Geometry
The "unit circle" of dual numbers consists of those with a = 1 or −1 since these satisfy z z* = 1 where z* = a − bε. However, note that
- ,
so the exponential function applied to the ε-axis covers only half the "circle".
If a ≠ 0 and m = b /a, then z = a(1 + m ε) is the polar decomposition of the dual number z, and the slope m is its angular part. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since (1 + p ε)(1 + q ε) = 1 + (p+q) ε.
The dual number plane is used to represent the naive spacetime of Galileo in a study called Galilean invariance since the classical event transformation with velocity v looks like:
- , that is .
Read more about this topic: Dual Number
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