DOCSLIB.ORG

Model Theory and Exponentiation David Marker

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Model Theory and Exponentiation David Marker Model Theory and Exponentiation David Marker odel theory is a branch of math- interested in showing that there is an algorithm ematical logic in which one studies for deciding first-order sentences about the real mathematical structures by con- numbers and hence showing the decidability of sidering the first-order sentences elementary Euclidean geometry. true of those structures and the Tarski’s decidability of the real numbers Msets definable in those structures by first-order stands in stark contrast to Gödel’s undecid- formulas. ability of the natural numbers. The great free- The fields of real and complex numbers have dom we have to solve polynomial equations in long served as motivating examples for model the real numbers insures that the real solution theorists. Many model theoretic concepts arose sets of systems of polynomial equations are by abstracting classical algebraic phenomena to topologically well behaved. In the natural num- a more general setting (see, for example, [29]). bers such solution sets can behave very badly. In the past five years significant progress has This wild behavior allows considerable latitude been made in the other direction. Model theoretic for coding complicated combinatorial phenom- methods have been used to develop new in- ena, while, as we will see below, in the reals we sights into real analytic geometry. In particular, cannot even define the integers. this has led to a generalization of semialgebraic The natural language for studying the real and subanalytic geometry to a setting in which numbers is the language of ordered rings one studies global exponentiation on the reals. = +, , ,<,0,1 . In this language the basic - L { − · } L Tarski’s Theorem formulas are of the form p(v1,...,vn)= q(v ,...,v ) and p(v ,...,v )<q(v ,...,v ) The logical study of the field of real numbers 1 n 1 n 1 n where p, q Z[X1,...,Xn]for some n N. The began with the work of Tarski. These results ∈ ∈ set of all -formulas is obtained by taking the were announced by Tarski in [33], but publica- L basic -formulas and closing under the Boolean tion was interrupted by the war, and the proofs L connectives , , and quantifiers and . For did not appear until [34]. Tarski was primarily ∧ ∨ ¬ ∃ ∀ example, (1) yx=y2 y David Marker is professor of mathematics at the Uni- ∃ − versity of Illinois at Chicago. His e-mail address is (2) x y (x = y2 x + y2 =0) [email protected]. ∀ ∃ ∨ This paper is based on an invited lecture presented in (3) w (x w 2 + xy =0 w<3y 4) October 1994 at the AMS Central regional meeting in ∃ − ∧ − Stillwater, Oklahoma. are legitimate -formulas. We say that an oc- L Partially supported by NSF grants DMS-9306159 and currence of a variable in a formula is free if it is INT-9224546, and an AMS Centennial Fellowship. not inside the scope of a quantifier; otherwise, JULY 1996 NOTICES OF THE AMS 753 we say it is bound. For example, in (1) x is a free Step 1: (Elimination of Quantifiers) If variable and y is bound, while in (2) both x and φ(v1,...,vn) is an -formula, then there is an L y are bound. If a formula has free variables, we equivalent quantifier-free -formula L often denote the formula in a way that explic- (v1, . , vn). (Here by equivalent we mean that itly shows which variables are free. For example, φ holds of a = (a1, . , an) if and only if holds n we might denote (3) by φ(x, y). We think of a for- of a for all a R .) Moreover, there is an algo- ∈ mula φ(x1, . , xn) as expressing a relation that rithm that transforms φ to . holds among the variables x1, . , xn. For exam- Note that quantifier-free formulas are just fi- ple, nite Boolean combinations of p(x1, . , xn) = q(x , . , x ) and p(x ,...,x ) > q(x ,...,x ) (4) w xw 2 + yw + z = 0 1 n 1 n 1 n ∃ where p, q Z[X1, . , Xn] for some n N. ∈ ∈ expresses the relationship “x, y, z are coeffi- We are familiar with some natural examples cients of a solvable quadratic polynomial”, while of quantifier elimination. For example, (4) is equivalent to the quantifier-free formula 2 2 (5) x y ux + vxy + zy 0 2 ∀ ∀ ≥ y 4xz 0. − ≥ expresses that u, v, z are the coefficients of a Also, while the formula in free variables positive semidefinite binary quadratic form. vi,j : 1 i, j n that expresses that the ma- A formula is a sentence if it has no free vari- { ≤ ≤ } trix (vi,j ) has an inverse involves existential ables. If is an ordered ring, then any -sen- A L quantifiers, it is equivalent to the quantifier- tence φ will either be true or false for . For A free formula that asserts that the determinant example, (2) is true for the field of real numbers of (vi,j ) is nonzero. R but false for the field of rational numbers Q. Tarski’s original proof relied on techniques Formulas that are not sentences may hold for from elimination theory and Sturm’s algorithm. some assignments of the free variables but not Other proofs are based on model-theoretic ideas for others. For example, in Q (4) holds for (1,5,6) (see, for example, [24]) or cylindric decomposi- but fails for (1,0,-2). Of course, (4) holds for (1,0,- tion (see, for example, [2]). 2) in R. The first-order theory of R, T h(R), is the The second step is just an easy observation. set of all -sentences that are true about R. L Many important properties of R can be expressed Step 2: Quantifier-free sentences are finite Boolean as -sentences. For example, (2) expresses that combinations of ones of the form m = n and L for any x either x or x is a square. For each n m > n where m and n are integers. These can − be decided easily. the sentence Thus, to decide if a sentence φ is true in R, a, b w0, . , wn [(a < b ∀ ∀ ∧ n n we first transform it into a quantifier-free sen- i i tence and then test to see if the quantifier-free ( wia )( wib ) < 0) → sentence is true. iX=0 iX=0 n While Tarski’s result proves the decidability i x(a < x < b wix = 0)] of the theory of the real numbers, the algorithm ∃ ∧ iX=0 obtained is far from feasible. The best-known al- gorithm for eliminating quantifiers is doubly expresses that the sign change property holds exponential in the number of quantifier blocks for polynomials of degree n. One sees that all [17], and it is known that even deciding first- of the axioms for real closed ordered fields (see order sentences for the ordered group (R, +, <) [21]) are contained in T h(R). On the other hand, is exponentially hard [14]. there is no first-order sentence that expresses It is also worth noting that for any formula the completeness of the ordering of R. Indeed, φ(v1, . , vn), the formula (v1,...,vn) ob- we cannot hope to characterize the real numbers tained by quantifier elimination is actually equiv- up to isomorphism by T h(R), as the Löwenheim- alent to φ(v1, . , vn) in any other real closed Skolem theorem (see [3]) insures there will be field as well. Since the decision made in Step 2 models of T h(R) of every infinite cardinality. depends only on Z, it follows that a sentence φ Tarski’s Theorem. The first-order theory of the is true in R if and only if it is true in every real closed field if and only if it is true in some real field of real numbers is decidable. That is, there closed field. This means that the axioms for real is an algorithm that will decide on input φ closed fields completely axiomatize T h(R). whether or not φ is a sentence true of R. In [35] Tarski asked if his results could be ex- The proof of Tarski’s theorem breaks into tended to include the exponential function. x two steps. Namely, let exp = e . Is the first-order L L ∪ { } 754 NOTICES OF THE AMS VOLUME 43, NUMBER 7 exp-theory of R decidable?1 We will give a par- to define the integers. Since we can define Z, L tial answer to Tarski’s problem in Section 4. Gödel’s results show that this theory is far from One might hope to generalize the arguments decidable. used for the ordered field, but serious difficul- ties arise. First, quantifier elimination fails. In [5], Definable Sets and o-Minimality We say that X Rn is definable using if there van den Dries showed that the formula ⊆ L is an -formula φ(x1, . , xn, y1, . , ym) and y > 0 w (wy = x z = yew ) L b1, . , bm R such that ∧ ∃ ∧ ∈ is not equivalent to a quantifier-free exp-for- n L X = (a1, . , an) R : mula. { ∈ φ(a1, . , an, b1, . , bm) holds . There are rather silly ways to extend the lan- } guage so that we do have elimination of quan- For example, let φ(x1, x2, y) be the formula tifiers, but the trick is to extend the language so 2 2 2 2 u (x1 u) + (x2 u ) < y . that we have quantifier elimination and the quan- ∃ − − tifier-free formulas are still natural and mean- Substituting the real number π for the variable ingful. y, we get the definable set 2 Even if we could eliminate quantifiers, there (x1, x2) : the distance from (x1, x2) to the is no obvious uniform algorithm for deciding { π quantifier-free sentences like parabola y = x2 is less than .
Load more

Recommended publications
  • The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
    To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition.
    [Show full text]
  • Grade 6 Math Circles Fall 2019 - Oct 22/23 Exponentiation
    Faculty of Mathematics Centre for Education in Waterloo, Ontario Mathematics and Computing Grade 6 Math Circles Fall 2019 - Oct 22/23 Exponentiation Multiplication is often thought as repeated addition. Exercise: Evaluate the following using multiplication. 1 + 1 + 1 + 1 + 1 = 7 + 7 + 7 + 7 + 7 + 7 = 9 + 9 + 9 + 9 = Exponentiation is repeated multiplication. Exponentiation is useful in areas like: Computers, Population Increase, Business, Sales, Medicine, Earthquakes An exponent and base looks like the following: 43 The small number written above and to the right of the number is called the . The number underneath the exponent is called the . In the example, 43, the exponent is and the base is . 4 is raised to the third power. An exponent tells us to multiply the base by itself that number of times. In the example above,4 3 = 4 × 4 × 4. Once we write out the multiplication problem, we can easily evaluate the expression. Let's do this for the example we've been working with: 43 = 4 × 4 × 4 43 = 16 × 4 43 = 64 The main reason we use exponents is because it's a shorter way to write out big numbers. For example, we can express the following long expression: 2 × 2 × 2 × 2 × 2 × 2 as2 6 since 2 is being multiplied by itself 6 times. We say 2 is raised to the 6th power. 1 Exercise: Express each of the following using an exponent. 7 × 7 × 7 = 5 × 5 × 5 × 5 × 5 × 5 = 10 × 10 × 10 × 10 = 9 × 9 = 1 × 1 × 1 × 1 × 1 = 8 × 8 × 8 = Exercise: Evaluate.
    [Show full text]
  • Meta-Complex Numbers Dual Numbers
    Meta- Complex Numbers Robert Andrews Complex Numbers Meta-Complex Numbers Dual Numbers Split- Complex Robert Andrews Numbers Visualizing Four- Dimensional Surfaces October 29, 2014 Table of Contents Meta- Complex Numbers Robert Andrews 1 Complex Numbers Complex Numbers Dual Numbers 2 Dual Numbers Split- Complex Numbers 3 Split-Complex Numbers Visualizing Four- Dimensional Surfaces 4 Visualizing Four-Dimensional Surfaces Table of Contents Meta- Complex Numbers Robert Andrews 1 Complex Numbers Complex Numbers Dual Numbers 2 Dual Numbers Split- Complex Numbers 3 Split-Complex Numbers Visualizing Four- Dimensional Surfaces 4 Visualizing Four-Dimensional Surfaces Quadratic Equations Meta- Complex Numbers Robert Andrews Recall that the roots of a quadratic equation Complex Numbers 2 Dual ax + bx + c = 0 Numbers Split- are given by the quadratic formula Complex Numbers p 2 Visualizing −b ± b − 4ac Four- x = Dimensional 2a Surfaces The quadratic formula gives the roots of this equation as p x = ± −1 This is a problem! There aren't any real numbers whose square is negative! Quadratic Equations Meta- Complex Numbers Robert Andrews What if we wanted to solve the equation Complex Numbers x2 + 1 = 0? Dual Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces This is a problem! There aren't any real numbers whose square is negative! Quadratic Equations Meta- Complex Numbers Robert Andrews What if we wanted to solve the equation Complex Numbers x2 + 1 = 0? Dual Numbers The quadratic formula gives the roots of this equation as Split-
    [Show full text]
  • Reihenalgebra: What Comes Beyond Exponentiation? M
    Reihenalgebra: What comes beyond exponentiation? M. M¨uller, [email protected] Abstract Addition, multiplication and exponentiation are classical oper- ations, successively defined by iteration. Continuing the iteration process, one gets an infinite hierarchy of higher-order operations, the first one sometimes called tetration a... aa a b = a b terms , ↑ o followed by pentation, hexation, etc. This paper gives a survey on some ideas, definitions and methods that the author has developed as a young pupil for the German Jugend forscht science fair. It is meant to be a collection of ideas, rather than a serious formal paper. In particular, a definition for negative integer exponents b is given for all higher-order operations, and a method is proposed that gives a very natural (but non-trivial) interpolation to real (and even complex) integers b for pentation and hexation and many other operations. It is an open question if this method can also be applied to tetration. 1 Introduction Multiplication of natural numbers is nothing but repeated addition, a + a + a + ... + a = a b . (1) · b terms | {z } Iterating multiplication, one gets another operation, namely exponentiation: b a a a ... a = a =: aˆb . (2) · · · · b terms | {z } Classically, this is it, and one stops here. But what if one continues the iteration process? One could define something like aˆaˆaˆ ... ˆa = a b . ↑ b terms | {z } But, wait a minute, in contrast to eq. (1) and (6), this definition will depend on the way we set brackets, i.e. on the order of exponentiation! Thus, we have to distinguish between the two canonical possibilities aaa..
    [Show full text]
  • LARGE PRIME NUMBERS This Writeup Is Modeled Closely on A
    LARGE PRIME NUMBERS This writeup is modeled closely on a writeup by Paul Garrett. See, for example, http://www-users.math.umn.edu/~garrett/crypto/overview.pdf 1. Fast Modular Exponentiation Given positive integers a, e, and n, the following algorithm quickly computes the reduced power ae % n. (Here x % n denotes the element of f0; ··· ; n − 1g that is congruent to x modulo n. Note that x % n is not an element of Z=nZ since such elements are cosets rather than coset representatives.) • (Initialize) Set (x; y; f) = (1; a; e). • (Loop) While f > 0, do as follows: { If f%2 = 0 then replace (x; y; f) by (x; y2 % n; f=2), { otherwise replace (x; y; f) by (xy % n; y; f − 1). • (Terminate) Return x. The algorithm is strikingly efficient both in speed and in space. Especially, the operations on f (halving it when it is even, decrementing it when it is odd) are very simple in binary. To see that the algorithm works, represent the exponent e in binary, say e = 2g + 2h + 2k; 0 ≤ g < h < k: The algorithm successively computes (1; a; 2g + 2h + 2k) g (1; a2 ; 1 + 2h−g + 2k−g) g g (a2 ; a2 ; 2h−g + 2k−g) g h (a2 ; a2 ; 1 + 2k−h) g h h (a2 +2 ; a2 ; 2k−h) g h k (a2 +2 ; a2 ; 1) g h k k (a2 +2 +2 ; a2 ; 0); and then it returns the first entry, which is indeed ae. Fast modular exponentiation is not only for computers. For example, to compute 237 % 149, proceed as follows, (1; 2; 37) ! (2; 2; 36) ! (2; 4; 18) ! (2; 16; 9) ! (32; 16; 8) ! (32; −42; 4) ! (32; −24; 2) ! (32; −20; 1) ! ( 105 ; −20; 0): 2.
    [Show full text]
  • The Strange Properties of the Infinite Power Tower Arxiv:1908.05559V1
    The strange properties of the infinite power tower An \investigative math" approach for young students Luca Moroni∗ (August 2019) Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. Paul Lockhart { \A Mathematician's Lament" Abstract In this article we investigate some "unexpected" properties of the \Infinite Power Tower 1" function (or \Tetration with infinite height"): . .. xx y = f(x) = xx where the \tower" of exponentiations has an infinite height. Apart from following an initial personal curiosity, the material collected here is also intended as a potential guide for teachers of high-school/undergraduate students interested in planning an activity of \investigative mathematics in the classroom", where the knowledge is gained through the active, creative and cooperative use of diversified mathematical tools (and some ingenuity). The activity should possibly be carried on with a laboratorial style, with no preclusions on the paths chosen and undertaken by the students and with little or no information imparted from the teacher's desk. The teacher should then act just as a guide and a facilitator. The infinite power tower proves to be particularly well suited to this kind of learning activity, as the student will have to face a challenging function defined through a rather uncommon infinite recursive process. They'll then have to find the right strategies to get around the trickiness of this function and achieve some concrete results, without the help of pre-defined procedures. The mathematical requisites to follow this path are: functions, properties of exponentials and logarithms, sequences, limits and derivatives.
    [Show full text]
  • Definition of a Vector Space
    i i “main” 2007/2/16 page 240 i i 240 CHAPTER 4 Vector Spaces 5. Verify the commutative law of addition for vectors in 8. Show with examples that if x is a vector in the first R4. quadrant of R2 (i.e., both coordinates of x are posi- tive) and y is a vector in the third quadrant of R2 (i.e., 6. Verify the associative law of addition for vectors in both coordinates of y are negative), then the sum x +y R4. could occur in any of the four quadrants. 7. Verify properties (4.1.5)–(4.1.8) for vectors in R3. 4.2 Definition of a Vector Space In the previous section, we showed how the set Rn of all ordered n-tuples of real num- bers, together with the addition and scalar multiplication operations defined on it, has the same algebraic properties as the familiar algebra of geometric vectors. We now push this abstraction one step further and introduce the idea of a vector space. Such an ab- straction will enable us to develop a mathematical framework for studying a broad class of linear problems, such as systems of linear equations, linear differential equations, and systems of linear differential equations, which have far-reaching applications in all areas of applied mathematics, science, and engineering. Let V be a nonempty set. For our purposes, it is useful to call the elements of V vectors and use the usual vector notation u, v,...,to denote these elements. For example, if V is the set of all 2 × 2 matrices, then the vectors in V are 2 × 2 matrices, whereas if V is the set of all positive integers, then the vectors in V are positive integers.
    [Show full text]
  • Efficient Modular Exponentiation
    Efficient Modular Exponentiation R. C. Daileda February 27, 2018 1 Repeated Squaring Consider the problem of finding the remainder when am is divided by n, where m and n are both is \large." If we assume that (a; n) = 1, Euler's theorem allows us to reduce m modulo '(n). But this still leaves us with some (potential) problems: 1. '(n) will still be large as well, so the reduced exponent as well as the order of a modulo n might be too big to handle by previous techniques (e.g. with a hand calculator). 2. Without knowledge of the prime factorization of n, '(n) can be very hard to compute for large n. 3. If (a; n) > 1, Euler's theorem doesn't apply so we can't necessarily reduce the exponent in any obvious way. Here we describe a procedure for finding the remainder when am is divided by n that is extremely efficient in all situations. First we provide a bit of motivation. Begin by expressing m in binary1: N X j m = bj2 ; bj 2 f0; 1g and bN = 1: j=0 Notice that N N m PN b 2j Y b 2j Y 2j b a = a j=0 j = a j = (a ) j j=0 j=0 and that 2 2 j j−1 2 j−2 2 2 a2 = a2 = a2 = ··· = a22 ··· : | {z } j squares So we can compute am by repeatedly squaring a and then multiplying together the powers for which bj = 1. If we perform these operations pairwise, reducing modulo n at each stage, we will never need to perform modular arithmetic with a number larger than n2, no matter how large m is! We make this procedure explicit in the algorithm described below.
    [Show full text]
  • Logarithms and Exponentiation Guide
    Skills Refresher: Exponentiation, Scientific Notation, Logarithms, and pH Logarithms and exponentiation are opposite mathematical functions involving two numbers, one a base and the other an exponent. This method of expressing numbers is important because many relationships found in the environment and in nature are logarithmic, such as the pH scale. Exponentiation Exponentiation expresses a base number (a) to the power of an exponent (n): x=an. This is also expressed as "a raised to the nth power" (where a and n are numbers). Here are some examples for different exponents (the * symbol indicates multiplication): Exponent Equivalent Examples positive number multiplying a by itself n times 64 = 6*6*6*6 = 1,296 negative number multiplying the reciprocal of a by itself n times 4-3 = 1/43 = 1/(4*4*4) = 1/64 1 (one) multiplying a by itself one time 171 = 17 0 (zero) called an 'empty product'; result for any non-zero base always 1 270 = 1 Common Exponents Exponents of 2 (squaring the number), used in the binary number system, are important in computer work, since computers can only choose between two options, commonly expressed as 0 and 1, represented by one bit. Commonly used to describe file sizes, there are 8 bits in one byte (computers show file sizes in kilobytes, megabytes, and gigabytes). One byte can be any of 28 (=2*2*2*2*2*2*2*2 = 256) values. The result of a number with the exponent 3 is the cube of the number: a3 = a*a*a. We use a base ten (decimal) number system, and exponentiation with base 10 (called scientific notation) is commonly used to describe very large and very small numbers.
    [Show full text]
  • Student Understanding of Large Numbers and Powers: the Effect of Incorporating Historical Ideas
    Student Understanding of Large Numbers and Powers: The Effect of Incorporating Historical Ideas Mala S. Nataraj Michael O. J. Thomas The University of Auckland The University of Auckland <[email protected]> <[email protected]> The value of a consideration of large numbers and exponentiation in primary and early secondary school should not be underrated. In Indian history of mathematics, consistent naming of, and working with large numbers, including powers of ten, appears to have provided the impetus for the development of a place value system. While today’s students do not have to create a number system, they do need to understand the structure of numeration in order to develop number sense, quantity sense and operations. We believe that this could be done more beneficially through reflection on large numbers and numbers in the exponential form. What is reported here is part of a research study that involves students’ understanding of large numbers and powers before and after a teaching intervention incorporating historical ideas. The results indicate that a carefully constructed framework based on an integration of historical and educational perspectives can assist students to construct a richer understanding of the place value structure. Introduction In the current era where very large numbers (and also very small numbers) are used to model copious amounts of scientific and social phenomena, understanding the multiplicative and exponential structure of the decimal numeration system is crucial to the development of quantity and number sense, and multi-digit operations. It also provides a sound preparation for algebra and progress to higher mathematics.
    [Show full text]
  • Complex Numbers
    MAT067 UniversityofCalifornia,Davis Winter2007 Introduction to Complex Numbers. Summary Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (January 14, 2007) 1 Definition of Complex Numbers Let R denote the set of real numbers. We will denote the set of complex numbers by C. Here is the definition. Definition 1.1. The set of complex numbers C is defined as C = (x, y) x, y R { | ∈ } For any complex number z =(x, y), we call Re(z)= x the real part of z and Im(z)= y the imaginary part of z. In other words, we are defining a new collection of numbers z by taking every possible ordered pair (x, y) of real numbers x, y R, and x is called the real part of the ordered ∈ pair (x, y) to imply that the set of real numbers R should be identified with the subset (x, 0) x R C. It is also common to use the term purely imaginary for any complex number{ | of∈ the} form ⊂ (0,y), where y R. In particular, the complex number (0, 1) is special, ∈ and it is given the name imaginary unit. It is standard to denote it by the single letter i (or j if i is being used for something else, such as for electric current in Electrical Engineering). Note that z =(x, y)= x(1, 0)+ y(0, 1) = x1+ yi. We usually write z = x + iy. It is often significantly easier to perform arithmetic operations on complex numbers when written in “x + iy” notation, rather than the ordered pair notation of the definition.
    [Show full text]
  • A Calculus Oasis on the Sands of Trigonometry
    φ θ b 3 A Calculus Oasis on the sands of trigonometry A h Conal Boyce A Calculus Oasis on the sands of trigonometry with 86+ illustrations by the author Conal Boyce rev. 130612 Copyright 2013 by Conal Boyce All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written prior permission of the author. Cover design and layout by CB By the same author: The Chemistry Redemption and the Next Copernican Shift Chinese As It Is: A 3D Sound Atlas with First 1000 Characters Kindred German, Exotic German: The Lexicon Split in Two ISBN 978-1-304-13029-7 www.lulu.com Printed in the United States of America In memory of Dr. Lorraine (Rani) Schwartz, Assistant Professor in the Department of Mathematics from 1960-1965, University of British Columbia Table of Contents v List of Figures . vii Prologue . 1 – Vintage Calculus versus Wonk Calculus. 3 – Assumed Audience . 6 – Calculus III (3D vector calculus) . 9 – A Quick Backward Glance at Precalculus and Related Topics . 11 – Keeping the Goal in Sight: the FTC . 14 I Slopes and Functions . 19 – The Concept of Slope . 19 – The Function Defined. 22 – The Difference Quotient (alias ‘Limit Definition of the Derivative’). 29 – Tangent Line Equations . 33 – The Slope of e . 33 II Limits . 35 – The Little-δ Little-ε Picture. 35 – Properties of Limits . 38 – Limits, Continuity, and Differentiability. 38 III The Fundamental Theorem of Calculus (FTC) . 41 – A Pictorial Approach (Mainly) to the FTC.
    [Show full text]