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 $\mathbb{H}$ […]

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 $\mathbb{C}$ […]

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 […]

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