New “Nobel” Prizes

Math Online Tom Circle

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

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Khan Academy

Math Online Tom Circle

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

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Facebook & Ranking Elo Formula

Math Online Tom Circle

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:

20130616-024218.jpg

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Math Chants

Math Online Tom Circle

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}…

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What is “sin A”

Math Online Tom Circle

What is “sin A” concretely ?

1. Draw a circle (diameter 1)
2. Connect any 3 points on the circle to form a triangle of angles A, B, C.
3. The length of sides opposite A, B, C are sin A, sin B, sin C, respectively.

Proof:
By Sine Rule:

$latex frac{a}{sin A} = frac{b}{sin B} =frac{c}{sin C} = 2R = 1$
where sides a,b,c opposite angles A, B, C respectively.
a = sin A
b = sin B
c = sin C

20130421-193110.jpg

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Vector Algebra

Math Online Tom Circle

Vector changes Geometry to Algebra

1. No complexity of Analytical Geometry
2. Remove the astute dotted (helping) line in Geometry
3. No need diagram: Use only 2 vector properties:
Head- to-Tail:
$latex vec{AC}=vec{AB}+vec {BC}$
Closed Loop:
$latex vec{DE}+vec{EF}+vec{FD}=0$
4. Enable Computer automated proof of Geometry via Algebra.

Example: 任意四边形 Quadrilateral ABCD with M,N midpoints of AB, CD, resp.
Prove: MN=1/2(BC+AD)
Proof: (by vector):

Consider MBCN:
MN=MB+ BC+ CN..(1)

Consider MADN:
MN=MA+ AD+ DN..(2)

(1) +(2):
2MN=(MB +MA) +
(BC +AD) +(CN +DN)

but (MB +MA) =0,
(CN +DN) =0 [same magnitude but different direction cancelled out ]

=> MN=1/2 (BC +AD)

Special cases:
1. A = B (=M)
=> triangle ACD
AN = 1/2 (AC +AD)
2. BC // AD
=> Trapezium ABCD
MN=1/2 (BC +AD)
=> MN // BC // AD

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Plato Solids

Math Online Tom Circle

Why only 5 Plato solids ?

Plato Solid is: Regular Polyhedron 正多面体

  • Each Face is n-sided polygon
  • Each Vertex is common to m-edges (m ≥ 3)

Only 5 solids possible:
Tetrahedron (n,m)=(3,3) 正四面体platonic_solids
Hexahedron (or Cube) (n,m)=(4,3) 正六面体
Octahedron  (n,m)=(3,4)正八面体
Dodecahedron  (n,m)=(5,3)正十二面体
Icosahedron  (n,m)=(3,5)正二十面体

Proof:
Since each Edge (E) is common to 2 Faces (F)
=> n Faces counts double the edges
nF = 2E …(1)

Since each Vertex has m Edges, each Edge has 2 end-points (Vertex).
=> m Vertex counts double the edges
mV = 2E …(2)

(1) : E= n/2 F
(2): V= 2/m. E = n/m. F
(1) & (2) into Euler Formula: V -E + F = 2
(n/m. F) – (n/2.F) + F = 2
F.(2m + 2n – mn) = 4m

Since F>0 , m>0
=> (2m + 2n – mn) >0
=> – (mn -2n -2m) >…
=> (mn -2n -2m) <…
=>…

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Indian Vedic Math

Math Online Tom Circle

Bharati Krishna Tirthaji @ early 19xx, a former Indian child prodigy graduating in Sanskrit, Philosophy, English, Math, History & Science at age 20.

16 sutras (aphorisms):
1. By one more than the one before
2. All from 9 and the last from 10
3. Vertically and cross-wise
4. Transpose and Apply
5. If the Samuccaya is the same it is Zero
6. If One is in Ratio the Other is Zero
7. By + and by –
8. By the Completion or Non-Completion
9. Differential Calculus
10. By the Deficiency
11. Specific and General
12. The Remainders by the Last Digit
13. The Ultimate and Twice the Penultimate
14. By One Less than the One Before
15. The Product of the Sum
16. All the Multipliers

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Vedic (Multiply)

Math Online Tom Circle

Vedic Math & 16 Sutras

[s2]: All from 9 and the last from 10
[s3a]: Vertically and
[s3b]: Cross-wise

Example: 872 x 997 = Y ?

Apply [s2]: (8-9) =-1 , (7-9)= -2 , last (2-10) = -8
872 -> [-128]

[s2]: (9-9) = & (9-9)= & last (7-10)=-3
997 -> [-003]

Arrange in 2 vertical columns as:
872 -> [-128]
997 -> [-003]

[s3a]: (Vertically):
[-128] x [-003] =384

[s3b]: (Cross-wise):
872 + [-003] = 869
=> Y = 869,384

Now, Quick Demo : Calculate 892,763 x 999,998 = Y

892,763 [-107,267]
999,998 [-2]
=> Y= 892,761,214,534

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Vedic (Factorize)

Math Online Tom Circle

Vedic Sutras:
[s1]: proportionally
[s2]: first by first and last by last

Example 1: E= 2x² + 7x +6

Split 7x = 3x+4x
First ratio of coefficient (2x²+3x) -> 2:3
Last ratio of coefficient (4x+6) -> 4:6=2:3
=> 1st factor = (2x+3)

2nd factor:
2x²/(2x) +6/(3)= (x+2)

=> E = (2x+3).(x+2)

Example 2: Factorize E(x, y, z) = x²+xy-2y²+2xz -5yz-3z²

1. Let z =…
E’= x²+xy-2y² = (x+2y)(x-y)

2. Let y=0
E’= x²+2xz-3z² = (x+3z)(x-z)

=> E(x, y, z) = (x+2y+3z)(x-y-z)

Example 3:  P(x, y, z) = 3x² + 7xy + 2y² +11xz + 7yz + 6z² + 14x + 8y + 14z + 8

1. Eliminate y=z=0, retain x:

P = 3x²+14x+8= (x+4)(3x+2)

2. Eliminate…

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Vedic (GCD Polynomials)

Math Online Tom Circle

G.C.D Polynomials by Vedic Math

Find G.C.D of P(x) & Q(x):

P(x) = 4x³ +13x²+19x+4
Q(x) = 2x³+5x²+5x -4

Vedic method:
1. Eliminate 4x³ in P(x):
P – 2Q = 3x² +9x+12

/3 => P-2Q = (x²+3x+4)

2. Q+P = 6x³+18x²+24x

/(6x) => Q+P = (x²+3x+4)

3. G.C.D. = (x²+3x+4)

P= (x² +3x+4).(ax+b) = 4x³ +13x²+19x+4
=> a=4, b=1
Similarly,
Q= (x² +3x+4).(2x+1) = 2x³+5x²+5x -4

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Amateur vs Professional

Math Online Tom Circle

Amateur versus Professional

1. Amateur is at liberty to study only those things he likes.
2. Professional must also study what he doesn’t like.
3. Conclusion: Most famous theorems are found by Amateurs.

Examples:
Fermat = Judge (Number Theory, Probabilty),
Venn = Anglican Pastor (Venn Diagram),
Ramanujan = Railway clerk (Number Theory)
Cayley = Lawyer (Group),
Leibniz = Diplomat (Calculus, Binary 0 & 1)

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Arabic Problem

Math Online Tom Circle

This is an old arabic problem:

An old man had 11 horses. When he died, his will stated the following distribution to his 3 sons:
1/2 gives to the eldest son,
1/4 for 2nd son,
1/6 for 3rd son.

Find: how many horses each son gets ?

There are 2 methods to solve: first using simple arithmetic trick without knowing the theory behind; the second method will explain the first method “from an advanced standpoint” – Number Theory (Felix Klein’s Vision )

1) Arithmetic trick:

11 is odd, not divisible by 2, 4 and 6.

Loan 1 horse to the old man:
11+1 = 12

1st son gets: 12/2 = 6 horses
2nd son gets:12/4 = 3 horses
3rd son gets: 12/6 = 2 horses

Total = 6+3+2=11 horses

Up to you if you want the old man to return the 1 loan horse 🙂

Strange! WHY ?

2)

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Differentiating under integral

Math Online Tom Circle

Prove: (Euler Gamma Γ Function)
$latex displaystyle n! = int_{0}^{infty}{x^{n}.e^{-x}dx}$

Proof:
∀ a>0
Integrate by parts:

$latex displaystyleint_{0}^{infty}{e^{-ax}dx}=-frac{1}{a}e^{-ax}Bigr|_{0}^{infty}=frac{1}{a}$

∀ a>0
$latex displaystyleint_{0}^{infty}{e^{-ax}dx}=frac{1}{a}$ …[1]

Feynman trick: differentiating under integral => d/da left side of [1]

$latex displaystylefrac{d}{da}displaystyleint_{0}^{infty}e^{-ax}dx= int_{0}^{infty}frac{d}{da}(e^{-ax})dx=int_{0}^{infty} -xe^{-ax}dx$

Differentiate the right side of [1]:
$latex displaystylefrac{d}{da}(frac{1}{a}) = -frac{1}{a^2}$
=>
$latex a^{-2}=int_{0}^{infty}xe^{-ax}dx$

Continue to differentiate with respect to ‘a’:
$latex -2a^{-3} =int_{0}^{infty}-x^{2}e^{-ax}dx$
$latex 2a^{-3} =int_{0}^{infty}x^{2}e^{-ax}dx$
$latex frac{d}{da} text{ both sides}$
$latex 2.3a^{-4} =int_{0}^{infty}x^{3}e^{-ax}dx$


$latex 2.3.4dots n.a^{-(n+1)} =int_{0}^{infty}x^{n}e^{-ax}dx$
Set a = 1
$latex boxed{n!=int_{0}^{infty}x^{n}e^{-x}dx}$ [QED]

Another Example using “Feynman Integration”:

$latex displaystyle text{Evaluate }int_{0}^{1}frac{x^{2}-1}{ln x} dx$

$latex displaystyle text{Let I(b)} = int_{0}^{1}frac{x^{b}-1}{ln x} dx$ ; for b > -1

$latex displaystyle text{I'(b)} = frac{d}{db}int_{0}^{1}frac{x^{b}-1}{ln x} dx = int_{0}^{1}frac{d}{db}(frac{x^{b}-1}{ln x}) dx$

$latex x^{b} = e^{ln x^{b}} = e^{b.ln x} $

$latex frac{d}{db}(x^{b}) = frac{d}{db}e^{b.ln x}=e^{b.ln x}.{ln x}= e^{ln x^{b}}.{ln x}=x^{b}.{ln x}$

$latex text{I'(b)}=int_{0}^{1} x^{b} dx=frac{x^{b+1}}{b+1}Bigr|_{0}^{1} = frac{1}{b+1}$
=>
$latex…

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Derivative Meaning

Math Online Tom Circle

The derivative of a function can be thought of as:

(1) Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.

(2) Symbolic: The derivative of
$Latex x^{n} = nx^{n-1} $
the derivative of sin(x) is cos(x),
the derivative of f°g is f’°g*g’,
etc.

(3) Logical:
$Latex boxed{text{f'(x) = d}} $
$Latex Updownarrow $
$latex forall varepsilon, exists delta, text{ such that }$
$latex boxed{
0 < |Delta x| < delta,
implies
Bigr|frac{f(x+Delta x)-f(x)}{Delta x} – d Bigr| < varepsilon
}$

(4) Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.

(5) Rate: the instantaneous speed of f(t), when t is time.

(6) Approximation: The derivative of a function is the best linear approximation to the function near a point.

(7)

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French Curve

Math Online Tom Circle

The French method of drawing curves is very systematic:

“Pratique de l’etude d’une fonction”

Let f be the function represented by the curve C

Steps:

1. Simplify f(x). Determine the Domain of definition (D) of f;
2. Determine the sub-domain E of D, taking into account of the periodicity (eg. cos, sin, etc) and symmetry of f;
3. Study the Continuity of f;
4. Study the derivative of fand determine f'(x);
5. Find the limits of fwithin the boundary of the intervals in E;
6. Construct the Table of Variation;
7. Study the infinite branches;
8. Study the remarkable points: point of inflection, intersection points with the X and Y axes;
9. Draw the representative curve C.

Example:

$latex displaystyletext{f: } x mapsto frac{2x^{3}+27}{2x^2}$
Step 1: Determine the Domain of Definition D
D = R* = R –…

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Prime Secret: ζ(s)

Math Online Tom Circle

Riemann intuitively found the Zeta Function ζ(s), but couldn’t prove it. Computer ‘tested’ it correct up to billion numbers.

$latex zeta(s)=1+frac{1}{2^{s}}+frac{1}{3^{s}}+frac{1}{4^{s}}+dots$

Or equivalently (see note *)

$latex frac {1}{zeta(s)} =(1-frac{1}{2^{s}})(1-frac{1}{3^{s}})(1-frac{1}{5^{s}})(1-frac{1}{p^{s}})dots$

ζ(1) = Harmonic series (Pythagorean music notes) -> diverge to infinity
(See note #)

ζ(2) = Π²/6 [Euler]

ζ(3) = not Rational number.

1. The Riemann Hypothesis:
All non-trivial zeros of the zeta function have real part one-half.

ie ζ(s)= 0 where s= ½ + bi

Trivial zeroes are s= {- even Z}:
s(-2) = 0 =s(-4) =s(-6) =s(-8)…

You might ask why Re(s)=1/2 has to do with Prime number ?

There is another Prime Number Theorem (PNT) conjectured by Gauss and proved by Hadamard and Poussin:

π(Ν) ~ N / log N
ε = π(Ν) – N / log N
The error ε hides in the Riemann Zeta Function’s non-trivial zeroes, which all lie on the Critical…

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Golden Ratio Φ

Math Online Tom Circle

A—————C———-B

$Latex frac {AB}{AC} = frac{AC}{CB}$
= 1.61803… = Φ
= $Latex frac {1+ sqrt{5}} {2}$

$Latex frac {6}{5} Phi^2$
= ∏ = 3.14159…

Donald Knuth (Great Computer Mathematician, Stanford University, LaTex inventor) noted the Bible uses a phrase like:
as my Father is to me, I am to you
=> F= Father = line AB
I (or me) = AC
U = You = CB
=> F/I = I/U = Φ
Note: Φ = 1.61803 = – 2 sin 666°

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What is i^i

Math Online Tom Circle

$Latex i^{i } = 0.207879576…$
$latex i = sqrt{-1}$

If a is algebraic and b is algebraic but irrational then $latex a^b $ is transcendental. (Gelfond-Schneider Theorem)

Since i is algebraic but irrational, the theorem applies.

1. We know
$latex e^{ix}= cos x + i sin x$

Let $latex x = pi/2 $

2. $latex e^{i pi/2} = cos pi/2 + i sin pi/2 $

$latex cos pi/2 = cos 90^circ = 0 $

$latex sin 90^circ = 1 $
$latex i sin 90^circ = (i)*(1) = i $

3. Therefore
$latex e^{ipi/2} = i$
4. Take the ith power of both sides, the right side being $latex i^i $ and the left side =
$latex (e^{ipi/2})^{i}= e^{-pi/2} $
5. Therefore
$latex i^{i} = e^{-pi/2} = .207879576…$

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Mystery numbers : 370 & 153

Math Online Tom Circle

Just can’t imagine how strange a plane MH370 could just disappear in the air, no explosion, no terrorists (?) although 2 Iranian passengers with stolen passports from an Italian and an Austrain.

Malaysian Flight: MH 370

Departure : Passengers, among them the majority are 153 Chinese, boarded on 3.7 (March 7) around 11 PM at Kuala Lumpur International Airport, disappeared 1 hour later in the air.

http://www.nst.com.my/latest/font-color-red-missing-mh370-font-timeline-of-flight-mh370-1.507516

Just notice 370 is a strange number:

$latex boxed { (3)^{3} + (7)^{3}+ (0)^{3} = 370}$

A lot of mystery numbers have such behaviors when decompose the digit, then each powered by 3, sum them up, you get back the mystery number itself.

Bible Math: 153 St. Peter Fish
[John 21:3-11]
3  So they went out and got into the boat, but that night they caught nothing.
6 He said, ”Throw your net on the right side of the boat and you will find some.” When…

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Bayesian Probability Could Help Search MH370 Missing Plane

Dr. Adrian Yeo Ning Hong’s Math Books

Math Online Tom Circle

Dr. Yeo Ning Hong was the former cabinet Minister of Singapore. He wrote a few Math books after retirement to teach his grand-daughters in primary schools – on Trigonometric Identity Proofs !!!
Yes!
He has wonderful tricks to make such difficult Secondary 3 Math easy for kids, and of course, also helps the weak Math teenagers.

Interesting !
Trig Or Treat: An Encyclopedia of Trigonometric Identity Proofs With Intellectually Challenging Games

The other books by him are:

The Pleasures of Pi,e and Other Interesting Numbers

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Ring Memorize Trick

Math Online Tom Circle

Memorize Trick For Ring:

1. Ring (R) is for Marriage with 2 operations: + (addition of kids), * (multiply asset).

2. (R , +) is Commutative Group
Analogy: your kids are also your wife’s kids, vice versa.

3. (R ,*) is Semi-Group (only closure & associative)
Analogy: your asset to multiply (*) is semi (50%) owned by your wife.

4. Fair distributive law for * with respect to +
=> distribute your asset (a) to your kids (k) & wife (w):
a*(k+w) = a*k + a*w

5. No division (/) operation => Ring can’t be broken.

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Our Daily Story #12: The Vagabond Mathematician Paul Erdős

Math Online Tom Circle

http://en.m.wikipedia.org/wiki/Paul_Erd%C5%91s

What is Erdős Number ?

image

Amazon Book: “My Brain Is Open

Paul Erdős (er-dish) was one of the greatest mathematicians in 21st Century, a Jewish Hungarian, single and no home. He traveled around the world in one small suitcase containing his mathematical papers. He would knock at the door of his former students or math collaborator impromptu, started working and published the papers, then moved on to next destination the moment his overstay became unwelcome by the host’s wife. In this way he published 1,500 articles in his lifetime.

image

Example:

Einstein had the Erdős Number 2 , through his assistant who co-wrote a paper with Paul Erdős.

Ref:

http://tomcircle.wordpress.com/2013/03/30/paul-erdos/

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Harvard Online Course: Abstract Algebra

For Junior College students after getting your A-level result this month, if you want to further study Math (or Science, Engineering) in the top universities overseas after your National Service, e.g. USA (Harvard, MIT, Stanford, Princeton, …), or France (where their modern math standard in French Baccalaureate – not the Singapore IB – is much higher than the GCE A level).
Attend this video up to the first 15 lectures will prepare you a good Modeen Math foundation, which is seriously lacking in our Singapore JC Math syllabus (regardless whether you get distinction in GCE A level Math)

Math Online Tom Circle

Prof Benedict Gross is one of the best Algebra professors I have seen – he can explain so well the abstract concepts, without injecting fear and confusion to the students.

As Prof Gross had brilliantly said in the beginning of this Lecture 1:

Algebra is the language of Math.

Since Math is the language of science,
therefore any serious Science needs to speak in Algebra language.

Today, if you read a research paper on any math (or Computer Science, Mathematical Physics…) topic, hardly you can avoid these “basic” algebraic lingoes: Group, Ring, Field, Vector Spaces, Quotient Group, Ideal, …

I strongly recommend to anyone who likes to study Modern Algebra but afraid of the abstractness, this is the course (free) for you. I can guarantee you by the halfway (15th lecture) you will have a solid foundation, and by the last lecture you will be able to follow high-level…

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Our Daily Story #11: The Anonymous Mathematician “Nicolas Bourbaki”

Math Online Tom Circle

The romantic gallic Frenchmen like to joke and give pranks. We have already seen the Number 1 Mathematical ‘prank’ in Our Daily Story #1 (The Fermat’s Last Theorem), here is another 20th century Math prank “Nicolas Bourbaki” – the anonymous French mathematician who did not exist, but like Fermat, changed the scene of Modern Math after WW II.

http://en.m.wikipedia.org/wiki/Nicolas_Bourbaki

image

André Weil (not to confuse with Andrew Wiles of FLT in Story #2 ) and his university classmates from the Ecole Normale Supérieure (Évariste Galois‘s alma mater which expelled him for involvement in the French Revolution), wanting to do something on the outdated French university Math textbooks, formed an underground ‘clan’ in a Parisian Café near Jardin du Luxembourg. They met often to brainstorm and debate on the most advanced Math topics du jour. Finally they decided to totally re-write the foundation of Math based on Set…

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In Conversation with Steve Ballmer at Oxford (4 Mar 2014)

Math Online Tom Circle

The 3 valuable  takeaways from this 1- hour interview with Steve Ballmer, who co-founded Microsoft with the richest man in the world Bill Gates:

1. Have a great idea before startup.
– The financial,  talent, etc come later.

2. Great company does (at least) one trick well.
– Microsoft has 2.5 tricks: PC operating system (Windows); Microprocessors in data center (Winservers); the half trick is X box game console. 

– Apple also has 2 tricks: Mac; Mobile computing in iPhone/iPAD/Appstore

– Facebook (Social Network) and Google (Search Engine)  have 1 trick.

– Sony has 1 trick : Audio TV

– HP has 1 trick: Tester Equipment (now in Agilent)

– IBM has 1 trick: Entreprise Data Centre

– Samsung has 1 trick: hardware manufacturer (LCD flat panel, Smartphone…)

– Amazon has 1 trick: online bookshop

– Alibaba.com has 1 trick: eCommerce

Many great companies degenerate into smallness or extinction, because…

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Our Daily Story #7: Algebraic Equation Owed to the Mathematical Thief

Math Online Tom Circle

From the previous O.D.S. stories (#3, #4) on Quintic equations (degree 5) by Galois and Abel in the 19th century, we now trace back to the first breakthrough in the 16th century of the Cubic (degree 3) & Quartic (degree 4) equations with radical solution, i.e. expressed by 4 operations (+ – × /) and radicalroots {$latex sqrt{x} , : sqrt [n]{x} $ }.

Example: Since Babylonian time, and in 220 AD China’s Three Kingdoms Period by 趙爽 Zhao Shuang of the state of Wu 吳, we knew the radical solution of Quadratic equations of degree 2 :
$latex ax^2 + bx + c = 0 $

can be expressed in radical form with the coefficients a, b, c:

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

Are there radical solutions for Cubic equation (degree 3) and Quartic equations (degree 4) ? We had to wait till the European Renaissance…

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Our Daily Story #6: A Subway Sandwich Mathematician Zhang Yitang 张益唐

Math Online Tom Circle

Zhang is the typical demonstration of pure perseverance of traditional Chinese mathematicians: knock harder and harder until the truth is finally cracked.

His work is based on the prior half-way proof by 3 other mathematicians “GPY”:

image

Gap between Primes:

Let p1 and p2 be two adjacent primes separated by gaps of 2N:

p1 – p2 = 2 (twin primes)
eg. (3, 5), (5, 7)… (11, 13) and the highest twin primes found so far (the pair below: +1 and -1)
image

p1 – p2 = 4 (cousin primes)
eg. (7, 11)

p1 – p2 = 6 (sexy primes)
eg. (23, 29)

p1 – p2 = 2N

Euclid proved 2,500 years ago there are infinite many primes, but until today nobody knows are these primes bounded by a gap (2N) ?

Zhang, while working as a sandwich delivery man in a Subway shop…

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Our Daily Story #5: The Prince of Math

Math Online Tom Circle

Carl Friedrich Gauss is named the “Prince of Math” for his great contributions in almost every branch of Math.

As a child of a bricklayer father, Gauss used to follow his father to construction site to help counting the bricks. He learned how to stack the bricks in a pile of ten, add them up to obtain the total. If a pile has only 3, for example, he would top up 7 to make it 10 in a pile. Then 15 piles of 10 bricks would give a total of 150 bricks.

One day in school, his teacher wanted to occupy the 9-year-old children from talking in class, made them add the sum:
1 + 2 + 3+ ….+ 98 + 99 + 100 = ?

Gauss was the first child to submit the sum within few seconds = 5,050.

He used his brick piling technique: add

1 + 100…

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Our Daily Story #3: The Math Genius Who Failed Math Exams Twice

Math Online Tom Circle

To prove the FLT, Prof Andrews Wiles used all the math tools developed from the past centuries till today. One of the key tool is the Galois Group,  invented by a 19-year-old French boy in 19th century, Evariste Galois. His story is a tragedy – thanks to the 2 ‘incompetent’ examiners of the Ecole Polytechnique (a.k.a. “X”), the Math genius failed in the Concours (Entrance Exams) not only once, but twice in consecutive years.
Rejected by universities and the ugly French politics and academic world, Galois suffered set back one after another, finally ended his life in a ‘meaningless’ duel at 20.

He wrote down his Math findings the eve before he died – “Je n’ai pas le temps” (I have no more time) – begged his friend to send them to two foreigners (Gauss and Jacobi) for review of its importance. “Group Theory”…

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Our Daily Story #9: The Indian Clerk Mathematician

Math Online Tom Circle

The story of Ramanujian:

http://en.m.wikipedia.org/wiki/Srinivasa_Ramanujan

image

image
http://mathworld.wolfram.com/Hardy-RamanujanNumber.html

We have seen how two 19th century greatest mathematicians Cauchy and Gauss who were not helpful to two young unknown mathematicians Galois and Abel, now let’s see an opposite example — the discovery of an unknown math genius Ramanujian by the greatest Pure Mathematician in 20th century Prof G.H. Hardy.

References:
1.
http://tomcircle.wordpress.com/2013/11/27/163-and-ramanujan-constant/

2.
http://tomcircle.wordpress.com/2013/07/04/ramanujian/

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Our Daily Story #10: A Shop Assistant Math Professor

Math Online Tom Circle

In the previous story (#9) we mentioned Ramanujan having the luck of being spotted by Prof G.H. Hardy as the treasure of mathematics, another Chinese Hua Luogeng 华罗庚, 20 years younger than Ramanujan,  was also coached by Prof Hardy, although Prof Hardy did not realize Hua’s potential later to the modern mathematics in China.

Hua dropped out of secondary school due to poverty, he worked in his father’s little grocery shop as the shop assistant. His talent was spotted by the French-educated mathematician Prof Xiong Qinlai ( 熊庆来) in Tsinghua University 清华大学 from a paper the young boy published – on Quantic Equation Solvability error made by a Math Professor Su. Hua was admitted to Tsinghua University as assistant math lecturer on exception. Later he was sent to Cambridge on 庚子赔款 Boxer Indemnity scholarship.

When Prof Hardy met Hua, he let Hua choose between:
1) Work on a PhD…

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