Editing Dual number (section)

==Division==
Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous to [[Complex number|complex division]] in that the denominator is multiplied by its conjugate in order to cancel the non-real parts.

Therefore, to evaluate an expression of the form
:<math>\frac{a + b\varepsilon}{c + d\varepsilon}</math>

we multiply the numerator and denominator by the conjugate of the denominator:
:<math>\begin{align}
  \frac{a + b\varepsilon}{c + d\varepsilon}
    &= \frac{(a + b\varepsilon)(c - d\varepsilon)}{(c + d\varepsilon)(c - d\varepsilon)}\\[5pt]
    &= \frac{ac - ad\varepsilon + bc\varepsilon - bd\varepsilon^2}{c^2 + cd\varepsilon - cd\varepsilon - d^2\varepsilon^2}\\[5pt]
    &= \frac{ac - ad\varepsilon + bc\varepsilon - 0}{c^2 - 0}\\[5pt]
    &= \frac{ac + \varepsilon(bc - ad)}{c^2}\\[5pt]
    &= \frac{a}{c} + \frac{bc - ad}{c^2}\varepsilon
\end{align}</math>

which is defined [[Division by zero|when {{mvar|c}} is non-zero]].

If, on the other hand, {{mvar|c}} is zero while {{mvar|d}} is not, then the equation
:<math>{a + b\varepsilon = (x + y\varepsilon) d\varepsilon} = {xd\varepsilon + 0}</math>
# has no solution if {{mvar|a}} is nonzero
# is otherwise solved by any dual number of the form {{math|{{sfrac|''b''|''d''}} + ''yε''}}.
This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially) [[zero divisors]] and clearly form an [[ideal (ring theory)|ideal]] of the associative [[Algebra over a field|algebra]] (and thus [[Ring (mathematics)|ring]]) of the dual numbers.