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Concept
Split-complex number
Summary
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying A split-complex number has two real number components x and y, and is written The conjugate of z is Since the product of a number z with its conjugate is an isotropic quadratic form.
The collection D of all split complex numbers for x,y \in \R forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.
A similar algebra based on \R^2 and component-wise operations of addition and multiplication, (\R^2, +, \times, xy), where xy is the quadratic form on \R^2, also forms a quadratic space. The ring isomorphism
relates proportional quadratic forms, but the mapping is an isometry since the multiplicative identity (1, 1) of \R^2 is at a distance \sqrt 2 from 0, which is normalized in D.
Split-complex numbers have many other names; see below. See the article Motor variable for functions of a split-complex number.
A split-complex number is an ordered pair of real numbers, written in the form
where x and y are real numbers and the hyperbolic unit j satisfies
In the field of complex numbers the imaginary unit i satisfies The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit j is not a real number but an independent quantity.
The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by
This multiplication is commutative, associative and distributes over addition.
Just as for complex numbers, one can define the notion of a split-complex conjugate. If
then the conjugate of z is defined as
The conjugate satisfies similar properties to usual complex conjugate. Namely,
These three properties imply that the split-complex conjugate is an automorphism of order 2.
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