I've also felt that real numbers aren't real ever since I tried doing algebra in floating point numbers and finding that some formulas that can be reversed in math cannot be reversed on a computer.

In the real numbers there exists a number that encodes the answer to the halting problem for every number, it's provable and also provable that you cannot write a program to print the digits.

Of course it's a computer scientist doing this. I'll have to read his papers. I wonder if he has anything about rational differential equations.