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
The square root of 2 is irrational,
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.
- 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.