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.
this post was submitted on 26 Jun 2025
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I'm studying EE in university, and have been surprised by just how much imaginary numbers are used
EE is absolutely fascinating for applications of calculus in general.
I didn't give a shit 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.
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
In quantum mechanics, there are times you divide two different complex numbers, and complex multiplication/division is the thing two real numbers can't really replicate. That's how the Bloch 2-sphere in 3D space is constructed from two complex dimensions (which maps to 4 real ones).
It's peripheral, though. Nothing in the guts of the theory needs it AFAIK - the Bloch sphere doesn't generalise much and is more of a visualisation. So, jury's still out on if it's us or if it's nature that likes seeing it that way.
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?
Would that make it less true? Complex numbers can be seen as a weird subgroup of the 2x2 real matrices. (And you can "stack" the two representations to get 4x4 real quaternions)
Furthermore, octonions are non-associative, and so can't be a subgroup of anything (although you can do a similar thing using an alternate matrix multiplication rule). They still show up in a lot of the same pure math contexts, though.
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.
That 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 in some 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).
Strangest? Functional analysis, maybe. I understand it's used pretty extensively in quantum field theory, although I don't actually know firsthand.
That's a body of mathematics about infinite-dimensional spaces and the operations on them. Even more abstract ways of defining those operations exist and have come up as well, like in Tseirlson's problem, which recently-ish had a shock negative resolution stemming from quantum information theory.
There's constructions I find weirder yet, but I don't think p-adic numbers, for example, have any direct application at this point.
As far as I know, matrices were a "pure math" thing when they were first discovered and seemed pretty useless. Then physicists discovered them and used them for all sorts of shit and now they're one of the most important tools in in science, engineering and programming.
Huge in 3d graphics and AI.
Imaginary numbers probably, they're useful for a lot of stuff in math and even physics (I've heard turbulent flow calculations can use them?) but they seem useless at first
The invention of the number 0, the discovery of irrational numbers, or l the realization that base 60 math makes sense for anything round, including timekeeping.
60 was chosen by the Ancient Sumerians specifically because of its divisibility by 2, 3, 4, and 5. Today, 60 is considered a superior highly composite number but that bit of theory wouldn’t have been as important to the Sumerians and Babylonians as the simple ability to divide 60 by many commonly used factors (2, 3, 4, 5, 6, 10, 12, 15) without any remainders or fractions to worry about.
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Integration.
Integration was literally developed to be useful
Having watched all the veritasium math videos I feel like all the major breakthroughs in math were due to mathemicians playing around with numbers or brain teasers out of curiosity without a concrete use case in mind.
It’s crazy how engaging and well done Veritasium videos are and they’re just free to watch on YouTube.
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The math fun fact I remember best from college is that Charles Boole invented Boolean algebra for his doctoral thesis and his goal was to create a branch of mathematics that was useless. For those not familiar with boolean algebra it works by using logic gates with 1s and 0s to determine a final 1 or 0 state and is subsequently the basis for all modern digital computing
Shoutout to Satyendra Nath Bose who helped pioneer relativity as a theoretical physicist because he didn't want to study something useful that would benefit the British.
Same thing with early studies on prime numbers
George Boole introduced Boolean algebra, not Charles. Charles, according to this site on the Boole family, he had a career in management of a mining company.
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