The Circle Group
Yesterday we saw that the unit-length complex numbers are all of the form , where measures the oriented angle from around to the point in question. Since the absolute value of a complex number is multiplicative, we know that the product of two unit-length complex numbers is again of unit length. We can also see this using the exponential property:
So multiplying two unit-length complex numbers corresponds to adding their angles.
That is, the complex numbers on the unit circle form a group under multiplication of complex numbers — a subgroup of the multiplicative group of the complex field — and we even have an algebraic description of this group. The function sending the real number to the point on the circle is a homomorphism from the additive group of real numbers to the circle group. Since every point on the circle has such a representative, it’s an epimorphism. What is the kernel? It’s the collection of real numbers satisfying
that is, must be an integral multiple of — an element of the subgroup . So, algebraically, the circle group is the quotient . Or, isomorphically, we can just write .
Something important has happened here. We have in hand two distinct descriptions of the circle. One we get by putting the unit-length condition on points in the plane. The other we get by taking the real line and “wrapping” it around itself periodically. I haven’t really mentioned the topologies, but the first approach inherits the subspace topology from the topology on the complex numbers, while the second approach inherits the quotient topology from the topology on the real numbers. And it turns out that the identity map from one version of the circle to the other one is actually a homeomorphism, which further shows that the two descriptions give us “the same” result.
What’s really different between the two cases is how they generalize. I’ll probably come back to these in more detail later, but for now I’ll point out that the first approach generalizes to spheres in higher dimensions, while the second generalizes to higher-dimensional tori. Thus the circle is sometimes called the one-dimensional sphere , and sometimes called the one-dimensional torus , and each one calls to mind a slightly different vision of the same basic object of study.