Both Methods (I) & (II) below does not use Algebra, which is not taught to primary school kids until secondary school (above 12 or 13 years old).

(I) Ancient Chinese Arithmetic Method:

例2：The sister is 13 years old this year, while the brother is 9 years old.
When the total of their ages is 40, how old will they be?

【这位妈妈太绝了！竟把小学6年奥数化为13句口诀】

Trick: Age difference does not change over time, they + or – in tandem. When 2 ages change, the multiple between them also changes.

https://m.toutiaocdn.cn/group/6636154201682477582/?iid=65171531104&app=news_article_lite&timestamp=1552573996&group_id=6636154201682477582&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(II) Singapore Modeling Math:

［例2］

Compare the Ancient Chinese Arithmetic (I) & Singapore Modeling Math (II):

• (II) is better for young kids to understand visually the logic behind,
• (I) is more a memorised trick but poor math education pedagogy. In [例 2] it is very hard for kids to understand why the 2 steps (40+4) & (40 – 4), although they lead to…

Different Maths:
1. English (UK/USA) Math is playfully tricky (Math Olympiad, Math Puzzles eg. this Numberpile video)

2. French Math is abstractly theoretical (Group theory explains Symmetry, Rubik’s cube game,… )

3. Chinese Math (Abacus, “Chicken-Rabbits Problem” 鸡兔问题, Singapore Modeling Math) is applied & procedural (明朝 13CE invented “Determinant” 行列式 before the 19th CE “Matrix” by UK Sylvester，Chinese Remainder Theorem 韩信点兵).

4. Indian Math (Vedic Math , Factorization, Ramanujian-Style Math) is mysteriously spiritual.

https://theconversation.com/i-build-mathematical-programs-that-could-discover-the-drugs-of-the-future-110689?from=timeline

Advanced Mathematical Methods with AI is a powerful tool:

• Algebraic Topology (Persistent Homology)
• Differential Geometry
• Graph Theory

https://sinews.siam.org/Details-Page/mathematical-molecular-bioscience-and-biophysics-1

Mathematics and quantitative finance, France

Since 1990, there have been 22 winners of the Fields Medal, widely regarded as the Nobel Prize of mathematics. Thirteen came from just two countries, Russia and France. Russia has more winners (seven), but more than twice the population, so the honours go to France, with six winners.

Cédric Villani, the 2010 Fields Medallist, cited national character. “Maths is an abstract way of looking at the world, which fits well with the French mentality. We apply algebra to everything.” Elite institutions help too. France’s brightest school leavers progress to the grandes écoles, which traditionally educate top scientists, administrators and presidents. For maths, you want Monsieur Villani’s alma mater, the École Normale Supérieure (ENS). All 10 French Fields Medallists learnt there. At ENS, no teacher can stay longer than 10 years. Instead of ancient dons, students get tutors at the forefront…

Urbain Le Verrier, Who Discovered Neptune

https://www.forbes.com/sites/startswithabang/2019/03/11/happy-birthday-to-urbain-le-verrier-who-discovered-neptune-with-math-alone/

Example:
4% of 75 = 75% of 4

Mental Steps:
75 x 4 – > 300 – > / 100 – > ＝ 3 (ANS. )
Is it easier ?

The Foundation of Modern Math which is based on the “Set Theory”, hit a crisis due to Bertrand Russell “Barber’s Paradox”, until the Zermelo-Fraenkel Axioms came to its rescue.

Here is a humorous example of an ISP Internet Plan violating the 9 Z-F Axioms:

How ISPs Violate the Laws of Mathematics:

Video:

Answer to Why can humans understand abstract mathematics if such things were never needed in our ancestral environment? Is it a fluke?

Challenge: Put ’15’ and ’14’ in the right sequence (Impossible! )

以下无解：群论

“Essential mathematical GIFs that will make mathematics finally make sense” by Ali Kayaspor https://link.medium.com/z1nWgHm9GU

https://m.toutiaocdn.com/i6662602678880698894/?iid=63376357991&app=news_article_lite&timestamp=1551357660&group_id=6662602678880698894&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

1. Lagrange Theorem:
Order of subgroup H divides order of Group G

Converse false:
having h | g does not imply there exists a subgroup H of order h.
Example: Z3 = {0,1,2} is not subgroup of Z6
although o(Z3)= 3 which divides o(Z6)= 6

However,
if h = p (prime number),
=>
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
ie.
$latex x^{p} = e $ ∀x∈ G

3. Sylow Theorem :
for p prime,
if p^n | g
=> G has a subgroup H of order p^n:
$latex h= p^{n}$

Conclusion: h | g
Lagrange (h) => Sylow (h=p^n) => Cauchy (h= p, n=1)

Trick to Remember:

g = kh (god =kind holy)
=> h | g
g : order of group G
h : order of subgroup H…

陈省身香港科技大学 演讲经典 (Highly Recommended – spoken in English & Chinese)

http://www.bilibili.com/video/av5473133

他的女婿是香港科大校长。现任校长是 Prof 陈繁星 Tony F. Chan , the ‘Search Algorithm Thesis’ supervising Prof of the 2 Google founders in Stanford University, Applied Math Department)

Key Points:

1. SS Chern won the “Wolf Prize” in the same year with the Hungarian “Vagabond” 流浪汉 Mathematician Erdos Paul.
2. He was appointed the Scientific Advisor of President Reagan, during which Chern recommended to build the USA Center of Math in Berkeley University. After retirement, Chern built a similar center in China at his alma mater Nankai University 南开大学 where he graduated in 1930.
3. SS Chern was an assistant lecturer of Prof Yang WuZhi 杨武之, who liked to invite Chern to his house for dinners, where he saw his 8-year old son Yang ZhenNing 杨振宁 (Nobel Prize Physics 1958).
4. SS Chern has guided great PhD students eg. James Simons (Billionaire Financial Investor applying…

Since Ancient Greek :

1. Euclid had proved there are infinite primes.
2. Sieve of Eratosthenes to enumerate the primes.
3. Recent time 3 Mathematicians GPY attempted another Sieve method to find the bounded gap(N) of primes in infinity, but stuck at one critical step.
4. Dr. YiTang Zhang 张益唐 (1955 -) spent 7 years in solitude after failure in academic career, in 2013 during a 10-min walk at the deer backyard of his friend’s house, he found an Eureka solution for the GPY’s critical step: $latex boxed { epsilon = frac {1} {168}}&fg=aa0000&s=2 $ which gave the first historical bounded Gap (N) from an infinity large number to a limit of 70 million.

Notes:

• Chinese love the number “8” ba which sounds like the word prosperity 发 fa (in Cantonese) . He could have instead used 160, so long as $latex epsilon $ is small.
• The UltimateGoal of the Bounded…

浙江临海市 算盘博物管主人 雷国华 收藏算盘20年，散尽储蓄，感慨涕下不已。

中国的算盘上面2珠 (2×5)，下面5珠(5)， 共15，加1 = 16 进位。古代是16位 (Hexadecimal) 制， 半斤 八两 (= 1/2 斤 x 16 两 = 8 两)。

日本改良成10位制: 上面1珠 (1×5)，下面4珠(4)， 共9，加1 = 10 进位。

《十二平均律》 是明朝 ＂布衣王子＂ 朱载堉 (1536年－1610年) 发明的，由当时在中国传教的意大利人 利马窦 (Matteo Ricci, 1552 – 1610), 传给欧洲的法国数学家 Marin Mersenne (Mersenne Prime ) 。现代音乐之父 巴哈 Bach 第一个采用，制作世界第一架钢琴有12黑白键，并作曲 《Bach 12-tone Equal Temperament》。

朱载堉把 “1” 到 “i” 的八度 (Octave) 分为等比 (ratio) 距离的12个半音 (half note), 每个音是前音的 $latex sqrt[12] {2 }$，第8高音频率 (frequency) “i” 巧好是第一音 “1”的2倍。他用81行的大算盘算出：

Side Note:

1977年法国大学数学教授在课堂好奇地问我，你们祖先如何解代数？是用算盘吗？当时计算机还不流行，复杂的算法只能用Log Table 或 计算尺 (Slide Rule) 。

朱载堉的算盤算法就是个例子。

Bach 12-tone Equal Temperament

https://zh.m.wikipedia.org/wiki/%E5%8D%81%E4%BA%8C%E5%B9%B3%E5%9D%87%E5%BE%8B

倍大吕：

Julius Wilhelmina Richard Dedekind (6 Oct 1831 – 12 Feb 1916)

– Last student of Gauss at Göttingen
– Student and closed friend of Dirichlet who influenced his Mathematical education
– Introduced the word Field (Körper)
– Gave the first university course on Galois Theory
– Developed Real Number ‘Dedekind Cut‘ in 1872
– Accomplished musician
– Never married, lived with his unmarried sister until death
– “Whatever provable should be proved.”
– By 1858: still yet established ?
$latex sqrt{2}.sqrt{3} = sqrt{2.3} $
– Gave strong support to Cantor on Infinite Set.

http://www-history.mcs.st-and.ac.uk/Biographies/Dedekind.html

Key Notes:

1. Rational Numbers Q eg. 1/2, … Complete but non Continuous.
2. Irrational numbers eg. $latex sqrt {2}$
3. Real number is Complete and Continuous
4. Dedekind Cut

The Proof of “0.999… = 1” by using Set Theory :

$latex A subset B$
and
$latex B subset C$
$latex implies A = B $

Bit Coin is only one application of Block Chain.

FINAL EPISODE – Riemann Complex Plane : 4 dimension but viewed in 3 dimensions

Episode 1 – 9 : History

Episode 10: Complex function

Function : 1 input mapped to 1 output

Multi-function : 1 input mapped to (n > 1 ) output

Simplest explanation by Cheh Wu:

(4 Parts Video : auto-play after each part)

Revision: Dot Product of Vectors

https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/vector-dot-product-and-vector-length

Gradient Descent (opposite = Ascent)

https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/gradient-and-directional-derivatives/v/why-the-gradient-is-the-direction-of-steepest-ascent