Most people are familiar with the idea that numbers can be negative or positive, whole or fractional. But what most people are not aware of is that this is not the end of the story. There are other, more advanced kinds of numbers too. In particular, complex numbers, the subject of this article.

Review of number types

Let us just take a moment to review the number types we already know about.

When human kind first started counting things, they probably used only positive whole numbers. (What are sometimes called the “natural numbers”.) And that’s just fine.

You can also perform addition on natural numbers. For example, if I have 3 pebbles in a bag, and I add 2 additional peddles, how many pebbles are in the bag now? 3 + 2 = 5.

So that’s just fine. But things get moderately more complicated if you try to perform the reverse of addition — subtraction. For as we all know, 5 − 3 = 2. But what is 3 − 5?

That’s like asking “if I have 3 pebbles in this bag and I take away 5 pebbles, how many pebbles are left?” Well, if the bag contains 3 pebbles, how are you going to remove 5 pebbles from it? That doesn’t make sense.

Today, we understand that the answers to questions like “3 − 5″ are negative numbers. Now for some things, a negative number is not a sensible answer. (How can a bag contain a negative number of pebbles?) But for some other things, negative numbers are a perfectly valid answer — and indeed, they’re very useful. (Probably the most obvious example is credit and debt…)

Notice that a negative number such as −2 is really just a shorthand for 0 − 2. Starting from 3 − 5, we simply “cancel down” until one of the numbers is a zero. So we have

• 5 − 3
• 4 − 2
• 3 − 1
• 2 − 0

telling us that 5 − 3 = 2, and we have

• 3 − 5
• 2 − 4
• 1 − 3
• 0 − 2

telling us that 3 − 5 = −2.

A similar story happens with multiplication and division. Multiplication causes no problems, but division requires us to invent a new kind of number: fractions. And again, for some things fractional answers don’t make any sense, but for another things they do make sense and they’ve very useful. And, yet again, a fraction is just a division that’s been “cancelled down” in a certain way.

More operations

The next operation in the mathematician’s arsenal is exponentiation. In particular, to square a number simply means to multiply it by itself. So 7 squared is 7 × 7 = 49.

As before, this presents no problems. It’s when you try to do the reverse operation that problems arise. The reverse of a square is a square root. The square root of 49, written √49, is 7. No problem there.

In fact, actually, it turns out that (−7) × (−7) = 49 as well, so the square root of 49 is also −7. In general, all numbers thus have two square roots, one negative and one positive. (Except zero; the square root of zero is just zero.)

The problem

The problem is, what on earth is the square root of minus 49? 7 × 7 = 49, (−7) × (−7) = 49, so ? × ? = −49.

The problem is that whenever you multiply negative by negative, you get positive. And when you multiply positive by positive, you get positive. So what the heck do you multiply by itself to get negative??

The answer

Clearly, the answer here is to invent some new kind of number. At first mathematicians didn’t believe that these new numbers really “existed”, so they became known as imaginary numbers (and all the other numbers we talked about were called real numbers). Those names have kind of “stuck” now, even though today everybody agrees that “imaginary” numbers are no less real than the “real” numbers…

The idea is quite simple. We use the symbol “i” (for “imaginary”) to stand for the square root of −1. So i = √−1, or i × i = −1. Now we can say that √−49 = 7i (which is really mathematical shorthand for “7 × i”). It goes something like this:

• (7 × i) × (7 × i)
• = 7 × 7 × i × i
• = 49 × −1
• = −49

Actually, by almost the same reasoning, we also have √−49 = −7i. So, once again, each number has two square roots.

7 is a real number, and 7i is an imaginary number. Basically, for every real number that exists, there is a single corresponding imaginary number, that’s just the real number multiplied by this magical “i” quantity.

A new problem

So we’re done, right?

Well, apparently not. You see, it turns out that since multiplying any two imaginary numbers gives a real number… well, what’s the square root of an imaginary number?

Oops. We appear to be back where we started! Now, we could write √i = j, and then we would have √144i = 12j. But then, what would the square root of a j-number be? We could invent k-numbers, but… where will this end?

A new solution

We do need a new kind of number, but it turns out not to be j-numbers. Actually we can do better. And the answer comes from another little problem.

If you add a real number to a real number, you get a real number. (E.g., 3 + 2 = 5.) If you add an imaginary number to an imaginary number, you get an imaginary number. (E.g., 3i + 2i = 5i.) But if you add a real number to an imaginary number… you get a quantity that is neither real nor imaginary. (E.g., 5 + 7i.) These are called complex numbers. (That’s “complex” as in “shopping complex”, not “complex” as in “complicated”!)

For every real number, there is an imaginary number. But that’s not quite true, since 0i = 0. So 0 is both real and imaginary. (Or neither, if you prefer.) If you imagine the real numbers as one line, and the imaginary numbers as another line, these two lines cross at 0. Rather like… the axes of a graph.

So a complex number is, in fact, a kind of 2D coordinate. By convention, the real part is the X-coordinate (left/right direction) and the imaginary part is the Y-coordinate (up/down direction).

Now, by performing a little basic algebra, we arrive at the following set of rules for calculating with complex numbers:

• (a + bi) + (c + di) = (a + c) + (d + b)i
• (a + bi) − (c + di) = (a − c) + (d − b)i
• (a + bi) × (c + di) = ((a × c) − (b × d)) + ((a × d) + (b × c))i

In particular, it turns out that, for example, √2i = 1 + 1i. Why is this? Well,

• (1 + 1i) × (1 + 1i)
• = ((1 × 1) − (1 × 1)) + ((1 × 1) + (1 × 1))i
• = (1 − 1) + (1 + 1)i
• = 0 + 2i

Indeed, it turns out that every complex number has two square roots (except 0, which has only one).

So, finally, with complex numbers, we can add, subtract, multiply, divide, square and square root all values. Indeed, it turns out that with complex numbers, we can now answer all sorts of other questions, like “what is the logarithm of −1?” or “what is the arcsine of 12?”