The French Mathematician and Physicist Joseph Fourier proved
e is irrational,

Another French mathematician Charles Hermite went further: e belongs to another mathematical world:
e is transcendental.

Hermite’s German student Lindermann followed the same method, proved:
pi is also transcendental.

Introduction to Category Theory 1:

Course Overview:

Category Theory = Abstract Algebra of Functions

Lambda Calculus = Calculus of Functions

Lambda Calculus = Category

History:

$latex cap bigotimes$

Introduction to Category Theory (2) Monoids 么群

Introduction to Category Theory (3)Real lecture begins from here: Categories, Functors, Natural Transformation:

1. Category Definition:

1a) Examples of Categories:

Excellent example on “Natural Transformation“:

Ref: Classic Textbook

范畴论 Category Theory :

Why is group theory useful for physics?

1. Matrix (M): stretch & twist space
2. Vector (v): a distance along some direction
3. M.v = v’ stretched & twisted by M

Some directions are special:-
a) v stretched but not twisted = Eigenvector;
b) The amount of stretch = constant = Eigenvalue (λ)

Let M the matrix, λ its eigenvalue,
v eigenvector.
By definition: M.v = λ.v
v = I.v (I identity matrix)
M.v = λI.v
(M – λI).v=0
As v is non-zero,
1. Determinant (M- λI) =0 => find λ
2. M.v = λ.v => find v

Note1: Why call Eigenvalue ?
From German: “Die dem Problem eigentuemlichen Werte”
= “The values belonging to this problem”
=> eigenWerte = EigenValue
Eigenvalue also called ‘characteristic values’ or ‘autovalues’.
Eigen in English = Characteristic (but already used for Field).

Note2: Schrödinger Quantum equation’s Eigenvalue = Maximum probability of electron presence at the orbit…

Relationship-Mapping-Inverse (RMI)
(invented by Prof Xu Lizhi 徐利治 中国数学家 http://baike.baidu.com/view/6383.htm)

Find Z = a*b

By RMI Technique:
Let f Homomorphism: f(a*b) = f(a)+f(b)

Let f = log
log: R+ –> R
=> log (a*b) = log a + log b

1. Calculate log a (=X), log b (=Y)
2. X+Y = log (a*b)
3. Find Inverse log (a*b)
4. ANSWER: Z = a*b

Prove:

$latex sqrt{2}^{sqrt{2}^{sqrt{2}}}= 2$

1. Take f = log for Mapping:
$latex logsqrt{2}^{sqrt{2}^{sqrt{2}}} $
$latex = sqrt{2}logsqrt{2}^{sqrt{2}}$
$latex = sqrt{2}sqrt{2}logsqrt{2} $
$latex = 2logsqrt{2} $
$latex = log (sqrt{2})^2 $
$latex = log 2$

2. Inverse of log (bijective):
$latex log sqrt{2}^{sqrt{2}^{sqrt{2}}}= log 2$
$latex sqrt{2}^{sqrt{2}^{sqrt{2}}}= 2$

New Geometry (新几何) invented by Zhang JingZhong (張景中) derived from 2 basic theorems:

1) Triangles internal angles =180º

2) Triangle Area = ½ base * height
=> derive all geometry
=> trigonometry
=> algebra
(These 3 maths are linked, unlike current syllabus taught separately)

The powerful Area (Δ) Proof Techniques:

1) Common Height:
Line AMB, P outside line
Δ PAM / Δ PBM = AM/BM

2) Common 1 Side (PQ):
Lines AB and PQ meet at M
Δ APQ /Δ BPQ = AM/BM

3) Common 1 Angle:
∠ABC=∠XYZ (or ∠ABC+∠XYZ = ∏ )
Δ ABC /Δ XYZ= AB.BC /XY.YZ

These 3 theorems can prove Butterfly and tough IMO problems.

Butterfly Theorem

In a circle draw a chord PQ with mid-point M. Through M draw 2 chords AB, CD. Join AD, BC cut PQ at X, Y resp. (Butterfly M)

1. Prove: M = mid-point of XY

http://gogeometry.com/GeometryButterfly.html

2. If circle changed to ellipse, still true?

Yes. Affine transformation from circle elongated to ellipse, like distorted image through funny mirror => still MX = MY

Butterfly theorem (Pho

http://m.dangdang.com/touch/product.php?pid=21120565&sid=bb1368df71a7809d&recoref=search

Do not remember these:
$latex boxed {
cos 3A = 4cos^{3} A – 3cos A
}$

$latex boxed {displaystyle
int frac {dx}{sec x}
=
int
frac {1}{sec x}
frac {sec x + tan x}{sec x + tan x}dx
}&fg=aa0000
$

However, it helps, though, to remember:
“Nine Zulu Queens Rule China”
$latex boxed {
mathbb{N}subset mathbb{ Z }subset mathbb{ Q }subset mathbb{ R} subset mathbb{ C }
}&fg=00bb00&s=3
$

How Much Mathematics Should a Student Memorize?

1. What do you call a rigorous demonstration that a statement is true?
   1. If “proof,” then you’re a mathematician
   2. If “experiment,” then you’re a physicist
   3. If you have no word for this concept, then you’re an economist

1. What do you call a slow, painful, computationally intense method of solving a problem?
   1. If “engineering,” then you’re a mathematician
   2. If “mathematics,” then you’re an engineer

Think of the 4th solution, if any, for this
“Monkey & Coconuts” Problem.

It was created by Nobel Physicist Prof Paul Dirac,  which he told another Chinese Nobel Physicist Prof Li ZhengDao (李政道)。
Pro Li wanted to test the Chinese young students in the first China Gifted Children University of 13 year-old kids, none of them could solve this problem (proved they are not so gifted after all for unknown problems :)

The first 2 solutions were solved by Prof Paul Richard Halmos,  the 3rd solved by myself using the Singapore Modelling Math (a modified version of Arithmetics from traditional Math taught in 1970s Chinese Secondary 1 “中学数学” in Singapore).

1st Solution: Higher Math: Sequence

https://tomcircle.wordpress.com/2013/03/30/monkeys-coconuts-problem/

2nd Solution: Linear Algebra: Eigenvalue and Eigenvector
https://tomcircle.wordpress.com/2013/03/30/solution-2-monkeys-coconuts/

3rd Solution: Singapore Modelling Math for PSLE (Primary 6)

https://tomcircle.wordpress.com/2013/03/30/solution-3-best-monkeys-coconuts/

4th Solution:
Any ?

UReddit Courses:

Modern Algebra (Abstract Algebra) Made Easy –

Part 0: Binary Operations

Part 1: Group

Part 2: Subgroup

Part 3: Cyclic Group & its Generator

Part 4: Permutations

Part 5: Orbits & Cycles

Part 6 : Cosets & Lagrange’s Theorem

Part 7 : Direct Products / Finite generated Abelian groups

Part 8: Group Homomorphism

Part 9: Quotient Groups

Part 10: Rings & Fields

Part 11: Integral Domain

Monster Group (code name “Moonshine”) is the largest group, discovered by two Cambridge Mathematicians John Conway and Simon Norton.

Monster Group – (1)

Monster Group (2):

John Conway: Life, Death and the Monster (3)

Ref:
1. Simon Norton (1952 -) – an eccentric mathematician who collects all British Railway Train Time Tables.
http://catalogue.nlb.gov.sg/cgi-bin/spydus.exe/ENQ/EXPNOS/BIBENQ?ENTRY=The%20genius%20in%20my%20basement&ENTRY_NAME=BS&ENTRY_TYPE=K&SORTS=DTE.DATE1.DESC%5DHBT.SOVR

2.
Finding Moonshine: A Mathematician’s Journey Through Symmetry by Marcus Du Sautoy

http://catalogue.nlb.gov.sg/cgi-bin/spydus.exe/FULL/EXPNOS/BIBENQ/6345422/5640834,2

Proof:

in Math logic,

$latex God subset Jesus text{ and}$
$latex Jesus subset God$
$latex implies $
$latex boxed {Jesus = God} $
[QED]

Tsung-tung Chang[張聰東] 1988:
“Indo-European vocabulary in Old Chinese: A new thesis on the emergence of Chinese language and civilization in the Late Neolithic Age”, Sino-Platonic Papers 7, Philadelphia.

This Chinese scholar wrote the 1988 paper on the Chinese language origin with the proto-Indo-European (proto IE).

Interestingly very similar ‘coincidence’ occurs in 1500 words between Chinese and proto IE:

Take -> 得 tek (ancient Chinese sound as in Fujian dialect today)
Mort -> 殁 mo
See -> 视 see
Cow -> 牛 gu
…

Click to access spp007_old_chinese.pdf

After the Tower of Babel, God confused the human into different languages, but by the linguistic ‘archaeology’ ‘Half Life’ Theory, we can deduce ~ 4,900 years ago the Chinese and the Germanic (English, Denmark, German …) shared the same common linguistic root.

The ancient Chinese scholar Xu Shen许慎(东汉 : 58 CE…

Einstein showed by Mathematics (the Riemann Geometry) to explain Space-Time curvature and the Gravity. It was later proved by the astrological discovery in Solar Eclipse.

Here, this professor made an experiment to demonstrate Einstein’s theory.

Gravity Visualized:

How did the NSA hack our emails?
Using Math in Elliptic curve…

NSA Surveillance (an extra bit) – Numberphile:

$latex zeta (-1) = 1 + 2 + 3 + 4 + … = – frac {1}{12} $

Why -1/12 is a gold nugget:

Natural Numbers (N) = {1,2,3, 4…}
1-dimension: a Line
2-dimension: a plane
n-dimensional flat space: a Vector Space

Now imagine in a world where we replace every natural number by vector space:
1 by a Line
2 by a Plane
n by a flat space Vector Space

Sum of numbers = Direct sum of vector space.
E.g. Add a 1-D Line to a 2-D Plane = 3-D Space

Product of numbers = Tensor Product (of two vector spaces of respective dimension m & n) with dimension m.n

This new world would be much interesting and richer than the Natural Number world: vector spaces have symmetries, whereas numbers are just numbers with no symmetries.
(Interesting): we can add 1 to 2 in only ONE way (1+2), but there are many ways to embed (add) a Line in a Plane (perpendicular, slanting in any angle, etc).
(Richer): the Lie Group SO(3)…

The young Russian doctor Sergei Arutyunyan was working with patients whose immune systems were rejecting transplanted kidneys.

The doctor has to decide whether to keep or remove it. If they kept the kidney, the patient could die, but if they remove it, the patient would need another long wait (or never) for another kidney.

The mathematician Edward Frankel helped him to analyze the collected data with ‘expert rules’ in a decision tree. (Note: this is like the Artificial Intelligence Rule-based Expert System, except no fuzzy math).

Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY

“…There’s a long history of high caliber mathematicians finding their experiences with school mathematics alienating or irrelevant. “
Read here:
http://lesswrong.com/r/discussion/lw/2uz/fields_medalists_on_school_mathematics/

In Récoltes et Semailles Fields Medalist Alexander Grothendieck describes an experience of the type that Alain Connes mentions:

I can still recall the first “mathematics essay” (math test, or Composition Mathématique) , and that the teacher gave it a bad mark. It was to be a proof of “three cases in which triangles were congruent.” My proof wasn’t the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and that it was in accord with the traditional spirit of “gliding this figure over that one.” It was self-evident that this man was unable or unwilling to think for himself in judging the worth of a train of…

This professor criticized the lack of rigor in today’s math education, in particular, there exists universally a prevalent ‘ambiguous’ gap between high school and undergraduate math education.

I admire his great insight which is obvious to those postwar baby boomer generation.

I remember I was the last Singapore batch or so (early 70s) taking the full Euclidean Geometry course at 15 years old, and strangely in that year of Secondary 3 Math (equivalent to 3ème in Baccalaureate) my (Chinese) school had 2 separate math teacher for Geometry and Elementary/Additional (E./A.) Math.

Guess what ? the Geometry teacher was an Art teacher. It turned out it was a blessing in disguise, as my class of average Math students who hated E./A. Maths all scored 90% distinctions in Geometry. We did not treat Geometry like the other boring maths. The lady Art teacher started on the first day from Euclid’s 5 axioms…

One fine day when we reach above 80 years old, if the doctor accuses us of having dementia, then prove the doctor wrong by shocking him with 100-digit Pi memory 🙂

With Chinese single-syllable sound for numbers, better still if can sing it as a song, memorizing 100-digit pi is easy!

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…