ABSTRACT
Of course real analysis is about the real numbers. The real numbers are a very subtle number system with many deep and mysterious properties. The complex numbers were initially developed so that people would have a number system in which all polynomial equations are solvable. One of the reasons, historically that mathematicians had trouble accepting the complex numbers is that they did not believe that they really existed and were just made up. This is, in part, how they came to be called "imaginary." Mathematicians had similar trouble accepting negative numbers; for a time, negative numbers were called "forbidden." The model for the real numbers is the set of all Dedekind cuts. This is a non-obvious construction that guarantees that there really is a number system that satisfies the important and nontrivial properties of the reals. Thus, this discussion concludes by recording the most important basic fact about the complex numbers.
