Split-complex number
relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of
The change of sign distinguishes the split-complex numbers from the ordinary complex ones.
On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum
The diagonal basis for the split-complex number plane can be invoked by using an ordered pair (x, y) for
so the two parametrized hyperbolas are brought into correspondence with S. The action of hyperbolic versor
The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a dilation by √2.
of the split-complex plane has only half the area in the span of a corresponding hyperbolic sector.
[2] For all real values of the hyperbolic angle θ the split-complex number λ = exp(jθ) has norm 1 and lies on the right branch of the unit hyperbola.
Multiplying by λ preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
This group consists of the hyperbolic rotations, which form a subgroup denoted SO+(1, 1), combined with four discrete reflections given by
sending θ to rotation by exp(jθ) is a group isomorphism since the usual exponential formula applies:
In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring
In fact there are many representations of the split-complex plane in the four-dimensional ring of 2x2 real matrices.
Any hyperbolic unit m provides a basis element with which to extend the real line to the split-complex plane.
More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a subring of M(2,R).
The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines.
[4] William Kingdon Clifford used split-complex numbers to represent sums of spins.
Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions.
He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the circle group.
Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane.
[5][6][7][8][9][10] In that model, the number z = x + y j represents an event in a spatio-temporal plane, where x is measured in seconds and y in light-seconds.
The model says that z can be reached from the origin by entering a frame of reference of rapidity a and waiting ρ nanoseconds.
expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities.
[11] The gamma factor, with R as base field, builds split-complex numbers as a composition algebra.
In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in Contribución a las Ciencias Físicas y Matemáticas, National University of La Plata, República Argentina (in Spanish).
These expository and pedagogical essays presented the subject for broad appreciation.
[13] In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in zz∗ = 1.
[14] In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Académie polonaise des sciences (see link in References).
In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.
Different authors have used a great variety of names for the split-complex numbers.
