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 2Edit

The square root of 2 is irrational,  

ProofEdit

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.

NotesEdit

  • 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.