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

2025/10

This contains the text of the popular science book, a tour of numbers. The book targets to present the historic introduction to numbers. As a central notion in mathematics, it connects varies of theories. We are going to introduce interesting stories and great mathematicians along with the history of numbers. There is zero content generated by AI, but 100% by human. We plan to release the book in both English and 中文 by 2026. We finished in 中文 (PDF file) as of October 2025.

• Preface
• Chapter 1, Numeral system
  Numeral systems in ancient Egypt, Babylonia, Rome, China, and Maya. Show how language influenced numbers. Introduce the widely used Hindu-Arabic (positional decimal) numeral system; why it has advantage in calculation. Binary numeral system and computer.
• Chapter 2, Zero
  The notion of zero in ancient India. Show how people rejected, debated, and finally accepted zero. The confusion and debating about negative numbers; how people accepted negative numbers. Explain why 'Two negatives make a positive'. Model the negative number through a tuple of natural numbers.
• Chapter 3, Fractions
  Fraction and Pythagoras music tuning. Egyptian fractions; Babylonia 60 based decimals; Arithmetic rules of fractions in ancient China. Indian fractions and the symbol of fraction bar. Fractions and decimals, including infinitely cyclic decimals. Extend numbers with fractions.
• Chapter 4, Irrationals
  (a). All is number., The school of Pythagorean. (b). Pythagoras's theorem. Varies of proofs to the Pythagoras's theorem. Number theory. Pythagorean triples, perfect numbers, Fermat numbers, and Mersenne numbers. (c) Irrationals. Irrationals from the straight edge and compass construction. (d). Euclidean algorithm. Euclidean algorithm and irrationals, continued fractions.
• Chapter 5, Reals
  (a) Geometric construction, Arithmetic of geometric construction. Coordinate geometry and geometric construction. Regular polygon construction and its limitation. (b) Pi. Pi in ancient time. Geometric approximation. Binomial theorem. The fundamental theorem of calculus. Series. Leibniz formula. Lambert's proof that Pi is irrational. Reals. Eudoxus's proportion theory. Dedekind cut.
• Chapter 6, Complex numbers
  (a) Cubic equation. Geometric and algebraic solutions to quadratic and cubic equations. The arising of imaginary numbers in cubic equation. (b) Meaning of complex number. Geometric meaning and Argand diagram. Algebraic meaning as a pair of numbers by Hamilton. The fundamental theorem of algebra. (c) e. History of e; definition as a limit. Euler's identity. e and the logarithmic spiral. (d) Pentagon. Geometric construction of pentagon with complex numbers.
• Chapter 7, Algebraic numbers
  (a) Fermat's last theorem and complex numbers. Infinite descent and Fermat's proof to n = 4; Euler's proof to n = 3 with the interpretation in complex numbers. (b) Ideal numbers. Fail of unique factorization in Lame's attempt to proof Fermat's last theorem. (c) Gaussian integers. Gaussian integers as an example of algebraic integers. (d) Quadratic numbers. Quadratic numbers and fail of unique factorization. (e) Ideals. The ideals as a way to restore unique factorization. (f) Infinite sets. One to one correspondence to compare infinite sets; Countable and uncountable infinite sets; reals as an uncountable set.
• Appendices and answers
  A. Answers to all exercises; B.1. Proof to the commutative of plus 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 Gauss-Wantzel theorem; B.5. A combinatorial proof to Fermat's little theorem; B.6. Some limits; B.7. Tangent function in continued fraction; B.8. Formula of Fibonacci series through generating function. B.9. Wilson's theorem. C. Greek letters.

You may use gitpod to build the PDF book from cloud desktop. (It takes about 15 ~ 20 min to build the workspace at the first time, while later access is quick.) For local build, you need TeXLive. We use LuaLaTeX, an extended version of TeX. Please refer to INSTALL for detail.

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