nLab
perplex number

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 $P$ or $\mathbb{P}$.

This can be thought of as:

• the vector space ${ℝ}^2$ made into an algebra by the rule

  $$(a,b)\cdot (c,d)=(ac+bd,ad+bc);$$(a, b) \cdot (c, d) = (a c + b d, a d + b c) ;

• ${ℝ}^2$ as a direct product $ℝ\times ℝ$ of rings;

• the subalgebra of those $2$-by-$2$ real matrices of the form

  $$\left(\begin{array}{cc}a& b\\ b& a\end{array}\right);$$\left(\array { a & b \\ b & a } \right);

• the polynomial ring $ℝ[\mathrm{x}]$ modulo ${\mathrm{x}}^2-1$;

• the hyperbolic $2$-dimensional algebra of hypercomplex numbers.

We think of $ℝ$ 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 $\overline{\mathrm{I}}=-\mathrm{I}$:

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

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

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

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

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

But this absolute value is degenerate, in that $\mid z\mid =0$ need not imply that $z=0$.

Some concepts in analysis can be extended from $ℝ$ 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.