Editing Dual number (section)

==Geometry==
The "unit circle" of dual numbers consists of those with {{math|''a'' {{=}} ±1}} since these satisfy {{math|''zz''* {{=}} 1}} where {{math|''z''* {{=}} ''a'' − ''bε''}}. However, note that
: <math> e^{b \varepsilon} = \sum^\infty_{n=0} \frac{\left(b\varepsilon\right)^n}{n!} = 1 + b \varepsilon,</math>
so the [[exponential map (Lie theory)|exponential map]] applied to the {{mvar|ε}}-axis covers only half the "circle".

Let {{math|''z'' {{=}} ''a'' + ''bε''}}. If {{math|''a'' ≠ 0}} and {{math|''m'' {{=}} {{sfrac|''b''|''a''}}}}, then {{math|''z'' {{=}} ''a''(1 + ''mε'')}} is the [[polar decomposition#Alternative planar decompositions|polar decomposition]] of the dual number {{mvar|z}}, and the [[slope]] {{mvar|m}} is its angular part. The concept of a ''rotation'' in the dual number plane is equivalent to a vertical [[shear mapping]] since {{math|(1 + ''pε'')(1 + ''qε'') {{=}} 1 + (''p'' + ''q'')''ε''}}.

In [[absolute space and time]] the [[Galilean transformation]]
:<math>\left(t', x'\right) = (t, x)\begin{pmatrix} 1 & v \\0 & 1 \end{pmatrix}\,,</math>

that is
:<math>t' = t,\quad x' = vt + x,</math>

relates the resting coordinates system to a moving frame of reference of [[velocity]] {{mvar|v}}. With dual numbers {{math|''t'' + ''xε''}} representing [[event (relativity)|event]]s along one space dimension and time, the same transformation is effected with multiplication by {{math|1 + ''vε''}}.

===Cycles===
Given two dual numbers {{mvar|p}} and {{mvar|q}}, they determine the set of {{mvar|z}} such that the difference in slopes ("Galilean angle") between the lines from {{mvar|z}} to {{mvar|p}} and {{mvar|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 {{mvar|z}}, a cycle is a [[parabola]]. The "cyclic rotation" of the dual number plane occurs as a motion of [[#Projective line|its projective line]]. According to [[Isaak Yaglom]],<ref name="yaglom"/>{{rp|92–93}} the cycle {{math|''Z'' {{=}} {''z'' : ''y'' {{=}} ''αx''<sup>2</sup><nowiki>}</nowiki>}} is invariant under the composition of the shear
:<math>x_1 = x ,\quad  y_1 = vx + y </math>

with the [[translation (geometry)|translation]]
:<math>x' = x_1 = \frac{v}{2a}  ,\quad  y' = y_1 + \frac{v^2}{4a}. </math>