Simple Proof Methods

Next, let's define what irrational number are, and use a proof by contradiction on them.

Definition. [Irrationality] An irrational number is one that can't be written as $\frac{m}{n}$, where $m, n \in \mathbb{Z}$ and $n \neq 0$ ($m$ and $n$ don't have common factors: $\frac{m}{n}$ is the most simplified fraction.)

Theorem. The number $\sqrt{2}$ is an irrational number.

Proof. To obtain a contradiction, assume that $\sqrt{2}$ is a rational number. This means that $\sqrt{2} = \frac{m}{n}$ for some $m, n \in \mathbb{Z}$ and $n \neq 0$. When squaring both side of the equation, we get: $2 = \frac{m^2}{n^2}$. Multiplying both side by $n^2$ gives us $2\cdot n^2 = m^2$. By the definition of even numbers, $m^2$ is even, and, therefore, $m$ is even. Moreover, if we substitute $m = 2k$, we get $2\cdot n^2 = (2k)^2 = 4k^2 = 2\cdot 2k^2$, so $n^2 = 2k^2$, so $n^2$, and therefore $n$, is even. Our conclusion that $m$ is even and $n$ is even is a contradiction to the fact that $m$ and $n$ don't share any common factors.
This means that our assumption that $\sqrt{2}$ is rational is false, so we conclude that $\sqrt{2}$ is an irrational number.◼