A Way to Think of Complex Numbers

When they first threw complex numbers and their arithmetic at me, there wasn’t a why given with it.

But by thinking of complex numbers as matrices, we can actually get a better feel as to how it all works.

A complex number is made up of the base units $1$ and $i$ and these are then scaled by the real numbers $a$ and $b$ . If can we find a 2×2 matrix that does the job of $1$ and another that does the job of $i$ , then we can ultimately find $a+bi$ as a 2×2 matrix.

Thus, to find a suitable $1$ , we think about the main property of $1$ , which is that anything you multiply by $1$ is itself. We know that the identity matrix in the set of 2×2 matrices has the same property, so we will take this to be the matrix equivalent of our number $1$ . We write

$\displaystyle 1=\text{I}_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}.$

Similarly, to find a suitable $i$ , we want a matrix that rotates 2D vectors by 90° anti-clockwise (this is what $i$ does to complex numbers!).

When rotating such a vector by 90° anti-clockwise, $[x,y]$ is turned into $[-y,x]$ . Thus, we can get our matrix $i$ by encoding this information, where

$\displaystyle i=\begin{bmatrix}0&-1\\1&0\end{bmatrix}.$

Interestingly, if we take the square of this $i$ , we see that

$\displaystyle i^2=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}0&-1\\1&0\end{bmatrix}=\begin{bmatrix}-1&0\\0&-1\end{bmatrix}=-1.$

This is the fundamental definition of $i$ that we are taught!

Now that we have a $1$ and an $i$ , our original complex number can be written as

$\displaystyle a+bi=\begin{bmatrix}a&-b\\b&a\end{bmatrix}.$

You might have learnt about the 2×2 rotation matrix before. The form is quite close to the numbers we made. By writing it as a complex number, we see that

$\displaystyle \begin{bmatrix}\cos\theta &-\sin\theta\\\sin\theta &\cos\theta\end{bmatrix}=\cos\theta + i\sin\theta,$

which we find to be the cis form that we see so often.

When multiplying the rotation matrix and a 2D vector, we get a vector which has been rotated by $\theta$ . We see that this is very similar to how cis $\theta$ works in how we multiply of complex numbers!

Finally, let’s have a brief look of the multiplicative inverse of the matrix

$\displaystyle a+bi=\begin{bmatrix}a&-b\\b&a\end{bmatrix}.$

The determinant of this matrix is $a^2+b^2$ which is the square of $|a+bi|$ ! We know that a matrix is only invertible if the determinant is non-zero.

The only way for $a^2 + b^2 = 0$ is for $a=b=0$ since $a$ and $b$ are real. This means that the only complex number that does not have a multiplicative inverse is 0, a key part in the axioms of a field!

This a nice confirmation that maths is amazingly interconnected! I’ll leave it to you to test and check the properties of complex numbers in these matrices.