[j2kun]

For reals, it's "Real numbers are the numbers you need to ensure every convergent sequence of rational numbers has a terminating point"

Convergence is determined in the "Cauchy" sense by having a vanishing distance between subsequent sequence entries, so as not to rely on the (potentially nonexistent) limit.

[chadcmulligan]

I thought you could have calculus using 'computable numbers' that is reals that have a rule. It's been on my list for a while to look into it - there's some notes here for the curious http://math.stackexchange.com/questions/963061/can-the-set-o...