(Warning: I am a computer guy not math guy!)

So I learned recently about transfinite cardinals, and that there are more real numbers than natural numbers. But this is bothering me, because I also learned about computable numbers, which are the numbers whose digits can be enumerated by a turing machine. (computer) Like pi. But any number that can be represented at all is computable. It seems to not be computable, a number must possess infinite digits, and they must be truly random so that any computer program that produce them would have to be as long as the number (infinite computer programs are not allowed so therefore its not computable)

But the computable numbers are countable, (by Godel numbering of Turing machines, so aleph null) the reals are not. So why do mathematicians need all these unrepresentable real numbers? What breaks if we get rid of them?

Just curious. Thanks.