In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε^{2} = 0 (ε is nilpotent). The collection of dual numbers forms a particular twodimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and b uniquely determined real numbers. The plane of all dual numbers is an "alternative complex plane" that complements the ordinary complex number plane C and the plane of splitcomplex numbers.
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Linear representation
Using matrices, dual numbers can be represented as
The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication; both operations are commutative and associative.
This procedure is analogous to matrix representation of complex numbers. Furthermore, the concept of the dual number is necessary when reading a matrix.
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 εaxis 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:
Cycles
Given two dual numbers p, and q, they determine the set of z such that the Galilean angle between the lines from z to p and q is constant. This set is a cycle in the dual number plane; since the equation setting the difference in slopes of the lines to a constant is a quadratic equation in the real part of z, a cycle is a parabola. In the Inversive ring geometry of dual numbers one encounters "cyclic rotation" as a projectivity on the projective line over dual numbers. According to Yaglom (pp. 92,3), the cycle Z = {z : y = α x^{2}} is invariant under the composition of the shear
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