Linear Algebra: A second course, featuring proofs and Python

Let’s quickly review some basic facts about complex numbers that are typically covered in an earlier course. First, we define the set of complex numbers by

where $i=\sqrt{-1}\text{.}$ We have a bijection $\mathbb{C}\to {\mathbb{R}}^2$ given by $x+iy\mapsto (x,y)\text{;}$ because of this, we often picture $\mathbb{C}$ as the complex plane, with a “real” $x$ axis, and an “imaginary” $y$ axis.

Arithmetic with complex numbers is defined by

The multiplication rule looks complicated, but it’s really just “FOIL”, along with the fact that $i^2=-1\text{.}$ Note that if $c=c+i0$ is real, we have $c(x+iy)=(cx)+i(cy)\text{,}$ so that $\mathbb{C}$ has the structure of a two dimensional vector space over $\mathbb{R}$ (isomorphic to ${\mathbb{R}}^2$).

Subtraction is defined in the obvious way. Division is less obvious. To define division, it helps to first introduce the complex conjugate. Given a complex number $z=x+iy\text{,}$ we define $\bar{z}=x-iy\text{.}$ The importance of the conjugate is that we have the identity

So $z\bar{z}$ is real, and non-negative. This lets us define the modulus of $z$ by

This gives a measure of the magnitude of a complex number, in the same way as the vector norm on ${\mathbb{R}}^2\text{.}$

Now, given $z=x+iy$ and $w=s+it\text{,}$ we have

And of course, we have $w\overline{w}\ne 0$ unless $w=0\text{,}$ and as usual, we don’t divide by zero.

One place where $\mathbb{C}$ is computationally more complicated is finding powers and roots. For this, it is often more convenient to write our complex numbers in polar form. The key to the polar form for complex numbers is Euler’s identity. For a unit complex number $z$ (that is with $|z|=1$), we can think of $z$ as a point on the unit circle, and write

If $|z|=r\text{,}$ we simply change the radius of our circle, so in general, $z=r(\cos (\theta )+i\sin (\theta ))\text{.}$ Euler’s identity states that

This idea of putting a complex number in an exponential function seems odd at first. If you take a course in complex variables, you’ll get a better understanding of why this makes sense. But for now, we can take it as a convenient piece of notation. The reason it’s convenient is that the rules for complex arithmetic turn out to align quite nicely with properties of the exponential function. For example, de Moivre’s Theorem states that

This can be proved by induction (and the proof is not even all that bad), but it seems perfectly obvious in exponential notation:

since you multiply exponents when you raise a power to a power.

Similarly, if we want to multiply two unit complex numbers, we have

But in exponential notation, this is simply

which makes sense, since when you multiply exponentials, you add the exponents.