On the Three Types of Complex Number and Planar Transformations

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 + i b (where i 2 = −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 + ∈b (where ∈ 2 = 0), and exp( ∈τ ) shears the plane parallel to the y -axis through the shear τ. The third type is the double number, a + j b (where j 2 = +1), and exp(jβ) represents another type of shear transformation, known as a Lorentz transformation, which shears the plane through the rapidity β.