Elements of mathematics : a problem-centered approach to history and foundations / Gabor Toth.

Format:

Book

Author/Creator:

Tóth, Gábor, 1964- author.

Series:

Undergraduate Texts in Mathematics

Language:

English

Subjects (All):

Mathematics--History.

Mathematics.

Physical Description:

1 online resource (534 pages)

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

This textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed. Beginning with a thorough treatment of the natural numbers via Peano's axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoff's axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics. Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details.

Contents:

Intro

Preface

Why This Book?

Audience

The Historical Context

In Closing: Gelfand's Teaching Legacy

Acknowledgment

Contents

0 Preliminaries: Sets, Relations, Maps

0.1 Sets

Exercises

0.2 Relations

Exercise

0.3 Maps and Real Functions

0.4 Cardinality

0.5 The Zermelo-Fraenkel Axiomatic Set Theory*

1 Natural, Integral, and Rational Numbers

1.1 Natural Numbers

1.2 Integers

1.3 The Division Algorithm for Integers

1.4 Rational Numbers

2 Real Numbers

2.1 Real Numbers via Dedekind Cuts

2.2 Infinite Decimals as Real Numbers

2.3 Real Numbers via Cauchy Sequences

2.4 Dirichlet Approximation and Equidistribution*

3 Rational and Real Exponentiation

3.1 Arithmetic Properties of the Limit

3.2 Roots, Rational and Real Exponents

3.3 Logarithms

3.4 The Stolz-Cesàro Theorems

4 Limits of Real Functions

4.1 Limit Inferior and Limit Superior

4.2 Continuity

4.3 Differentiability

5 Real Analytic Plane Geometry

5.1 The Birkhoff Metric Geometry

5.2 The Cartesian Model of the Birkhoff Plane

5.3 The Cartesian Distance

5.4 The Triangle Inequality

5.5 Lines and Circles

5.6 Arc Length on the Unit Circle

5.7 The Birkhoff Angle Measure

5.8 The Principle of Shortest Distance*

5.9 π According to Archimedes*

6 Polynomial Expressions

6.1 Polynomials

6.2 Arithmetic Operations on Polynomials

6.3 The Binomial Formula

6.4 Factoring Polynomials

Exercises.

6.5 The Division Algorithm for Polynomials

6.6 Symmetric Polynomials

6.7 The Cauchy-Schwarz Inequality

7 Polynomial Functions

7.1 Polynomials as Functions

7.2 Roots of Cubic Polynomials

7.3 Roots of Quartic and Quintic Polynomials

7.4 Polynomials with Rational Coefficients

7.5 Factoring Multivariate Polynomials

7.6 The Greatest Common Factor

8 Conics

8.1 The General Conic

8.2 Parabolas

8.3 Ellipses

8.4 Hyperbolas

9 Rational and Algebraic Expressions and Functions

9.1 Rational Expressions and Rational Functions

9.2 The Partial Fraction Decomposition

9.3 Asymptotes of Rational Functions

9.4 Algebraic Expressions and Functions, Rationalization

9.5 Harmonic, Geometric, Arithmetic, Quadratic Means

9.6 The Greatest Integer Function

10 Exponential and Logarithmic Functions

10.1 The Natural Exponential Function According to Newton

10.2 The Bernoulli Numbers*

10.3 The Natural Logarithm

10.4 The General Exponential and Logarithmic Functions

10.5 The Natural Exponential Function According to Euler

11 Trigonometry

11.1 The Unit Circle S vs. the Real Line R

11.2 The Sine and Cosine Functions

11.3 Principal Identities for Sine and Cosine

11.4 Trigonometric Rational Functions

11.5 Trigonometric Limits

11.6 Cosine and Sine Series According to Newton

11.7 The Basel Problem of Euler*

11.8 Ptolemy's Theorem

Further Reading

Index.

Notes:

Includes index.

Description based on print version record.

ISBN:

3-030-75051-5

OCLC:

1269094774