This method of proof can easily be generalised to prove that $\sqrt{n}$ is irrational when $n$ is not a square number.

What is the length of the diagonal of a square with sides of length 2?
How do we find the value of $\sqrt{2}$?
What number has 2 as its square?
What is the side of a square which has area 2?
Now $(1.4{)}^2=1.96$, so the number $\sqrt{2}$ is roughly $1.4$. To get a better approximation divide $2$ by $1.4$ giving about $1.428$, and take the average of $1.4$ and $1.428$ to get $1.414$. Repeating this process, $2÷1.414\approx 1.41443$ so $2\approx 1.414\times 1.41443$, and the average of these gives the next approximation $1.414215$. We can continue this process indefinitely getting better approximations but never finding the square root exactly.

If $\sqrt{2}$ were a rational number, that is if it could be written as a fraction $p/q$ where $p$ and $q$ are integers, then we could find the exact value. The proof sorter shows that this number is IRRATIONAL so we cannot find an exact value.