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 and and these are then scaled by the real numbers and . If can we find a 2×2 matrix that does the job of and another that does the job of , then we can ultimately find as a 2×2 matrix.
Thus, to find a suitable , we think about the main property of , which is that anything you multiply by 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 . We write
Similarly, to find a suitable , we want a matrix that rotates 2D vectors by 90° anti-clockwise (this is what does to complex numbers!).
When rotating such a vector by 90° anti-clockwise, is turned into . Thus, we can get our matrix by encoding this information, where
Interestingly, if we take the square of this , we see that
This is the fundamental definition of that we are taught!
Now that we have a and an , our original complex number can be written as
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
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 . We see that this is very similar to how cis works in how we multiply of complex numbers!
Finally, let’s have a brief look of the multiplicative inverse of the matrix
The determinant of this matrix is which is the square of ! We know that a matrix is only invertible if the determinant is non-zero.
The only way for is for since and 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.