Do you really not understand? I assumed you were playing riverrat for this thread. I'm still not convinced, but whatever, anything for doubt.
The definitions are made up after the fact and rely on some work by Cantor (set theory.) Initially, real numbers are a set devised to include both rational and irrational numbers. What they do not include are things like imaginary and undefined numbers. These are products of the square root of -1 and of 1/0 respectively. To the extent that new formulations are made up that do not seem to constitute rational or irrational numbers in the same way, the definitions will be expected to show why. For example, the square root of negative 2 could be considered irrational, but it isn't, specifically because it is imaginary.