Split-complex

In mathematics, a number

z = x + yj

is called split-complex when x and y are real numbers and

j² = + 1.

The collection of all such z is called the plane of split-complex numbers D. In abstract algebra terms, this is therefore the quotient ring

R[t]/(t² − 1).

James Cockle invented D when he revealed his Tessarines of 1848. William Kingdon Clifford used D to represent sums of spins in 1882. Clifford called the elements "motors", so D is also the "motor plane".

In the twentieth-century D became a common platform to describe the Lorentz transformations of special relativity, in a spacetime plane because its structure is precisely that used for the physical theory.

Other authors using D include I.M. Yaglom, Walter Benz, and Garett Sobczyk.

The split-complex plane D has these features:

1. If z = x + y j and z* = x - y j , then z z* = x x - y y . Thus for every constant d in R , H(d) = {z : z z* = d } is a hyperbola.
2. exp(aj) = cosh(aj) + sinh(aj) = cosh a + j sinh a because the power series for cosh has only even powers while that for sinh has odd powers. Therefore {exp(a j): a in R } lies on the hyperbola H(1). Furthermore, exp:R --> H(1) is a group isomorphism.
3. For u = exp(a j) = cosh a + j sinh a , the mapping z' = u z is a hyperbolic rotation that moves points along their hyperbolas H(d).
4. Split-complex numbers z = x + yj and w = u + vj are hyperbolically orthogonal when Re(z w*) = 0 where Re() takes the real part of the product, and w* = u - vj.
5. Elements e = (1 + j)/2 and f = (1 - j)/2 are idempotents because ee = e and ff = f.