# Number and the Nature of Mathematics

## Contents

About the Author

Introduction

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1 Mathematics: Misconceptions

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2 Early Counting – Speculation

First Steps

Then, The Babylonians, Egyptians, and The Greeks

And Elsewhere

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3 Natural Numbers and Axiom Systems

Axioms

Definitions and Properties

Addition

Properties of Addition – Proof Based on Axioms

The Associative Law

The Law of Substitution

The Commutative Law

What is Mathematics: A First Look

The Law of Cancellation

Summary So Far

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4 An Introduction To Sets And A Bit Of Logic

Sets – Definitions and Properties

Simple Logic

A Note About Infinity

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5 Zero. A First Extension – The Whole Numbers

Definitions and Properties

A new relationship: Less–Equals

Some Terminology

Multiplication

What is Easy, What is Hard, and Mathematical Insight

Algebra, Variables, Unknowns, and How Mathematics is Not Computer Science

Some Interesting Sums

Representing Whole Numbers

Summary So Far

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6 The Integers

Equivalence Relations and Equivalence Classes

Definitions and Properties

Generalization and the Nature of Mathematics – Foreshadowing Abstract Algebra

Applied Mathematics: A Simple Example

Subtraction

A Note on Proof

Substitution and Cancellation Laws in Z

Integer Division and Remainders

A Note on Mathematical Insight

Summary So Far

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7 The Rational Numbers

Definitions and Properties

Denseness and Distance in the Rational Numbers

Cardinality of the Rational Numbers

Decimals and the Rational Numbers

Summary So Far

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8 Infinite Sequences

Definition and Properties

Infinite Series

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9 The Real Numbers

Cauchy Sequences

Definition and Properties

Cardinality of the Real Numbers

Foreshadowing: Topology

Foreshadowing – Calculus and Analysis

Summary So Far

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10 Polynomials

Definition and Properties

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11 The Complex Numbers

Definition and Properties

Foreshadowing – Trigonometry

Foreshadowing – Matrix Theory

Summary So Far

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12 Mathematics

What Do Mathematicians Do?

A Simple Example

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13 Ideas for Studying Mathematics

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14 Further Reading

Index