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Getting down to brass tacks, complex numbers -- represented by i -- are real numbers, but they're not real numbers. Got it? ... No? Well, let's try something else.

First of all, let's see what the deal is with Real numbers -- i.e., with really real numbers.  You may know that real numbers -- capital r, or Real, numbers -- are the rational and irrational numbers.

Oi vei! Out of the frying pan, into the fire!

It's hard to imagine numbers taking up room -- because they aren't objects. But if you think of them as filling up a space, you might see that Real numbers form a dimension line of a graph. In other words, think of numbers as models -- concepts. They are one-dimensional and express distance. That's their measure. It takes too long to write down more than just a few Real numbers, and I could never write all of them, anyway. But you can imagine adding more and more until the cows come home, and not make a dent in the total.

You might notice, too, that the irrationals, like the square root of two, or five times the square root of 3, can not be written as ratios of integers. And there are more irrationals than rationals, by a mile! In fact, the rationals don't take up any distance at all. Literally, if you first put in all the rationals -- and I do mean all of them, if you could -- they still wouldn't take up any room on a line. The line would be empty. And I do mean empty. But that's more or less what you might expect of numbers, since they're abstract, and not solid objects. Strictly speaking, they are forms, or concepts, or models.

On the other hand -- and here's the really odd thing -- irrationals do take up space (i.e., line). If you pull all the rational numbers out of the line, the line would still be full! It's as if nothing had happened. In other words, the irrationals, alone, fill the entire line. So irrationals don't act like abstractions at all; there are so many of them, they actually take up space -- or line!

But if the real numbers, alone, fill the line, what room is left for imaginary numbers? Do we just imagine they're in there?

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Let's start this business without preamble. We're talking about an imaginary number, called i. (Some people use j, but that's just convention!) Yes, plain old i. That's the name, and it stands for the square root of minus one!  Yikes!

So we have:

i = Ö -1

This is the way the number was introduced in 1777 by Leonhard Euler, one of the brightest stars in the galaxy of math stars, though I understand another bright star thought of it earlier in history, if that's important to you.

Why in the world would a smart guy like Euler get involved with such a thing as i? Especially considering that the square of any real-life number is positive, not negative. (For instance, -2 times -2 is +4.) Yet in the case of i it's negative. That is, (Ö -1) times (Ö -1) is -1.

Ah! Could the use of this new model -- i is a model, you see -- be one of the reasons we think Euler was so smart?

It happens that in Euler's time the quadratic equation (pardon the algebra) written as:

x² + 1 = 0

showed up as a puzzler -- just when math guys were getting used to the idea that all quadratic equations should have two solutions. They wanted it so bad they could taste it!

Let me go on a tangent here to give you a quick review of quadratic equation solving. First, a quadratic equation is a form with expressions like x². This says a number is raised to the second power -- like 3 to the second power is 9. Quadratic equations have second degree expressions like that. If they don't have, ... well, they're just not quadratic. They may or may not have plain old first degree terms, like 7y. But they definitely can not have terms of degree higher than 2, like x⁴ or z³. In that event they would still be polynomials, as it is said, but not quadratics. I hope that makes sense.

To solve the equations is to find numerical values of the variables of the quadratic that satisfy the equations -- i.e., that make the quadratic statement true. It had already been shown that all quadratic equations have two solutions. But this was before someone uncovered the silly quadratic equation:

x² + 1 = 0,

which must have been found by some radical!

This quadratic equation had the audacity -- I might say even the effrontery -- not only not to have two real solutions, but not even to have one! There is no real number you can plug into the equation for x that makes the equation true. In other words, when taken over the set of Real numbers, the equation has no solution. Whatever x is, it has to be a number whose square is -1, because x² + 1 goes to zero only if x² goes to -1, and Real numbers don't allow this. All squares are positive.

The problem this equation created is that it destroyed the artistic harmony of mathematicians who were pleased to know all quadratic equations had two solutions. Yet, here comes this stupid little equation with absolutely no solution! Well, that just wasn't tolerable!

So along came Euler and showed us a way out of the dilemma. In so doing, Euler invented a new kind of number (used someone else's old number?). He enlarged the solution space for equations -- he let more members into the club. We call the number a complex number. And with it we get a whole new complex number system! It extends the range of numbers and thus our solution space, so our equation has the desired two solutions, though they're not real, only imaginary.

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What's so intriguing about Euler's insight is that it was a new way of looking at numbers. Indeed, it set off a new way of modeling the world. Just as the jump in thought from natural numbers to integers, or from integers to rational numbers, or from rationals to Real numbers, stirred great interest in, and added new meaning to integers and rational numbers, so too the jump to complex numbers gave new meaning to the Reals. They were seen in a new light. They provided a new framework for numbers and new information, if you can imagine!

With complex numbers, we no longer need to think in terms of a line of numbers. Instead, numbers have two-dimensional quality. Rather than points on a line, numbers are points on a plane -- on a flat, two-dimensional surface. Indeed, you can extend the range of complexity to more than two dimensions, and distinguish other orders of the imaginary, but we won't get into that -- the step from one to two is the big move.

The form of two-dimensional complex number has two parts: a real part and an imaginary part, defined as follows:

ax + byi

or just

x + yi.

The Real numbers are now a sub-set of the complex numbers. That is, Reals make up part of the whole class of complex numbers. This is the case when the coefficient, b, is zero (or when y is zero), because then the complex number reduces to the real number, ax -- or just plain x.

On the coordinate system of its 2-space, the real part, x, is presented on one coordinate -- usually the horizontal coordinate -- and the imaginary part, yi, is given on the other -- the vertical coordinate. Now we have a real and an imaginary dimension, and our equation:

x² + 1 = 0

has two solutions. These solutions are +i and -i.

Let's test them.

First, say x is +i. In that case, x² = (+i)² = (+Ö -1)² = (Ö -1)² = -1. So x² + 1 = 0 is true.

Now say x is -i. Then x² = (-i)² = (-Ö -1)² = (-1)(-1)(Ö -1)² = -1. So again x² + 1 = 0 is true.

We now have two solutions, and harmony once more reigns in the world.

That is, all is harmonious for the guys who lost sleep because they weren't happy to have a quadratic equation with no solutions. But now there's another bunch of insomniacs. For this bunch the invention isn't such a happy one, because these guys don't especially like i. They don't think i is a real number.

Oh well, I guess you can't please everyone. That's for real!

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What, indeed, is i? They say it's a number, but it's a strange bird for a number, wouldn't you think? Even so, if it walks like a number and talks like a number, maybe that's what it is. So let's see how it walks and talks.

First of all, though, how are numbers supposed to behave? That is, what do numbers have to be able to do to be numbers? What is their dynamics?

Well, mainly, don't numbers just add and subtract, multiply and divide? If that's what numbers do, then imaginary numbers, if that's what they are -- numbers, I mean -- should do those things too. Let's see.

We know i is the square root of -1, or Ö -1. And we know we can multiply an imaginary number, i, by a real number, a, and get another imaginary number, ai. The same is true for dividing by a real number, giving us i/a. And finally we can add or subtract a real number to an imaginary to get still another imaginary, (a + bi) or (-a + bi). So our imaginary number has the general form, a + bi, where a and b are any real numbers.

Given the imaginary form, we ought to be able to have it do all the things that numbers do: add, subtract, multiply, and divide. So take any two imaginaries, (a + bi) and (c + di), and perform the operations. We get the following results for addition, subtraction, and multiplication:

This should be pretty convincing that the imaginary number, (a + bi), is indeed a number. What walks like a duck and talks like a duck ... .

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Curiously, the complex number model can be used in a significant way as an aid in solving difficult problems by representing and computing two variables as one. The technique is to let the non-interacting real and imaginary components of the complex number be holding places for different problem variables and perform a single simultaneous complex computation for them. This was pointed out to me by Anand Shirahatti (anand_gs@usa.net), who referred me to the book, The Scientist and Engineer's Guide to Digital Signal Processing, written by Steven W Smith.

Here's what Smith has to say:

The approach treats one of the variables as the real part of a complex number, and the other variable as the imaginary part, and manipulates them as one variable. When the computation is completed, the two components of the complex number are separated into the real and imaginary parts, which again correspond to the original variables. Very clever!

You might also like to see how i is used in quantum mechanics. That, too, is very clever.

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