At last, after all that work, we can finally make the real numbers out of rationals! This time, the method being used actually has a name; they’re called Dedekind cuts, after the mathematician Richard Dedekind, among the first mathematicians to actually give a rigorous definition of real numbers. One of the other methods is to simply define reals by their decimal expansion; this has the advantage of being simpler in some cases, but subtraction and division are much harder to define explicitly due to borrowing and the difficulty of explicitly dividing infinitely long numbers.

A real number is simply a nonempty set, A, of rational numbers with these properties:

1. There is a rational number x not in A.
2. If x is in A, and y < x, then y is in A.
3. If x is in A, then there is some z also in A such that z > x.

We can then identify each rational q with the set of all rational x < q; this set is nonempty, it has a number not in it, if x < q and y < x, then y < q, and if x < q, then there is a z with x < z < q.

We first define comparison: A < B if and only if A is a subset of B. Addition is straightforward: A + B is just the set of all a + b, where a and b are in A and B. Subtraction is also fairly simple; define A – B to be the set of all a – b, where a < 0 and a is in A and b is in Q\B, the set of all rational numbers not in B. Negation is then just 0 – B. Multiplication is a little harder; if A and B are both non-negative, then AB is just the set of all x that are less than some ab for a, b > 0 and a, b in A, B respectively. On the other hand, if at least one of A and B is negative, we define AB = (-A)B = A(-B) = (-A)(-B) and use these relations to compute a positive product. Division is similar; for B > 0, define B^-1 to be the set of all 1/b where b is nonzero and in Q\B, and define A/B to be AB^-1. If B < 0, then B^-1 = -(-B)^-1 and A/B = A(-B)^-1.

And there you have it; we now have enough structure to get the rest of the operations we have on the reals, such as exponentiation, logarithms, and functions. As an aside, Dedekind cuts technically consist of two sets, A and Z, with Z = Q\A, but Z is very rarely used, as it’s uniquely determined by A. On the other hand, the surreal numbers are defined via an analogous construction: each surreal corresponds to at least one ordered pair of sets of surreal numbers (L, R). However, the only requirement for surreals is that if x is in L and y is in R, then x < y. At first, this definition of surreals as sets of surreals appears to be circular; if we need surreals to make surreals, how can we make one? But I’ll leave that for you to ponder.