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Comment by dhosek | Hacker News Reader

I first encountered quaternions in the concept of abstract algebra. It's interesting to think of it as part of a spectrum starting with reals and continuing to octonions (and beyond). Interestingly with each extension we lose something:

Complex numbers (2-dimensional): No more ordering

Quaternions (4-dimensional): No more commutative multiplication

Octonions (8-dimensional): No more associative multiplication

Sedenions (16-dimensional): We lose the alternative property, i.e., with octonions its still true that x(xy) = (xx)y but with sedenions it is not.

We can continue this process indefinitely, although I know nothing of the characteristics of trigintaduonions or what comes later. I suspect that it might be the loss of the flexible identity next, a(ba) = (ab)a but I don't know for sure.

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Here's a short article that gave me some insight into the matter:

https://nautil.us/blog/the-strange-numbers-that-birthed-mode...

"Place your phone face-up on a flat surface, for example. Spin it 90 degrees to the left, and then flip it away from you. Note which way the camera points.

Returning to the original position, flip it away from you first and then turn it to the left second. See how the camera points to the right instead?

This initially alarming property, known as non-commutativity, turns out to be a feature the quaternions share with reality "

It's worth noting that one of the early finite groups that is taught in abstract algebra are the dihedral groups which model exactly this sort of behavior (For your phone example this would be D₄. Knowing that mattress flipping is modeled with D₂ makes it trivial to prove that that there is no combination of flip and rotate that will allow you to always do the same move to your mattress to get it into the four possible positions for optimal wear.

Interesting Applied Mathematics

Interestingly, if we go in the opposite direction we also loose something:

Real numbers (1-dimensional): No more algebraic closure [1]

though as you suggest we do gain a nice ordering where "nice" means "compatible with operations". BTW, you can always impose some ordering on all of these sets. It is the orderings' compatibility with arithmetic that we lose going from real numbers to complex numbers.

This can be generalized a little. Real numbers fail to be algebraically closed because some real polynomials lack real solutions, e.g. xx=-2. Complex numbers "fix" this problem since there every n degree polynomial has exactly n roots (counting multiplicity). In a sense, quaternions overdo this, e.g. xx=-2 has an infinite number of solutions (exercise for the reader). However, since quaternion multiplication is not commutative, polynomials as traditionally understood are ill-defined. For example, axx, xax and xxa are different degree two quaternion "polynomials". Ugh.

[1]: https://en.wikipedia.org/wiki/Algebraically_closed_field

I used to teach the expansion of the concept of numbers from ℕ to ℂ via a fictionalized history of the numbers (most notably, expanding ℕ to ℤ before introducing ℚ, but historically, rational and even irrational numbers preceded zero and negative numbers in mathematical understanding. It also elides over the various subdivisions of all the different kinds of irrational numbers that exist in ℝ\ℚ which could be a whole class in itself, but this was a math for liberal arts majors class and I had to make sure not to have their eyes glaze over too much.

One neat characteristic is that the derivations of these higher Cayley-Dickinson algebras stabilize!

[1] https://projecteuclid.org/journals/pacific-journal-of-mathem...

First 4 (reals, complex, quaternions, octonions) are normed division algebras.

Normed means that each number has associated nonnegative real number that describes the "size" of the number, also |xy| <= |x||y|. Number having a size (a norm) is very useful concept.

Sedions and the rest are less useful because they are not normed algebras.

I don't think it makes sense to say you "lose" something. Yes, the properties of different spaces _are_ different, but these differences are intentional so that they can be used to study something useful. For instance, complex analytic functions have a number of properties that don't hold for real analytic functions, so you actually "gain" something. An example is the residue theorem. One interesting thing about the residue theorem is that you can use it to calculate the integral of a real-valued function that wouldn't be calculable using conventional calculus techniques. So not only do complex functions have a number of interesting properties on their own, they can also be used to expand what's possible in real functions.

Another interesting property from which I learnt about quaternions is that all the proper subgroups of the quaternion group are normal and commutative, but the quaternion group itself is non commutative. [0]

[0] https://www.cs.cas.cz/portal/AlgoMath/AlgebraicStructures/St...

Why do you think the dimensions are in powers of two?

Because the construction generating this sequence of algebras doubles the dimension every time.

https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_const...

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