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, $\sqrt{2}\notin \mathbb{Q}$

Proof

This is a proof by contradiction, so we assume that $\sqrt{2}\in \mathbb{Q}$ and hence $\sqrt{2}=\frac{a}{b}$ for some a, b that are coprime.

This implies that $2=\frac{a^2}{b^2}$. Rewriting this gives $2b^2=a^2$.

Since $b^2\in \mathbb{Z}$, we have that $2|a^2$. Since 2 is prime, 2 must be one of the prime factors of $a^2$, which are also the prime factors of $a$, thus, $2|a$.

So we may substitute a with $2k,k\in \mathbb{Z}$, and we have that $2b^2=4k^2$.

Dividing both sides with 2 yields $b^2=2k^2$, and using similar arguments as above, we conclude that $2|b$.

Here we have a contradiction; we assumed that a and b were coprime, but we have that $2|a$ and $2|b$.

Hence, the assumption was false, and $\sqrt{2}$ 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 $x^2-2$ is an irreducible polynomial in the field of rationals using Eisenstein's criterion.