transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The algebra obtained from the generalization of the Cayley-Dickson construction applied on the complex numbers applied to the real numbers with parameter $\gamma=-1$.
A split-complex number $z$ may be represented as
where $j^2=1$ (in contrast with the imaginary unit $i^2=-1$ in the complex numbers). Conjugation is similarly given by
A consequence is that the product $z z^*$ is not non-negative anymore, since
meaning in particular that zero-divisors exist (for example $(1 - j)(1 + j)=1 - j^2=0$). Using the diagonal basis
of idempotent elements, hence for which $e^2=e$ and $(e^*)^2=e^*$, and fulfilling $e e^*=e^* e=0$ results in multiplication given by
which yields that the algebra $\mathbb{R}[j]=\mathbb{R}[e]$ of split-complex numbers is isomorphic to the algebra $\mathbb{R}\oplus\mathbb{R}$ with pointwise multiplication.
Last revised on February 3, 2024 at 20:43:25. See the history of this page for a list of all contributions to it.