Edition:
2025/04
This project is to collaboratively draft 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 Chinese by 2027. We'll draft in Chinese first (and release the PDF file for preview), then follow with the English version.
- Preface
- Chapter 1, Numeral system
Typical numeral systems of ancient Egypt, ancient Babylonia, ancient Rome, ancient China, and Maya. Show how language influenced numbers. Introduce the widely used Hindu-Arabic (positional decimal) numeral system; why it has advantage in calculation. The story of al-Khwarizmi and Fibonacci. Method of counting rods as a calculation tool in ancient China. Binary numeral system and computer. - Chapter 2, Zero
Ancient Indian people developed the notion of zero. How did people reject, debate, and finally accept it. The confusion and debating about negative numbers; how did people accept negative numbers. Explain why 'Two negatives make a positive'. Model the negative number through a tuple (a, b) by van der Waerden. - Chapter 3, Fractions
Fraction and Pythagoras music tuning. Egyptian fractions; Babylonia 60 based decimals; Arithmetic rules of fractions in ancient China. Indian fractions and fraction bar. Fractions and decimals, including infinitely cyclic decimals. Fraction as a typical example that extends numbers. - Chapter 4, All is number
(a). Pythagoras, the mathematician and philosopher in ancient Greece; the school of Pythagorean. (b). Number, shape, and music. Figurate numbers and some sequences. How Pythagoras developed the theory of music through numbers, hence believed all is number. (c). Number and geometry. Straight edge and compass construction and Plato. Arithmetic in straight edge and compass construction. (d). Axiomatic method. Euclid and his Elements, its influence. - Chapter 5, Irrational numbers
(a) Commensurable, concept of common measure; The unreal legend that Hippasus discovered the irrational number. The classic construction puzzle in ancient Greece: double the cube, trisect the angle, and square the circle. (b) Euclid's algorithm. How ancient Greeks solve the greatest common divisor with Euclid's algorithm. Define irrational number with Euclid's algorithm. (c) The treatment of irrational numbers. Odoxos's treatment.$\pi$ and Archimedes; Apollonius and conics sections. (d)$e$ . (1) Decartes and coordinate geometry; (2) Napier and logarithm; (3) The development of calculus, Newton and Leibniz; (4) Euler - Chapter 6, Real numbers
(a) Rigor of calculus. Criticism from Bishop Berkeley to Calculus. Cauchy; Weierstras, father of modern math analysis. (b) the Sword of thought. Dedekind cut; the number axis and how arithmetic operations act on it. (c) number and the set theory. Infinity and infinitesimal; Cantor and the naive set theory. (d) Paradox and logic. Russel and his paradox; Hilbert's 23 questions in 1900; Gödel's incompleteness theorems. Gödel and Hao Wang. Turing machine and computable problem. Turing and artificial intelligence. - Chapter 7, Complex numbers
(a) Solving equation. Varies of ancient civilizations developed solution to quadratic equation. Italian mathematicians broke through cubic and quartic equations and encountered intermediate imaginary solution. (b) The fundamental theorem of algebra. Story of Gauss, constructed 17-gon at age of 17. Example of pentagon with straight edge and compass. Gauss proved the fundamental theorem of algebra in four ways in his life. (c) Radical solution. Quintic equation and Abel-Ruffini theorem; Galois theory and abstract algebra. - Chapter 8, Algebraic numbers
(a) Fermat's last theorem (FLT). Story of Fermat and his last theorem; Pythagorean theorem and Pythagorean triple. (b) Kummer and the ideal numbers. Sophie Germain's contribution to FLT; The fundamental theorem of number theory, Unique Factorization, Kummer, and ideal numbers. (c) From ideal number to ideal. Dedekind and ideal; Emmy Noether and Emil Artin. (d) Transcendental numbers. Abstraction of numbers; Liouville numbers; The transcendent of$\pi$ and$e$ . Cordiality of transcendental of algebraic numbers by Cantor's diagonal method. - Chapter 9, Numbers with structure
(a) School of Bourbaki. Story of André Weil and school of Bourbaki; Story about Grothendieck. (b) Proof to Fermat's last theorem. Andrew Wiles proved FLT, an example of continuous effort by generations of mathematicians; a great result from multiple different mathematics domains. (c) Who are making the history of mathematics? Mathematics evolution as the effort of the entire community. - Appendices and answers
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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