How would one find other real numbers that aren't in the rational field of numbers? For example, $\sqrt2$ isn't a rational number, since there is no rational number whose square equals two. And I see this example of a real number all the time and I'm just curious about how you can find or determine other numbers like so. Or better yet, how was it found that if $\mathbb Q$ is the set of all rational numbers, $\sqrt2\notin\mathbb Q$?
I appologize if the number theory tag isn't appropriate, I'm not really sure what category this question would fall under.

You might be interested in this earlier post, where the question being answered is "How do we prove that for any $a \in \mathbb{N},\;\sqrt{a}$ is an integer, or else irrational"? Some of the answers generalize even further.
You'll get a lot of helpful answers and links to additional resources.  
