nLab perplex number

Idea

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 $\mathbb{P}$.

Definition

This can be thought of as:

• the vector space $\mathbb{R}^2$ made into an algebra by the rule
  $$(a, b) \cdot (c, d) = (a c + b d, a d + b c) ;$$
• $\mathbb{R}^2$ as a direct product $\mathbb{R} \times \mathbb{R}$ of rings;
• the subalgebra of those $2$-by-$2$ real matrices of the form
  $$\left(\array { a & b \\ b & a } \right);$$
• the polynomial ring $\mathbb{R}[\mathrm{x}]$ modulo $\mathrm{x}^2 - 1$;
• the hyperbolic $2$-dimensional algebra of hypercomplex numbers.

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 $\bar{\mathrm{I}} = -\mathrm{I}$:

$$\overline{a + \mathrm{I} b} = a - \mathrm{I} b .$$

$\mathbb{P}$ also has an absolute value:

$${|a + \mathrm{I} b|} = \sqrt{a^2 - b^2} ;$$

notice that the absolute value of a perplex number is a complex number, with

$${|z|^2} = z \bar{z}.$$

But this absolute value is degenerate, in that ${|z|} = 0$ need not imply that $z = 0$.

Properties

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.