### The first shoe: Multiplying

Adding complex numbers is a straightforward task. Given two numbers, \(a + bi\) and \(c + di\), the sum is the sum of the real portion and the sum of the imaginary portion: \((a + c) + (b + d)i\). When working with the standard form, multiplication (and division) is a trickier matter: \[(a + bi)(c + di) = (ac – bd) + (bc + ad)i \\ \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2}\]

The mathematics is straightforward enough, and certainly nothing that can’t be handled by someone dealing with complex numbers in the first place. And this is how I recall learning it: As a mechanical process, with no clear connection to what the numbers actually mean.

Just as the real numbers can be represented with a one-dimensional number line, the complex numbers can be represented with a two-dimensional number plane. And just as adding on the number line means moving the specified distance along the line, adding on the number plane means moving the specified distance. In terms of the plane, we can see a complex number \(a + bi\) as a vector with the length \(\sqrt{a^2 + b^2}\) and a specific rotation from the positive x-axis ray (which I shall call the “x-ray”).

For instance, these numbers are all five units from the origin: {5, 4 + 3i, 3 + 4i, 5i, -3 + 4i, -4 + 3i, -5, -4 – 3i, -3 – 4i, -5i, 3 – 4i, 4 – 3i}. They each represent a different rotation: {0°, tan^{-1} 4/3, tan^{-1} 3/4, 90°, tan^{-1} -3/4, …}

We tend to talk about the number line, and we tend to talk about the positive reals as if they’re the same as the absolute reals, but they’re not. The absolute values are the distance from the origin, regardless of direction. I think it would be more proper to think of the complex number plane as being the third step in the expansion, not the second: First we have the unsigned real number ray, then we have the real number line, then we have the complex number plane. Another way to look at it: The real number line is a special case of the number plane.

When adding numbers, we can quickly determine the new distance, but not the new rotation. We’re really adding two distances: One along the real number (x) axis and one along the imaginary number (y) axis. Unless the two numbers being added are along the same vector from the origin, the rotation won’t be the same.

If you multiply two real numbers, you wind up moving along the x-axis. Multiply a positive real number and another (positive or negative) number, and your movement is fairly predictable. Multiply two negatives, and for reasons not immediately obvious, you wind up with a positive. But you still stay on the number line.

If you multiply two complex numbers, the movement isn’t as obvious, particularly when using the standard (x, y) method of writing complex numbers.

But when we convert complex numbers to polar coordinates, the effect of multiplication becomes much more patterned.

Take \((1 + \sqrt{3}i)(\sqrt{2} – \sqrt{2}i) = \sqrt{2} + \sqrt{6} + (\sqrt{6} – \sqrt{2})i\). Not particularly intuitive. Let’s consider these in terms of polar co-ordinates instead. \(1 + \sqrt{3}i\) represents a point 2 units from the origin, with a rotation of 60° from the x-ray. \(\sqrt{2} – \sqrt{2}i\) is a point 2 units from the origin, with a rotation of -45° from the x-ray. \(\sqrt{2} + \sqrt{6} + (\sqrt{6} – \sqrt{2})i\) is 4 units from the origin, with a rotation of 15°.

What just happened?

If we represent our points in polar coordinates, where (r, θ) represents the distance and rotation, we have (2, 60°)(2, -45°) = (4, 15°). Much cleaner than the multiplication with the standard representation.

To multiply complex numbers: Multiply the distances, add the rotations. That’s it.

Now we have a clean explanation for why a negative times a negative is a positive: Negative numbers represent a half-rotation around the complex plane. Multiply a positive and a negative, and you’re adding a half-rotation to a non-rotation. Multiply a negative and a negative, and that’s two half-rotation, or a full rotation.

Now we have a clean explanation for why every number except 0 has n nth roots. Let’s take the complex number (a, 20°) (where a a distance greater than 0). How many fourth roots could it have? 20/4 = 5, so there’s a root at 5°. 380°/4 = 95°, so there’s another one. There are two more at 185° and 275°. In general, given a complex number with a rotation of θ, there are nth roots at (2kπ + θ)/n, where k is a positive integer and the angle is in radians. These will begin repeating existing rotations when k = n (so (2kπ + θ)/n = θ/n + 2π).

Zero is a degenerate case: We could still argue that 0 has n nth roots, but because we don’t care about rotation with no distance, they’re irrelevant.

For the standard student of mathematics, I don’t see the utility of actually doing multiplication and division this way, particularly since it doesn’t help with addition. Having one system for adding and another system for multiplication seems cumbersome. However, I think seeing that multiplication works so elegantly this way helps explain what multiplication is.

### The second shoe: Bridging arithmetic and geometry

The first two operations, and arguably the two “core” operations, in arithmetic are addition and multiplication.

The first two measurements, and arguably the two “core” measurements, in geometry are distance and rotation. I’ve argued previously that rotation operates on a zero-powered unit; distance is a one-powered unit, while area and volume are two- and three-powered, respectively.

Addition is combining distances. We’re either moving closer to or farther from some arbitrary origin. Rotation is a sidebar: When dealing with complex numbers, we add the real part and we add the imaginary part, and let the rotation be what it may be.

Multiplication is conceptually more difficult: We’re still moving with relation to the origin. We move closer by multiplying by a number less than one; we move farther away by multiplying by a number more than one. With addition, we’re moving an absolute amount: 4 + 3 and 7 + 3 move us the same amount away from the origin. With multiplication, we’re moving by a scale: 4 * 3 involves less a movement than 7 * 3 does, but in both cases we’re moving by a factor of 3. There are times when we want to move an absolute amount; there are times when we want our movement to be based on a scale factor. These are different, equally important concepts.

On the one hand, it feels like the analogue to multiplication within geometry is area. And for some purposes, this is true. It’s certainly the easiest way to model multiplication of non-integers. Multiplication of integers is usually first taught as repeated addition (much to Keith Devlin’s chagrin). This model doesn’t work so well with, say, 4.5 * 3.2. What on Earth is 3.2 piles?

Modeling multiplication with area works well enough to explain the concept: 4.5 * 3.2 is the area of a rectangle with a length of 4.5 and a height of 3.2. We can have students draw a rectangle, see that there’s a strip that sits outside the 4×3 rectangle that can be calculated to have an area of 2.4, and there you go.

Okay, so what’s 5.2 * -1.2? How do we have a rectangle that has a side length of -1.2?

And how about 5.2 * 6.1 * -3.1 * -1.2? Is it the hypervolume of a rectangular hyperprism with two negative side lengths? Visualize that.

Instead, if we can visualize 5.2 * 6.1 * -3.1 * -1.2 as the repeated scaling of 5.2 by factors of 6.1, 3.1, and 1.2, rotated twice around the origin (for one full rotation)… that’s much harder to get to, certainly, but for the student that gets there, that’s what multiplication really is, as an abstract mathematical concept: Scaling and rotating.

I cannot stress enough that I’m not recommending that this complex number model be used for beginning students. But, at the same time, when do we admit that the models we use are models, not the actual concept? Historically, perhaps, multiplication was repeated addition. I have also seen it argued that multiplication was actually originally used to calculate area, and it was later noticed that multiplication and repeated addition happened to yield the same results.

Either way, though, neither model works for how we use multiplication. Multiplication is scaling; addition is fixed movement.

Looking at it from that perspective, informed by the complex number explanation but simplifying it: Multiplication by a negative number results in a scaling of the same degree as multiplying by a positive, but it flips over the origin. If we multiply by five, we make our original length five times as long; if we multiply by negative five, we still make our original length five times as long, but we reflect it over the origin. Hence, multiplying a negative by a positive results in a negative (either because the original was already negative, and just gets stretched, or because the original positive was flipped); multiplying two negatives results in a positive.

I started with some mathematics that would be at the edge or beyond for the typical high school senior. I’ll end with something more basic, as a summation: Addition and multiplication are inherently different operations that happen to produce compatible results when integers are involved. Addition involves movement towards or away from the origin in a fixed amount. Multiplication involves scaling a value by a factor. On the complex plane, this difference produces an interesting result: The two operations are different enough that the ideal representation for adding complex numbers is different than the ideal representation for multiplying them.