Complex numbers

Every time a calculation did not work out, the set of numbers was extended. $3 - 5$ had no answer until negative numbers existed. $1 : 3$ had none until fractions existed. $\sqrt{2}$ had none until irrational numbers existed. Now $\sqrt{-1}$ is on the table, and the trick is the same as always: give the missing answer a name and see whether you can calculate with it sensibly.

The number i

You define $i^2 = -1$. So $i$ is the number whose square is $-1$, and it is called the imaginary unit. The name is historical and really rather unfair, because $i$ is just as usable as any other number. With it, every root of a negative number can be written down: $\sqrt{-9} = 3i$, because $(3i)^2 = 9 \cdot i^2 = -9$.

Two parts, one number

A complex number consists of an ordinary part and a multiple of $i$, for example $z = 3 + 4i$. The 3 is called the real part, the 4 the imaginary part. Together they form a single number, just as 3.5 is a single number and not two. The set of all these numbers is called $\mathbb{C}$, and the real numbers sit inside it completely: they are simply the complex numbers with imaginary part 0.

You calculate as if $i$ were a variable, and at the very end you replace every $i^2$ by $-1$. So adding means: real parts together, imaginary parts together. $(3 + 4i) + (1 - 2i) = 4 + 2i$. Multiplying works by expanding brackets as usual: $(3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i^2$. The last two steps: $-6i + 4i = -2i$, and $-8i^2 = +8$. What remains is $11 - 2i$.

The number plane

Real numbers live on a number line. For complex numbers a line is not enough, because they carry two pieces of information. So you take a plane: the real part goes to the right, the imaginary part upwards. This picture is called the complex plane, and every complex number is a point in it. The distance from the origin is called the modulus and is worked out with Pythagoras: $|3 + 4i| = \sqrt{3^2 + 4^2} = 5$.

What this is good for

With complex numbers every quadratic equation has a solution. $x^2 + 1 = 0$ has none over the reals, but two over the complex numbers: $x = i$ and $x = -i$. When the discriminant is negative, you simply take the root of its size and write an $i$ next to it. This is not a useless trick: alternating current, signal processing and quantum physics all calculate with complex numbers, because oscillations are far easier to describe that way.