“Math is nothing but a history of Group”

The Math teaching from primary schools to secondary / high schools should begin from the journey of Symmetry.

After all, the Universe is about Symmetry, from flowers to butterflies to our body, and the celestial body of planets. Mathematics is the language of the Universe, hence
Math = Symmetry

It was discovered by the 19th century French tragic genius Evariste Galois who, until the eve of his fatal death at 21, wrote about his Mathematical study of ambiguities.

Another French genius of the 20th century, Henri Poincaré, re-discovered this ambiguity which is Symmetry : Group, Differential Equation, etc.

Only in university we study the Group Theory to explore the Symmetry.

This addictive game “2048” is better than any other violent game like “The World of Warcraft”. At least it improves your math!

The video explains its principle and why you will never exceed 131,072.

It is binary arithmetic, or power of 2 = $latex 2^{n} &s=3$

Notice the rule of 0 & 1:
$latex 32 = 1underbrace {00000}_{5 : zero}$

Minus 2:
$latex 30 = 1111underbrace {0}_{1 : zero}$

Minus 4:
$latex 28 = 111underbrace {00}_{2 : zero}$

The maximum scenario whereby all 15 boxes are filled with the power of 2:

Final score (Maximum)
$latex 131,072 = 1underbrace {00,000,000,000,000,000}_{17: zero} = 2^{17} $

Case 1: The 16th box: – 2
$latex 131,070= underbrace {1111111111111111}_{16 : one} underbrace {0}_{1: zero}$

Case 2 (Maximum) : The 16th box: – 4
$latex 131,068 = underbrace {111,111,111,111,111}_{15 : one} underbrace {00}_{2: zero}$

Israel Gelfand, the student of Kolmogorov (the Russian equivalent of
Gauss), created in 1964 the famous VZMSh, a national Math Correspondence School.

He wrote: “4 important traits which are common to Math, Music, and
other arts and sciences:
1st Beauty
2nd Simplicity
3rd Precision
4th Crazy ideas.
”
The Russian mathematicians also built special Math-Physics schools:
Moscow School #7, #2, #57 (one of the best high school in the world, http://www.sch57.msk.ru) Leningrad Schools #30, #38, #239 (Perelman studied here)

Rukshin at 15 was a troubled russian kid with drink and violence, then a miracle happened: He fell in love with Math and turned all his creative, aggressive, and competitive energies toward it.

He tried to compete in Math olympiads, but outmatched by peers. Still he believed he knew how to win; he just could not do it himself.

He formed a team of schoolchildren a year younger than he and trained them.
At 19 he became an IMO coach who produced Perelman (Gold IMO & Fields/Clay Poincare Conjecture). In the decades since, his students took 70 IMO, include > 40 Golds.

Rukshin’s thoughts on IMO:

1. IMO is more like a sport. It has its coaches, clubs, practice sessions, competitions.

2. Natural ability is necessary but NOT sufficientfor success: The talented kid needs to have the right coach, the right team, the right kind of family…

The great Mathematician Israel Gelfand used to say:

“People think they don’t understand math, but it’s all about how you explain it to them.

If you ask a drunkard what number is larger , 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question:
What is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.”

– Extract: “Love and Math”
by Edward Frenkel

http://books.google.com.sg/books?id=dc7yZKRizNQC&pg=RA1-PA129&lpg=RA1-PA129&dq=timothy+gowers+continuous+path+from+high+school+calculus&source=bl&ots=suLvmFtsJT&sig=yF7s9f5UozP25OO9zgDE_nPtRgI&hl=en&sa=X&ei=BZ1KU7y4EJSfiQf0v4DYCg&ved=0CA0Q6AEwAQ

Extract:

Timothy Gowers (Professor of Math, Cambridge, Fields Medalist)

The Basel Problem is:
$latex displaystyle sum_{k=1}^{infty} frac {1}{k^2} = frac {{pi}^2}{6}$

Euler was 28 years old when he proved that it converged.

The Basel Problem is also called the Riemann Zeta function: ζ(2).

He studied the function sin x which has zeroes,
i.e. sin x= 0 for
$latex x=npi, n = 0,pm1,pm 2, pm 3…$

In other words, we can factor sin x this way:
$latex sin x = x.(1+frac {x}{pi}) .(1-frac {x}{pi}).(1+frac {x}{2pi}). (1-frac {x}{2pi}).(1+frac {x}{3pi}). (1-frac {x}{3pi})…
&s=3$

Note: the right side any factor = 0 when
$latex x=npi, n = 0,pm1,pm 2, pm 3…$

$latex frac {sin x}{x}
= (1-frac {x^2}{1^{2}{{pi}^2}}).
(1-frac {x^2}{2^{2}{{pi}^2}}).
(1-frac {x^2}{3^{2}{{pi}^2}})…
&s=3
$

Note: $latex (1+a).(1-a)= 1 – a^{2}$

From Taylors series,
$latex
sin x = +frac {x}{1!} – frac {x^3}{3!} + frac {x^5}{5!} – frac {x^7}{7!} +…
&s=2$

$latex
frac {sin x}{x} = 1 – frac {x^2}{3!} + frac {x^4}{5!} –…

The Powerful Brains:

1. Spot the odd magic cube
2. “Drain man” Arithmetics
3. The mental Hanzi (汉字笔画) strokes
4. The ‘Breathing and Smelling’ (气息触觉) cognitive power of a blind.

I found (3) fantastic but the judge Dr. Wei disqualified her for being an “Asperger’s Syndrome” rather than a skill.

Which answer is the correct box ?

Ah Beng was asked to make a sentence using 1, 2, 3, 4, 5, 6, 7, 8, 9,10.

Not only did he do it 1 to 10, he did it again from 10 back to 1. This is what he came up with…..

1 day I go 2 climb a 3 outside a house to peep.  But the couple saw me, so I panic and 4 down. The man rushed out and wanted to 5 with me. I ran until I fell 6 and threw up. So I go into 7-eleven and grabbed some 8 to throw at him. Then I took a 9 and try to stab at him. 10 God he run away.

10 I put the 9 back and pay for the 8 and left 7-eleven.  Next day I called my boss and told him I was 6.  He said 5 , tomorrow also no need to…

Prime Secret: ζ(s).

There is a common proverb in my Chinese dialect Fujian spoken today in China Fujian province, Taiwan, Singapore and Malaysia, which says
“A nephew is like his maternal uncle”  外甥像母舅
In modern Biology we know mother passes some genes to her children. Some disease like colorblind is carried by mother down to her sons, the mother herself is immune but her brothers are colorblind as the nephews.
Interesting behavior, intelligence are also similarly inherited from mother and maternal uncles.

Two greatest mathematicians in the history, Newton and Gauss, were the lucky nephews from their maternal uncles who were highly educated to spot the nephew’s genius, although the boys’ parents were uneducated.

Newton’s father died early, mother Hannah Ayscough had a brother William Ayscough educated in Cambridge. William convinced Hannah to send the talented boy Newton to Cambridge.

Gauss’s father was a bricklayer, mother Dorothy Benz had a younger brother Friedrich…

Newton on how he made his discoveries:
“I keep the subject constantly before me and wait until the first dawnings open little by little into the full light.”

Newton was Lucasian prof of math at Cambridge. It was not obvious to
his students that he would become the greatest scientist in history.

His students wrote:
“… So few went to hear him, and fewer yet understood him, that
oftimes he did in a manner, for want of Hearers, read to ye Walls. ”

“He always kept close to his studies, very rarely went a visiting, &
had as few visitors… I never knew him take any Recreation or
Pastime, either in Riding out to take ye Air, Walking, Bowling, or any
other Exercise whatever, thinking all Hours lost, that was not spent
in his studies… He very rarely went to Dine in ye Hall…& then, if
He has not been minded, would…

First discovered by the Chinese Mathematician Minggatu 明安图 (清 康熙, 1730), later by the French (né Belgian) Ecole Polytechnique mathematicianEugène Charles Catalan (1814 – 1894).

Ref:

1. History:

2. Proof of Minggatu Catalan numbers

3. Monoid

4. Kleene Star

Extract from Catalan’s Retirement Speech:

Definition of Combination:
$latex displaystyle boxed {
{_n}C_k = frac {n!}{k!(n-k)!}
= binom{n}{k}
}$

Example:
$latex displaystyle
{_5}C_3 = frac {5!}{3!(5-3)!}
= frac {5!}{3!2!}
= frac {5.4.3.2.1}{3.2.1.2.1}
= frac {5.4.3}{3.2.1}
= binom{5}{3}
$

Combinations are even simpler to write with ‘Falling Factorial’ $latex x^{underline {n}}$

$latex x^{underline {n}} = underbrace {(x-0)(x-1)(x-2)… (x-(n-1))}_{n factors}$

$latex n! = n^{underline {n}} $

$latex displaystyle
binom{n}{k}
= frac {n!}{k!(n-k)!}
= frac {(n-0).(n-1)… (n-(k-1))}
{ (k-0).(k-1)… (k-(k-1)) }
= frac { n^{underline {k}}}
{k^{underline {k}}}
$

$latex displaystyle boxed {
binom{n}{k}
= frac { n^{underline {k}}}
{k^{underline {k}}}
}$

QED = Quod Erat Demonstrandum
or
Quite Easily Done

The order is important!

(a) & (b) have different meanings:

(a): For all M in R, there exists n in Nsuch that …

(b): There exists n in Nsuch that for all M in R, …

See also example below:
http://tomcircle.wordpress.com/2013/05/30/sequence-limit/

Ref: “The Math Girls”

$latex (2^0 +2^1 + 2^2 +…). (3^0 +3^1 + 3^2 +…)
$
$latex = (2^{0}3^{0})+ (2^{0}3^{1}+ 2^{1} 3^{0}) + (2^{0} 3^{2} + 2^{1} 3^{1} + 2^{2} 3^{0} ) + …
$
$latex displaystyle
= sum_{n=0}^{infty}
sum_{k=0}^{n}
2^{k} 3^{n-k}
$

Let the sequence $latex left { a_{n} right }$ convolved with another sequence $latex left { b_{n} right }$

$latex boxed {
left { a_{n} right } = left { a_{0}, a_{1}, a_{2}, …, a_{n}, … right }
}$
Its correspondence $latex leftrightarrow $ the generating function:
$latex displaystyle boxed {
a(x) = sum_{k=0}^{n}a_{k}x^{k}
}$

$latex boxed {
left { b_{n} right } = left { b_{0}, b_{1}, b_{2}, …, b_{n}, … right }
}$
Its correspondence $latex leftrightarrow $ the generating function:
$latex displaystyle boxed {
b(x) = sum_{k=0}^{n}b_{k}x^{k}
}$

The convolution is $latex displaystyle boxed { left { a_{n}* b_{n} right } =
left { sum_{k=0}^{n} a_{k}b_{n-k}right }
&fg=aa0000&s=1}$

“…to discover the equation in the first place, using the important method of generating functions, which is a valuable technique for solving so many problems.” – “The Art of Computer Programming Volume I”

Example:
Fibonacci sequence: {0, 1, 1, 2, 3, 5, 8, 13…}

Definition of the Fibonacci sequence as a recurrence relation:
$latex
boxed{
F_{n}=
begin{cases}
0, & text{for }n=0
1, & text{for }n=1
F_{n-2} + F_{n-1} , & text{for } n geq { 2}
end{cases}
}
$

This definition is not so useful in computation, we want to find a general term formula $latex F_{n}$ in terms of n.

Step 1: Find the generating function F(x)

The correspondence below:
Sequence $latex leftrightarrow $ Generating…

Chapter 3 on Rotation is excellent ! He combines Analytic Geometry, Linear Algebra (Matrix) , and Physics (Rotation) into “one same thing” to show the beauty of Mathematics:

The following matrix represents a rotation $latex rho (theta)$ by an angle $latex theta$:

$latex begin{pmatrix}
cos {theta} & -sin {theta}
sin {theta} & cos {theta}
end{pmatrix}
$

Rotate by $latex 2theta $ will be:
$latex begin{pmatrix}
cos {2theta} & -sin {2theta}
sin {2theta} & cos {2theta}
end{pmatrix}
$

Which is equivalent to rotate 2 successive angle of $latex theta $:
$latex rho (theta) .rho (theta) = rho^2 (theta) $:

$latex begin{pmatrix}
cos {theta} & -sin {theta}
sin {theta} & cos {theta}
end{pmatrix}^2
$
= $latex begin{pmatrix}
cos {theta} & -sin {theta}
sin {theta} & cos {theta}
end{pmatrix} $ $latex begin{pmatrix}
cos {theta} & -sin {theta}
sin {theta} & cos {theta}
end{pmatrix}$
= $latex begin{pmatrix}
cos ^2 {theta} – sin ^2 {theta…

Answer = 40

My nephew gets confused drawing exponential curves with many variations:
$latex e^{ax} $

If $latex -e^{ax} $, then reverse the y values, that is flip the curve over x-axis.

The 3 ancient Greek Problems are:
1. Trisect a Triangle
2. Square a circle
3. Doubling a Cube

Why restrict using only unmarked ruler ?
Answer: Using a Straight line: $latex boxed{ y = mx + c } $

Why a compass?
Answer: Using a Circle: $latex boxed{ x^2 + y^2 = r^2 }$

The 3 Greek Problems have been proven by 19th Century impossible to solve with only a straight line and a circle.

Calculate:
$Latex displaystyle
2^{2^{2^{2^{2^{ -infty} }}}}
$

Answer : 16

[Source: ]
D. Knuth “Digital Typography”

Whoever created this was either a math genius who tried to trick you or a failed mathematician

This is the 3rd post in the series. Previous ones:

• Basics and overview
• Use of mathematical symbols in formulas and equations

Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup.

You can present equations with several lines, using the array statement. Inside its declaration you must :

• Define the number of columns
• Define column alignment
• Define column indentation
• Indicate column separator with & symbol &

Example: {lcr} means: 3 columns with indentations respectively left, center and right

begin{array}{lcl} z & = & a f(x,y,z) & = & x + y + z end{array}

$latex begin{array}{lcl} z & = & a f(x,y,z) & = & x + y + z end{array} &fg=aa0000&s=1 $

begin{array}{rcr} z & = & a f(x,y,z) & = & x + y + z end{array}

$latex begin{array}{rcr} z & = &…

If we were to choose only 3 greatest scientists in the entire human history, who excelled in every field of science and mathematics, they are:
1) Archimedes
2) Issac Newton
3) Carl Friedrich Gauss

Let’s see how Gauss became a great scientist in his formative years in the university, it would give us a clue by knowing what kind of books did he read ?

Carl Friedrich Gauss was awarded a 3-year ‘overseas’ scholarship to study in Göttingen University (located in the neighboring state Hanover) by his own state sponsor the Duke of Brunswick.

Gauss chose Göttingen University because of its rich collection of books.
During the 3 years, he read very widely on average 8 books in a month.

Below was his student days’ library records:

1795-1796 (1st semister): total 35 books
Math (M) :1 ,
Astrology (A):2,
History/Philosophy (H): 1,
Literature/ Language (L): 15,
Science Journal (S): 16

Recent years, there are more newly created “Nobel” Prizes with much bigger prize amounts than the Nobel prize:

BREAKTHROUGH PRIZE IN LIFE SCIENCE (2013)
Donated by:
Yuri Milner (Russian Internet Billionaire)
Mark Zuckerberg (Facebook Founder)
Sergey Brin (Google co-founder)
US$ 3 million
Award Frequency: Every year
Status: 9 scientists had been awarded

FUNDAMENTAL PHYSCIS PRIZE (2012)
Donated by Yuri Milner
US$ 3 million

TANG PRIZE 唐奨 (2013)
Donated by Samual Yin 尹衍梁 (Taiwan Property Tycoon) for Asian countries.
US$ 1.675 million
Frequency: Every 2 years

QUEEN ELIZABETH ENGINEERING PRIZE (2013)
US$ 1.5 million

NOBEL PRIZE (1901)
US$ 1.2 million

SHAW PRIZE 邵逸夫奨 (2004)
Donated by Run Run Shaw (Hong Kong Movie Producer Billionaire)
US$ 1 million

LASKER AWARD (1946)
US$ 250,000

BLAVATNIK YOUNG SCIENTIST AWARD (2013)
Donated by Len Blavatnik (Billionaire Investor)
US$ 250,000

FIELDS MEDAL (1936)
US$ 14,700

I find Khan Linear Algebra video excellent. The founder / teacher Sal Khan has the genius to explain this not-so-easy topic in modular videos steps by steps, from 2-dimensional vectors to 3-dimensional, working with you by hand to compute eigenvalues and eigenvectors, and show you what they mean in graphic views.

If you are taking Linear Algebra course in university, or revising it, just go through all the Khan’s short (5-20 mins) videos on Linear Algebra here:

In 138 lessons sequence:

http://theopenacademy.com/content/linear-algebra-khan-academy

or random revision:

http://m.youtube.com/playlist?list=PLqXgtGPHph5ZLP-XMktk0Ggzte-xAxvI9

Eduardo Saverin (now a Singaporean billionaire investor) gave the wrong Elo formula to his Facebook co-founder Mark Zuckerburg, both of them became ‘accidental’ billionaire. Watch the video clip in the movie “Social Network”:

http://m.youtube.com/#/watch?v=BzZRr4KV59I

The Elo formula is based on the theory of Normal Distribution with Logarithm function, from base of exponential e to base of 10.
The correct Elo Formula should be :
$Latex boxed
{
E_a =frac{1}
{1+ frac{1}{400}.Huge 10^{(R_b – R_a)}
}
}$

$Latex boxed
{
E_b =frac{1}
{1+ frac{1}{400}.Huge 10^{(R_a – R_b)}
}
}$

Eduardo had missed the power ^ below:

Rules of congruence:
SSS
SAS
ASA (or AAS, SAA)
but not: AAA, ASS, SSA, why ?

Danica McKellar tells you why…

Check this out on AMZN: Girls Get Curves: Geometry Takes Shape

中国的小学离校考试 (PSLE) 「神题」: 猜成语
1) 20 除 3
2）1 除100
3）9寸+1寸=1尺
4）12345609
5）1,3,5,7,9

答案：:
1) 20/3= 6.666 六六大顺
2）百中挑一
3）得寸進尺
4）七零八落
5）举世无双

Math Chants make learning Math formulas or Math properties fun and easy for memory . Some of them we learned in secondary school stay in the brain for whole life, even after leaving schools for decades.

Math chant is particularly easy in Chinese language because of its single syllable sound with 4 musical tones (like do-rei-mi-fa) – which may explain why Chinese students are good in Math, as shown in the International Math Olympiad championships frequently won by China and Singapore school students.

1. A crude example is the quadratic formula which people may remember as a little chant:
“ex equals minus bee plus or minus the square root of bee squared minus four ay see all over two ay.”

$latex boxed{
x = frac{-b pm sqrt{b^{2}-4ac}}
{2a}
}$

2. $latex mathbb{NZQRC}$
Nine Zulu Queens Rule China

3. $latex boxed {cos 3A = 4cos^{3}…