## Math Rigor

Rigorous proof (O-level Math but Modern Math technique) by reductio ad absurdum (反证法).

【一道题检验你是否具有数学思维(视频版) – 今日头条】

https://m.toutiao.com/is/e4QeQFQ/

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## 金庸武林江湖与数学江湖

1）华山正邪二派 ： 气宗 (正) vs 剑宗 (邪)

2）虚竹 ：忘掉以前的少林功夫才能学 消遥派

3) 少林寺僧好高鹜远：还没学精本派“一阳指”，就想去换 印度鸠摩罗的功夫。

https://v.ixigua.com/e5Y62ye/

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## Feymann Calculus Trick

Feymann (Nobel Physicist) has many funny speedy Math tricks for Calculus eg. Differentiate an Integral (Applied Fundamental Theorem of Calculus) , and this one below.

https://m.toutiaocdn.com/i6933218183247626755/?app=news_article&timestamp=1614316375&use_new_style=1&req_id=2021022613125401019410001308000E75&group_id=6933218183247626755&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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## Princeton 《Calculus Lifesaver》

This is a free video lecture series of the Princeton Math Textbook 《Calculus Lifesaver》(link Amazon.com) for university non-math majors.

He explains the “funny” but smart Calculus basics : First & 2nd Fundamental Theorems of Calculus (by Newton’s co-inventor Leibniz), some how never taught in the 1970s GCE A level (now also ? ) bcos not Newton’s Anglo-saxxon invention. Neither the French textbooks teach so (bcos Leibniz was German ?). They only appear in pure American/German Calculus books.

“Funny” bcos by “1st Theorem” you could ‘d/dx’ any intergral to get the anti-derivative . Richard Feymann self-learnt this trick in High school to solve complex integrations.

Also why the “Coset” (+ C) is explained by the 2nd Theorem in this video.

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## Coset & Index 陪集，指数

https://v.ixigua.com/eU7pf3V/

｜Z : nZ｜Index is easy ?

0Z，1Z, 2Z,… (n-1)Z 个陪集,
Total = n个

｜Z : nZ｜＝ n

Note: Coset in Calculus ( Indefinite Integration) :

Integral Solution “+ C” 就是 Right Coset ＝ solution的右陪集：意思所有 “+C” 都是solution Set 的 右陪伴。

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## 函数概念并不难，理解“函”字是关键——函数概念如何理解】

The unique 1 single output of a function becomes very important for subsequent development in Math & IT:
functions are composable, associative, identify function，etc (distributive,… ) => it can be treated like vector => structure of a Vector Space “Vect”

Extended to..

“Vect” is a bigger structure “Category” in which “function of functions” is a
Functor” （函子）F：F(f)

Example : F(f) = fmap (in Haskell)

fmap (+1) {2,7,6,3}

=> {3,8,7,4}

here F = fmap, f = +1

The Math branch in the study of functions is called “functional” 泛函。

IT : Functional Programming in Lisp, Haskell, Scala, ensure safety of guaranteed output by math function property. Any unexpected exception (side effects: IO, errors) is handled by a special function called “Monad” (endo-Functor).

(想看更多合你口味的内容，马上下载 今日头条)

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## 吴文俊 Wu WenJun

https://v.ixigua.com/efrQhx8/

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## French Baccalaureate Math Paper 2 (Analysis)

【「给我进来做题啦!」法国高考数学题难度如何？居然只有四道题？】

Baccalaureate Scientifique Math 2019 (Analyse).

Yet to see the Baccalaureate (S) Paper 1 “Algèbre” which is tougher on Abstract Algebra (Group, Ring, Field, Vector Space / Linear Algebra,…)

https://m.ixigua.com/video/6867459333400035851/?app=video_article&timestamp=1621316174&utm_source=native_share&utm_medium=android&utm_campaign=client_share

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## Modern Math Education : Time for Reform

Modern Math Foundation : Set + Logic (Equivalence Relation).

A level (H2 level) / International Baccalauréat Math should study the big red square box (Number System, Set Theory, Logic).

All STEM university Year 1 & 2 study these 3 Math foundations : Algebra, Analysis & Vector Space.

The Math major university Year 3&4 study more : real /complex analysis, Topology, Galois Theory, graph theory, etc…

https://m.toutiaocdn.com/i6743479520755450380/?app=news_article&timestamp=1615286356&use_new_style=1&req_id=202103091839150102120751990F03C964&group_id=6743479520755450380&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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## 新数启蒙 (1-5) – 给16岁以上的中学生

This “Modern Math” Introductory Course is based on the French Baccalaureate “Modern Math” for 16+ years old, simplified & customized for local students in Chinese.

https://v.ixigua.com/e6mRFnP/

（1）集合： 等价关系 Equivalence Relation

https://v.ixigua.com/e6mDFhw/

（2）集合与集合的关系：映射 Mapping

https://v.ixigua.com/e6mfC4C/

(3) 新数 历史： Bourbaki 学派

https://v.ixigua.com/e6mjedR/

（4）由外向内看代数结构 : 域Field，环 Ring，群 Group，向量空间 Vector Space

https://v.ixigua.com/e6mj5sd/

(5) 域 Field : 足球场

https://v.ixigua.com/e6mkau7/

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

Please explain the Number Theory behind this trick :\$latex boxed{frac {a } {b}= frac {frac {a}{b-a}}{frac {b}{b-a}}} &fg=aa0000&s=4\$

Example: \$latex 246 – 205 = 41\$

\$latex boxed {frac {205} {246}= frac {frac {205}{41}}{frac {246}{41}}=frac{5}{6}&fg=0000aa&s=3}\$

Example:

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 )

\$latex boxed {frac {10227} {27759}= frac {frac {10227}{1461}}{frac {27759}{1461}}=frac{7}{19}&fg=00aa00&s=3}\$

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

https://zhidao.baidu.com/question/109475024.html

Bézout’s Theorem :

For a, b CO-PRIME, ie gcd (a, b) = 1
There exist integersxandysuch thatax+by= 1

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## 为啥法国数学这么强，平民的算数这么烂 ？

{?€ 6.5 是这样算的 ：313.5+0.5 [凑整]＝314.0, 然后 314＋6＝320，得[找] 0.566.5}

https://m.toutiaocdn.com/i6896389485832389134/?app=news_article&timestamp=1607277607&use_new_style=1&req_id=202012070200070100140540151610B97E&group_id=6896389485832389134&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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## Function Continuous =>? Differentiable

【高等数学】变限积分与变量分离

This young Chinese Math student is quite rigorous :

Given f(x) CONTINOUS function,
Is it wrong to f'(x) right away?

https://m.toutiaoimg.cn/a6888102052447126028/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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## Differential Theorem

Differential Theorem

If f differentiable at point a => f continuous at point a

Converse not true !

‘Differentiability’ stronger than ‘Continuity’
Are all Continuous functions Differentiable ? False!
Counter-example (by Weierstrass):

\$Latex f(x)=Sigma{b^n}cos(a^npi x)\$ n ∈[0,8], a= odd number, b∈[0,1], ab > 1+3Π/2

f(x) Continuous everywhere (cosine), but non-differentiable everywhere!
Note: Weierstrass Function is the first known fractal. (e.g. Snowflake Koch’s curve).

Plot of Weierstrass Function (Photo credit: Wikipedia)

Note: What it means a curve (function) is :
1. Continuous = not broken curve
2. Differentiable = no pointed ‘V‘ or ‘W‘ shape curve

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## 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？

https://m.toutiaocdn.com/group/6771961322671178247/?app=news_article_lite&timestamp=1576803223&req_id=201912200853420100140400930DC57A0B&group_id=6771961322671178247

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).

Jewish / French Math vs German Math

The French “Bourbaki” School of Abstract Math (based on Set Theory as Foundation of ALL Math) influenced the…

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## 日本战后数学教育家 远山启

https://m.toutiaocdn.com/i6895034925657883144/?app=news_article&timestamp=1605514504&use_new_style=1&req_id=2020111616150301020405504612025E05&group_id=6895034925657883144&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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## 丘成桐：我研究数学的经验

‘路漫漫其修远兮，吾将上下而求索‘ – 屈原 《离骚》(340BC)

1. 在香港中文大学遇到美国人年轻数学讲师，引导他进＂数学之大门＂，并推荐他去美国拜他的教授 陈省身 ( 微分几何 的 “Gauss II” 宗师) 读PhD。
2. 14岁前父亲给他打下古文学/哲学的基础，”真/善/美” 艺术的直觉 Intuition， 对高深数学理解有大帮助。
3. “Read directly from Masters” (Abel 说的) . 读经典原著。
4. 多听别人的讲座，多看 journal papers. “抛砖引玉”，“他山之石可以攻玉”。Get inspiration from others.
5. 高中/ undergrad 基础功夫要打好，Algebra, Analysis, Geometry, Physics ，都要 “懂” （只有多作习题才知道）。

https://m.toutiaocdn.com/i6888616174393754123/?app=news_article&timestamp=1603895425&use_new_style=1&req_id=20201028223025010202055140190F8C1C&group_id=6888616174393754123&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

https://m.toutiaocdn.com/i6888616174393754123/?app=news_article&timestamp=1603895425&use_new_style=1&req_id=20201028223025010202055140190F8C1C&group_id=6888616174393754123&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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## 如何学数学 How to Learn Math

https://m.toutiaoimg.com/group/6783678419574981120/?app=news_article_lite&timestamp=1579537218&req_id=2020012100201801001404702306D0DBEB&group_id=6783678419574981120

He has redefined the abstract “Cauchy ε-δ definition of a limit” in terms of only elementary maths (trigonometry, inequality, gradient).

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## 化学：金属活动性顺序表

K Na Ca Mg Al Zn Fe Sn Pb (H)

[嫁 倩 (轻)]

Cu Hg Ag Pt Au

[统 金]

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## 数学历史大纲by丘成桐

28 Dec 2019 @中国科学技术大学研究生院

Key Points:

1. 中国数学从古到今只注重计算和解题没有提出自己的问题。 (这是 科举 /Concours / 高考 / IMO Math Olympiad 的心态mindset)。大问题： eg. 法国人 Group Symmetry, Cauchy’s Limit, Poincare Topology， 英国Newton / Leibniz’s Calculus, Russell’s Logic, 德国 Cantor’s Set Theory, Infinity,…
2. 清朝数学家花200+年 考古研究古代中国数学 (《九章算数》… ), 这时西方文艺复兴开始搞近代数学 (意大利 虚数 i, Fermat 数论，Descartes 解析几何，Pascal 或然率, Newton Calculus,… ) 。中国数学从此落后西方。
3. 要读数学历史，尤其是17/18CE Fermat/Newton 之后的近代数学史， 才有宏观思想，继承他们的精神，从而开拓新的领域 (eg. AI 数学: HoTT = Homotopy Type Theory?)
https://m.toutiaoimg.cn/a6864500678341952013/?app=news_article_lite&is_hit_share_recommend=0
4. 他选出 41位 17/18世纪之后至今的伟大数学家, 属于开创新领域的宗师：第一个是 Fermat, 第二个是 Newton…. GALOIS, ABEL,… LIE, KLEIN, HILBERT, Minkowski, Riemann, Euler, Gauss, Cauchy, POINCARÉ, Noether, Weyl, E. Cartan, André Weil, 陈省身，丘成桐, Von Neumann, Turing, … [没有 Bertrand Russell, Ramanujian, Hardy, Terence Tao,… ? ]
5. 提到重要的数学：Lie Group, Lie Algebra, Guage Theory, Homology, Co-homology, Homotopy, Riemann Geometry, Zeta Function, etc.
6. 他将出书写这些数学家的历史。
7. [Q&A 1] : 当明白数学来龙去脉，才知道历史，看未来发展。司马迁写《史记》，司马光 写《资治通鉴》的目的：以古鉴今！
8. [Q&A 2] : 数学读得越多，一切数学就变得简单。(这跟学厨艺，钢琴等一样，起初一切都难，但练习煮/弹得多，就越来越容易上手，而且能够创新，举一反三)。40年前在法国大学读抽象代数(Abstract Algebra) 满头雾水，考试心生恐惧。15年前温习并读其他更多的数学书, 顿然开窍，现在爱不释手。

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## 《Galois Theory》 – Cousera by Ecole Normale Supérieure

Complete Free COUSERA Course 《Galois Theory》 (French) from Ecole Normale Supérieure (ENS 巴黎师范) , the university which kicked out Galois as a student in1830, but apologized to him 150 years later.

Today ENS ranked top 3 globally (after Harvard, Princeton IAS) in Math research & education, producing 1/3 of the world’s Fields Medalists by a single university.

For English Text translation (choose Google translate)

https://www.coursera.org/learn/theorie-de-galois

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## IMO Coach : Drill vs Think Style

The USA IMO coach Prof Loh is the son of 2 Singaporean Math Prof (dad) & JC Math teacher (mom).
He uses the Chinese “drill” style + American “think” style to beat China IMO Team last few years.

https://m.toutiaocdn.com/group/6764245433339347460/?app=news_article_lite&timestamp=1574935791&req_id=201911281809500100100542211C369EF1&group_id=6764245433339347460

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## Music = Math + History

To really appreciate music, you must understand this Math & history :
5 notes to 7 notes to 12 notes.
& fractions: 3/2, 3/4, 2/1 (octave)

Music = Math + History

1. Ancient China 600 BC 管仲 (5 notes 宫商角徵zhǐ羽）

2. Pythagoras 450 BC (7 notes, Chords 和弦)

3. Modern Music: 十二平均律 (明朝 1600 AD 朱载堉- > Bach 1700 AD)

https://m.toutiaoimg.com/group/6683391962764018183/?app=news_article_lite&timestamp=1581773659&req_id=20200215213418010194100050312DF0B9&group_id=6683391962764018183

Pythagoras did not believe in Irrational numbers (sqroot 2) , so only 7 integer notes.

17CE Chinese 明朝 Prince invented 12 notes music, using a giant 81-row Abacus to compute the12th root of 2 (picture below) ：

https://chinoiseries2014.wordpress.com/2019/02/07/%e5%8d%81%e4%ba%8c%e5%b9%b3%e5%9d%87%e7%8e%87/

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## Ideal in Ring

Historical Background:

Ideal (理想) was a by-product by mathematicians in the 350-year proof of the 17CE Fermat’s Last Theorem, wherein they found a violation of the existing “Fundemental Law of Arithmetic” (Unique Prime Factorization) . Since it is a Law, there must be an alternative ideal number to satisfy it, hence the birth of the “Ideal”.

Read here: the raison d’être of Ideal : What is an Ideal ?

Note:

Why Integer (Z) is called “Ring” (Dedekind coined it using the German word “Der Ring”) ? because
{1, 2, … , 11, 12 = 0} is clock number “Z/12Z” like a Ring-shaped Clock 🕜

Application:

The ancient “Chinese Remainder Theorem” (aka 韩信点兵 ) since 200 BCE is explained by 19CE Ideal Theory.

[Solve] : “The Problem of 6 Professors

Whatever inside multiplies outside, still comes back inside.

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## 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 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 )

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

Differential Equation:

https://m.phys.org/news/2019-09-car-math.html

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.

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## 11 of the Best YouTube Channels for Understanding Mathematics

Below are some of the best and Free Math channels from Secondary School to University to Public Math Enthusiasts.

2. Numberphile (Math Curiosity)

3. WooTube (School Math)

5. Rigourous French Math (Classe Prepa, Undergrad Math)

Set Theory Basics (Baccalaureate Math Revision)

Espace Vectorielle (French Abstract Version of Linear Algebra)

Rigorous French Math (Classe Prepa)

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## Data Science: 5 Statistics & Math tools

1. MEAN / MEDIUM

2. PERCENTILE

3. SKEWNESS

4. STANDARD DEVIATION

5. PCA = Pricinpal Component Analysis

1. Probability & Statistics

2. Linear Algebra (Matrix, Eigenvectors, Eigenvalues)

3. Optimisation : eg. Gradient Descent, etc.

4. Discrete Math in Computing

(想看更多合你口味的内容，马上下载 今日头条)

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## 林群院士：怎样学好数学？How to learn Math well ?

【中国科学院林群院士告诉你，怎样才能学好数学？】复制这条信息€80cKc€ZjlaWx€后打开👉今日头条极速版👈

…把简单道理讲得很复杂:

…其实一个日常生活的案例(火车 Case), 就可以解释清楚。

Note: 林群院士是第一个把复杂吓人的Cauchy Epsilon-Delta Analysis (aka Advanced Calculus) 用中学数学简单化。

Video: 大自然的悟性发现数学:

… 文学 《The War & The Peace》： 托尔斯泰用Calculus 证明Napoleon 必败。

＂不是你听不懂数学，是数学老师教得不好。＂

https://baike.baidu.com/wikisecond/videoflow?secondId=332699&lemmaId=11364&from=lemma&feedInlemma=1

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