Imaginary numbers always confused me. Like understanding e, most explanations fell into one of two categories:
∴<span>It’s a mathematical abstraction, and the equations work out. Deal with it.
</span>∵<span>It’s used in advanced physics, trust us. Just wait until college.</span>
Gee, what a great way to encourage math in kids!
x²= 9
The answers are 3 and -3. But suppose some wiseguy puts in a teensy, tiny minus sign:
i²= -1
Uh oh. This question makes most people cringe the first time they see it. You want the square root of a number less than zero? That’s absurd! (Historically, there were real questions to answer, but I like to imagine a wiseguy.)
It seems crazy, just like negatives, zero, and irrationals (non-repeating numbers) must have seemed crazy at first. There’s no “real” meaning to this question, right?
Wrong. So-called “imaginary numbers” are as normal as every other number (or just as fake): they’re a tool to describe the world. In the same spirit of assuming -1, .3, and 0 “exist”, let’s assume some number i exists where:
That is, you multiply i by itself to get -1. What happens now?
Well, first we get a headache. But playing the “Let’s pretend i exists” game actually makes math easier and more elegant. New relationships emerge that we can describe with ease.
You may not believe in i, just like those fuddy old mathematicians didn’t believe in -1. New, brain-twisting concepts are hard and they don’t make sense immediately, even for Euler. But as the negatives showed us, strange concepts can still be useful.
I dislike the term “imaginary number” — it was considered an insult, a slur, designed to hurt i‘s feelings. The number i is just as normal as other numbers, but the name “imaginary” stuck so we’ll use it.
