Why do $n$-dimensional numbers follow a geometric sequence?

Question

I'm new here so please forgive my poor formatting and understanding.

I'm trying to understand why $n$-dimensional numbers follow a sequence. For example, octonions are $8D$ numbers that (I assume) correspond to $4D$ space, quaternions are $4$-dimensional numbers that correspond to $3D$ space, etc.

My question is: why is this the case? Why do $n$-dimensional numbers follow this sequence?

Answer 1

There are reasons which conspire for composition algebras to have dimension of the form $2^n$.

Given any composition algebra $H$, we can pick $a\in H\setminus\Bbb R$ and form $\mathbb{R}[a]$, then $b\in H\setminus \mathbb{R}[a]$ and adjoin again to form $\mathbb{R}[a,b]$, and so on. This will eventually terminate to yield $H$ (I'm assuming our algebra is finite-dimensional). Thus, it ultimately boils down to what adjoining an element does.

Suppose $K\subsetneq H$ is a proper composition subalgebra. Pick $x\in H\setminus K$. It can be written as $x=a+u$ where $a$ is the component parallel to $\mathbb{R}$ and $u$ the perpendicular component (with respect to the inner product associated with the quadratic form or "composition" of the algebra). Purely imaginary elements like $u$ can be shown to satisfy $u^2=-|u|^2$ (like $i^2=-1$ for complex numbers). Moreover, perpendicular elements anticommute ($uv=-vu$ if $u\perp v$ are purely imaginary and orthogonal), so $uy=\overline{y}u$ for $y\in K$. Thus, $x$ is quadratic over $K$; that is, $uK=Ku$ and $K[x]=K\oplus Ku$ as vector spaces. This means

$$ \dim K[x]=2\dim K. $$

This is the reason that the dimensions of the (Euclidean) composition algebras, $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}$ exhibit the doubling up pattern $\dim = 1,2,4,8$ (although going $8\to16$ doesn't quite work out).

Answer 2

For algebras related to real numbers that can reasonably be called numbers there's no sequence to speak of. The possible dimensions are $1$, $2$, $4$, and $8$, and that's it. There are no more finite dimensional division algebras over the real numbers. As for the reason, this is complicated and usually proved with topology.