Complex Numbers

A complex number is simply an ordinary number plus some amount of one new number, $i$, whose square is $-1$. "Complex" here means "made of parts," not "complicated": $a + bi$ has two simple parts, $a$ (the ordinary, real part) and $bi$ (some amount of $i$). The multiplication formula looks scary, but it's just FOIL with the rule $i^2 = -1$ applied at the end, and $|a + bi|$ is just the distance from $0$, found with the Pythagorean theorem. Engineers use them constantly for anything that rotates or oscillates, so it's less mysterious than the word "imaginary" suggests.

Why invent them? The equation $x^2 + 1 = 0$ has no real solution, since no real number squares to $-1$. So we name one, $i$, and ask what happens if we just keep doing algebra. The answer is that nothing breaks, and now every quadratic has roots: $x^2 + 1 = 0$ has $\pm i$.

Adding. Treat $i$ like a variable: add real parts and imaginary parts separately, the way you'd combine like terms.

Multiplying. FOIL, exactly as with polynomials, then replace $i^2$ with $-1$. That's the only new step.

Dividing. This is the same idea as rationalizing a denominator. Multiply top and bottom by the conjugate $\overline{c + di} = c - di$, because $(c + di)(c - di) = c^2 + d^2$ is real. It's the difference-of-squares pattern again, with $i^2 = -1$ turning the minus into a plus.

Powers of $i$ cycle with period four: $i, -1, -i, 1$, and then back to $i$. To find $i^{23}$, all you need is the remainder of $23$ divided by $4$.

Multiplication is rotation. Plot $a + bi$ as the point $(a, b)$. Multiplying by $i$ turns any point a quarter turn about the origin. Do it twice and you've made a half turn, which sends $1$ to $-1$, and that's a nice way to see why $i^2 = -1$. In general, multiplying by a complex number multiplies lengths and adds angles. There's a video further down that shows this.

Electrical engineers write AC voltages and impedances as complex numbers; signal processing, quantum mechanics and the 2D rotations in graphics all use them.