Could you explain why? For someone without a math background, it seems indeed like a pretty arbitrary thing to define.
(I can understand the idea behind complex numbers and how the multiplication rules followed from the desire to define the square root of a negative number - however, so far, I don't get the motivation of introducing even more "special" elements)
> the multiplication rules followed from the desire to define the square root of a negative number
That's a bit reductionist. You don't just get the square root of a negative number, you get the Fundamental Theorem of Algebra (an Nth degree polynomial has N roots), which is a mathematical power tool if ever there was one.
Complex numbers dramatically simplify a bunch of proofs in linear algebra, give us tons of nifty integration techniques in complex analysis (the techniques are relevant for real numbers, they just use C), provide a representation of 2D rotations that can be manipulated using the rules of algebra (this is the most relevant to the thread), and give physicists, electrical engineers, and signal processing people an abstraction to represent oscillations (energy sloshing between two buckets = two elements of a complex number, which you can then do algebra with). They're a workhorse.
Quaternions are an attempt to do that in 3D. The dot and cross product of vector calculus are other pieces of those efforts. Unfortunately, vector calculus escaped the "math lab" before it was complete and got written into other fields and engineering books, so even though the underlying concepts were eventually sorted out (it's called Geometric Algebra), everybody just uses the half-baked abstractions (quaternions, dot product, cross product) which are Good Enough. It's a perfect example of "worse is better" affecting something other than software engineering.
I guess the question is, why does it then stop. Why not a 4D alternative. Or if you look at it going by scalars needed in a single value, it goes from 1 to 2 to 4. Why not 8 or 16 (or some other growth)? Why does it stop there?
Also, is there as easy of a problem to understand introducing the 3D technique (be it quarternions or be it Gemoetric Algebra) that works as well as using sqrt(-1) for imaginary numbers?
It can be generalized, but doing so requires some subtlety. The naive approach (the Cayley-Dickson construction) can be repeated ad infinitum, but it doesn't continue to yield useful results for representing geometric interactions like rotations in high dimensions.
Thankfully, this is a solved problem. The correct generalized structure for doing geometry is called a Clifford algebra. For n-space and any nonnegative integers p,q satisfying p+q=n, there is a corresponding real Clifford algebra Cl(R,p,q). Cl(R,0,1) turns out to be isomorphic to C (the complex numbers), and Cl(R,0,2) is a four-dimensional algebra that turns out to be isomorphic to Q (the quaternions).
This is actually not that surprising, because the signature (p,q) more or less means the algebra is built by adjoining p generators that square to +1 and q generators that square to -1 in the base field. This is formalized by taking a quotient of the tensor algebra of the field. You might wonder though why we have (p,q) = (0,2) for the quaternions. That's because if the two generators that square to -1 are i and j, then we can build the third as k = ij, so we get it for free.
A real Clifford algebra is known as a geometric algebra, and these give rise to objects called rotors. Rotations in an arbitrarily high-dimensional space can then be written as conjugation by a rotor.
> I guess the question is, why does it then stop. Why not a 4D alternative.
It doesn't stop. That's what motivated geometric algebra, which works in any dimension. Quaternions are a sub-algebra of geometric algebra. They represent 3D rotations, which makes them interesting in their own right.
Asterisk: I believe there's a sign convention issue in mapping between quaternions and the even subalgebra of the 3D geometric algebra, so they aren't identical, just isomorphic.
> Also, is there as easy of a problem to understand introducing the 3D technique (be it quarternions or be it Gemoetric Algebra) that works as well as using sqrt(-1) for imaginary numbers?
That's an extraordinarily high bar. I don't believe anything reaches it. Part of the problem is that complex numbers are one of the most successful concepts in all of mathematics. The other part of the problem is that most of the useful facets of geometric algebra escaped the field of abstract mathematics under their own name before the unifying structure was discovered. The dot and cross product, quaternions, differential forms and the general Stokes' theorem are all examples. The remaining value proposition of geometric algebra lies mostly in getting rid of minor annoyances that come from this half-baked nature of traditional vector calculus tools:
* Cross products break in more than 3 dimensions and they break if you reflect them (see: pseudovectors). Bivectors have no such issues. They represent rotations in any dimension, reflected or not.
* Vector algebra with dot and cross products involves memorizing lots of new identities and applying creativity to work around the absence of division, while geometric algebra just has division and the same bunch of algebra tricks you already know. The geometric product isn't commutative, so it isn't perfect in this sense, but learning to deal with non-commutative algebra is a much more fundamentally useful thing than learning a bunch of 3D-specific identities.
* Dot and Cross with one argument fixed "destroy information" mapping from their input to their output. If you put them into an equation, the equation does not fully constrain the free vector, so you are often going to need more than one equation to represent any single geometric concept. Not so with geometric algebra. Many concepts map to a single equation. Including Maxwell's Equation (I use the singular intentionally)!
do you have a good reference for this? i've looked into GA bit but don't remember seeing anything like this. e.g. what would dividing a bivector by a vector mean?
I still haven't wrapped my head around quaternions, but 3Blue1Brown on Youtube has a good series of videos justifying and explaining the complex numbers and quaternions in terms not of sqrt(-1) but of transformations of space.
I think the answer to that question is that it doesn't "stop", but I'll try to offer a reason that isn't "octonions exist", but instead goes in a different direction.
Geometric Algebra can capture the structure of both complex numbers and quaternions, and also the structure of the dot products, cross products, and the different kinds of vectors that arise from those operations.
To be clear, matrix multiplication can also capture the structure of complex numbers [1] and quaternions [2]. There might also be a concise reference to matrix representations of some geometric algebras, but I didn't find one. So matrices are kind of one way to not "stop at 3D", but the structure is almost too uniform (which on one hand makes it too general, and on the other hand makes it not general enough), I'd say). Sure, with a matrix you can represent rotations in 4D, but you still need to operate on vectors only. Geometric algebra, if it does have a matrix representation, gives names to special kinds of matrices and special kinds of vectors.
By the Frobenius theorem, there are only three possible structures for a real finite-dimensional associative division algebra. Those structures correspond to the real numbers, the complex numbers, and what are called the quaternions. So essentially the above definition is not arbitrary because it's the only other possible way (besides R and C) to get that sort of algebraic system. Of course, this is not obvious at all. C famously is algebraically closed as a field, which makes it a ripe playground for much of topology, algebraic geometry, and analysis. There are some nonobvious generalizations of algebraic closure for the quaternions. (Naively, the quaternions are not algebraically closed in the classic sense because, evidently, ix + xi - j has no root.)
As for why one might want to consider such a noncommutative division algebra in the first place, the answer I suppose is just that it manages to pop up in a variety of areas in mathematics. We've already seen the connection with rotations in 3-space (the topic of this post). Here's another. The 3-sphere (that is, a sphere in 4-dimensional space whose surface is itself 3-dimensional) can be realized as the multiplicative group of unit quaternions spanned by {1,i,j,k}. Consider the circle H = {cos(theta) + i * sin(theta)} for real values of theta; H is a subset of the 3-sphere. If r is any unit quaternion, then the coset rH is another circle. But given a subgroup H of any group G, the left cosets of H in G form a partition of G. Therefore, these circles just described form a partition of all of the 3-sphere (the Hopf fibration).
Speaking of rotations, the involvement of quaternions should not be surprising. Indeed, complex numbers are intimately involved in rotations in 2-space (multiplication by a unit complex number e^(i*theta) corresponds to rotation about the origin by theta). Quaternions can similarly express rotations in 3-space, but one cannot just left- or right-multiply but must instead use conjugation. In general, one can generalize this using the techniques of geometric algebra.
When I took abstract algebra as an undergrad, we did a brief bit on the quaternions. Bursting with curiosity I asked the professor if 8 and 16 dimensional structures existed. "Of course! But just as you lose commutivity with Q, when you go to the octonions, you lose associativity, and the sedonions lack "alternativity" (had to look that up -- I didn't remember) and they're basically algebraic novelties with out any application."
Right, but while the Cayley-Dickson construction mostly provides novelties (though I remember reading something about octonions and string theory[1]), Clifford algebras are derived differently; they are isomorphic to complex numbers and quaternions for two and three base vectors respectively, but they produce something else after quaternion. This "something different" can be used to represent, you guessed it, reflections and rotations in a 4D space. Because they are not obtained from the Cayley-Dickson construction they are not division algebras, however.
As other replies have said, the math is kind of important. The idea of a tattoo harkens to the story of the discovery of quaternions: Rowan Hamilton was out for a walk in Dublin, trying to figure out how to generalize complex numbers. He was walking under a bridge when he came up with that equation, and carved the equation on the bridge.
His carving, if it ever existed, is gone. But there is a plaque on the bridge commemorating the event. It reads:
Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i² = j² = k² = ijk = −1
& cut it on a stone of this bridge.
My PhD advisor was a stickler for citing original sources. Really, really original sources. He made me cite some papers written by Lagrange in the 17th century in French, when neither he nor I nor nearly anyone else who would ever read my dissertation could speak French.
I got to the point where I needed to cite an original source for the quaternion equations, so I cited the bridge.
For a summary of William Rowan Hamilton's life (including the bridge story), see this amazingly clever video based on the song from Hamilton:
https://www.youtube.com/watch?v=SZXHoWwBcDc
There's also a great book "A History of Vector Analysis: The Evolution of the Idea of a Vectorial System" by Crowe which covers vectors from Complex numbers to Gibbs vectors and includes Hamilton and the competitor at the time Grassman Algebra, both the basis for geometric algebra.
Its one of the only maths history books I couldn't put down.
This invention/discovery is a fundamental development in abstract algebra, not a terminal one. Quaternions are just a jumping-off point, and I've always found the Caley-Dickenson construction that pauldraper explains[2] absolutely beautiful.
Why would I want it specifically as a tattoo? jfengel points out the special history of that specific equation[3]: it was (allegedly) carved into a bridge in Dublin when Hamilton stumbled onto it, but the carving is gone. Kinda fitting to give it new permanence.
So, putting it all together: it's a fundamental development in abstract algebra, which is my jam. It's could have been permanently inscribed in a bridge, but that's been lost to time, so giving it new permanency seems fitting.
Also, it's practical. My first thought was actually the Cayley table for the Klein four-group[4], but that would be a lot harder to get tattooed in a nice visible way. How I went from there to Hamilton's quaternion equation is left as an exercise to the reader. (If you're new to Cayley tables, they're just fancy times tables. Replace "e" with 1.)
I'm not a mathematician, but I think it's about extending the idea of a scalar and a single rotation (Complex numbers) into a scalar + 3 rotations (Quaternions). The idea can be extended further to a scalar with 7 rotations - https://en.wikipedia.org/wiki/Octonion, but no further, for reasons I don't understand.
You can actually go as far as you want to with the Cayley–Dickson construction [1] of algebras.
1. Complex numbers have associativity and communitivity of multiplication. (That is, (ab)c=a(bc) and ab=ba).
2. Quaternions have associativity but not communitivity.
3. Octonions have neither.
4. Sedenions [2], trigintaduonions, and not associative, commutative, nor even alternative [3]. (Alternative is associative specifically when the middle value is equal to one of the other's; i.e. a(ab)=(aa)b.)
Could you explain why? For someone without a math background, it seems indeed like a pretty arbitrary thing to define.
(I can understand the idea behind complex numbers and how the multiplication rules followed from the desire to define the square root of a negative number - however, so far, I don't get the motivation of introducing even more "special" elements)