Computable Calculus

Computable Calculus treats the fundamental topic of calculus in a novel way that is more in tune with today's computer age. Comprising 11 chapters, the book presents mathematical analysis that has been created to deal with constructively defined concepts. The book's "show your work" approach makes it easier to understand the pitfalls of various computations and, more importantly, how to avoid these pitfalls.

Preface

xi

Acknowledgments

xiii

Introduction

1

(6)

What Is Computable Calculus?

1

(2)

What Are the Advantages of Computable Calculus, If Any?

3

(1)

A Brief Description of Computable Calculus

4

(3)

The Real Numbers

7

(14)

Definition of a Real Number

7

(3)

An Ordering of Intervals

10

(4)

Interval Arithmetic

14

(1)

The Real Number Operations +, -, x, ÷

15

(3)

The Absolute Value of a Real Number

18

(3)

Solvable Problems and Nonsolvable Problems

21

(14)

Introduction

21

(1)

Turning's Resolution of the Halting Problem

22

(2)

A Certain Computation Problem

24

(2)

Deciding Whether a Number is Zero

26

(1)

A List of Nonsolvable Problems

27

(1)

Solvable Problems

28

(2)

Key Nonsolvable Problems

30

(1)

Deciding Whether a Number is Rational or Irrational

31

(1)

Deciding Which of Two Real Numbers is Larger

32

(3)

Sequences and Functions

35

(16)

Sequences of Real Numbers

35

(1)

The Cantor Counting Theory

36

(1)

Functions

37

(3)

The nth Root of a Real Number

40

(1)

An Algebra of Functions

41

(1)

The Function sgn(x)

42

(2)

Functions Defined on Intervals

44

(2)

Semifunctions

46

(1)

Other Calculus Concepts

47

(4)

The Ideal Computer

51

(42)

The Goal of This Chapter

51

(1)

The Various Methods of Proof

52

(2)

Definition of the Ideal Computer

54

(1)

The Ideal Computer Steps

55

(4)

Viewing, Compiling, and Executing Ideal Computer Programs

59

(3)

More Ideal Computer Details

62

(2)

Programming the Ideal Computer to Add Natural Integers

64

(4)

The Addition of Two General Integers

68

(2)

The Subtraction, Multiplication, and Division of Integers

70

(1)

Rational Number Arithmetic

71

(2)

Interval Arithmetic

73

(1)

The Retrieval of a Program from Disk Memory

74

(2)

The Real Number π

76

(1)

Changing and Approximation Algorithm

77

(2)

Rational Numbers Converted to Real Numbers

79

(1)

nth Roots of Real Numbers

79

(1)

Arithmetic Operations on Real Numbers

80

(1)

Rational Sequence Arithmetic

81

(1)

Function Programs

82

(1)

The Exponential Function ex

83

(1)

Two Semifunction Examples

84

(2)

The Beginning of the Nonsolvability Proof for Problem 3.1

86

(2)

Programs That Decide Whether a Real Number is Zero

88

(5)

Limits

93

(26)

Limit of a Sequence

93

(2)

Monotone Sequences

95

(2)

The Specker Theorem

97

(2)

Consequences of Specker's Theorem

99

(1)

Limit of a Function

99

(6)

Using Limits to Extend a Function's Domain

105

(2)

A Standard Result for Functions

107

(3)

Unbounded Continuous Functions on [a, b]

110

(2)

Limits of Sequences of Functions

112

(2)

Limits of Functions with More Than One Variable

114

(5)

Uniformaly Continuous Functions

119

(12)

Introduction

119

(5)

Bounds of Uniformly Continuous Functions

124

(7)

The Derivative

131

(12)

A Difficulty with the Derivative Definition

131

(4)

Rules of Differentiation

135

(2)

A Computation Problem

137

(1)

The Mean Value Theorem

138

(5)

The Riemann Integral

143

(8)

Riemann Sums

143

(1)

The Integration of Uniformly Continuous Functions

144

(2)

Properties of Integrals

146

(2)

Defining a Function by Means of an Integral

148

(2)

A Mean Value Theorem for Integrals

150

(1)

Functions of Two Variables

151

(12)

Partial Derivatives

151

(3)

The Chain Rule

154

(3)

Equality of Cross Derivatives

157

(6)

The Differential Equation y'=f(x,y)

163

(12)

Introduction

163

(2)

The Lipschitz Condition

165

(5)

The Possibility of No Solution

170

(5)

Ideal Computer Simulation

175

(10)

The Extended Ideal Computer

175

(1)

Call Loops

176

(2)

Composing and Deleting a Program

178

(2)

Managing the Ideal Computer's Input and Output

180

(2)

Possible Projects

182

(3)

References

185

(4)

About the CD

189

(2)

Index

191

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