algebra of split-complex numbers forms a composition algebra since for any numbers and . From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring of the cyclic group over the real numbers Matrix representations One can easily represent split-complex numbers by matrices. The split-complex number can be represented by the matrix Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The modulus of is given by the determinant of the corresponding matrix. In fact there are many representations of the split-complex plane in the four-dimensional ring of 2x2 real matrices. The real multiples of the identity matrix form a real line in the matrix ring M(2,R). Any hyperbolic unit m provides a basis element with which to extend the real line to the split-complex plane. The matrices which square to the identity matrix satisfy For example, when a = 0, then (b,c) is a point on the standard hyperbola. 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 number can be represented by the matrix 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. Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane. In that model, the number represents an event in a spatio-temporal plane, where x is measured in nanoseconds and in Mermin's feet. The future corresponds to the quadrant of events , which has the split-complex polar decomposition . The model says that can be reached from the origin by entering a frame of reference of rapidity and waiting 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 ; is the line of events simultaneous with the origin in the frame of reference with rapidity