# Transcendental Numbers: e, pi

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.

# Draw e^ax

My nephew gets confused in drawing exponential curves with many variations: $\boxed {y = e^{ax} }$

If $-e^{ax}$, then reverse the y values, that is flip the curve over x-axis.

Math Joke:

Note:

$\frac {dy}{dx} e^{x} = e^{x}$
$\int e^{x} dx = e^{x} + c$

# Transcendental Number

Transcendental numbers: e, Π, L…

What about $e^{e}, \pi^{\pi} ,\pi^{e}$ ?

Aleksander (Alexis) Osipovich Gelfond (1906-68):

Gelfond-Schneider Theorem

$a^ b$ transcendental if
a is algebraic, not 0 or 1
b irrational algebraic number

Examples:
$\sqrt{6}^{\sqrt{5}}, 3^{\sqrt{7}}$
Hilbert Number: $2^{\sqrt {2}}$ (Hilbert Problem proven by Gelfond}

Is log 2 transcendental ?
[log = logarithm Base 10]

Proof:
$10^{log 2} = 2$

1) Sufficient to prove log 2 irrational
Assume log 2 rational
log 2= p/q, p and q integers
$10^ {log 2} = 2 = 10^ {p/q}$
raise power q
$2^{q} = 10^{p} = (2.5)^{p}$
$2^{q} = 2^{p}.5^{p}$

Case 1: p>q
$1 = 2^{p-q}.5^{p}$
=> False

Case 2: q>p
$2^{q-p}= 5^{p}$
Left is even : $2^{m} \text { = even}$
Right is odd: $5^{n} \text {= ....5}$
=> False

Therefore p,q do not exist,
=> log 2 irrational

Reference: Top 15 Transcendental Numbers: http://sprott.physics.wisc.edu/pickover/trans.html

# Irrational ‘e’

In secondary school, we know how to prove √2 is irrational, how about e ?

e= 1 + 1/1!  +1/2!  + 1/3!  + 1/4!  +…

Assume e= p/q as rational
multiply both sides by q!
LHS: e. q!= (p/q) .q! = p.(q-1)!  => integer

RHS:  q!+q! + (3.4…q)+ (4.5…q) +…1 + 1/(q+1) +…. => fraction