Possibly off-topic: why does practically any description of quaternions include the anecdote about somebody carving something into some bridge in Ireland?
I’ve learned about plenty of mathematical concepts while having no idea who discovered them or under what circumstances. Why are quaternions the exception?
My doctoral dissertation used quaternions to describe spacecraft attitudes. My advisor was a stickler for not only appropriately citing references, but citing the most original (in the sense of oldest) references. This was something of a problem because the original references to some of the mathematical techniques I used were several hundred years old and not in English, and under the theory that a big part of the purpose of a citation is to help the reader understand the backgroud material this seemed a bit excessive. Nobody is going to learn French in order to read about Lagrange multipliers from Lagrange's original papers.
But for quaternions, it was easy: I actually cited the Brougham Bridge inscription. One cannot, of course, check the bridge out of the engineering library to check the citation, but clearly this was the original “publication” of quaternion multiplication.
If someone learned French to read about Lagrange multipliers from Lagrange's original papers, then they would be sorely disappointed: it was actually Euler who first used the Lagrange method! (And by some curious coincidence, the L/lambda could just as easily be a homage to Leonhard as to Lagrange...)
It used to be common in hard sciences and mathematics for English-speaking doctoral candidates to be required to show some aptitude for German or Russian. Maybe still is, dunno.
I think the story is told as a sort of cultural marker of what a Big F-ing Deal this discovery was/is. Mathematics, collectively, struggled for a long time to find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers. (The cross and dot products are unsatisfying, because they don't have division.) The shock that this can be done in four, -- not three -- dimensions is still sort of reverberating, and that's what I think this story marks. It's a short stand-in for the longer story I've just summarized, and it evokes (or is meant to evoke) the mind-shattering thrill of discovery.
I don't necessarily think the story accomplishes this -- your question is but one piece of evidence that it doesn't -- but I think for those who spend a good amount of time with these kinds of algebra questions, it comes to take on that role, and that's why I think it's repeated.
(Teaser -- if you want to know more about these kinds of questions, Google for "real division algebras". There are not very many, and they way they are organized is not, I think, something one would expect.)
Also, in Hamilton's time they didn't have the view of mathematics as axiom systems that could be played with. Math was a way of finding Truth, so the idea of making up multiplication rules at your convenience probably seemed like an extremely non-obvious move.
The discovery of complex numbers and quaternions probably played a big part in getting people to question what math is, leading to Hilbert's program to study Foundations etc. Hamilton's story is a nice, rare single instance we can point to, symbolizing this discovery.
> To find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers
What does that mean? My understanding was that Hamilton was searching for a way to make the manipulation of points in space easier, such as rotation, and noticed that the imaginary part of the complex numbers could be manipulated in the way he wanted. He then created a rather artificial tool in the form of the quaternions that allowed this.
By "extends" here we formally mean "normed division algebra". Basically, we want any 2D slice of our space to be equivalent to the complex numbers. This is analogous to how the reals embed into the complex plane or how slices of vector spaces are still vector spaces. We don't want things to depend on a particular basis (i.e. implementation).
Anyway, it's pretty easy to make up some multiplication on 3D vectors, like multiplying their components. However, in general, it won't play nicely with such arbitrary 2D slices. As it turns out, this slicing property is equivalent to having multiplication play nicely with vector norms:
|ab| = |a| |b|.
that is, multiplication of vectors multiplies their lengths. Getting a multiplication with this property is the hard part, per se, and is only possible in dimensions 1, 2, 4, and 8.
It wasn’t when we learned it in high school, but it’s just that it’s a story that’s easy to tell. Just like the Königsberg reference for Eulerian graphs; Kekule and the Benzene ring; Lorenz, the weather, the butterfly, and deterministic chaos.
Thanks - out of all the replies to my comment, I think this is one of the most plausible hypotheses. I indeed think the Konigsberg and benzene stories are as commonly trotted out as the Hamilton one.
- quaternions maybe have a more interesting backstory than other constructs. Not everything was carved into stone.
- quaternions never really became part of mainstream mathematical education. This makes them more niche and strange, and worth telling a story about. It's not as interesting to tell a story about something commonplace.
People should read up on Galois theory and the story behind it. It involves a french youth, the french revolution, groundbreaking maths, and tangentially Poisson, Fourier and Cauchy, Gauss, Jacobi.
And why do descriptions of quaternions include a rigorous proof? I trust quaternions work, so skip the proof and show me how to use them in a practical, efficient way.
The fact that the quaternions “work” (are a finite-dimensional real vector space with an associative real-bilinear multiplication, and a division operation) is quite special. In fact there are only three objects satisfying this criteria: the real numbers themselves, the complex numbers, and the quaternions.
In mathematics often proofs lead to understanding why something works (on the other hand, many proofs aren’t written in a way that make the main ideas clear or easy to tease out from details)
There are millions of abstractions in mathematics, and the truth is that the human brain has a much better time remembering stories than it does math. You teach the stories so people have some context to associate complex ideas with.
I’ve learned about plenty of mathematical concepts while having no idea who discovered them or under what circumstances. Why are quaternions the exception?