Best way to study abstract math is to use concrete examples, visualizable 2- or 3- dimensional mathematical ‘objects’.

Groups:
(Do not need to search too far…) Best example is the Integer Group (Z) with + operation, denoted as {Z, +}. (Note: Easy to verify it satisfies the group 4 properties: CAN I“.

It has infinite elements (infinite group)

It is a Discrete Group, because it jumps from one integer to another (1,2,3…, in digital sense).

The group of rotation of a round table, which consists of all points on a circle, is an infinite group. We can change the angle of rotation continously between 0 and 360 degrees, rotating around a geometrical shape – a circle. Such shapes are called Manifolds (流形).

All points of a manifold forms a Lie group.

Example: The group of rotations of a sphere around a central…

Durian & Group

Hi guys here the comes posters of Application of Math in real life .

Source: http://mathigon.org/

Prove:  Any line L will cut a circle at most 2 points:

Factorize C(x,y) : (x+iy) (x-iy) = 1 in the complex plane.
So C  = {L1} U {L2}
where L1 and L2 are two lines

We have compiled a list of Top 5 Best selling and Top rated Singapore Math Books on Amazon. This list is more targeted towards parents and students living outside Singapore, like in the United States. Students in Singapore are already breathing and living Singapore Math!

Hope this list will help you in finding the Best Singapore Math Books for your child. The reviews are from actual customers on Amazon.
1)

Singapore Math Practice, Level 1A, Grade 2

This math practice book contains wonderful teaching strategies from the Singapore math program including number bonds and counting on. This would be a good book for homeschooling. We use it as an enrichment tool when we have a little extra time during vacations or on weekends.
I would recommend it to parents who would like to teach their struggling kids math, because it tells you how to teach these concepts.

2)

Why Before How: Singapore Math Computation…

Examples:

Dim (Circle) in 2-dim plane = 1

As we approach near the neighborhood of the tangential point on the circle, the curvature of the circle disappears, there is no difference between the circle and the tangent line (dim = 1).

Hence, Dim (Circle) = 1

A point on a circle is determined by one independent variable only, which is the polar angle.

Note:
The dimension of the ambient space (2-dim plane) is not relevant to the dimension of the circle itself.

Dim (Sphere) in 3-dim Space = 2

The 2 variables (longitude, latitude) determine a position on the globe. Therefore dimension of a sphere is 2.

Interesting note:
Four Dimension Space (x, y, z, t): what we get if the 4th dimension time is fixed (frozen in time) ? We get a…

Shimura and Tanyama are two Japanese mathematicians first put up the conjecture in 1955, later the French mathematician André Weil re-discovered it in 1967.

The British Andrew Wiles proved the conjecture and used this theorem to prove the 380-year-old Fermat’s Last Theorem (FLT) in 1994.

It is concerning the study of these strange curves called Elliptic Curve with 2 variables cubic equation:

Example:
$latex boxed {y^{2} + y = x^{3} – x^{2}
} &fg=aa0000&s=3 $ — (I)

There are many solutions in integers N, real R or complex C numbers, but solutions in modulo N hide the most beautiful gem in Mathematics.

For modulo 5, the above equation has 4 solutions:
(x, y) = (0, 0)
(x, y) = (0, 4)
(x, y) = (1, 0)
(x, y) = (1, 4)

Note: the last solution when y=4,
Left side = 16 + 4 = 20 = 4×5 = 0…

“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.