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.