The complex unit circle, a group under mutiplication (corrected on 22 May)

 

Most of my posts so far have been dealing with finite groups, and in the last month, I have been concentrating on non-abelian groups. When I first learned the topic, I was fascinated that a math structure could be so easily defined, but that we could have a system where ab ≠ ba in every case. Today, we will learn about an infinite group, all numbers of the form a+bi, where sqrt(a² + b²) = 1 and  i² = -1. When we add real and imaginary numbers, we call the result complex numbers, and we represent them on a plane where the x-axis is purely real and the y-axis is purely imaginary. They are still complex, since 0+bi and a+0i still fit the form a+bi. 

 

The set we have defined is known as the complex unit circle. In the illustration above, let r = 1, which means the diameter d = 2 and the circumference = 2π. Because this is the unit circle, radians make more sense than angles, so we will write these points in the form cos(𝜃) + isin(𝜃). Our group operation will be multiplication, and complex multiplication has the lovely property that the product will be multiplying the lengths together and adding the angles. Since all lengths are 1, 

[cos(𝜃) + isin(𝜃)]*[cos(𝜶) + isin(𝜶)] = cos(𝜃+𝜶) + isin(𝜃+𝜶) and

[cos(𝜃) + isin(𝜃)]/[cos(𝜶) + isin(𝜶)] = cos(𝜃-𝜶) + isin(𝜃-𝜶).

 

When we add or subtract angles, the results could be greater than 2π or less than 0, but this is not a problem. If we divide by 2π and take the remainder, we will get a number 𝛽 such that 0 ≤ 𝛽 < 2π. 

 

While we have defined the group as a multiplicative group, we see that this multiplication is mapped onto the addition of angles, and additive groups are always abelian. More than this, any finite cyclic group is a subgroup of the complex unit circle. We will discuss this in greater detail tomorrow.

I came back to this post to correct it. The complex unit circle does wrap around on itself but it is NOT cyclic because it does not have a single generator. If some number a+bi generates an infinite subgroup of the complex unit circle, it can only be dense on the circle, and an infinite number of points will be missing. This is analogous to the rationals being dense among the reals.