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.
Transcendental numbers: e, Π, L…
What about ?
Aleksander (Alexis) Osipovich Gelfond (1906-68):
a is algebraic, not 0 or 1
b irrational algebraic number
Hilbert Number: (Hilbert Problem proven by Gelfond}
Is log 2 transcendental ?
[log = logarithm Base 10]
1) Sufficient to prove log 2 irrational
Assume log 2 rational
log 2= p/q, p and q integers
raise power q
Case 1: p>q
Case 2: q>p
Left is even :
Right is odd:
Therefore p,q do not exist,
=> log 2 irrational
Reference: Top 15 Transcendental Numbers: http://sprott.physics.wisc.edu/pickover/trans.html
In secondary school, we know how to prove √2 is irrational, how about e ?
e= 1 + 1/1! +1/2! + 1/3! + 1/4! +…
Prove by Reductio ad absurdum (contradiction):
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
Therefore e is irrational.
Google = Googol means ‘1 followed by 100 zeroes‘
Google raised Capital at IPO = $2.718281828 Billion = e Billion
Google’s 3 building names: e, pi, ø (Golden ratio)