist students with the basic math skills most often used in the Auto shop. Whether they need a little extra help or just need to refresh their math skills, this course covers what they need to know to work successfully in the Auto Shop. All math examples are expressed in terms associated with typical auto shop tasks or jobs.
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...hat is Algebra And why is it important What Algebra is A method of thinking that Uses symbols usually letters of the alphabet represent numbers or quantities Expresses general relationships that hold for the specific application What Algebra is NOT A bunch of letters and numbers with no meaning...
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[ ps ] ...his is page 212 Printer Opaque this CHAPTER 5 Algebra The Search for an Elusive Formula 5 1 Introduction On the night of February 12 1535 Niccol o Tartaglia of Brescia found the solution to the following vexing problem A man sells a sapphire for 500 ducats making a profit of the cube root of his...
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0198507178
9780198507178. Carefully structured, coherent and comprehensive, each chapter contains tailored exercises and solutions to selected questions, and miscellaneous exercises are presented throughout. This is an invaluable text for students seeking a clear introduction to discrete mathematics, graph theory, combinatorics, number theory and abstract algebra. «Show less.... Show more»
Rent Discrete Mathematics 2nd Edition today, or search our site for other Biggs
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MATH43012 - 2009/2010
General Information
Title: Computation and Complexity
Unit code: MATH43012
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Once you've mastered the algebra, geometry, and coordinate geometry topics covered in Pre-GED Mathematics, you'll be effectively prepared for this comprehensive review. If you're math-phobic, or just need a quick refresher, put your mind at ease. This unique program helps you prepare for even the most challenging problems on the GED Mathematics Test. Work along as your friendly math professor guides you through many practice examples on the chalkboard, offering step-by-step solutions and clear explanations along the way. This easy-to-follow review course includes sample problems in all of the following areas: number operations; probability; statistics; data analysis; algebra; geometry; and coordinate geometry. You'll also learn about the structure of the exam and the rules for filling in the grids; plus, you'll learn how to judge when to use the calculator – and when not to!
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Easy Solutions and Visualizations: Exploring a System of Three Equations in Three Variables Using Mathcad
In scientific calculator was considered high tech. Last year I taught Grade 8 algebra at The Park School in Brookline, MA. Students used graphing calculators, laptops, iPhone Apps, Java Applets, etc. Despite the incredible change in available technology, many people presume that the key skills required in an Algebra 1 course remain the same as they were when I took the course.
In recent years, as I have learned more about the use of mathematics in the workplace, I have grown increasingly disenchanted with the traditional Algebra 1 course that I received in 1979 (and I still see in existence in many schools). Topics like factoring, which I was quite good at in Grade 8, seem unconnected to any real vocation. Tools like matrices, which were left out of the course, seem to be ubiquitous in many modern, high paying STEM fields. I can think of two defensible reasons for this fact: (a) matrices can be time consuming to do by hand and (b) even with a graphing calculator they are difficult to visualize in more than the two variables.
Recently I was exploring some problems involving systems of three equations in three variables. I used Mathcad to solve the problems using matrices and then to visualize the solution on a 3-D plot. A portrait of my experiences follows. I propose that in 2010, matrices are a much more accessible topic for school algebra.
Solving a System and Visualizing a Solution
The first problem that I explored had a unique solution. Below I show how I defined matrices F and G to represent the system of three equations. Of course, I also checked the determinant of F to see if the system would yield a solution. Since |F| = 70, I solved the system by multiplication of F-1*G. In pursuing this strategy I asked Mathcad to solve the system using both numeric and symbolic evaluation. Note how the results are equivalent, but the symbolic result is a fraction instead of a decimal. Next, I used the lsolve function to see if Mathcad would replicate the same results and wrote an explanation of my findings. (Ref Fig. 1)
Some may consider the above work to be a black box solution. Mathcad has completed the calculations without showing any work. Yet, I have determined the results efficiently and explained them in writing using Mathcad's math and text capabilities. In addition, below I use Mathcad's graphing capabilities to further illustrate the problem. It is easy to create a 3D quick plot in Mathcad. The graph below clearly illustrates the intersection of the three planes at a single point. (Ref Fig. 2)
For contrast, I also explored a system that did not have a solution. In this case, Mathcad's determinant function helped me to identify that the system was unsolvable. However, the more enlightening exercise was graphing the system in order to observe how the planes intersect in three-space.
When Mathcad evaluated the determinant at zero, I knew that there would not be a solution. The symbolic results confirmed this, identifying the solution as undefined. The lsolve command, however, produced an error (shown in red text beneath my explanation). Thus, working with Mathcad to solve systems of equations offers a number of opportunities for mathematical explanation. First, the user needs to interpret the results intelligently. Do we need to debug the lsolve error or interpret its meaning? Second, there is a rich opportunity for thinking about the system by interpreting the graph. How is this graph different from the previous graph that depicted a solution? Comparing and describing the pair of graphs above can help students ground their understanding of the nature of the solution to a system of three equations in three variables.
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Matrices are challenging, but they are really important in applied mathematics – they are a critical STEM topic. Engineers and scientists use matrices to solve challenging problems in many, many dimensions. Mathcad's matrix and graphing tools offer capabilities that can help students' explore matrices early in their school experience so that they are both prepared to use and aware of the importance of matrices. With currently available technologies matrices can be used, explored, and visualized effectively in Algebra 1 class. Systems of equations in three variables need not be avoided any longer. Matrices can be an efficient and powerful way to solve systems, with increased clarity now that we have tools to graph 3D plots. (Ref Fig. 4).... (Show more)(Show less)
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MathMagic Personal Edition 7.4.3.48: Free Download
MathMagic Personal Edition is a comprehensive and powerful piece of software designed to assist you in learning any mathematical symbols and expressionsMathMagic Personal Edition is a stand-alone equation editor for editing any mathematical expressions and symbols with easy-to-use graphical interface and various powerful features. [read more >>]
NOTE: If you have problems downloading MathMagic
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Course Description:This course is a
functional approach to algebra that incorporates the use of appropriate
technology. Emphasis will be placed on the study of functions, and their
graphs, inequalities, and linear, quadratic, piece-wise defined, rational,
polynomial, exponential, and logarithmic functions. Appropriate applications
will be included.
Topics:Functions and Their Graphs, Polynomial and
Rational Functions, Exponential and Logarithmic Equations, and Systems of
Equations.
Attendance Policy:Students are expected to attend all
classes.If a class is missed, the
student is responsible for material and assignments.Tests and quizzes can't be made up.
Sections Covered:Review R.1-R.6 1.1-1.7, 2.1-2.6, 3.1-3.6
and 4.1-4.6
Recommendations:It is recommended that students review new
material before class, take notes, ask questions, complete all assignments,
review material weekly and work chapter reviews.If you have any questions, comments, or
concerns during the semester, please feel free to contact me.
Team Goal—Mathematics: The
student will be able to demonstrate the ability to apply mathematical thought
and methods.
Related Team Outcomes:
·Students will be able to model concrete problems
and arrive at solutions.
·Students will be able to graph relationships
other than functions.
·Students will be able to demonstrate algebraic
skills in solving inequalities.
·Students will be able to graph a real-life
function.
·Students will be able to demonstrate algebraic
skills in solving equations.
·Students will be able to graph an abstract
function.
·Students will be able to use appropriate technology
to enhance mathematical thinking and understanding.
·Students will be able to interpret a real-life
function.
Withdrawal:
Students who find that they cannot
continue in college for the entire semester after being enrolled, because of
illness or any other reason, should complete an official withdrawal form.
Forms may be obtained from the Office of the Registrar.
Students who officially withdraw
from the university with the approval of the registrar before mid-semester
(including registration days) will be assigned grades of "W", which will not
affect their grade point average. Students who officially withdraw after
mid-semester (and before the last three weeks of the semester) will receive a
"WF," which will be counted as an "F" in the calculation of
the grade point average. Those students who stop attending classes
without notifying someone will be assigned failing grades, which jeopardize
their chances of future academic success.
Students may, by means of the same
withdrawal form and with the approval of the university Dean, withdraw from
individual courses while retaining other courses on their schedules. This
option may be exercised up until June 29, 2007.
This is the date to withdraw
without academic penalty for Summer Term, 2007 classes. Failure to
withdraw by the date above will mean that the student has elected to receive
the final grade(s) earned in the course(s). The only exception to those
withdrawal regulations will be for those instances that involve unusual and fully
documented circumstances.
Academic Integrity:
Every KSU student is responsible
for upholding the provisions of the Student Code of Conduct, as published in
the Undergraduate and Graduate Catalogs. Section II of the Student Code
of Conduct addresses the University's policy on academic honesty, including
provisions regarding plagiarism and cheating, unauthorized access to University
materials, misrepresentation/falsification of University records or academic
work, malicious removal, retention, or destruction of library materials,
malicious/intentional misuse of computer facilities and/or services, and misuse
of student identification cards. Incidents of alleged academic misconduct
will be handled through the established procedures of the University Judiciary
Program, which includes either an "informal" resolution by a faculty
member, resulting in a grade adjustment, or a formal hearing procedure, which
may subject a student to the Code of Conduct's minimal one semester suspension
requirement.
Classroom Etiquette:
1.Come
to class prepared, having attempted all homework problems and having previewed
the material that will be covered during the class period.
2.Be
punctual.People coming in late disrupt
the flow of the class and places them behind in material covered in class.Occasionally, being late cannot be helped,
and that is fine.On these occasions, I
would rather have you enter the classroom late rather than miss the entire
class.
3.Be
attentive.If you need to engage in
other activities (e.g., studying for another course), please do not do them
during class.If you get drowsy during
class, please feel free to leave the classroom and get a drink of water.Simply leave and return quietly.
4.No
side conversations.Discussions with
your neighbors while I am presenting material or while one of your classmates
is speaking are disrespectful to us all.If you missed something that was said, let me know and it can be
repeated.
5.Behave
in a dignified and respectful manner toward your fellow students and the
instructor.
6.Turn
off all cell phones and pagers.Cell
phones must not be on the desk.
7.Please ask questions when they occur to
you.No questions are "dumb"
questions.Failure to ask your question
can hamper the learning process.If I
must move ahead with additional material at that time, I will address your
question at the end of the class period.
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Synopsis
Many colleges and universities require students to take at least one math course, and Calculus I is often the chosen option. Calculus Essentials For Dummies provides explanations of key concepts for students who may have taken calculus in high school and want to review the most important concepts as they gear up for a faster-paced college course. Free of review and ramp-up material, Calculus Essentials For Dummies sticks to the point with content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical two-semester high school calculus class or a college level Calculus I course, from limits and differentiation to integration and infinite series. This guide is also a perfect reference for parents who need to review critical calculus concepts as they help high school students with homework assignments, as well as for adult learners headed back into the classroom who just need a refresher of the core concepts.
The Essentials For Dummies Series Dummies is proud to present our new series, The Essentials For Dummies. Now students who are prepping for exams, preparing to study new material, or who just need a refresher can have a concise, easy-to-understand review guide that covers an entire course by concentrating solely on the most important concepts. From algebra and chemistry to grammar and Spanish, our expert authors focus on the skills students most need to succeed in a
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Math and Science Workout for the ACT, 2nd Edition
If you need to know it, it's in this book. This revised second edition of Math & Science Workout for the ACT includes: • 3 full-length practice sections (2 for Math and 1 for Science) • Numerous drills with detailed answer explanations for each question • Comprehensive advice on the Math and Science tests from our ACT experts • Techniques for mastering the most common types of Math questions • Guidance for identifying easier types of Science passages to help plan out the best order for attacking the Science testMath & Science Workout for the ACT contains all the information you'll need to learn where your weaknesses lie—and how to overcome them.
show more show less
List price:
$19.99
Edition:
N/A
Publisher:
Random House Information Group
Binding:
Trade Paper
Pages:
224
Size:
9.00" wide x 10.00 and Science Workout for the ACT, 2nd Edition - 9780307945952 at TextbooksRus.com.
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Find a Sunrise, FL Geometry
...Linear Algebra is about very practical application of mathematics. It is a view of the many discrete applications in our reality. As a professor of applied math at Concordia University, I taught math for the decision sciences.
...However, as soon as students experience the simple methods used and techniques of solving these Math problems, then they will become successful in this subject. Example 1 : Solve to find the value of y. Solution :
2y + 7 = 19
2y = 19 - 7
2y = 12
y = 12/2
y = 6 Geometry is a very fascinating section to study in mathematics.
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Calculator this calculator different
is its simplicity. It is much easier to use than its competitors and any input can
immediately be converted to a graph.
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MATHEMAtICS WITH EDUCATION
This is a course for students who are considering a career in mathematics teaching. It contains high-level mathematical and statistical modules from our other mathematics degrees. This gives graduates both a very solid background and also flexibility in their future career choices. Furthermore the degree provides a foundation of concepts and skills to support the teaching of mathematics in the National Curriculum and at A-level.
There is a severe shortage of mathematics teachers in this country. A recent survey revealed that one in four secondary mathematics lessons were given by teachers who were not mathematics specialists. The career prospects for good mathematics teachers are therefore excellent.
The video below offers an introduction to studying mathematics with us at Plymouth.
Mathematics with Education with Plymouth University
This course covers the central ideas and developments of pure and applied mathematics and offers opportunities to work in schools. It thus offers an ideal start for anyone wanting to pursue a mathematics teaching career. Throughout the degree students have the opportunity to work with school children on the mathematics enrichment courses run by our Centre for Teachings Mathematics. The degree is designed to develop your analytical and strategic thinking skills as well as equipping you with experience with professional software and in applying mathematics to solve a wide variety of problems.
The first year develops and strengthens topics previously met at A-level, such as calculus, but also introduces completely new material such as the number theory underlying internet security. Professional software such as Maple, a computer algebra system, and R, a programming language and software environment are also introduced in the first year. This experience ensures that our students are prepared to tackle numerical mathematics and mathematical programming. In the second year our students can master more advanced mathematical techniques. After this sound base is established, the final year features include a year-long schools based module working alongside an experienced mathematics teacher and a project module in mathematics education.
This is a highly practical degree which lets you master many technical skills as well as exploring the beauty of mathematics. It also gives you the essential school experience required to be accepted onto a mathematics teacher training (PGCE) course. We are very proud of the support we offer to our students. Our lecturers are at the forefront of their subjects and the degree is strongly informed by their research. Graduate positions include advanced skills teacher, post-16 network leader and head of the mathematics department in a school.
TECHNOLOGY SUPPORTED LEARNING - iPad mini
As a mathematics student at Plymouth, we use technology to support your learning. Our first year students have already received an individual Apple iPad mini. This will give you access to various additional resources that support your modules (e.g. podcasts, online videos and electronic copies of lecture notes), as well as enabling you to participate in interactive activities such as in-class voting and feedback, and to access various University online systems such as module sites, the electronic library, and of course email.
What our students say
"The course at Plymouth was superb. I was able to consolidate and extend my mathematics skills whilst at the same time gaining enough experience in schools to decide if teaching was a career I wanted to pursue. The education side of the degree involved both the practical side of teaching and the theoretical aspects of the profession. I was able to go into a school on a regular basis, observe lessons and gradually build up to leading a class. At the same time I was also looking in detail at the theory behind how teachers teach and how pupils learn, a vital aspect of teaching.
For me, the Mathematics with Education degree was an excellent route into teaching, ensuring that my subject knowledge was sound and giving me a rounded view of what a career in teaching would be like. The experience I gained was invaluable during my PGCE year and the ideas and knowledge I gained during the course are used every day whilst teaching my classes."
Projects
Teaching Mathematics at Secondary School: A Comparative Study between France and England
A Case Study: Using Tactile Resources as a Kinaesthetic Approach to Algebra at Key Stage 3
An Investigation into the Problem-Solving Strategies Adopted by Children
What Engages Children in the Mathematics Classroom?
How Can the Move From Primary to Secondary School be Made Easier?
Leading Students to the Target Concept Through Questioning
Support for Our Students
We are very proud of the support we offer to our students. In two recent National Student Surveys we scored 100% for student satisfaction.
We have an open-door policy and a personal tutor system where your tutor is a lecturer whom you will also be seeing regularly in the lecture theatre. Additionally our SumUP mathematics and statistics drop-in centre is open in the main library from 10am to 4pm Monday to Friday during term.
Academic Excellence Awards
Applicants are eligible for one of the two following awards. There is no need to make an application for them.
Dean's Award for Academic Excellence: £2,000 for any applicant for this course who has made Plymouth University their first choice by Wednesday 8 May 2013 and who achieves the equivalent of ABB in three A Levels (or equivalent) and enrols here in September 2013. Students are eligible if they have at least one of the following combinations listed in the qualification/grade table as well as satisfying all our entrance criteria.
Mathematics Scholarship: students who do not obtain the Dean's Award are eligible for a £500 automatic scholarship if they have a grade A* in A Level Mathematics plus £500 for an A in Further Mathematics up to a total of £1,000. To be eligible for this scholarship, students must put us as their first choice before 1st August. The scholarship is paid in term one of the first year. There are additional prizes and certificates to reward high marks in later years.
Optional Placement Year
Students on this degree have the opportunity to take a 48-week placement year with a suitable organisation. This is done between the second and final years. Recent placements have been in the Department of Communities and Local Government, leading IT services company Fujitsu, Europe's largest accountancy firm KPMG and in Jagex Ltd a computer gaming company.
Placement salaries are currently typically around £16-17,000 for the year. Placements offer you an opportunity to apply some of the skills from your degree in a commercial or industrial setting. After a placement your CV demonstrates that you have experience in industry and it can help you obtain the job you want.
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Mathetes Solutions, named for the Greek word meaning "disciple," is a set of instructional DVDs to accompany the upper-level Saxon math curriculum. The set of twelve DVDs I have is for Algebra I. (Sets for Algebra ½ and Algebra II are also available.) Kenneth Updike strongly believes Saxon is one of the best math programs (he is not affiliated with Saxon), and he has created Mathetes Solutions as a help to parents who are using Saxon with their children. Mr. Updike is a homeschooling father of six children. He is also a certified math teacher with 18 years of teaching experience at the middle and high school level. Mathetes Solutions is a family business.
It is recommended that you use the third edition of the Saxon Algebra I textbook, but the DVD gives references for the second edition if that is what you are using. Each segment includes a lesson as well as a demonstration of how to do some of the problems. Up to lesson ten, the instructor (Kenneth Updike) does half of the problems. After that, he works about seven of the thirty problems per lesson. The lessons, including problems, are about twenty minutes each, and there are 120 lessons in this course.
You just pop in the DVD, select the lesson you want, and away you go. The instructor addresses the students, whom he assumes are being homeschooled. In the first lesson the student is told to take notes and shown how to do it. At the beginning of each lesson, there is a short introduction in which you see Mr. Updike. After that, all you see is his hand as he writes on white paper with a black Sharpie. All distractions are thus eliminated. This benefits the student who has trouble staying focused.
I love Mr. Updike's positive attitude. He believes in the importance of his subject, and this enthusiasm comes through. His voice is pleasant, and he is easy and interesting to listen to. I especially liked the way he connected math to Christianity. For example, he compares the infinity of a line, with no beginning or end, to the character of God. Mr. Updike has a gift for clear explanation, and he moves at a good clip to keep the student engaged. I think it would be important to make clear to the student that he or she should stop and rewind and/or ask the parent/teacher for clarification if something isn't making sense. That is one of the advantages of the DVD format.
Mr. Updike shares some real gems in the way of math "tricks." Using "product of the primes" to find the least common multiple is one example. Even my engineer husband learned some new shortcuts while viewing the DVDs!
In the first lesson, the instructor practically insists the student do all of the assigned problems unless two situations are true: the student is getting 90% of the problems correct and the parent permits it. I liked him better before he issued this exception, but I'm a pretty strict teacher. Ninety percent is a low "A" after all.
By and large, the DVDs are excellent, but I was disappointed by a couple of things. In solving equations, Mr. Updike uses the phrase "kill the constant" as a step in isolating the variable. Some parents may object to this phrasing. Also, I wish he had used two colors of ink, especially when illustrating equal fractions as pieces of a pie. There were some little "misspeaks" and "miswrites", some of which were self-corrected, but some of which were not ("3" instead of "13"). I wonder if these errors could have been edited.
The last lesson ends just as all the others do. It seems abrupt for the completion of a DVD, instructional or not. I wish there had been some sort of closure or wrap-up, like "Congratulations upon completing Saxon Algebra I. Good job! I hope you've enjoyed learning algebra as much as I've enjoyed teaching it. I hope you will continue to work hard and do well with your future math studies." The end of the DVD would be a great place to advertise additional products.
At $119.95, Mathetes Solutions is somewhat expensive, but I think for any teaching parent who doesn't have a solid grounding in algebra or who wants the student to be more independent, it is worth the money.
Product review by Kathy Gelzer, The Old Schoolhouse Magazine, LLC, May 2007
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…
Designed for mathematics majors and other students who intend to teach mathematics at the secondary school level, College Geometry: A Unified Development unifies the three classical geometries within an axiomatic framework. The author develops the axioms to include Euclidean, elliptic, and …
Classroom-tested, Advanced Mathematical Methods in Science and Engineering, Second Edition presents methods of applied mathematics that are particularly suited to address physical problems in science and engineering. Numerous examples illustrate the various methods of solution and answers to the …
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Master the tools of MATLAB through hands-on examplesShows How to Solve Math Problems Using MATLAB
The mathematical software MATLAB® integrates computation, visualization, and programming to produce a powerful tool for a number of different tasks in mathematics. Focusing on the MATLAB toolboxes …
Brings Readers Up to Speed in This Important and Rapidly Growing Area
Supported by many examples in mathematics, physics, economics, engineering, and other disciplines, Essentials of Topology with Applications provides a clear, insightful, and thorough introduction to the basics of modern topology. …
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