Famous Theorems of Mathematics/Proof style

This is an example on how to design proofs. Another one is needed for definitions and axioms. Irrationality of the square root of 2 This result uses the following: Definition of rational number. Definition of prime and coprime. Definition of square root. The square root of 2 is irrational, Proof This is a proof by contradiction, so we assume that and hence for some a, b that are coprime. This implies that . Rewriting this gives . Since , we have that . Since 2 is prime, 2 must be one of the prime factors of , which are also the prime factors of , thus, . So we may substitute a with , and we have that . Dividing both sides with 2 yields , and using similar arguments as above, we conclude that . Here we have a contradiction; we assumed that a and b were coprime, but we have that and . Hence, the assumption was false, and cannot be written as a rational number. Hence, it is irrational. Notes As a generalization one can show that the square root of every prime number is irrational. Another way to prove the same result is to show that is an irreducible polynomial in the field of rationals using Eisenstein's criterion.