4.3 Section summary
The modulus function provides us with a measure of distance that turns the set of complex numbers into a metric space in much the same way as does the modulus function defined on R. From the point of view of analysis the importance of this is that we can talk of the closeness of two complex numbers. We can then define the limit of a sequence of complex numbers in a way which is almost identical to the definition of the limit of a real sequence. Another analogue of real analysis arises with the Nested Rectangles Theorem. This will play a role in complex analysis similar to the least upper bound property (or, equivalently, the Nested Intervals Theorem) of real analysis.
The modulus function also gives us a way of describing certain sets of points in the complex plane in a convenient algebraic form. These descriptions are the subject of some of the problems in the next section.