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In this article, the set of dual-complex numbers is denoted <math>\mathbb {DC}</math>. A general element <math>q</math> of <math>\mathbb {DC}</math> has the form <math display="inline">A + Bi + C\varepsilon j + D\varepsilon k</math> where <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> are real numbers; <math>\varepsilon</math> is a [[dual number]] that squares to zero; and <math>i</math>, <math>j</math>, and <math>k</math> are the standard basis elements of the [[quaternions]].
Multiplication is done in the same way as with the quaternions, but with the additional rule that <math display="inline"> \varepsilon </math> is [[nilpotent]] of index <math>2</math>, i.e. <math display="inline"> \varepsilon^2=0 </math>, which in some circumstances makes <math display="inline">\varepsilon</math> comparable to an [[infinitesimal]] number. It follows that the multiplicative inverses of dual-complex numbers are given by
: <math> (A + Bi + C\varepsilon j + D\varepsilon k)^{-1} = \frac{A - Bi -C\varepsilon j - D\varepsilon k}{A^2+B^2}</math>
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