Motor Variable - Bireal Variable

Bireal Variable

In 1892 Corrado Segre recalled the tessarine algebra as bicomplex numbers. Naturally the subalgebra of real tessarines arose and came to be called the bireal numbers.

In 1946 U. Bencivenga published an essay on the dual numbers and the split-complex numbers where he used the term bireal number. He also described some of the function theory of the bireal variable. The essay was studied at University of British Columbia in 1949 when Geoffry Fox wrote his master’s thesis "Elementary function theory of a hypercomplex variable and the theory of conformal mapping in the hyperbolic plane". On page 46 Fox reports "Bencivenga has shown that a function of a bireal variable maps the hyperbolic plane into itself in such a manner that, at those points for which the derivative of a function exists and does not vanish, hyperbolic angles are preserved in the mapping".

G. Fox proceeds to provide the polar decomposition of a bireal variable and discusses hyperbolic orthogonality. Starting from a different definition he proves on page 57

Theorem 3.42 : Two vectors are mutually orthogonal if and only if their unit vectors are mutually reflections of one another in one or another of the diagonal lines through 0.

Fox focuses on "bilinear transformations" are bireal constants. To cope with singularity he augments the plane with a single point at infinity (page 73).

Among his novel contributions to function theory is the concept of an interlocked system. Fox shows that for a bireal k satisfying

(a - b)2 < |k| < (a + b)2

the hyperbolas

| z | = a2 and | z – k | = b2

do not intersect (form an interlocked system). He then shows that this property is preserved by bilinear transformations of a bireal variable.