Complex Numbers and the Unit Circle

When I first talked about complex numbers there was one perspective I put off, and now need to come back to. It makes deep use of Euler’s formula, which ties exponentials and trigonometric functions together in the relation

where we’ve written for and used the exponential property.

Remember that we have a natural basis for the complex numbers as a vector space over the reals: . If we ask that this natural basis be orthonormal, we get a real inner product on complex numbers, which in turn gives us lengths and angles. In fact, this notion of length is exactly that which we used to define the absolute value of a complex number, in order to get a topology on the field.

So what happens when we look at ? First, we can calculate its length using this inner product, getting

by the famous trigonometric identity. That is, every complex number of the form lies a unit distance from the complex number .

In particular, is a nice reference point among such points. We can use it as a fixed post in the complex plane, and measure the angle it makes with any other point. For example, we can calculate the inner product

and thus we find that the point makes an angle with our fixed post , at least for . We see that traces a circle by increasing the angle in one direction as increases from to , and increasing the angle in the other direction as decreases from to . For values of outside this range, we can use the fact that

to see that the function is periodic with period . That is, we can add or subtract whatever multiple of we need to move within the range . Thus, as varies the point traces out a circle of unit radius, going around and around with period , and every point on the unit circle has a unique representative of this form with in the given range.