The Journey of Numbers

Edition: $\displaystyle \phi = \frac{\sqrt{5} - 1}{2} = 0.618$

2026/06

This repository contains the text of the popular math book, The Journey of Numbers. The book provides a historical introduction to numbers. As a central notion in mathematics, numbers connect various areas and theories. This book includes 33 interesting stories and biographies of great mathematicians in the history of numbers. The book is available in English (PDF) and Chinese (中文) (PDF).

Contents

• Preface
• Chapter 1, Numeral systems
  Covers numeral systems in ancient Egypt, Babylonia, Rome, China, and Maya. Shows how language influenced numbers. Introduces the widely used Hindu–Arabic (positional decimal) numeral system and explains why it has advantages for calculation. Briefly introduces the binary numeral system and computers.
• Chapter 2, Zero
  Introduces the notion of zero in ancient India and how people rejected, debated, and finally accepted it. Covers confusion and debates about negative numbers and their acceptance. Explains why "two negatives make a positive" and presents how to model negative numbers as pairs of natural numbers.
• Chapter 3, Fractions
  Pythagoras music tuning and fractions. Egyptian fractions; the Babylonian base-60 fractional system; arithmetic rules of fractions in ancient China. Indian fractions and the fraction bar symbol. Fractions and decimals, including repeating decimals. Extension of numbers with fractions.
• Chapter 4, Irrational numbers
  (a) All is number. The Pythagorean school. (b) Pythagoras's theorem. Various proofs of the Pythagorean theorem. Number theory. Pythagorean triples, perfect numbers, Fermat numbers, and Mersenne numbers. (c) Irrational numbers. Irrational numbers from geometric constructions. (d) Euclidean algorithm. The Euclidean algorithm and its relation to irrational numbers and continued fractions.
• Chapter 5, Real numbers
  (a) Geometric constructions. Arithmetic of geometric constructions. Coordinate geometry and geometric construction. Regular polygon construction and its limitations. (b) Pi. Pi in ancient times. Geometric approximations. Binomial theorem. The fundamental theorem of calculus. Series, and Leibniz formula. Lambert's proof that Pi is irrational. Real. Eudoxus's theory of proportion and Dedekind cuts.
• Chapter 6, Complex numbers
  (a) Cubic equations. Geometric and algebraic solutions to quadratic and cubic equations; the emergence of imaginary numbers in cubic equations. (b) Interpretations of complex numbers. Geometric interpretation and the Argand diagram. Algebraic interpretation as a pair of real numbers by Hamilton. The fundamental theorem of algebra. (c) e. History of e and its definition as a limit. Euler's identity and the relation between e and the logarithmic spiral. (d) Regular Pentagon. Geometric construction of the regular pentagon with complex numbers.
• Chapter 7, Algebraic numbers
  (a) Fermat's Last Theorem and complex numbers. Infinite descent and Fermat's proof for n = 4; Euler's proof for n = 3 with interpretations using complex number. (b) Ideal numbers. Failure of unique factorization in Lamé's attempt to prove Fermat's Last Theorem. (c) Gaussian integers. An example of algebraic integer. (d) Quadratic numbers. Quadratic numbers and failure of unique factorization. (e) Ideals. Ideals as a way to restore unique factorization. (f) Infinite sets. One-to-one correspondences for comparing infinite sets; countable and uncountable infinities; R as an uncountable set.
• Appendices and answers
  A. Answers to all exercises; B.1. Proof of the commutativity of addition and multiplication of natural numbers; B.2. Algorithm to search the "best" Egyptian fraction decomposition. B.3. Even perfect numbers; B.4. Proof scheme of the Gauss–Wantzel theorem; B.5. A combinatorial proof of Fermat's little theorem; B.6. Some limits; B.7. Tangent function in continued fractions; B.8. Formula for the Fibonacci sequence via a generating function. B.9. Wilson's theorem. C. Greek letters.

Claim use of AI: The content is 100% human-written but not generated by AI. I use AI only in these tasks: drawing figures in Tikz, and the first round of translation from Chinese to English from section 5.2.2 to 5.3.2, and from B.3 to B.7.

Install

I use LuaLaTeX, an extended version of TeX provided in the TeXLive distribution, to build the book in PDF format. Please refer to INSTALL for details.

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