ABSTRACT
Every rational number is representable as a pair of integers, the numerator and the denominator. Are all numbers rational? If not, can at least all numbers be represented as a finite collection of integers? The answer to both questions turns out to be no, but a satisfactory explanation is far from simple. There are a lot more numbers than the rationals, and the numbers that are representable as a finite collection of integers are still extremely rare (but also extremely interesting). Yet, based on the limited evidence we have gathered so far, it is hard to be convinced that almost all numbers are not rational. For instance, if we represent the rationals as points on a line, we can find them as close as we please to any given point.
