Newton π

计算圆周率π的神奇算法,比“割圆术”更快的无穷级数!

Newton method (Taylor Series coefficient ).

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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.

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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)

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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。

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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

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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。

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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.

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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

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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).

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“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

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“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} }

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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。

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《万物皆数》

[中文书评]

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

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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 ) .

CAS (Computer Algebra System)

CAS or YaCAS (free).

https://www.r-bloggers.com/doing-maths-symbolically-r-as-a-computer-algebra-system-cas/

[Manual Proof :]

[Notes]

1.

That’s why French Math insists must test rigorously the Integrand is CONTINOUS over interval [0,1], else integration is meaningless & wrong !

2. CAMBRIDGE GCE A-level Calculus still never insists the first Rigorous Test step : “Interval” (aka Domain of Definition) + Continuous

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}}

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 )

\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

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

更相减损术证明

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

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