How imaginary numbers are useful
Imaginary numbers don’t exist (well, duh…as the name suggests…). Yet, many students spend hours solving equations involving them in maths classes — the italicised letter i still gives me… less than pleasant flashbacks. This begs the question: If imaginary numbers don’t exist, why learn them?
Turns out, they have important applications in physics and engineering. Instead of using equations, here’s an explanation on the utility of imaginary with a story about the history of mathematics.
A long, long time ago, before numbers were invented…
John was at the market to sell his cows. “I’d like to trade your cows with my chicken, good sir!” David shouted over the noise.
“Sure! I don’t mind the trading my this amount of cows with your this amount of chicken.” replied John, gesturing at the animals. David considered for a while and answered “That’s not a fair deal — I prefer this amount of my chickens to that amount of your ducks!”
Not having numbers seems awfully inconvenient, doesn’t it? And hence, natural numbers were created — 1, 2, 3, 4 and so on. Instead of this and that, we can use numbers to quantify the amount of something. Now record-keeping and commerce are so much easier.
But this set of numbers weren’t enough.
“Fine, I’ll trade 5 cows for 50 of your chickens. But I only have 3 cows with me right now. So I will owe you the remainder” John acquiesced.
“We have a deal!”
With that, John flipped open his neat record-keeping notebook and wrote “Owe 2 cows”. And as the day went on, John owed more and more cows.
This seems rather inconvenient for accounting purposes, John thought. So he invented a new way of representing “owe” — the minus sign (-). And that’s how negative numbers are created, lengthening our existing number line.
Notice how negative numbers don’t actually exist — it’s impossible to have literally -5 cows. We use them because they are useful for representing or calculating things.
Think about all the maths problems we in high school, many of which involving tangible entities like animals and vehicles. Negative numbers can never be the answer to such questions, yet we often use them in our calculations. Of course, we can omit them and still get the final answer, but they do make calculations easier.
Moral of the story
Negative numbers are don’t exist, but they are useful — and imaginary numbers are exactly the same. When we use them to study waves (i.e. Fourier transform) and electrical circuits, imaginary numbers are never the final answer. They are used to make calculations simpler.
In fact, negative and imaginary numbers are not so different. At the end of the day, both are expansions of our number system — the former expands the x-axis, while the latter creates a y-axis. In other words, the Argand diagram.
But negative and imaginary numbers seem different because of the misleading denotation of the word “imaginary”, that the latter doesn’t exist but negative and positive numbers do. Fundamentally, all numbers are imaginary — simply concepts trapped within our minds. We can use it to describe things (e.g. 5 cows), but the number 5 itself does not exist in reality. It would be absurd to point at anything around us and say “That’s 5”.
The unfortunate naming of imaginary numbers creates misconceptions about the nature of them, so much so that Gauss recommended to change how we name the number system.
“That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.”
~Carl Friedrich Gauss
Conclusion
Imaginary numbers are simply another part of our number system. And while they don’t exist in reality, we use them to make calculations simpler, the same way negative numbers do.
If you want to go deeper into the exact use cases of imaginary numbers, check out this video by Zach Star.
