Galois's 1st Public Lecture

[caption id="" align="alignright" width="300"] Évariste Galois (1811–1832) (Photo credit: Wikipedia)[/caption]

19-yr-old Galois's 1st public Math Tuition class after being expelled from Ecole Normale Superieure:

"My work proceeds from a tendency to think intuitively. Dispensing the details, I prefer to leap from foresight to idea to concept...
I propose an analysis of analysis : a new, terser (abstract) algebra, where the most elevated calculations will be considered as particular case, which ultimately must be put aside for greater, more general investigations... so my foresight will be used by generations of mathematicians to advance new theories."

Regression To The Mean

2002 Nobel Prize Economics was won by Daniel Kahneman, a Psychologist, not an economist.
He studied the cognitive factors behind decision-making, and the statistics 'regression to the mean' on Officer Cadet Training. He found: "Praise is more effective than punishment for making cadets learn."

"Regression to the Mean" (Math): means 'An extreme event is likely to be followed by a less extreme event.'
=> eventually forms a Normal distribution centred around the Mean.

Example:
Sports: sportsman suffers a marked drop after appearing in the sport magazines - 'Curse of Sports Illustrated' phenomenon.

Erdös and Posa: coprime

Paul Erdös asked the 12-yr old Hungarian Math prodigy Posa:

"Take 51 numbers from 1 to 100, why there will be at least 2 numbers relative prime?"

Posa took few coffee cups, imagine 50 of them.
1. Put consecutive numbers (1, 2) in 1st cup, (3, 4) in 2nd cup,... (99, 100) in 50th cup.
2. Take out all odd numbers (1, 3, 5, k..., 99) from each cup => total 50 odd numbers
3. Take 51st number from any remaining (say kth) cup => the number = k+1
4. (k+1) & k from (step 2) co-prime    [QED]

Two consecutive numbers co-primes

Prove by reductio ad absurdum:
Let a, b consecutive numbers: a – b = 1
Assume a, b not co-prime,
There is g.c.d. (a, b) = c divides both a, b
=> a = c.a', b = c.b'
=> a – b = c(a'-b')
Since a-b = 1 (consecutive),
c(a'-b') = 1
=> c |1 impossible
=> a , b must be co-prime [QED]

Isomorphism = Congruence, Homomorphism = Similar

New Math <=> Old Math

1. Isomorphism of Groups (or any structures)
<=> Congruence Triangles
(Faithful Representation)

2. Homomorphism of Groups (or any structures)
<=> Similar Triangles
(unFaithful Representation)

Confucius Proof

孔子 Confucius (551–479 BC)

http://en.wikipedia.org/wiki/Confucius

謙虚的孔子曰:“三人行, 必有我師焉”
"Among 3 persons, surely there will be a teacher of mine."  
師六艺 (Teach 6 subjects): {琴 Music ,棋 Chess,书 Reading, 画 Painting, 骑 Horse riding,射Archery}

Prove: if Confucius is right ? "Among 3 persons, surely there will be a teacher of mine."  
“三人行, 必有我師” in 6 subjects

Proof:

1. Case A: among the 3 persons, Confucius  is THE teacher for all 6 subjects, the Probability P(A):
=> P(A)=1/3.1/3.1/3.1/3.1/3.1/3
= 1/729 = 0.1%

2. Case B: 2 other persons can be Teacher of Confucius in any subject(s), the probability P(B):

P(B)= 1- P(A)
= 728/729
= 99.9%

=> Confucius is 99.9% sure: Among 3 persons, surely there will be a teacher of mine !(必有) 我師 !

[caption id="" align="alignright" width="240"] Confucius (Photo credit: JayPLee)[/caption]

'Million' by Marco Polo

Do you know the word 'million' came from China?

Italian Marco Polo lived in Yuan dynasty Mongolian China. He saw so many 'milli-one' Chinese ('mille' = 'thousand' in Latin / French/ Italian, 'one'='many')
=> milli-one => million(e) = 10,000 [万]

Language 'Half-Life'

Language 'Half-Life' Formula

T = log N / 2 log 85%
(N= % of vocabulary with common root)

Modern English & German:
N=60%
=> T =1.561 (K-Yrs)
=> 1,561 yrs ago the 2 languages shared the common root @ 449 CE. [confirmed by historians]

Genesis 11:1-9 Tower of Babel (Confusion)l

God confused Noah's descendants from their common language root 'Proto-Canaanite' to different languages:
Phoenician
Greek
Aramaic -> Arabic / Jewish
Chinese
...
1. Chinese / 日耳曼語系 Germanic (English) have common root T = 4,951 yrs ago [尧/舜]

2. Chinese / Latin (French)
T= 3,700 yrs ago [夏/商]

3. Chinese / Persian (Iran) T= 3,100 yrs ago [商/周]

Proof: [Chinese / Germanic / Latin Common Roots]

背bei -> Back

爸ba -> papa

鳳feng -> [Phoin] = Phoenix [Greek]

殁mo (=死) -> Mort (English) / Mort (French)

民min -> man (eng) / mann (German)

核he /ker -> Heart / kerd (Kernel)

酸suan -> sour