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.

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