[phaemon]

I was under the impression that irrational numbers are numbers that simply can't be represented as a ratio. This is independent of the number system used. I'm not familiar with the Taylor series: can you use it as a number system? Isn't it simply a representation of functions (in which case, you might as well just write "sqrt(2)" as invoke anything else)?

It's tough to Google for "Taylor series number system" as you just get pages about the guitars. ;-)

[mtdewcmu]

I don't know if you're still reading this, but... Using Taylor series as a number system is just a hypothetical. Any systematic way of describing numbers could be a number system. Irrationals can't be represented as ratios, but beyond that, some have other finite representations, and others have none.