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?
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?