day 2012

Also this year, day March 14th hits Harvard spring break and no -eating contest takes place.

Pi in the news

Pi on the web

Theorem: is irrational.

Proof: by Ivan Niven (1915-1999), Bull.Amer.Math.Soc. 53 (1947),509):
Assume = a/b with positive integers a and b. We will deduce a contradiction. For an integer n>0 define

f(x) = xn (a-b x)n/n!

It satisfies f(x) = f(-x) and the inequality

0 < f(x) <n an/n!

for all x in [0,]. For all j, smaller or equal to n, the j-th derivative of f is zero for x=0 and x=. For j larger or equal to n, the j-th derivative of f is an integer at 0 and . The function

F(x) = f(x) - f(2)(x) + f(4)(x) - ... + (-1)n f2n(x)

now has the property that F(0) and F() are integers and F + F'' = f. Therefore,

(F'(x) sin(x) - F(x) cos(x))' = f sin(x)

By the fundamental theorem of calculus, f(x) sin(x) dx is an integer. By the inequality however, this integral is strictly between 0 and 1 for large n. QED.