in base N, any the number N is written "10"
in base ten, the number ten is written 10
in base sixteen, the number sixteen is written 10
in base two, the number two is written 10.
cool, eh?
I posted this on the web a long time ago and never documented where I got it from. I think it was from a photocopied page from some book, given to me by a professor who thought I would be interested in it. If anyone can inform me as to the author of the proof, I'd gladly make note of it here.
Colin Wright, points out that this proof is "fundamentally identical to the one in book IV of the series 'Elements of Mathematics', by Nicola Bourbaki (perhaps) and is called 'Functions of one real variable'. The proof appears in chapter III, section 2.4, exercise 8."
Theorem: pi is irrational
Proof: Suppose pi = p / q, where p and q are integers. Consider the
functions f_n(x) defined on [0, pi] by
f_n(x) = q^n x^n (pi - x)^n / n! = x^n (p - q x)^n / n!
Clearly f_n(0) = f_n(pi) = 0 for all n. Let f_n[m](x) denote the m-th
derivative of f_n(x). Note that
f_n[m](0) = - f_n[m](pi) = 0 for m <= n or for m > 2n; otherwise some integer
max f_n(x) = f_n(pi/2) = q^n (pi/2)^(2n) / n!
By repeatedly applying integration by parts, the definite integrals of
the functions f_n(x) sin x can be seen to have integer values. But
f_n(x) sin x are strictly positive, except for the two points 0 and
pi, and these functions are bounded above by 1 / pi for all
sufficiently large n. Thus for a large value of n, the definite
integral of f_n sin x is some value strictly between 0 and 1, a
contradiction.
