Nine, New

{Nine, New} Why their spelling so similar?

English: {Nine, New}
French: {Neuf, Neuf}
Spanish: {Nueve, Nuevo}
German: {Neun, Neu}
Norway: {Ni, Ny}

Irish: {Naoi, Nua}

Italian: {Nove, Nuovo}

It was a long-forgotten legacy Base-8 system {0,1,2,...8}
NINE is the first of a NEW set of eight.

Indian Vedic Math

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

Google Math

Google = Googol means '1 followed by 100 zeroes'

Google raised Capital at IPO = $2.718281828 Billion = e Billion
(e= 2.718281828...)

Google’s 3 building names: e, pi, ø (Golden ratio)

Structure Beyond C

Ask: NZQRC...X?
(Nine Zulu Queens Rule China ...)
Is there a number system X beyond Complex C?

We know that Z is extended of N
because of solving equation like :
y+2=0
=> y= -2 ∈Z

Similarly Q extends Z:
3y-2 =0
=> y= 2/3 ∈ Q

R extends Q:
y² = 2
=> y = √2 ∈ R

C extends R:
y² = -1
=> y = √-1 ∈ C

What about something extends C ?
Answer: No such number system !

Why ?

Let the Polynomial equation
P(x) = xⁿ +...+ dx³+ cx² + bx + a

P(x) has n solutions in C (by Gauss Fundamental Law of Algebra): z0, z1, z2, ...zn-1
P(x) can be factorized as:
P(x) = (x-z0).(x-z1).(x-z2)....(x -zn-1)

P(x)=0 has all n complex solutions still in C
=> closed in C or Complete in C
=> no need to extend C like the previous number systems (NZQR)

Sylvester, Matrix, Nightingale

Sylvester, James Joseph
(1841-1897)
Cambridge, St John's college. He coined the word 'Matrix'.
His private math tuition student was Florence Nightingale (Founder of Nursing), who later applied her statistics math in Nursing.

Eigenvector & Eigenvalue

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 outside nucleus.

Note3: Excellent further explanation of the eigenvector and eigenvalue:

http://lpsa.swarthmore.edu/MtrxVibe/EigMat/MatrixEigen.html

Calculus Fundamental Technique

$latex D first, then \int$

This is just a simple but powerful application of Calculus, behind which lies the philosophy of Leibniz:

1. D (=dy/dx) is the inverse function of $latex \int$
2. Calculus Fundamental Technique: $latex D first, then \int$
E.g. Sherlock Holmes example:
1. D first:
dT/dt = k(T-Ts)
=> can't solve directly
2. Take D's inverse:
$latex \int {dT/(T-Ts)} = k.dt$
=> can solve now !