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}…

Axiom ( Greek ): meant request . The reader is requested to accept the axioms unquestioningly as the rules of the game.

Euclid’s “Element” built the whole Geometry with only 5 axioms.
The 5th axiom “Parallel line” was not challenged for 3,000 years until 19th CE Gauss & Riemann developed the Non-Euclidian Geometry.

C Continuous
I Integrable
D Differentiable

The CID Relation:

$latex D to C, C to I$
=>
$latex D to I$

What it means a curve (function) is :
1. Continuous = not broken curve
2. Differentiable = no pointed ‘V’ or ‘W’ shape curve
3. Integrable: can compute the area under the (unbroken) curve.

Visualize by mapping Exp e (red and pink arrows):

$latex e^{ipi} = -1$

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

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

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
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) <…
=>…

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

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

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…

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

Bi-quadratic Equation by Vedic Math

Solve
$latex (x+7)^4+(x+5)^4=706$
Let y = x + 6 = average of (x+5, x+7)
$latex (y+1)^4+(y-1)^4=706$
Cancel terms y, y³:
$latex 2y^4+12y^2+2=706$
$latex y^4+6y^2-352=0$
$latex y^2=16$ or
$latex y^2=-22$
y= ±4 or ±$latex sqrt{-22}$
y=x+6
x=-2, -10, ± $latex sqrt{-22}-6$

“Transire suum pectus mundoque potiri”
[To transcend human limitations & to master the universe]

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)

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)

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…

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)

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 –…

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…

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°

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

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
$latex i^{i} = e^{-pi/2} = .207879576…$

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…