# The Evolution Of A Mathematician

This is a parady of "The Evolution of a Programmar".
I don't know about manager things so I omited that part.


## Middle School

a^x = a * a * ... * a (x times)


## High School

a^(-n) = 1 / a^n, a^(m/n) = nth-root(a^m)


## First year in College

e^x = 1 + x + x^2 / 2! + x^3 / 3! + ...
e^(ix) = cos(x) + i * sin(x)


## Second year in College

We can define a^x for all real numbers with this method:

Ex. a^(sqrt(2))

Let c_n be

c0 = 1
c1 = 1.4
c2 = 1.41
...

Since |c_{n+1} - c_n| < 10^(-n), c_n is a Cauchy sequence. c_n ->
sqrt(2) since |c_n - sqrt(2)| -> 0 (To say exactly, it is less tha
n epsilon).

Thus
|a^(c_{n+1}) - a^(c_n)|
= |a^(c_n)| * |a^[c_{n+1} - c_n] - 1|
< M * |a^{10^(-n)} - 1| for some M>0.

Hence a^(c_n) is a Cauchy sequence. Therefore it converges since R
is complete.
Let a^(sqrt(2)) be that number.


## New professional

a^x is an eigenvector of the differential operator.
If we define e^x by setting its eigenvalue 1, then a^x will have i
ts eigenvalue log a.


## Seasoned professional

M: A Riemannian manifold
TM: The tangent bundle of M
h: The geodesic with h'(0)=v

exp: TM -> M,
exp(v) = h(1)


## Master Mathematician

G, H: Lie groups
g, h: Their Lie algebras
Phi: G -> H, A Lie group homomorphism
Phi*: g -> h, Its derivative at the identity

exp makes the following diagram commute:

         Phi*
       g --> h
       |     |
exp(g) |     | exp(h)
       v     v
       G --> H
         Phi


## Apprentice Computer Scientist

a^n = a * a^(n-1)
a^0 = 1


## Experienced Computer Scientist

a^n
= a^(n/2) * a^(n/2) if n is even
= a * a^[(n-1)/2] * a^[(n-1)/2] if n is odd

a^0 = 1


## Seasoned Computer Scientist

A = {1,...,a}
a^n = |A^n|


## Guru Computer Scientist

2^n = 1 << n