Complex numbers

Some of my following posts might use complex numbers so I want to properly introduce them. To understand why we need them we will first take a look at other, more simple, sets of numbers.

In elementary school one is first introduced to the natural numbers 1,2,3,… which are fine for counting things but as soon as you start doing subtractions like 3-7 they stop giving any answers. This was easy enough to fix by introducing negative numbers (and zero if that was not part of the natural numbers already). But there is another problem: dividing 9 by 3 works fine but dividing it by 2 is impossible. A quick fix is the remainder (1 in the case of 9 divided by 2) but once you know fractions you might be comfortable leaving it as 9/2 or 4.5. Considering only the four basic operations those so called rational numbers (aka fractions) do not pose any immediate problems any more.

Many of you will know that the square root of 2 is irrational, it cannot be represented as a fraction. But we could see this before ever learning about square roots by thinking about equations, namely x^2 = x*x = 2. So the question is if there is any rational number that, multiplied by itself, gives two. If you are interested, I might give a proof that there isn’t some day. In the real numbers, those most often used, this can be solved.

The real numbers also give up after only a little change to the equation: x^2 = -1. At this point we have no choice but to make up some solution and we will call it i. This might seem to make little sense but if you consider the introduction above we actually did the same thing with negative numbers and fractions. There is no reason for -4 to be there except that we say so. 9/2 is no more real than i (except in the literal sense that it is a “real” number wile i isn’t).

If we want to make i part of our numbers we also have to allow for it to be multiplied by other numbers as well as added to them. Because of this anything written x + y * i with real numbers x and y has to be part of the complex numbers. It turns out that by this we did not only solve the problem of x^2 = -1 but actually every equation of the form z^n = w with fixed n and w has a solution z.

This is a wonderful showcase of the core concept of maths that we make up our own rules and then try to figure out what is possible within them and what isn’t – and if we are not satisfied we can always add or remove rules to obtain other results.