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)?
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 mean, they didn’t get.
Things stayed that way for quite a while. Various others, 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 next step was in Italy, over 1000 years later. During the 16th century, people were searching for solutions to the cubic equations – the same thing that the 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 they were *real*, *meaningful* numbers: it was viewed as a useful artifact of the process of solving the equations, but it wasn’t accepted.
It got its name as the *imaginary number* as a result of a diatribe by Rene Descartes, who believed it was a phony artifact of sloppy algebra. He did not accept that it had any meaning at all: thus it was an “imaginary” number.
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 bizzare 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.
eiθ = cos θ + i sin θ
And what *that* really means is:
eiπ = -1
That’s just astonishing. The fact that there is *such* a close relationship between i, π, and e is just shocking to me.
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. If you grabbed the wire when the voltage is supposedly zero on that graph, *you’d still get a shock*! Take the moment marked “t1” on the graph above. 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.
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).