IMO Coach : Drill vs Think Style

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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.

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Music = Math + History

Math Online Tom Circle

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)

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) :

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Ideal in Ring

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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 ?


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 🕜


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

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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:

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

Math Online Tom Circle

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

1. Khan Academy (School Math, Sponsored by Bill Gates)

2. Numberphile (Math Curiosity)

3. WooTube (School Math)

4. Linear Algebra (Undergrad 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

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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

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林群院士:怎样学好数学?How to learn Math well ?

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林群院士很谦虚,自称很笨, 看到200页的数学教科书头也会晕:

他用学习Calculus例子,批评"教科书不教真理,只教证明 (Proof)" :


…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。


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Singlish Math : Toa-Cah-Soh大脚嫂

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In Chinese school Sec 3 we had to memorize by hard these 3 Trigonometric formula:

正弦 sin, 馀弦cos, 正切 tan,
对边比斜边= sin,
邻边比斜边= cos,
对边比邻边= tan,


她说学高数”三角” (Trigonometry) 第一步就是死记这个公式。我最反感无趣味的学习,违背”数学精神” (Mathematical Spirit) 。尤其讨厌 “三角恒等式” (Trigonometric Identities Proof) 证明,用死记的公式纠缠地 (messy) 变来变去,"剪不断, 理还乱",考试差点不及格 (60%)。

80年代 英校数学老师发明一个记忆妙招 :Singlish TOA CAH SOH 大脚嫂👍

Who says Singlish is useless? Our Sec 3 Math students are Top PISA (#1) in the World !

Another smart Singlish Math Chant:

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Math High-Pay Career in Data Science

Math Online Tom Circle

Gone are the days where Math graduates were destined to low-pay teachers in the 60s to 90s.

Now Math career is the hottest high pay job in the 2 key engines of the “4th Industrial Revolution” , ie Mind Automation by Machine Learning and its ‘Mother’ Artificial Intelligence.

The 3 top jobs (by order of pay) :

1. Data Scientist (US $120K) :

Skills : Math (Stats, Linear Algebra, Calculus, Probability, & potentially Algebraic Topology / Homological Algebra) + A. I.

Mathematician PhD preferred.

2. Data Architect / Engineer (US $100K)

Skill: IT, especially in Big Data technologies

3. Data Analyst / (US $65K)

Skills : IT, Business

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A Programmer’s Regret: Neglecting Math at University – Adenoid Adventures

Math Online Tom Circle

Advanced Programming needs Advanced Math: eg.

Video Game Animation: Verlet Integration

AI: Stats, Probability, Calculus, Linear Algebra

Search Engine : PageRank: Linear Algebra

Abstraction in Program “Polymorphism” : Monoid, Category, Functor, Monad

Program “Proof” : Propositions as Types, HoTT

Abstraction: Monoid, Category


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IMO 2019

Math Online Tom Circle


USA Team (2016 & 2018 World Champion) :

China IMO Team :

【2019国际数学奥赛结果公布 中美两国并列团体冠军】

Two observations:
1. Why almost all Chinese in top China, USA, Canada, NZ, Singapore teams?
Either Chinese chase the wrong Math (IMO) education, or the western Math Power countries (UK, France, Germany…) are RIGHT to ignore IMO education?
2. Singapore team skewed heavily in 2 schools: RI (5) + HCI (1). Also 0 girl.

【解读中美数学奥赛并列第一,敬请期待对话美国队总教练 | 袁岚峰】

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Math Online Tom Circle


10 Math Videos:

  1. The Joy of Math
  2. The Joy of Stats
  3. The Joy of Logic
  4. The Code
  5. A Brilliant Madness
  6. The Great Math Mystery
  7. How Long is a piece of string
  8. Hunting the hidden dimension
  9. Dimension: A walk thru mathematics
  10. The secret life of chaos

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New Math applied in Modern Physics

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【文小刚:物理学的新革命——凝聚态物理中的近代数学 | 众妙之门】


  • Quantum Entanglement (QE) : 量子纠缠
  • Topological Order (Long-range QE) : 拓扑序
  • Topological Insulator (Short-range QE) : 拓扑绝缘体
  • Rigid State : 凝聚态
  • Cohomology : 上同调
  • Tensor : 张量
  • Category Theory : 范畴学
  • Group Theory : 群论
  • Algebraic Topology : 代数拓扑


1. Four “Physics” Revolutions aided by Math

Math is applied to explain the Nature (Physics) :

Calculus (Differential Equation) : Newtonian Physics

Riemann Geometry : Einstein General Relativity

Partial Differential Equation: Maxwell Electro-Magnetic Field

Linear Algebra (Matrix) : Quantum Mechanics

Group Theory: Physics / Chemistry Symmetry


Category Theory : Topological Order of the long-range Quantum Entanglement

Algebraic Topology: Topological Insulator (2016 Nobel Prize Physcis)

2. The 5th Physics Revolution aided by New Math tool : Category Theory

3. Category Theory = “Structural Relationship” Math

4. Algebraic Topology

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5G Polar Codes Dr. Erdal Arikan

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Dr. Erdal Arikan (1958 -) of Bilkent University, Turkey, recieved his PhD in MIT at the age of 27.

Huawei Award: The Father of Polar Codes (极化码)

Industry watchers believe that the selection of polar coding as the channel coding technique for control channels for 5G communications system may have put Huawei at the forefront in the 5G race(27 Gbps in 2016).

1 Gbps = 1000 Mbps

5G Basic Explained:

Part 1: The Math behind 5G Polar Codes


Part 2:

Part 3:



$latex displaystyle F = begin {bmatrix}
1 &…
1 & 1

$latex A =
a_{11} & a_{12}& ldots & a_{1n}
a_{21} & a_{22} & ldots & a_{2n}
vdots & vdots & ddots & vdots
a_{n1}& a_{n2} &ldots & a_{nn}
$$latex B =
b_{11} & b_{12}& ldots & b_{1n}
b_{21} & b_{22} & ldots &…

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“Poisson Distribution Intuition (and derivation)”

Math Online Tom Circle

“Poisson Distribution Intuition (and derivation)” by Aerin Kim 🙏

Poisson was a 19CE French mathematician, or Statistician, who invented the Poisson Distribution.

Note: “Applied” Math vs “Pure” (aka Modern / Abstract) Math

Poisson, being an Applied Math expert in the French Science Academy, did not understand the “abstract” Math paper submitted to him by the 19-year-old Evariste Galois in “Group Theory” for the “Insolvabilityof Quintic Polynomial Equations“. Poisson rejected the paper commenting it was too vague, destroying the final hope of Galois to make his invention the “Modern Math” known to the world. The dejected poor Galois died at 21 after a duel during the chaotic era of the French Revolution.

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China Gaokao 2019 Interesting Math Question

Math Online Tom Circle

Gaokao = Baccalaureate = GCE A Level

…[Solution scrolled below]

Let Ф = 0.618

… Golden Ratio Ф = 0.618

defined as:

“Head to Throat” vs “Throat to Navel” = a/b = Ф ,


“Head to Navel” vs “Navel to bottom of Legs” =(a+b) / c= Ф

Knowing :

Head to Neck = 26 cm &

Legs = 105 cm

Find the height (H) ?

Note: The tricky part is to know “the neck is below the throat” & “the legs start some distance below the navel”, hence 2 inequalities :

a <…

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Math Online Tom Circle


Abel / Gauss 自修数学经验:

Read Directly from the Masters – 自接读大师的书 (而非读其他人注解的书)

下面这2本数学经典 "葵花宝典"(有中译版) 是WW1&2 前 世界第一流 宗师 (Noether’s ‘Abstract Algebra’ , Hardy’s ‘Number Theory’ ) 的杰作,值得阅读,收藏。

1. Modern Algebra 史上第一本抽象代数的书: by Van der Waerden (Noether的学生: Artin, Waerden)

2. Number Theory 史上第一本数论启蒙书:

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Simple groups, Lie groups, and the search for symmetry

Math Online Tom Circle

Math History | NJ Wildberger


  1. Normal subgroup
  2. Quotient Group
  3. Finite Simple Group: Z5, D5,…
  4. Tesselation (regular)
  5. Platonic solids

… Two families of Group :

  1. Symmetric Groups: Sn
  2. Linear Group: GL(n) = M, det M ≠0

An = Alternating Group

[even permutation, det = +1]

n>=5, An is a Simple (non-commutative) Group

Lie Groups :



Simple Lie Groups : Matrix Group M

  1. Special Linear Group: SL(n), det M = 1
  2. Orthogonal Group: O(n), $latex M. M^T = I$
  3. Symplectic Group : Sp(2n)

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Alibaba “Damo” Global Math Competition

Math Online Tom Circle

Jack Ma is a kongfu fan, his Alibaba offices all named under the Kongfu heroes, eg. Damo Academy.

Damo” 达摩 (Bodhidharma) was an indian Buddhist monk who came to China in 5th Century, established ShaoLin Monastery, also invented Chinese Kongfu.

Problem 1

Hint: Answers in the above link.

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Homotopy Type Theory: Vladimir Voevodsky

Math Online Tom Circle

A self-study Math genius Vladimir Voevodsky, Fields Medalist, Russian.

Voevodsky borrowed idea from Computer Science (“Interface” =hide implementation details) to Math, and vice-versa, from Math to Computer Science :
Homotopy Type Theory (HoTT)

Result: Computer to prove math theorems.

Computer Science Idea:

  • Hiding Implementation Detail
  • Black Box “Interface”
  • Type (类) : Boolean, Integer, String, … (stronger structure than "Set" 集合)

Math Idea (Abstract Geometry : Algebraic Geometry) :

  • Homotopy (同伦)
  • eg. Cup ~ Donut, both are homotopic or same “Type” (in Computer)
  • Isomorphism is Equality “=”
  • Homotopy Type Theory (HoTT):
    • Univalent Theorem: both objects are considered “=” if they are isomorphic (prove if 1-to-1 correspondence 对应)

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Learn Calculus With These Four Online Courses

Math Online Tom Circle

Four Online Free Calculus Courses:

2. “Calculus Made Easy” (1910 )

3. “Essence of Calculus” (12 short videos, with Chinese subtitle)

4. [Best👍] “Calculus” (MIT Prof Gilbert Strang)

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Chinese Math Olympiad (Primary) Trick vs Singapore Modeling Math

Math Online Tom Circle

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?


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

(II) Singapore Modeling Math:


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…

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English Fun Math

Math Online Tom Circle

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.

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AI with Advanced Math helps in discovering new drugs

Why French excel in math ?

Math Online Tom Circle

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…

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The Laws of Mathematics Zermelo-Fraenkel Axioms Set Theory

Math Online Tom Circle

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:


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Group Theorems: Lagrange, Sylow, Cauchy

Math Online Tom Circle

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

if h = p (prime number),
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
$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…

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陈省身 SS Chern – “The 2nd Gauss”

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

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…

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70-million Bounded Gap Between Primes

Math Online Tom Circle

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.


  • 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…

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小算盘 大乾坤 Abacus

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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。

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十二平均律 12-tone Equal Temperament

Math Online Tom Circle

《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) 。


Bach 12-tone Equal Temperament


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Richard Dedekind

Math Online Tom Circle

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.

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