Articles of sedenions

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I’m starting to come around to an understanding of hypercomplex numbers, and I’m particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ dimensions. I think I understand the first $n<4$ instances: As we move from $\mathbb{R}$ to $\mathbb{C}$ we lose ordering From $\mathbb{C}$ to $\mathbb{H}$ […]

What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions} $$ or: “Reals” $\subset$ “Complex” $\subset$ “Quaternions” $\subset$ “Octonions” $\subset$ “Sedenions” With the following “properties”: From $\mathbb{R}$ to $\mathbb{C}$ […]

Why are properties lost in the Cayley–Dickson construction?

Motivating question: What lies beyond the Sedenions? I’m aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$ “Reals” $\subset$ “Complex” $\subset$ “Quaternions” $\subset$ “Octonions” $\subset$ “Sedenions” $\subset$ $\ldots$ and that at each step you’re given a multiplication table […]