Famous Theorems of Mathematics/Proof style

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This is an example on how to design proofs. Another one is needed for definitions and axioms.

Irrationality of the square root of 2[edit]

The square root of 2 is irrational,

Proof[edit]

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[edit]

  • 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.