On the three types of complex number and planar transformations

Abstract

This paper compares the three possible types of complex number. These provide a very concise means for representing certain geometric transformations of the points of a plane. The first type considered is the ordinary complex number, a + ib (where i2 = -1). This is used in the exponential form exp(iα) to rotate the plane through the angle α. The second type is the dual number, a + εb (where ε2 = 0), and exp(ετ) shears the plane parallel to the y-axis through the shear τ. The third type is the double number, a + jb (where j2 = +1), and exp(jβ) represents another type of shear transformation, known as a Lorentz transformation, which shears the plane through the rapidity β.