Split-complex Number - History

History

The use of split-complex numbers dates back to 1848 when James Cockle revealed his Tessarines. 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. Extending the analogy, functions of a motor variable contrast to functions of an ordinary complex variable.

In the twentieth century the split-complex multiplication is commonly seen as a Lorentz boost of a spacetime plane. In the model the number z = x + y j represents an event in a spacio-temporal plane where x is measured in nanoseconds and y in Mermin’s feet. The future corresponds to the quadrant of events {z : |y| < x } which has the split-complex polar decomposition . The model says that z can be reached from the origin by entering a frame of reference of rapidity a and waiting ρ nanoseconds. The split-complex equation

expressing products on the unit hyperbola, illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity a :

is the line of events simultaneous with the origin in the frame of reference with rapidity a. Two events z and w are hyperbolic-orthogonal when z* w + z w* = 0. Canonical events exp(aj) and j exp(aj) are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to j exp(aj).

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.

In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in zz* = 1.

In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Academie Polanaise des Sciences (see link in References). He identified an interval with the split-complex number

and called it an "approximate number". D. H. Lehmer reviewed the article in Mathematical Reviews.

In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.