Prep questions:
- Suppose x is even. If we multiply it by something, is the product always even, always odd, or does it depend?
- Suppose x is even. Is x^2 always even, always odd, or does it depend?
- Suppose x is odd. Is x^2 always even, always odd, or does it depend?
- Suppose p/q is a fraction in lowest terms. Can both the top and bottom be even? Can they both be odd? Can one be even, and one odd?
- What happens when we square a fraction?
- What happens when we square a product, say, pq?
Rules:
- "x is even" means "For some integer k, x = 2k."
- Squaring an even number always gives another even number.
- Squaring an odd number always gives another odd number.
- Squaring a fraction is the same as squaring its top and bottom.
Suppose sqrt(2) was some fraction p/q, where p and q are integers.
We can assume this is in lowest terms: p and q have no common factors. In particular, p and q can't both be even.
- So 2 = p^2 / q^2.
- So 2 q^2 = p^2
- So p^2 is even.
- So p must be even. Suppose it is 2k for some k.
- So 2 q^2 = (2k)^2 = 4 k^2
- So q^2 = 2 k^2
- So q^2 is even.
- So q is even.
But, if p and q are both even, then p/q wasn't in lowest terms, as we assumed.