​How do you tell if a number is rational or irrational?

To prove a number is irrational, you could assume that it can represented by a ratio of integers and then prove that would be a contradiction if that were to happen.

For example with \(\sqrt{2}\), you could assume it can be represented by a ratio of integers, but if you tried to do that, you'd find that one of the integers would have to be both odd and even at the same time, which is a contradiction, proving \(\sqrt{2}\) is irrational.

All square roots of non perfect squares are irrational by a version of this proof.

All an irrational number is is a number which cannot be represented as a ratio of integers and has no repeating patterns in its infinitely continuing decimal expansion.

There's probably more on this topic, but I can't really go any deeper right now.