Chinese 3AD Arithmetics 东晋. 刘徽 《九章算术》更相减损术

Please explain the Number Theory behind this trick :\boxed{\frac {a } {b}= \frac {\frac {a}{b-a}}{\frac {b}{b-a}}}

Example: 246 - 205 = 41

\boxed {\frac {205} {246}= \frac {\frac {205}{41}}{\frac {246}{41}}=\frac{5}{6}}


27759 – 10227 = 17532 = 2 x 8766 = 2 x (2 x 4383) = 2 x 2 x (3 x 1461) = 2 x 2 x 3 x (3 x 487 )

\boxed {\frac {10227} {27759}= \frac {\frac {10227}{1461}}{\frac {27759}{1461}}=\frac{7}{19}}

Explanation:This method is from《九章算术》295AD 刘徽(曹魏/东晋),he invented the “Limit” 割圆法 method with 95-polygons to get the world’s best pi = 3.1416


Bézout’s Theorem :

For a, b CO-PRIME, ie gcd (a, b) = 1
There exist integers x and y such that ax + by = 1

书法”九宫格” 的”均” =黄金分割

书法是”字如人品” 。

大自然的美表现在 “黄金比率” (Golden Ratio) = , 暗藏在 唐初 欧阳询 发明的楷书”九宫格” : 一个比率 “均” (即:均称,对称 Symmetry) 。

1, 1, 2, 3, 5, 8,… (Fibonacci Series)

黄金分割= 0.618 ~ 1:2 ¦ 2:3 ¦ 3:5 ¦ 5:8

Note : Golden Ratio 黄金比率 =1.618 (长:短)

欧阳询 楷书 《九成宫》

晋/隋/唐的行书, 草书是在汉朝的楷书 (aka 真书) 的”“基础上”速度化” : “” (角度), (笔画突) ,尚保存”均” 的美。

东晋. 王羲之 《兰亭序》行书

唐.张旭 草书

毛泽东 草书沁园春· 雪

Math (Stats) applied in Coronaviruses Pandamic

Math (Stats) applied in Coronaviruses pandamic, by controlling the Reproduction number Ro < 1 via :
1) self protection : frequent wash hands, don’t shake hands / touch face / eyes / nose…
2) wear mask in enclosed environment (meeting room, church, shop, cinema, office… ) eg. SG COVID19 major Clusters in 2 SG churches, Hyatt Hotel Conference room, Wizlearn Technologies, CNY Loy Hei party,…

Malady math: The key statistic that determines your chances of getting coronavirus


1978年 我的法国大学”数学班” Classe Préparatoire “Trinome” (三人学习小组) 之一的 Pierre邀请我去家里午餐并作devoir (功课) 。

他的妈妈第一句话不是问我测验成绩如何, 而是:

Çava les Maths? (数学学习得如何 ? )

傍晚,他的工程师爸爸放工回家,刚踏进门还没去掉领带,就坐下来帮Pierre 解 Abstract Algebra 作业题。

巴黎有100多条路名是法国数学家的名字(Poincare, Laplace, Descartes, Lagrange, Charles Hermite, Sophie Germain, Fermat, …)


张景中院士发明的几何证明用"面积法"把"点"(points) 消失, 更容易解难题。华罗庚用20行来证明1978几何(华罗庚出的)赛题, 张只用2行。




20CE 中学数学教育 (陈建功)

20世纪(1950s -1970s)的中国数学教育改革是迎合WW2 战后工业化, 影响我们中学数学 Secondary School Math (13-18 岁) 。

21世纪 (2000s – 2030s) 迎合的是 工业 4.0 (AI, Big Data, IoT, Fintech… ),中学数学改革 是什么呢?

陈建功 (1893-1971) 和 师弟 苏步青是二战前 留学日本派的数学教育家 (微分几何),建立浙江大学和复旦大学数学系 – 与 陈省身 的 留法/德/美 “南开”派,华罗庚的留英/俄 “清华” 派 三派鼎立,融合成特有的 中国数学派 。

这三本数学(中三/中四)书是我们 70s年代 新加坡华文中学的课本 (文言文) :

Note [*]: Henry Fine was the Founder of the Princeton Institute of Advanced Study (IAS) , who hired the first professor  – Albert Einstein.

]This 《葛氏平面三角》(Granville Trigonometry) text book was arcane and translated in ‘old’ Chinese (文言文).

德国 Gottingen 学派 Felix Klein 引入 Function (函数) 进 中学(Secondary 3)数学主题 (main theme) :

Jewish / French Math vs German Math

The French “Bourbaki” School of Abstract Math (based on Set Theory as Foundation of ALL Math) influenced the post-WW2 French University Math Syllabus and 1970s Baccalaureate (High School) Math, led by the French Jewish Mathematicians Andre Weil et al, and (last batch) Jewish mathematician Grothendieck (who once proposed – but rejected – Category Theory to replace Set Theory) .

The leaders of the German ‘Gottingen School’ of Mathematicians, although learned from the French school masters (French Charles Hermite’s German student Lindemann, French Camille Jordan’s German student  Felix Klein), were more inclined to concrete math, eg. Gauss, Riemann, Hilbert, Dedekind, Noether, etc.


中国的中学数学教育:为了删掉抽象 (Abstract)概念,却同时也丢掉 "近代数学" (Modern Math)。可惜!

改革提议]:应该学习 法国中学数学(16 -18岁) 一部分 Modern Math 教材:
  1. 基本代数结构 Basic Algebraic Structures: Group 群, Ring 环, Fields 域, Vector Space 向量空间.
  2. Analysis 分析: Epsilon-Delta Calculus.
  3. Affine Geometry 仿射几何
  4. 减少一些老旧的中国数学教材(立体几何,复杂的 Trigonometry, … ),减轻学生负担。

Changing Money 找钱数学

买物 $46, 我给售货员$50:
Buy goods at $46, I give the cashier $50 note, she changes money as follow:

英式 [UK Style Direct Subtraction] :找 $ 4 (50-46)
Mental calculation : Change $4 (50-46)

美国/法国式 [USA/French Style Add Back]补$4 凑 $50. Add $4 to make $50.

Another example: you bought something for $46.55, and paid with $50 note.

The cashier gives change by returning nickel (5 cents) and say $46.60, then 4 dimes (40 cents) and say $47, then $3 notes and say $50. Total Change is now $3.45.

中国式 [Chinese Style Round-Up Subtraction] :向你讨 多$1, 还整数 $ 5 (50+1-46=5)
Ask you to give 1 more $1, then change $5 note (50+1-46=50)


UK style: Mental calculation 心算

French style : Reverse Calculation 逆算

Chinese style: “Abacus” Calculation 算盘

PSLE Singapore Math for 12 year-old kids

The “Singapore Math” , an invention by Singapore Prof. Lee Peng Yee 李秉彝 of ex-Nantah (now NTU-NIE) in the 1980s, is a combination of :

  1. The ancient Chinese Arithmetics (算数) Technique,
  2. Polya Problem Solving Methodology,
  3. Aided by Visual Modeling Diagram.

All Singapore kids learn this Singapore Math from 7 to 12 (‘PSLE’ Primary School Leaving Exams before secondary school). Singapore scores Top #1 in World’s Pisa Math Test (for all 13-year-old kids), beating China, Finland, Korea, Japan, Taiwan… and all Europe and USA.

Note: However, there are pros & cons of this Singapore Math, revealing its weaknesses in post-secondary / university Math education with too applied less theoretical foundation.

France has adopted Singapore Math since 2018 in the French primary schools, as recommended by the President Macron’s Education Reform Team headed by the Fields Medalist Cedric Villani, who praised the Singapore Math as ‘Une Méthode Miracle‘ . France is very strong in abstract Modern Math – the 19th century invention by the French prodigy 天才神童 Évariste Galois (Group Theory) . France wins 1/3 of the world’s “Math Nobel Prize equivalent ” Fields Medals , second only behind the USA, but is strangely weak in applied math (eg. Arithmetics, Euclidean Geometry, IMO Math Olympiad…).

[No Algebra please!]

Answer : Try before scrolling down

…Method 1:

…Method 2:

Popular Math Writer 谈祥柏

《十万个为什么》(数学) 的作者之一。

这三个科普作家都是中国数学院士 :张景中,李毓佩,谈祥柏。

Where to park your car, according to math

This “Simple Car Parking” is the Decision Making Strategy demonstrated in our daily life, even in the micro-cell biological level in nature.

In the current USA-China Trade War , China Xi (习近平) is taking the Prudent (中庸) approach of a partial 80% deal, while Trump is too Optimistic (or gung-ho) , insisting to accept either a “Full 100%” deal or None. Math tells you the smartest man’s approach is China Xi “Prudent” strategy.

Opportunity Score:

Meek : 1/5 score = ‘Kiasu’ (怕输 )

Prudent : 3/5 score = (中庸 )

Optimistic : 5/5 or 0/5 score = (Gung-Ho)

Differential Equation:

Apply to Huawei Mate30 Phone counters Google Ban of Android Google Mobile Services (Appstore, YouTube, Google Map, Gmail) :

  • M (Meek) : Coward
    • Wait for Trump’s reprieve or never.
  • P (Prudent) : Smart
    • Work-around solution (sideloading) from alternate apps gallery (3rd party).
  • O (Optimistic) : No Choice Last Resort
    • Release HongMengOS & build Huawei AppGallery.
    • Costly Time & 50% Sales (USA/Europe) Opportunity missed: wait next 2-3 years.
    • $1.5 billion investment to entice developers to Huawei HongMengOS (15% cut of apps profit sharing than Google Appstore’s 30%).

Warfare :

In 三国演义 Three Kingdoms, the smartest military strategist Zhuge Liang (诸葛亮) was attacking the biggest northern enemy Wei (魏). He chose the Prudent (‘P’) route longer distance but less ambushes, rejected his General 魏延 ‘s Optimistic (‘O ‘) approach of taking the risky “子午谷” mountainous but shortest route. History proved that few hundred years later some generals took that risky route, all ended being wiped out by enemies.

少年学习"去"抽象化数学 Concretized De-Abstract Math

"抽象化 =Concretized De-Abstract Math


抽象数学 (Abstract Math) 是现代科学的工具, 通常是大学 理工 &非数学系 本科生undergraduate (大一/大二) 的必修数学。

然而, 高中之前的数学是从古希腊 (Ancient Greek) 到 牛顿时代(17 – 18世纪)的传统数学 (几何, 代数, 三角,牛顿微积分 ) 。

这是世界上第一本给中国和海外华人少年(介于 16 – 18岁, 初/高中)学习 19世纪后"去抽象化"的近代数学 (Modern Math) 。希望能够 帮助 高中生 “搭桥”(bridge) 顺利过渡学习大学数学的困难。

本书不采用数学教科课本的"定律-推论-证明"(Theorem-Lemma-Proof) 的形式, 而是 先从"动机"(Motivation)出发 ,讲诉数学家发现定理的历史背景,并举出大自然界中的具体(concrete) 例子,从而引导学生的兴趣和领悟。

读者必备知识 Prerequisites :

  • 初中数学 (O-Level Math)
  • 中文数学名词
  • 基础英文
  • 法文(optional. 有中/英翻译)

编书计划 : 2020 – 2023


第一部:代数 Algebra

0. 历史背景: 什么是抽象代数 (Abstract Algebra) ?

  1. “数” 的宇宙 (The Universe of Numbers) “NZQRC” (Nine Zulu Queens Rule China).
    1. 实数(Real numbers) 分割Dedekind Cut
    2. 虚数 (Complex Number)
    3. 代数基本定律 (Fundamental Theorem of Algebra)
    4. Cantor 无限大 Infinity Paradox
    5. 素数 Prime :
    1. Twin Primes & Gap,
    2. Fermat Little Theorem
    3. Riemann Conjecture
  2. 数学的"地基"Math Foundation:集合 (Sets) ->范畴(Categories)
    1. Mapping 映射
    2. Surjective (On-to) 满射
    3. Injective 单射
    4. Bijective 全射
  3. 域环群与向量空间:Field-Ring-Group-Vector Space
    1. 计算机上的 (+-*/) 四运算:域
    2. 不能除的时钟数 (Clock Numbers) :环
      1. Ideal (理想)
      2. 应用:韩信点兵
    3. Symmetry 对称美的数学:群
      1. 陪集CoSet
      2. 正子群 Normal Sub-Group
      3. 五次方程解 :Galois Group
      4. 群的”作用” Group Action
      5. 群解释几何:Klein Erlangen Program
  4. 结构的各种"态"&"构"(Morphisms) : homo- (同态) , endo-(自同态) , iso-(同构) , auto-(自同构)
  5. 空间 I : 线性 Linear Space
    1. 距阵 Matrix 与
    2. 线性代数 Linear Algebra
      1. 特征向量/特征值 Eigen-vector & Eigen-values
      2. Google Search
  6. 空间 II :仿射 Affine Space
    1. 没有原点坐标的仿射几何 Affine Geometry:平移 Translation,旋转Rotation, 平翻 Flip…

第二部:分析 Analysis

  1. 函数连续 (Continuous Function)
  2. 极限 (Limit)
  3. 有序列 Sequence
    1. 猴子和椰子Monkeys & Coconuts
  4. 级数 Series
  5. 微积分 (Calculus)
    1. 微积分基本定律 (Fundamental Theorem of Calculus)
  6. 微分方程 (Differential Equation)

体材 (Format) :

  1. 语言: 中,英,法文(翻译)
  2. 有声影 Audio-Video
  3. 图像动片 Animation Graphic


  1. 法国 Baccalaureate (Terminale) 数学课本
  2. Popular Math Books 科普书
  3. Blogs
  4. YouTube



Singlish Math : Toa-Cah-Soh大脚嫂

In Chinese school Sec 3 we had to memorize by hard these 3 Trigonometric formula:

正弦 sin, 馀弦cos, 正切 tan,
对边比斜边= sin,
邻边比斜边= cos,
对边比邻边= tan,


她说学高数”三角” (Trigonometry) 第一步就是死记这个公式。我最反感无趣味的学习,违背”数学精神” (Mathematical Spirit) 。尤其讨厌 “三角恒等式” (Trigonometric Identities Proof) 证明,用死记的公式纠缠地 (messy) 变来变去,"剪不断, 理还乱",考试差点不及格 (60%)。

80年代 英校数学老师发明一个记忆妙招 :Singlish TOA CAH SOH 大脚嫂👍

Who says Singlish is useless? Our Sec 3 Math students are Top PISA (#1) in the World !

Another smart Singlish Math Chant:

A Programmer’s Regret: Neglecting Math at University – Adenoid Adventures

Advanced Programming needs Advanced Math: eg.

Video Game Animation: Verlet Integration

AI: Stats, Probability, Calculus, Linear Algebra

Search Engine : PageRank: Linear Algebra

Abstraction in Program “Polymorphism” : Monoid, Category, Functor, Monad

Program “Proof” : Propositions as Types, HoTT

Abstraction: Monoid, Category