Dual Number - Division

Division

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

Therefore, to divide an equation of the form:

We multiply the top and bottom by the conjugate of the denominator:

= {(a+b\varepsilon)(c-d\varepsilon) \over (c+d\varepsilon)(c-d\varepsilon)} = {ac-ad\varepsilon+bc\varepsilon-bd\varepsilon^2 \over (c^2+cd\varepsilon-cd\varepsilon-d^2\varepsilon^2)} = {ac-ad\varepsilon+bc\varepsilon-0 \over c^2-0}

Which is defined when c is non-zero.

If, on the other hand, c is zero while d is not, then the equation

  1. has no solution if a is nonzero
  2. is otherwise solved by any dual number of the form
.

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 of the associative algebra (and thus ring) of the dual numbers.

Read more about this topic:  Dual Number

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