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 $\mathbb{C}$ you gain “algebraic-closure”-ness (but you throw away ordering).
  • From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity.
  • From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity.
  • From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.

The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ?

Answer

What you are talking about is precisely the Cayley-Dickson construction.

Remark: I am left wondering what is gained by going past Octonions. The the first 4 are very special as they are the unique 4 normed divison algebras over $\mathbb{R}$. Perhaps someone with more knowledge can point out the possible uses of the Sedenions and their higher counterparts.

Attribution
Source : Link , Question Author : Willem Noorduin , Answer Author : Eric Naslund

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