Split-complex number

Definition

The binary numbers (English split -complex numbers or hyperbolic numbers ( for justification see below) ) form a two-dimensional hypercomplex algebra over the field of real numbers; The complex numbers as this algebra is generated by two basic elements 1 and a non- real unit, i is the complex numbers referred to herein E for differentiating the imaginary unit. Each binary number can be so clear as

Represent with a, b ​​∈, ie as a linear combination of 1 and E. The definition of a general multiplication for binary numbers is completed by a definition of the square of the non- real unit, by

In addition, as with the complex numbers, the conjugate of z to

Defined.

Properties

Like all hypercomplex algebras also satisfy the binary numbers, the right - and left- distributive. Like the complex numbers, they are also commutative and associative, and indeed necessarily, since there is only one different from the one base element, namely.

Thus, the binary numbers form a commutative ring with identity, but - in contrast to - not a body, but a principal ideal ring with two nontrivial ideals, the reellzahligen multiples of and those of, vividly so the line passing through the origin diagonals of the complex plane. Principal ideals are they, since they are each generated by a single element. They are both zero divisor, because 0 is the product of any element of an ideal with any member of the other:

A standard or an amount is not defined for binary numbers, but there are two properties that are so " further bequeath" in multiplication as is the norm in complex numbers or the determinant for matrices (in the sense of " standard / determinant of the product equal to product of the standards / determinants of factors "):

Which always results in a real number. This is

Like all complex numbers with a certain amount lie on a circle, are all binary numbers whose product has a fixed value with its conjugate, on a hyperbola; therefore they are called in English " hyperbolic numbers". Thus, the binary numbers follow a Minkowski metric, such as time ( = real axis) and spatial direction ( = non-real axis) in the special theory of relativity.