What is the strangest math that turned out to be useful?
What is the strangest math that turned out to be useful?
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?
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.
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.
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 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 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).
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.
Integration.
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
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.
And on spotify nowadays
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
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.
A brain teaser about visiting all islands connected by bridges without crossing the same bridge twice is now the basis of all internet routing. (Graph theory)
freaking freaky little Russian outpost that one is. Bridges galore
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.
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.
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.
IIRC quaternions were considered pretty useless until we started doing 3D stuff on computers and now they're used everywhere
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.
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.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.
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.
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.
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.
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.
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.
Not math but the discovery of Thermus aquaticus was seemingly useless but later had profound applications in medicine. There's a good Veritasium video on it
Donuts were basis of the math that would enable a planned economy to be more efficient than a market economy (which is a very hard linear algebra problem).
Basically using that, your smart phone is powerful enough to run a planned economy with 30 million unique products and services. An average desktop computer would be powerful enough to run a planned economy with 400 million unique products and services.
Odd that knowledge about it has been actively suppressed since it was discovered in the 1970s but actively used mega-corporations ever since…
It's funny that you're saying this is "actively suppressed" while not naming this field or providing links for further readings.
Donut mathematics is the name of the field; I literally named it. The writings on it are dense and only available in Russian and Mandarin Chinese. Further I provided the name of an author on the subject.
What would you call the purposeful prevention of English/French/German/etc translations of the material?
Maybe they're scared that project Cybersyn would actually work
That's pretty interesting. Do you happen to have any introductory material to that topic?
I mean, it might even have applications outside of running a techno-communist nation state. For example, for designing economic simulation game mechanics.
How do you define "pure math discovery"?
A math discovery unmotivated by research in other fields; just discovering math to see if it works out
With that as the definition I would say Boolean Algebra.
