The Greeks knew about incommensurable lengths by the fifth century BCE. The diagonal of a unit square has length √2, and √2 — the Pythagoreans discovered, to their reported horror — cannot be written as a ratio of whole numbers. Two and a half thousand years later, working analysts were still casually using "real numbers" without anyone having a precise definition of what one was. Calculus rested on a foundation that nobody had built. The repair came in a thirty-year burst at the end of the nineteenth century — Cauchy, Bolzano, Weierstrass, Dedekind, Cantor — and required, finally, an honest answer to the question what is the continuum?
Two equivalent constructions emerged. Dedekind cuts (1872): a real number is a partition of the rational numbers into a lower set and an upper set, where every rational below the cut is in the lower set and every rational above is in the upper. The cut at √2 puts rationals whose square is less than 2 below, and the rest above; the real number is the cut. Cauchy sequences (Cantor, same year): a real number is an equivalence class of rational sequences whose terms eventually get arbitrarily close to each other. Either definition gives the same object — an ordered field with the completeness property: every set of reals that is bounded above has a least upper bound. Completeness is the axiom that distinguishes the reals from the rationals, and it is exactly what calculus needs. Without it, sequences could converge "almost" without converging anywhere; intermediate-value statements could fail; the foundations of analysis simply do not work. Cantor also showed that the reals are uncountable: there is no way to list them in a sequence indexed by the natural numbers — his diagonal argument from 1891 is one of the most cited proofs in mathematics, and it implies that almost every real number cannot be described by a finite expression. Most reals are, in this sense, unspeakable: they exist, the theory needs them, but no human or computer can ever write one down.
Computers cannot store real numbers; they store floating-point approximations — the IEEE 754 standard is, in a sense, a working compromise with the impossibility of representing the continuum exactly. Numerical analysis is the study of how badly the approximation hurts and what to do about it. Constructive mathematics (Brouwer, Bishop) rebuilds analysis allowing only reals you can compute, sacrificing some classical theorems for tighter algorithmic content. The continuum hypothesis — whether there is an infinity strictly between the naturals and the reals — was shown by Gödel and Cohen to be independent of ZFC: it can be assumed true or false without contradiction, an unsolved problem promoted to an unsolvable one.