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?
Riemann went nuts working on higher dimensional mathematics and linear algebra. At the time there was not a clear use case for math higher than like 3 or 4 dimensions, but he drove himself crazy discovering it anyways. Today, this kind of math underlies all of artificial intelligence
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’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 just think complex vector spaces are a great place to stop your abstraction
Stopping while we’re ahead? Never!
/s, but also I’m sort of in this picture.
Well who wants constraints anyway? The most inconvenient constraints in the wrong place can make certain things much more complicated to deal with… Now a nice, sensible normal Hilbert space, isn’t that lovely?
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).
Non-Euclidean geometry was developed by pure mathematicians who were trying to prove the parallel line postulate as a theorem. They realized that all of the classic geometry theorems are all different if you start changing that postulate.
This led to Riemannian geometry in 1854, which back then was a pure math exercise.
Some 60 years later, in 1915, Albert Einstein published the theory of general relativity, of which the core mathematics is all Riemannian geometry.
This won’t make any sense to any of you right now, but: E = md3
That’s a perfect example of a typical interaction between a Technology Management Consultant and somebody from a STEM area.
Techies with an Engineering background who are in Tech and Tech-adjacent companies are often in the receiving end of similar techno-bollocks which makes no sense from such “Technology” Management Consultants, but it’s seldom quite as public as this one.
Oh god, the cringe.
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.
12 is the most based number in that respect IMO.
But then…hey, we use that for hours!
and in parts of the world for inches to a foot. pretty useful for carpentry for example
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.
Man, I’ve had the wrong Boole in my head this whole time? Guess I have a 0 somewhere that I should have a 1
Was he trying to dunk on his professors?
Yes and no
So no.
But also yes
I work with a guy who is a math whiz and loves to talk. Yesterday while I was invoicing clients, he was telling me how origami is much more effective for solving geometry than a compass and a straight edge.
I’ll ask him this question.
My disclaimer: I don’t know what any of this means, but it might give you a direction to start your research.
First thing he came up with is Number Theory, and how they’ve been working on that for centuries, but they never would have imagined that it would be the basis of modern encryption. Multiplying a HUGE prime number with any other numbers is incredibly easy, but factoring the result into those same numbers is near impossible (within reasonable time constraints.)
He said something about knot theory and bacterial proteins, but it was too far above my head to even try to relay how that’s relevant.
Tell him I would like to subscribe to his blog
The following aren’t necessarily answers to your question, but he also mentioned these, and they are way too funny to not share:
The Hairy Ball theorem
Cox Ring
Tits Alternative
Wiener Measure
The Cox-Zucker machine (although this was in the 70s and it’s rumored that Cox did most of the work and chose his partner ONLY for the name. 😂)
Based Cox.
I am pretty sure that the first thing you mentioned (multiplying being easy and factoring being hard) is the basis of public key cryptography which is how HTTPS works.
Somewhat related fun fact: One of the most concrete applications for quantum computers so far is breaking some encryption algorithms.
Origami can be used as a basis for geometry:
http://origametry.net/omfiles/geoconst.html
IIRC, you can do things that are impossible in standard Euclidean construction, such as squaring the circle. It also has more axioms than Euclidean construction, so maybe it’s not a completely fair comparison.
Does this count? Because it really is wtf.
Doom absolutely counts!
Quake, not Doom. Doom didn’t use true 3D rendering and had almost no dynamic lighting.
Oops. I thought that weird approximated constant was somewhere in the doom sources… Thanks I guess for correcting me.
Here’s some math-related Doom content for you: John Romero accidentally coded in the wrong digit of pi in the 10th position, and this guy explores how the game rendering changes when pi is increasingly wrong
The exact example I also thought of from the question! Well done
Don’t put that cursed shit on mathematicians, lol.
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.
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
IIRC quaternions were considered pretty useless until we started doing 3D stuff on computers and now they’re used everywhere
This talk by Freya Holmer on Quarternions is awesome and worth anybody’s time that like computer graphics, computer science, or just math.
That was a cool watch. Thanks.
I wonder if complex numbers predate the discovery of electromagnetism
Yes, mathematicians first encountered equations which could only be solved with complex numbers in the 16th century.
If I recall correctly, one mathematician in the 1800s solved a very difficult line integral, and the first application of it was in early computer speech synthesis.
the man you’re thinking of is, I believe, George Boole, the inventor of Boolean algebra.
Integration.
Integration was literally developed to be useful
Complex numbers. Also known as imaginary numbers. The imaginary number
iis the solution to√-1. And it is really used in quantum mechanics and I think general relativity as well.It’s used extensively in electronic circuit design (where it’s called “j”, as "i’ already meant electronic current).
Also signal processing has i or j all over it.
I’m the akshually guy here, but complex numbers are the combination of a real number and an imaginary number. Agree with you, just being pedantic.
Sure, but 1 is a real number. 😜
Yes, and 1 is also a complex number.
Of course, but 1 is the loneliest number.
2 is as bad as 1: it’s the loneliest number since the number 1.
A complex number is just two real numbers stitched together. It’s used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.
In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn’t, then no one would use complex numbers, because computers can’t process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.
I don’t think this is really an accurate way of thinking about them. Yes, they can be mapped to a 2d plane, so you can represent them with their two real-numbered coordinates along the real and imaginary axes, but certain operations with them (eg. multiplication) can be done easily with complex numbers but are not obvious how to carry out with just grid points. (3,4) * (5,6) isn’t well-defined, but (3+4i) * (5+6i) is.
That’s not quite accurate because the two numbers have a relationship with each other. i^2 = - 1, so any time you square a complex number or multiply two complex numbers, some of the value jumps from one dimension to the other.
It’s like a vector, where sure, certain operations can be treated as if the dimensions of the vector are distinct, like a translation or scale. But other operations can have one dimension affecting the other, like rotation.
That’s like saying negative numbers or fractional numbers is just two while numbers stitched together because that’s how computers deal with it
Electromagnetics as well.
Non-linear equations have entered the chat.
Chaos and non-linear dynamics were treated as a toy or curiosity for a pretty long time, probably in no small part due to the complexity involved. It’s almost certainly no accident that the first serious explorations of it after Poincare happen after the advent of computers.
So, one place where non-linear dynamics ended up having applications was in medicine. As I recall it from James Gleick’s book Chaos, inspired by recent discussion of Chaotic behavior in non-linear systems, medical doctors came up with the idea of electrical defibrillation- a way to reset the heart to a ground state and silence chaotic activity in lethal dysrhythmias that prevented the heart from functioning correctly.
Fractals also inspired some file compression algorithms, as I recall, and they also provide a useful means of estimating the perimeters of irregular shapes.
Also, there’s always work being done on turbulence, especially in the field of nuclear fusion as plasma turbulence seems to have a non-trivial impact on how efficiently a reactor can fuse plasma.
A good friend of mine from high school got his physics PhD at University of Texas and went on to work in the high energy plasma physics lab there with the Texas Petawatt laser, and a lot of the experiments it was used for involved plasma turbulence and determining what path energetic particles would take in a hypothetical fusion reactor.
Be honest, how many unofficial experiments were there?
You ever just start lasering shit for kicks?
Probably not as many as we’d like to think. I recently got to run a few days of tests at Lawrence Livermore National Labs with an absurdly massive laser. At one point we needed to bring in a small speaker for an audio test. It took the lab techs and managers about two hours and a couple phone calls to some higher ups to make sure it was ok and wouldn’t damage anything. There’s so much red tape and procedure in the way that I don’t think there’s an opportunity to just fuck around. The laser has irreplaceable parts that people aren’t willing to jeopardize. Newer or smaller lasers are going to be more relaxed. This one is old enough to be my father, and it’s LLNL’s second biggest single laser iirc. And they are the lab using lasers for fusion, so they have big lasers.
Reminds me of that scene from Ali G
I’ve read that all modern cryptography is based on an area (number theory?) that was once only considered “useful” for party tricks.
prime number factorization is the basis of assymetric cryptography. basically, if I start with two large prime numbers (DES was 56bit prime numbers iirc), and multiply them, then the only known solution to find the original prime numbers is guess-and-check. modern keys use 4096-bit keys, and there are more prime numbers in that space than there are particles in the universe. using known computation methods, there is no way to find these keys before the heat death of the universe.
DES is symmetric key cryptography. It doesn’t rely on the difficulty of factorizing large semi-primes. It did use a 56-bit key, though.
Public key cryptography (DSA, RSA, Elliptic Curve) does rely on these things and yes it’s a 4096-bit key these days (up from 1024 in the older days).
RSA mostly uses 4096 bit keys nowadays. DSA is no longer used (or shouldn’t be lol). Ed25519 uses 256 bit keys.
thank you
