My All 2020 Mathematics A to Z: Complex Numbers


Mr Wu, author of the Singapore Maths Tuition blog, suggested complex numbers for a theme. I wrote long ago a bit about what complex numbers are and how to work with them. But that hardly exhausts the subject, and I’m happy revisiting it.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Complex Numbers.

A throwaway joke somewhere in The Hitchhiker’s Guide To The Galaxy has Marvin The Paranoid Android grumble that he’s invented a square root for minus one. Marvin’s gone and rejiggered all of mathematics while waiting for something better to do. Nobody cares. It reminds us while Douglas Adams established much of a particular generation of nerd humor, he was not himself a nerd. The nerds who read The Hitchhiker’s Guide To The Galaxy obsessively know we already did that, centuries ago. Marvin’s creation was as novel as inventing “one-half”. (It may be that Adams knew, and intended Marvin working so hard on the already known as the joke.)

Anyone who’d read a pop mathematics blog like this likely knows the rough story of complex numbers in Western mathematics. The desire to find roots of polynomials. The discovery of formulas to find roots. Polynomials with numbers whose formulas demanded the square roots of negative numbers. And the discovery that sometimes, if you carried on as if the square root of a negative number made sense, the ugly terms vanished. And you got correct answers in the end. And, eventually, mathematicians relented. These things were unsettling enough to get unflattering names. To call a number “imaginary” may be more pejorative than even “negative”. It hints at the treatment of these numbers as falsework, never to be shown in the end. To call the sum of a “real” number and an “imaginary” “complex” is to warn. An expert might use these numbers only with care and deliberation. But we can count them as numbers.

I mentioned when writing about quaternions how when I learned of complex numbers I wanted to do the same trick again. My suspicion is many mathematicians do. The example of complex numbers teases us with the possibilites of other numbers. If we’ve defined \imath to be “a number that, squared, equals minus one”, what next? Could we define a \sqrt{\imath} ? How about a \log{\imath} ? Maybe something else? An arc-cosine of \imath ?

You can try any of these. They turn out to be redundant. The real numbers and \imath already let you describe any of those new numbers. You might have a flash of imagination: what if there were another number that, squared, equalled minus one, and that wasn’t equal to \imath ? Numbers that look like a + b\imath + c\jmath ? Here, and later on, a and b and c are some real numbers. b\imath means “multiply the real number b by whatever \imath is”, and we trust that this makes sense. There’s a similar setup for c and \jmath . And if you just try that, with a + b\imath + c\jmath , you get some interesting new mathematics. Then you get stuck on what the product of these two different square roots should be.

If you think of that. If all you think of is addition and subtraction and maybe multiplication by a real number? a + b\imath + c\jmath works fine. You only spot trouble if you happen to do multiplication. Granted, multiplication is to us not an exotic operation. Take that as a warning, though, of how trouble could develop. How do we know, say, that complex numbers are fine as long as you don’t try to take the log of the haversine of one of them, or some other obscurity? And that then they produce gibberish? Or worse, produce that most dread construct, a contradiction?

Here I am indebted to an essay that ten minutes ago I would have sworn was in one of the two books I still have out from the university library. I’m embarrassed to learn my error. It was about the philosophy of complex numbers and it gave me fresh perspectives. When the university library reopens for lending I will try to track back through my borrowing and find the original. I suspect, without confirming, that it may have been in Reuben Hersh’s What Is Mathematics, Really?.

The insight is that we can think of complex numbers in several ways. One fruitful way is to match complex numbers with points in a two-dimensional space. It’s common enough to pair, for example, the number 3 + 4\imath with the point at Cartesian coordinates (3, 4) . Mathematicians do this so often it can take a moment to remember that is just a convention. And there is a common matching between points in a Cartesian coordinate system and vectors. Chaining together matches like this can worry. Trust that we believe our matches are sound. Then we notice that adding two complex numbers does the same work as adding ordered coordinate pairs. If we trust that adding coordinate pairs makes sense, then we need to accept that adding complex numbers makes sense. Adding coordinate pairs is the same work as adding real numbers. It’s just a lot of them. So we’re lead to trust that if addition for real numbers works then addition for complex numbers does.

Multiplication looks like a mess. A different perspective helps us. A different way to look at where point are on the plane is to use polar coordinates. That is, the distance a point is from the origin, and the angle between the positive x-axis and the line segment connecting the origin to the point. In this format, multiplying two complex numbers is easy. Let the first complex number have polar coordinates (r_1, \theta_1) . Let the second have polar coordinates (r_2, \theta_2) . Their product, by the rules of complex numbers, is a point with polar coordinates (r_1\cdot r_2, \theta_1 + \theta_2) . These polar coordinates are real numbers again. If we trust addition and multiplication of real numbers, we can trust this for complex numbers.

If we’re confident in adding complex numbers, and confident in multiplying them, then … we’re in quite good shape. If we can add and multiply, we can do polynomials. And everything is polynomials.

We might feel suspicious yet. Going from complex numbers to points in space is calling on our geometric intuitions. That might be fooling ourselves. Can we find a different rationalization? The same result by several different lines of reasoning makes the result more believable. Is there a rationalization for complex numbers that never touches geometry?

We can. One approach is to use the mathematics of matrices. We can match the complex number a + b\imath to the sum of the matrices

a \left[\begin{tabular}{c c} 1 & 0 \\ 0 & 1 \end{tabular}\right] + b \left[\begin{tabular}{c c} 0 & 1 \\ -1 & 0  \end{tabular}\right]

Adding matrices is compelling. It’s the same work as adding ordered pairs of numbers. Multiplying matrices is tedious, though it’s not so bad for matrices this small. And it’s all done with real-number multiplication and addition. If we trust that the real numbers work, we can trust complex numbers do. If we can show that our new structure can be understood as a configuration of the old, we convince ourselves the new structure is meaningful.

The process by which we learn to trust them as numbers, guides us to learning how to trust any new mathematical structure. So here is a new thing that complex numbers can teach us, years after we have learned how to divide them. Do not attempt to divide complex numbers. That’s too much work.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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