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+Iba + \mathrm{I} b, where aa and bb are real numbers and I 2=1\mathrm{I}^2 = 1 (but I±1\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\mathbf{P} or \mathbb{P}.


This can be thought of as:

  • the vector space 2\mathbb{R}^2 made into an algebra by the rule
    (a,b)(c,d)=(ac+bd,ad+bc); (a, b) \cdot (c, d) = (a c + b d, a d + b c) ;
  • 2\mathbb{R}^2 as a direct product ×\mathbb{R} \times \mathbb{R} of rings;
  • the subalgebra of those 22-by-22 real matrices of the form
    (a b b a); \left(\array { a & b \\ b & a } \right);
  • the polynomial ring [x]\mathbb{R}[\mathrm{x}] modulo x 21\mathrm{x}^2 - 1;
  • the hyperbolic 22-dimensional algebra of hypercomplex numbers.

We think of \mathbb{R} as a subset of \mathbb{P} by identifying aa with a+0Ia + 0 \mathrm{I}. \mathbb{P} is equipped with an involution that maps I\mathrm{I} to I¯=I\bar{\mathrm{I}} = -\mathrm{I}:

a+Ib¯=aIb. \overline{a + \mathrm{I} b} = a - \mathrm{I} b .

\mathbb{P} also has an absolute value:

|a+Ib|=a 2b 2; {|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=zz¯. {|z|^2} = z \bar{z}.

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