Themes > Science > Mathematics > Calculus > Real Number > Complete fields

We have used the real numbers in some of our preceding discussions. For instance, the complex numbers are ordered pairs of real numbers, and our example of infinitesimals involved rational functions with real coefficients. In effect, we "borrowed" the real numbers -- we used the reals in examples, even though we hadn't formally defined them yet; we just relied on the informal and intuitive understanding that students already have, based on the geometric line. Trust me, there is no circular reasoning here -- I won't use the "borrowed" concepts when I finally get around to defining the real numbers. You'll see that if you actually work through all the details. (I'm not claiming that this web page is more than an outline.)

The definition of the reals depends on two more theorems, both of which are difficult to prove.

 Theorem 1. There exists a Dedekind-complete ordered field.

The literature contains many different proofs of this theorem. I think three are simple enough to deserve mention here:

• Proof using decimal expansions. Let Y be the set of all infinite decimal expansions -- i.e., expressions such as 3.682951... and -17.311897... . Adopt the convention that 2.719999... is the "same" as 2.7200000..., etc. Use the usual operations of addition and multiplication. Then Y is a complete ordered field, but verifying that fact is extremely tedious. It generally isn't worked out in full detail. One place that you can find it in fairly complete detail is in J. F. Ritt, Theory of Functions, 1946. It is also sketched in M. Rosenlicht, Introduction to Analysis, reprinted by Dover.

• Proof using Dedekind cuts. Let Q be the set of rational numbers; we assume that we already have a good understanding of those. By a Dedekind cut we mean a pair of sets (A,B), where A and B are nonempty subsets of Q, A is the set of all lower bounds of B (in Q), and B is the set of all upper bounds of A (in Q). An example of such a cut is

A = {r Î Q: r < Ö5},   B = {r Î Q: r > Ö5}.

It is possible to describe such cuts without mentioning square roots (mention squares of rational numbers instead). The set of all cuts can be made into a complete ordered field, if we define addition and multiplication the right way. Again, it's tedious; you can find some of the details worked out in W. Parzynski and P. Zipse, Introduction to Mathematical Analysis.

• Proof using Cauchy sequences. Again start from the rational numbers. Say that a sequence r1, r2, r3, ... of rational numbers is a Cauchy sequence if it has the property that

 for each positive integer p there exists a positive integer m (which may depend on p and on the particular sequence being studied) such that, whenever i and j are greater than m, then |ri - rj| < 1/p.

Now, say that two Cauchy sequences r1, r2, r3, ... and s1, s2, s3, ... of rationals are equivalent if they have the property that

 for each positive integer p there exists a positive integer m (which may depend on p and on the particular sequences being studied) such that, whenever i is greater than m, then |ri - si| < 1/p.

By an equivalence class we mean the set of all the sequences that are equivalent to some particular sequence. Now, it can be shown that the set of all equivalence classes is a complete ordered field, if we define addition and multiplication on it in the right fashion. This proof, due to Cantor, is a slight modification of a proof that can be found in many analysis or topology books, showing that every metric space has a metric completion.

The other theorem is harder to prove, and I won't even sketch a proof here. In fact, this theorem is even difficult to state:

 Theorem 2. Any two Dedekind-complete ordered fields are isomorphic i.e., there exists a one-to-one correspondence between them that preserves, in both directions, the orderings and the arithmetical operations. Thus, any two Dedekind-complete ordered fields are essentially "the same"; one is simply a relabeled copy of the other.

In particular, the decimal expansions, the Dedekind cuts, and the equivalence classes of Cauchy sequences, though they appear to be entirely different, all turn out to have the same arithmetic and algebraic structure -- they are really the "same" object. It is that object which we call the real number system.