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.

I dont know if this would answer your question.
We take all cauchy sequences whose elements are in rationals. now there are some sequences whose limit is not in the rationals(We call this property as rationals are not complete.) and we add all the limits to a number system and we call this number system as reals.
I don't know your background and so i am unable to explain all the technicalities.
You can refer Analysis-1 by Terence Tao for more details. Or you can check Dedekind cuts if you can understand (Dedekind cuts is another way to create reals from rationals.)
