complex numbers

Counting numbers answer one question: how many? That works until someone asks "three take away five?", and we invent numbers below zero. That works until "split three loaves between two people?", and we invent fractions.

Every new kind of number exists because the old ones couldn't answer something. So what question forced complex numbers into existence?

Turning.

a line can't turn

A line can only go forwards and backwards. But the world spins. Wheels, waves, orbits, the current in the wall. So we swap the line for a flat surface.

what a complex number is

A complex number is a dot on that surface, with an arrow from the middle out to it. Two numbers instead of one. The flat part is written x, and the up-and-down part is counted in i's, so the same number is written (x, iy). And i is the number you cannot find anywhere on the line: multiply it by itself and you get minus one, so i = √−1.

adding and multiplying

Adding is walking. Walk your number, then walk the second arrow from wherever you landed, and the result is where you end up. Multiplying is turning and stretching: the second arrow's lean is the turn, its length is the stretch.

why it is a turn

Multiplying by i does not make the arrow longer, and it does not slide it up a second axis. The length stays the same and the direction rotates, so the arrow stays on the same circle.

That is what makes it a turn instead of one more place to stand. Two quarter turns land you facing the opposite way, which is what minus one has always meant: owe five dollars and you are five dollars in the other direction. So i × i = −1.

Four turns and you are home, which is why it goes round and round forever, like a wheel. Everyone calls that arrow i.