We say a real number x is rational if it can be written as the quotient of two integers, x=m/n. We say a number is irrational if it cannot be written in this form.

Note that any rational number can be written in lowest terms, that is, m and n can be chosen so that GCD(m,n)=1.

You may assume that all positive integers have a unique factorization as a product of primes.

So here's the problem: Prove that the square root of two is irrational.

This is another of the classic examples of an indirect proof, wherein one proves a statement by assuming it is false and deriving a contradiction. You might wish to use this style of proof for some of the many problems we'll encounter involving linear independence.

If you want criticism from me, you can write up your proof in the space below and submit it; I'll return it with comments at the next section meeting. (You get absolutely no credit for doing this, but the feedback might prove beneficial.)