Students often meet mathematics with skepticism, some even to the point where they might declare “This is all made up!”. This doesn’t ring more true for the students who hear their teachers mention so called ‘imaginary numbers.’ It is often only as a sly tangential remark and never something that receives serious treatment in the high school classroom. However, imaginary numbers are real after all — and they have a more immense impact on your life than you might think.
The history of imaginary numbers — which mathematicians normally refer to as complex numbers — starts in the same context you might have encountered them: algebra class.
You might recall being given a polynomial like y=x² + x -2 with instructions asking you to find its roots: when the equation equals zero. For this example, the answer is 1 and -2.
But what about a polynomial like y=x²+1? Drawing a quick picture below, we see that the polynomial never crosses the x-axis, i.e. it never equals zero. You might think this is where the story would stop. This particular equation never equals zero, so what? Well mathematicians were not satisfied with that answer, and hence complex numbers where born.
We do this by expanding the rules for square roots, to include the square roots of negative numbers. In particular, letting i be equal to the square root of negative one. With this new addition we see that when x = i (or -i) then y = x²+1 is in fact zero.
What’s more astonishing is that with our new invention, i, all polynomials with real (or complex) coefficients have the same number of roots as their degree. This means all quadratic equations, like those we have seen here, have two roots, cubic equations have three, and so on.
Though i is a particular complex number we can easily create infinitely many more complex numbers. Anything of the form x+iy, where x and y are normal real numbers, is a complex number. The complex numbers are just as vast as the real numbers we are more familiar with. But the complex numbers have an additional neat property the real numbers don’t have: they are what mathematicians call an algebraically closed field.
It isn’t important to fully unpack this definition, but in essence it means exactly what we said before: all polynomials with real (or complex) coefficients have the same number of (complex) roots as their degree.
You may now be thinking, this is all well and fine, but I certainly don’t see why I should care, or even why this shouldn’t further convince me that math is something we have made up to torment students.
To do this, I will show you how complex numbers are useful, beautiful, and seemingly connected to the fabric of our very universe.
First, we start with the useful. Complex numbers pop up all throughout engineering and physics, but in one place they are absolutely crucial: modern communication.
The fact that we can instantaneously communicate by video and voice with someone half-way around the world is astonishing. Sending a signal around the world whether by deep-sea fiber optic cable or satellite relay leaves lots of room for messages to be garbled up and distorted. Complex numbers are used in signal processing techniques to make sure your video gets to your friend crystal clear. Similar methods are used to remove background noise when you are calling your friend from a busy coffee shop.
Now, onto the beautiful. I present to you the Mandelbrot Set. The Mandelbrot Set is a very specific grouping of complex numbers with a fascinating property.
You can zoom into the Mandelbrot set forever and you will keep seeing the same structures ad infinitum. Structures with this property are called fractals. Fractals give way to an entire family of beautiful visualizations of this sort.
Finally, though their invention (or discovery) seems largely happenstance and arbitrary, complex numbers appear everywhere we look in the universe. The formulation of quantum mechanics, which seeks to describe the behavior of the universe at the smallest scales, uses complex numbers at its very foundation. Whether we like to think of them as imaginary or not, there is no escaping imaginary numbers — they just might be baked into the universe itself.
