I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to have time to write while I’m away, I’m taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised.
After the amazing response to my post about zero, I thought I’d do one about something that’s fascinated me for a long time: the number i, the square root of -1. Where’d this strange thing come from? Is it real (not in the sense of real numbers, but in the sense of representing something real and meaningful)? What’s it good for?
The number i has its earliest roots in some of the work of early arabic mathematicians; the same people who really first understood the number 0. But they weren’t quite as good with i as they were with 0: they didn’t really get it. They had some concept of roots of a cubic equation, where sometimes the tricks for finding the roots of the equation just didn’t work. They knew there was something going on, some way that the equation needed to have roots, but just what that really meant, they didn’t get.
Things stayed that way for quite a while – people could see that there was something wrong, something missing in a variety of mathematical concepts, but no one understood how to solve the problem. What we know as algebra developed for many years, and various scholars, like the Greeks, encountered them in various ways when things didn’t work, but no one really grasped the idea that algebra required numbers that were more than just points on a one-dimensional number-line.
The first real step towards i was in Italy, over 1000 years after the Greeks. During the 16th century, people were searching for solutions to the cubic equations – the same thing that the early arabic scholars were looking at. But getting some of the solutions – even solutions to equations with real roots – required playing with the square root of -1 along the way. It was first really described by Rafael Bombelli in the context of the solutions to the cubic; but Bombello didn’t really think that i was something real or meaningful in the field of numbers; he just viewed it as a peculiar but useful artifact in the process of solving cubic equations. But beyond its use in finding tricks for solving cubic equations, it wasn’t accepted as an actual number.
It got its unfortunate misnomer, the “imaginary number” as a result of a diatribe by Rene Descartes. Descartes was disgusted by it; he believed it was a phony artifact of sloppy algebra. He did not accept that it had any meaning at all: thus he termed it an “imaginary” number, as part of an attempt to discredit the concept.
They finally came into wide acceptance as a result of the work of Euler in the 18th century. Euler was probably the first to really, fully comprehend the complex number system created by the existence of i. And working with that, he discovered one of the most fascinating and bizarre mathematical discoveries ever, known as Euler’s equation. I have no idea how many years it’s been since I was first exposed to this, and I still have a hard time wrapping my head around why it’s true.
And what that really means is:
That’s just astonishing. The fact that there is such a close relationship between i, π, and e is just shocking.
What i does
Once the reality of i as a number was accepted, mathematics was changed irrevocably. Instead of the numbers described by algebraic equations being points on a line, suddenly they become points *on a plane*. Numbers are really two dimensional; and just like the integer “1” is the unit distance on the axis of the “real” numbers, “i” is the unit distance on the axis of the “imaginary” numbers. As a result numbers in general become what we call complex: they have two components, defining their position relative to those two axes. We generally write them as “a + bi” where “a” is the real component, and “b” is the imaginary component.
The addition of i and the resulting addition of complex numbers is a wonderful thing mathematically. It means that every polynomial equation has roots; in particular, a polynomial equation in “x” with maximum exponent “n” will always have exactly “n” complex roots.
But that’s just an effect of what’s really going on. The real numbers are not closed algebraically under multiplication and addition. With the addition of i, multiplicative algebra becomes closed: every operation, every expression in algebra becomes meaningful: nothing escapes the system of the complex numbers.
Of course, it’s not all wonderful joy and happiness once we go from real to complex. Complex numbers aren’t ordered. There is no < comparison for complex numbers. The ability to do meaningful inequalities evaporates when complex numbers enter the system in a real way.
What i means
But what do complex numbers mean in the real world? Do they really represent real phenomena? Or are they just a mathematical abstraction?
They’re very real. There’s one standard example that everyone uses: and the reason that we all use it is because it’s such a perfect example. Take the electrical outlet that’s powering your computer. It’s providing alternating current. What does that mean?
Well, the voltage – which (to oversimplify) can be viewed as the amount of force pushing the current – is complex. In fact, if you’ve got a voltage of 110 volts AC at 60 hz (the standard in the US), what that means is that the voltage is a number of magnitude “110”. If you were to plot the “real” voltage on a graph with time on the X axis and voltage of the Y, you’d see a sine wave:
But that’s not really accurate. That implies that the current is basically shutting on and off really quickly! But it’s not – there’s a certain amount of energy in the alternating current, and the amount of energy is actually constant over time. But it’s a system in motion: it’s constantly changing. The voltage at time t1 on the complex plane is a point at “110” on the real axis. At time t2, the voltage on the “real” axis is zero – but on the imagine axis it’s 110. In fact, the magnitude of the voltage is constant: it’s always 110 volts. But the vector representing that voltage is rotating through the complex plane. When it’s rotated entirely into the imaginary plane, the energy is expressed completely as a magnetic field. This is really typical of how i applies in the real world: it’s a critical part of fundamental relationships in dynamic systems with related but orthogonal aspects, where it often represents a kind of rotation or projection of a moving system in an additional dimension.
You also see it in the Fourier transform: when we analyze sound using a computer, one of the tricks we use is decomposing a complex waveform (like a human voice speaking) into a collection of basic sine waves, where the sine waves added up equal the wave at a given point in time. The process by which we do that decomposition is intimately tied with complex numbers: the fourier transform, and all of the analyses and transformations built on it are dependent on the reality of complex numbers (and in particular on the magnificent Euler’s equation up above).