There have been a number of Scientific discoveries that seemed to be purely scientific curiosities that later turned out to be incredibly useful. Hertz famously commented about the discovery of radio waves: “I do not think that the wireless waves I have discovered will have any practical application.”
Are there examples like this in math as well? What is the most interesting “pure math” discovery that proved to be useful in solving a real-world problem?
It’s imaginary numbers. Full stop. No debate about it. The idea of them is so wild that they were literally named imaginary numbers to demonstrate how silly they were, and yet they can be used to describe real things in nature.
I mean, quaternions are the weirder version of complex numbers, and they’re used for calculating 3D rotations in a lot of production code.
There’s also the octonions and (much inferior) Clifford algebras beyond that, but I don’t know about applications.
Yeah but aren’t quaternions basically just a weird subgroup of 2x2 complex matrices?
I don’t really get 'em. It seems like people often use them as “a pair of numbers.” So why not just use a pair of numbers then?
They also have a defined multiplication operation consistent with how it works on ordinary numbers. And it’s not just multiplying each number separately.
A lot of math works better on them. For example, all n-degree polynomials have exactly n roots, and all smooth complex functions have a polynomial approximation at every point. Neither is true on the reals.
Quantum mechanics could possibly work with pairs of real numbers, but it would be unclear what each one means on their own. Treating a probability amplitude as a single number is more satisfying in a lot of ways.
They don’t exist is still a position you could take, but so is the opposite.
I totally get your point, and sometimes it seems like that. Why not just use a coordinate system? Because innsome applications the complex roots of equations is relevant.
If you square an imaginary number, it’s no longer an imaginary number. Now it’s a real number! That’s not something you can accomplish with something like a pair of numbers alone.
Because the second number has special rules and a unit. It’s not just a pair of numbers, though it can be represented through a pair of numbers (really helpful for computing).
I’m studying EE in university, and have been surprised by just how much imaginary numbers are used
From what I’ve seen that’s one example where you could totally just use trig and pairs of numbers, though. I might be missing something, because I’m not an electrical engineer.
You can, they map, but complex numbers are much much easier to deal with
EE is absolutely fascinating for applications of calculus in general.
I didn’t give a shkt about calculus and then EE just kept blowing my mind.
I was gonna ask how imaginary numbers are often used but then you reminded me of EE applications and that’s totally true.