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 ?

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/

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…

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mRNA剪接Splicing 原理 – 施一公

3 Step Splicing : first & last steps are linear, 2nd step 3D syructure (non-linear) .

施一公 returned from USA, established the private 西湖大学 (funded by Alibaba, Baidu, Huawei etc), a first pure research university modeled on Caltech. His team first decoded the COVID19 genetic code sequence just 1 month after the Wuhan outbreak & announced the result FREE to the world.

The Hungarian lady scientist Katalin Karikó jumped on his newly published DNA code to invent the mRNA PfizerBioNTech vaccines.

https://m.toutiaoimg.cn/a6936774993665589790/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

数学与物理的关系

(小学生与诺奖得主杨振宁的对话)

Math was a tool for Physics (Newtonian Physics)

Math & Physics were independently developed ( Physics Gauge Field = Math Fibre Bundle )

Physics is a tool for discovering Math (Quantum Physics : String Theory)

杨振宁 Yang-Mills Conjecture is one of the unsolved Millenium Math Problems.

https://m.toutiaoimg.cn/a6936190545714741798/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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

林群:先通过案例学道理,后才证明

中科院院士著名数学家林群为大家分享数学怎么学

林群:先通过案例学道理,后才证明

中科院数学院士 林群:"如何学高深 抽象 数学":
先通过案例 (case / motivation) 学道理,后才学证明 (Theorem Proofs) 。

eg. Learn “Group Theory”
Case / Motivation: Symmetry

https://m.toutiaoimg.cn/a6783678419574981120/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Ellipse Proof by Circle : Affine Transformation

Cumbersome Ellipse proof below :

By Affine Transformation the Ellipse to a Circle, the corresponding ratios preserved. Hence it is easier to prove in the circle below:

Similar triangles CQV ~ CTQ

数学中的冷知识:圆幂定理,现在初中课本中都不讲了,但是非常有趣?

圆幂 定理 II: The Affine Transformation FROM Circle to Ellipse. Also applicable to all other 2 Theorems.

https://m.toutiaoimg.cn/a6580683485986423309

Proof : Ellipse Area from Circle :

函数 Function 概念

【数学科普】:

Function = f : R – > R
(with 1& only 1 image)

Mapping 映射 (l’Application) , can be non-value to non-value

仿射变换 Affine Transformation (解析几何 Analytic Geometry 的 Modern Math).

Affine 是 德文 (coined by Euler)

https://m.toutiaoimg.cn/a6791714355483247119/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

丘成桐 的雄心计划: 培养800个数学天才苗子

当年汉朝 霍去病 带 800 骑士 深入漠北驱赶匈奴。

丘成桐 (属牛)的雄心计划: 培养800个数学天才苗子, 每年从中国&世界选16岁起的100个 华人,在清华少年班 8年培训:本科4年+硕2年+博士 2年.

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