It has been known since the Greeks that there is no ratio of numbers, a/b, that is equal to the square root of 2. But there is no maximum ratio whose square is less than 2, and no minimum ratio whose square is greater than 2. The German mathematician Richard Dedekind in 1872 pointed out that each real number corresponds to a ‘cut’ like this in the class of ratios. This means that if we are given the set of rationals, we can construct reals in terms of sets of them: the set of rationals whose square is less than 2 is an open set that can represent the square root of 2. Mathematicians including Russell and Whitehead were thus able to identify a real with the class of all ratios less than it. This has the disadvantage that rational numbers are no longer a subset of the reals, but there are other ways of using Dedekind's insight and preserving a uniform treatment for rational and irrational reals.