68 Chapter^1
In either case the system is a real algebra of dimension two with unit
element u = (1, 0).
For the general complex number z = a + bi, with i^2 = a + (:Ji, the
conjugate z of z is defined to be
z = (a + b(:J) - bi
and the square of the modulus of z is defined as follows:
lzl^2 = zz = a^2 +(:Jab - ab^2
If this is considered as the square of the distance from the origin to z, the
locus of points at a distance r from the origin is given by
x^2 + (:Jxy - ay^2 = r^2
which represents a conic section, the type of a conic depending on .D. =
(:J^2 + 4a. For .D. < 0 the conic is an ellipse (which reduces to a circle of
radius r in the case a= -1, (:J = 0). For .D. = 0 it reduces to two straight
lines x +^1 / 2 (:Jy = ±r, and for .D. > 0 the conic becomes a hyperbola.
It is clear that the inverse of z is given by z-^1 = z/lzl^2 ' provided that
z -=fa 0. Upon setting
lzl^2 = x^2 + (:Jxy - ay^2 = 0
we obtain
Thus, in the case .D. < 0, x is real iffy = 0, and then x = 0 also. Hence
in the elliptic case z = 0 is the only number that does not have an inverse,
and the corresponding algebraic system is a field. However, in the case
.D. = 0 all numbers z = x + iy with x = -^1 / 2 (:Jy (i.e., on the straight line
2x+(:Jy = 0) have no inverses, and in the case .D. > 0 all numbers z = x+iy
with x = %(-(:J ± ,/fS.)y [i.e., on the lines 2x + (f:J =i= ,/fS.)y = 0] have no
inverses either. In the preceding two cases the corresponding algebraic
systems are rings with proper divisors of zero. For example, for complex
numbers written in the form a + bj, with j2 = 0, all numbers of the form
bj (b -=fa 0) are proper divisors of zero since (bj)(b'j) = bb'j2 = 0.
Parabolic complex numbers were first considered by E. Study (1862-
1930) and hyperbolic complex numbers by W. Clifford (1845-1879). They
have applications in the theory of numbers, mechanics, and in non-
Euclidean geometries. For further information concerning these numbers
the reader is referred to the publications [2a], [6], [9], and [24] listed at the
end of the chapter.