Wednesday, April 1, 2009

√2 is irrational

Here's an old proof that √2 is irrational. It is a proof by contradiction, and the fundamental theorem of arithmetic (which says that factorization into primes is unique). Suppose that √2 = p/q, where p and q are relatively prime. Then
(p/q)2 = 2
so
p2 = 2 q2
so p2 has 2 as a factor, which means that p must have 2 has a factor. So p=2r for some r.

We now have
(2r)2 = 2 q2
so
4r2 = 2 q2
so
2r2 = q2
and therefore q has 2 as a factor.

So both p and q have 2 as a factor, but we assumed that they were relatively prime, which is a contradiction.

The kids didn't really think much of this one. They seem take it for granted that √2 is irrational, so why do they need a proof. Or maybe they don't like proof by contradiction. (Maybe they are constructionist mathematicians...)

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About Me

I'm research faculty at MIT, and Chief Architect at Tokutek.