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 […]

Intereting Posts

Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables
The Riesz representation theorem on locally compact Hausdorff spaces-alternate proof
How to rotate n individuals at a dinner party so that every guest meets every other guests
A “clean” approach to integrals.
Number of $\sigma$ -Algebra on the finite set
How to calculate $\lim \limits_{x \to 0} \frac{x^2 \sin^2x}{x^2-\sin^2x}$ with $\lim \limits_{x \to 0} \frac{\sin x}{x}=1$?
Prove the triangle is equilateral given that a quadrilateral related to its circumcircle is a kite
Automorphism of algebraic closure $\overline{{\bf F}}_p$.
Isomorphisms between $(\mathbb Z_4,+)$ and $(U_5,*)$
Probability that a Brownian motion with drift hits +1 before hitting -1 before time 1, and similar events
drawing at least 90% of colors from urn with large populations
Is this two-dimensional version of the Intermediate Value Theorem correct?
Unique expression of a polynomial under quotient mapping?
Definition and example of a partition
Derivation of Lagrange-Charpit Equations