The Math Power of Number 4

Despite Chinese & Japanese don’t like ‘4’ for the same sound as \si (death), ‘4’ is the magic number for all ideal compositions:

1) All TCM 药方 has 4 compositions : 君臣佐使
eg. “银翘解毒片” (Chinese Panadol) :
银花 (君药)有毒, 但可以杀flu virus,
连翘(臣药),中和neutralise 银花的毒。
其他的
(佐药 Assistant ) 甘草,..
(使药 Smoothening, sweetening ) 蜂蜜

2)烹饪 Cuisine :
eg. 炒饭
君:冷饭
臣:蛋 x2
佐:mix veggies, 腊肠,烧肉,…
使:香料 (葱花,胡椒粉,garlics,… ), 麻油/鱼露

3) Music
eg. Quartet
君:violin
臣:viola
佐:flute, guitar,…
使:Cello,… Double base

4)Maths
Eg. “4-Color” Problem : the only one Math impossible by mathematicians but proven only by computer.

5) Exemplary Nation Racial Harmony
Eg. Switzerland / Singapore : 4 races & official languages.

6) War against Evils:
Eg. WW2 Allied :
USA, UK, France, China, against evils Japan & German (+Italy).

7) Four seasons :
春夏秋冬

8) Healthy Meal:
Eg.
Fill 1/4 plate with wholegrains.
• Fill 1/4 plate with good sources of protein.
• Fill 1/4 plate with fruit and 1/4 plate of vegetables.

That’s why the 19-year-old Math genius Galois (1832 AD) proved by his invention Abstract Algebra the “Group Theory” that Polynomial equations with radical solution (+ – */,nth root) ONLY possible up to maximum degree 4.
That explains why the “A5-Group” Icosahydron (二十面体) structured SARs-like viruses (eg. COVID-19) WILL NEVER have “radical” drug SOLUTION, only possble killed by our own Antibody pre-trained by Vaccines (eg. mRNA or Sinovac… ).

Also humans impossible to have more than 四代同堂。

The 5th Generation Computer “Prolog Machine” by Japan in 1980s failed miserably.

Anything more than 4G has only “particular case” solution with physical limit beyond which we have to shift paradigm / technology to restart on another track.
Eg 5G by Huawei facing so much difficulty by Western sabotage, only by jumping to new “Quantum Communication Technology” to overcome these physical & artificial human resistance, whether they like it or not by Western countries.

IMO 2020 (6th Question)

2020 IMO (6th problem) only one Chinese 李 student scored full 7 marks.

https://v.ixigua.com/emUQpn1/

This question is similar to Analysis finding epsilon-delta value of N:

(Epsilon-Delta Analysis, aka Advanced Calculus ) definition for “Limit of Series” :
It exists N such that, for all n > N…

[Solution] :

[See also] IMO 2020 (4th Question) : https://m.toutiaoimg.cn/i6881188536880464384/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Leibniz’s original proof : Integration by Parts

莱布尼茨微积分——

Leibniz’s original proof : Integration by Parts (分部积分公式”)

https://m.toutiaocdn.com/i6965860180663976452/?app=news_article&timestamp=1621922350&use_new_style=1&req_id=202105251359090102120610721E3A8184&group_id=6965860180663976452&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

FTA Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic | by Maths and Musings | Cantor’s Paradise

https://www.cantorsparadise.com/the-fundamental-theorem-of-arithmetic-37470aa1a7a0

Note:

The Fundamental Theorem of Arithmetic (FTA):

2300 years ago Euclid 《Elements》proved prime factorisation :
If p | x. y then p | x or p |y

19CE Gauss proved FTA. Uniqueness of prime factorisation.

French 《Bezout Theorem》 is the useful tool: if p, q co-prime, then there exist n, m integers such that
np + mq = 1
eg. Prove 5, 7 co-prime,
(3) *5+ (-2) *7 = 15-14=1

Strange, why Bezout Theorem not taught in A level ?

Feymann Calculus Trick

理查德·费曼非常聪明的求导 differentiate 技巧

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

吴文俊:中学数学教育

吴文俊:数学教育(上)

要点:

1. 文化大革命 / 数学教育革命 灾难

2. 东方数学 : 小学 – 中学/高中数学 (机械化)eg. 四则运算

3. 西方数学 :大学”新”数学 (非机械化)

https://m.toutiaocdn.com/i6917519470810366472/?app=news_article&timestamp=1610825278&use_new_style=1&req_id=202101170327580100080431050833ACBD&group_id=6917519470810366472&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

吴文俊 批评法国的1970s Bourbaki 数学进入中学/Bac 课本。美国也同时推行 New Math。

问题出在 数学老师的训练不够,太仓促。比如 Singapore Math 1980s 刚推行 也是很"乱",后来新加坡教育部 (MOE) 要求家长们也去学习,才能在家里辅导孩子。

吴文俊:数学教育(下)

他提倡中学数学教育 2 要点:
1) 教少一点 Euclidean Geometry, 用 Analytic Geometry取代, 才可以 computerise (机械化).

2) 中国古代是有 Geometry, 但 用天元(x), 地元(y) 代数(Algebraic) 化。

https://m.toutiaocdn.com/i6917528352391807499/?app=news_article&timestamp=1610902867&use_new_style=1&req_id=20210118010107010204051088185887A1&group_id=6917528352391807499&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

IMO Geometry Techniques 几何理论基础:分角定理、张角定理,推理证明

IMO Math usually contains 1 or 2 Geometry questions.

France, UK, Singapore, and some countries which reduce Secondary school syllabus in Euclidien Geometry, are disadvantaged in scoring Gold.

数学竞赛几何理论基础:分角定理、张角定理,推理证明

https://m.toutiaoimg.cn/a6910816390312100360/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

中国古代数学的三个高峰和落没

三高峰时期:

1. 秦汉:张苍&耿寿昌 编 《九章算术》, 3AD 东汉 刘徽 (注解)

2. 4AD 南北朝:祖冲之父子 圆周率π

3. 13-16世纪 元/明朝:珠算盘

落没

只是Applied 应用, 没有 希腊Deductive Theory 理论。

https://m.toutiaoimg.cn/a6909028612117103117/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

CRT Chinese Remainder Theorem

x= (a+1)+(a+2)+… (a+9) = 9a + 45

x = 9(a +5) +0

x = (b+1)+(b+2)+… (a+10) = 10b + 55

x = 10 (b +5) + 5

x = (c+1)+(c+2)+… (c+11) = 11c + 66

x = 11 (c+6) +0

CRT : (9,10), (9,11), (10,11) are pair-wise coprimes.

x = 0 (mod 9)

x = 5 (mod 10)

x = 0 (mod 11)

Notes: R = Z integer Ring

{X, r_9, r_{10} , r_{11} } \in R

Eg. (9) = 9Z = Ideal

X = 9*11*5 + 9*10*0 + 10*11*0 (mod 9*10*11)

[/ 10… ]

X = 9*11*5 +0+0 = 495 = 490+5 = 5 (mod 10)

[/9… ] X = 0+0+0= 0 (mod 9)

[/11… ] X = 0+0+0=0 (mod 11)

X = 495 (mod 9*10*11)

Minimum X = 495

Note :

China Covid19 Test 20 millions in 5 days : Use Math!

惊艳世界的大规模检测,凭啥只有中国做的如此完美?详解混检技术

Why Chinese hospitals can test Covid19 at the speed of 20 million people in 5 days ?

Use Math !

Pre-requisite conditions :
1) <= 0.1% infected Covid19 patients in local population, then save 90% testing effort. If 9% infected, then only save 50%.

2) 10-in-1 混检 mixed test samples is the optimum algorithm : safety + accurate + efficient.

https://m.toutiaoimg.cn/a6902367747532194317/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Chinese Remainder Theorem 《韩信点兵 》& Ideal

听说过“韩信点兵”的歌谣吗?李永乐老师讲中国古代数学定理

Chinese Remainder Theorem

By 1930, even Bourbaki founder Dieudoné didn’t know “Ideal” when he read 《Algebra》from Noether’s lecture-note compiled by her student Van der Waeden

https://m.toutiaoimg.cn/a6559856147912720909/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

IMO 2020 Solutions

IMO 2020 : China World Team Champion ( 5 Golds + 1 Silver), the only one in the World with Perfect Score.

This Chinese IMO coach comments :

IMO 2020 is the easiest in the past 10 years, compared to 2015 (tougher) & 2018 (toughest).

https://m.toutiaoimg.cn/a6881188536880464384/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

“充分性”与“必要性”的三种判定方法

高中数学:“充分性”与“必要性”的三种判定方法


Necessary 必要 &/or Sufficient 充分 Conditions

充分 :条件 => 结论
eg. 出生地在SG => SG公民

必要: 条件 <= 结论
eg. SG 公民 “不必要” 出生在SG (可以是外国converted )

充(份必) 要 : 条件

https://m.toutiaoimg.cn/a6775009396981236238/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

“辗转相除法”巧妙求解不定方程

什么是“辗转相除法”,如何利用“辗转相除法”巧妙求解不定方程

Find all x, y integers :

\boxed{37x+107y = 25}

Step1) Find x, y such that

37x+107y = 1

Solution 1: x = 26, y= 9

Step 2) multiplied {x, y} solutions by 25

Solution 2:

x = 26×25=650, y= 9×25=225

Step 3: generalize {x, y} solution sets.

Solution 3:

\boxed {\begin{cases} x= 650 +107k,& \text {for }  k \in \mathbb{Z}, \\ y =225-37k ,  \end{cases} }

https://m.toutiaoimg.cn/a6875258049439728132/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

The French co-prime Theorem is called Bézout’s Theorem:

https://en.m.wikipedia.org/wiki/B%C3%A9zout%27s_identity

辗转相除法 因式分解

林群: 数学考试100分 不如会思考问题

Prof 林群 :“数学考100分,不算什么”。他在厦门大学读数学(同陈景润):一天做100题,考100分,但自己成就 比不上 同辈的 屠呦呦 ( Nobel Medicine ).

林群是过于 谦虚。他把Cauchy / Wierstrass 的 深奥吓人的 "Epsilon-Delta Limit" 定义用简单的 中学 Trigonometry 改写。

https://m.toutiaoimg.cn/a6763129174438707720/?app=news_article_lite&is_hit_share_recommend=0

《万物皆数》

[中文书评]

https://m.toutiaoimg.cn/i6857428347115766286/?app=news_article_lite&timestamp=1596647454&use_new_style=1&req_id=202008060110540100160281340A44B4BF&group_id=6857428347115766286

Audio book ]: It all adds up – the story of people & Mathematics

https://m.toutiaoimg.cn/i6857428347115766286/?app=news_article_lite&timestamp=1596647454&use_new_style=1&req_id=202008060110540100160281340A44B4BF&group_id=6857428347115766286

Kinematics (French Method in Baccalaureate)

Polar Reference:
The vector Velocity V in Polar form:

\overrightarrow {OM} = r(t). \vec e_r

\boxed{\displaystyle {\vec v_{(M)} = \dot r \vec e_r + r \dot \theta \vec e_{\theta}}}

\boxed{\displaystyle {\frac {d. {\vec e_r}} {dt} =\dot{ \theta}\vec e_{\theta}}} \boxed{ \displaystyle {\frac {d. \vec e_{\theta}} {dt} = - \dot{ \theta}\vec e_{r}}}

Acceleration:

\boxed{ \displaystyle { \vec a_{(M)} = \begin{pmatrix} \ddot r - r {\dot{\theta}} ^2\\ 2\dot r \dot \theta + r \ddot \theta \end {pmatrix} \begin{pmatrix} \vec e_r \\ \vec e_{\theta} \end {pmatrix}} }

This Prof uses the “Polar Reference” for circular movement with (r, ϴ), in vector form.

Frenet ReferenceVideo from 3:25m Circular Movement, why easier to use “Frenet Reference” , not X-Y Reference or Polar References (without the troublesome sine & cosine ) .