Richard Dedekind (1831–1916) asked a question so basic it sounds silly until you try to answer it: what, exactly, is an irrational number? Everyone happily used \sqrt{2} and \pi, but nobody had said what they really are. Dedekind's answer was breathtakingly simple: an irrational number is the gap it leaves in the rationals.

Cutting the number line in half

His trick — a Dedekind cut — is to slice the rational numbers into two pieces, a lower set A and an upper set B, with everything in A below everything in B. For \sqrt{2}:

There is no largest rational in A and no smallest in B — the cut falls in a gap, and Dedekind simply declared that gap to be the number \sqrt{2}. Do this for every cut and you have built the entire real line, completeness and all, out of nothing but fractions. It's the rigorous foundation under the real number system.

The last student of Gauss — and his own obituary

Dedekind was the final doctoral student of the legendary Gauss, and a close friend of Riemann. He was modest and unhurried — he also gave the first clean definition of an infinite set (one that can be matched one-to-one with a proper part of itself). A famous mathematical calendar once printed his date of death years too early; Dedekind cheerfully wrote to point out that he had spent that supposed last day having a perfectly pleasant mathematical conversation with a friend.

Read more

The full story (with far fewer jokes) is on Wikipedia: Richard Dedekind — Wikipedia.