Motivation for complex numbers

I understand that the complex numbers have a geometric representation as ordered pairs of real numbers. From what I understand complex numbers are very useful for working in two dimensions mathematically. The fact that $i^2 = -1$ follows from the definition of multiplication of complex numbers:

$(a,b)(c,d) = ac-bd+(bc+ad)i$

My question is this: If one was simply working with ordered pairs of real numbers with no knowledge of $i=\sqrt{-1}$, would this definition of multiplication ever follow logically from a practically arising situation?

Solution 1:

Yes, at least in a way.

For the longest time the only "numbers" considered fully philosophically sound by the mathematical establishment were the positive integers. But by the time Caspar Wessel invented the complex plane, it was becoming respectable to consider numbers that would represent any geometric length at all -- that is, what we call real numbers today.

What is the connection between geometry and arithmetical multiplication of real numbers? "You multiply the sides of a rectangle to find its area," say modern school children. But back in the time, this was a rather iffy kind of connection, because lengths and areas are different kinds of thing, so from a geometry point of view this connection doesn't explain why multiplication produces a real number of the same kind as the factors.

A much more orthodox connection derives from the theory of proportions: $$ \frac{ab}{a} : \frac{b}{1} $$ or in words, $ab$ has the same ratio to $a$ that $b$ has to unity. In this view the definition of what we do when we multiply is to solve the equation $$ \frac{X}{a} = \frac{b}{1} $$ (in stark contrast to the more contemporary view where multiplication is fundamental and division or ratios are derived concepts).

If we seek a 2-dimensional analogue of ratios, what naturally shows up is similarity. In two similar triangles, matching sides are in proportion to each other.

Thus, enter Wessel: Once we have gotten the idea to try to treat "length and direction" as a kind of number, rather than just "length", we're faced with deciding what multiplication ought to mean: $$ \frac{?}{z} : \frac{w}{1} $$ The product $zw$ ought to be the lenght-and-direction whose relation to $z$ is similar to the relation $w$ has to the length-and-direction we have chosen as our unit.

When we work with length and direction, we get a triangle just by specifying two of its sides -- just like the vectors, it is left undetermined where in the plane the triangle is, but all its sides, angles, and directions are fixed once we know two neighboring sides. So let's try:

The product $zw$ is the length-and-direction such that the triangle with sides $zw$ and $z$ is similar to the triangle with sides $w$ and $1$.

or, perhaps slightly more understandably,

The product $zw$ is the point such that the triangle with corners at $0,zw,z$ is similar to the triangle with corners at $0,w,1$.

And what do you know -- if we add the additional requirement that the two triangles must not just be similar but also oriented the same way (they can be turned relative to each other, but not mirrored), then what this definition gives us is exactly multiplication of complex numbers!

It takes a bit of footwork to show that this gives the same result as "$ac-bd+(ad+bc)i$", but it does.

But is this practical? you ask. To Wessel, it certainly was. He wasn't even a mathematician by trade, but a surveyor and cartographer -- and his goal with the complex plane was to help organize and simplify geometric calculations he had to do in the cause of his work.

(Though note that, perhaps not coincidentally, surveying is the original, literal meaning of the Greek word "geometry"!)

In today's setting, all this is not very important -- in fact, in most of the cases where we want to treat points in the plane as numbers, our go-to tools are linear algebra, vectors and so forth. They have the same addition and scalar multiplication as complex numbers, but don't have an internal multiplication. Many of the applications Wessel and his followers had for multiplication for are instead dealt with by matrices instead.

To a modern mathematician, the chief property of $\mathbb C$ is that it is the algebraic closure of $\mathbb R$, and a lot of applications follow from that. Compared to this, it is just a scattered few places in mathematics where complex numbers are still a useful way to approach geometric problems. But they're there.