First of all, unless you are already quite familiar with complex magnitudes, you must watch the following video, which covers the content presented here. This page will be helpful for clearing up any other uncertainties. You can jump ahead to time 7:10 to get to the beginning of the specific pedagogy.
The interactive animation used is available here.
First, we'll look at simple numbers. Ordinarily, when numbers aren't equal, we can say that one is lesser than the other. In the following examples, even if it is tricky, it is possible to say that A is greater than B, equal to B, or lesser than B:
The last, sixth image was very tough, since the lines really are the same length, with the only difference being rotation. They certainly aren't the same, but now "more" and "less" aren't adequate words for describing their relationships to each other. What if numbers didn't exist only as lengths on a ruler, but as motions on a plane? Such numbers are no longer simple, but complex.
Working with Complex Numbers
We can add these complex numbers, these motions, by simply adding up the motions and seeing where you wind up in the end. As you see in this short video, adding a and b is as simple as performing the motion of b while starting at a. We will then have performed both actions, giving us a+b. Addition works in either order, so if we perform the action of a while starting at b, we'll get the same result, drawn in here as the diagonal of the parallelogram:
We can also multiply them. Multiplying is harder than adding, since, as you saw in the first video, you must have a "one" before multiplying makes sense. Multiplication is like a combined comparison. Take as an example, 2 × 3. Instead of 2 and 3 as specific lengths, think of them as changes, in relation to 1. That is, rather than 2 inches, think of the process of doubling one inch. Similarly, 3 is the act of tripling, not the number three itself. When we multiply two numbers, they are verbs rather than nouns. So to do 2 × 3, think of what the action of doubling would do to 3. We say that the relation between 2 and 1 is the same as that between 6 and 3. Vice versa, 3 stands to 1 as 6 stands to 2, so we could have either 1:2::3:6 or 1:3::2:6.
When we use complex magnitudes rather than simple numbers, we’ll continue to think of them not as magnitudes in themselves, but in relation to the original unit one. In this video, you'll see two magnitudes a and b in comparison to the unit 1. To multiply a×b, we'll replicate the action from 1 to a, but starting at b, giving us the product a·b. Similar triangles show the relation from 1 to a being the same as that between b to a·b.
Squaring in particular, gives us an interesting magnitude. If motion to the right is 1, and we call motion upwards by the roman numeral i, let's ask what we get when we square i. That is, if we take the change from 1 to i, and then apply that change to i itself, what do we get?
As we see, i2 equals motion to the left, which is -1. So i could be called the square root of -1, just as 2 is the root of a square whose area is 4. Tripling twice gives nine, so tripling is half of multiplying by nine — the square root of nine — since the area of a square is multiplied in two dimensions when you increase its side. Similarly, i-ing twice gives -1, so it is half of -1-ing, or the square root of -1.
When we look at squaring in general, we'll find something interesting about the space that it implicitly occurs in.
Before you go on to the next page, it is required that you have a sense of multiplying complex numbers. The video at the top of the page provides the background. If you've seen it, and would like another refresher, watch the following 12-minute video (the Q+A from that video) before proceeding.