1. Can a and b both be whole numbers? similar triangles so

2. Proof that  2 is irrational by origami Suppose 1712 5 5 7 So We have found another fraction equal to  2, but with a smaller denominator 45˚ THE IDEA

3. Proof that  2 is irrational by origami a b b b a-b 2b-a So 45˚ THE PROOF Suppose it's rational We have found another fraction equal to  2, but with a smaller positive integer denominator note: b < a < 2b So if  2 is rational, we find an infinite decreasing sequence of integers less than b

4. But an infinite decreasing sequence of integers less than b is impossible! so our original assumption that  2 is rational must be wrong i.e. we have proved  2 is irrational