symmetric monoidal (∞,1)-category of spectra
A perplex number (also known as a split-complex number or a hyperbolic number or a Lorentz number or myriad other such synonyms varying from author to author) is an expression of the form $a + \mathrm{I} b$, where $a$ and $b$ are real numbers and $\mathrm{I}^2 = 1$ (but $\mathrm{I} \ne \pm{1}$). The set of perplex numbers (in fact a topological vector space and commutative algebra over the real numbers) may be denoted $\mathbf{P}$ or $\mathbb{P}$.
This can be thought of as:
We think of $\mathbb{R}$ as a subset of $\mathbb{P}$ by identifying $a$ with $a + 0 \mathrm{I}$. $\mathbb{P}$ is equipped with an involution that maps $\mathrm{I}$ to $\bar{\mathrm{I}} = -\mathrm{I}$:
$\mathbb{P}$ also has an absolute value:
notice that the absolute value of a perplex number is a complex number, with
But this absolute value is degenerate, in that ${|z|} = 0$ need not imply that $z = 0$.
Some concepts in analysis can be extended from $\mathbb{R}$ to $\mathbb{P}$, but not as many as work for the complex numbers. Even algebraically, the perplex numbers are not as nice as the real or complex numbers, as they do not form a field.
Last revised on November 15, 2020 at 10:03:51. See the history of this page for a list of all contributions to it.