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Let <math display="block">q = A + Bi + C\varepsilon j + D\varepsilon k</math> be a unit-length dual-complex number, i.e. we must have that <math display="block">|q| = \sqrt{A^2 + B^2} = 1.</math>
The Euclidean plane can be represented by the set <math display="inline">\Pi = \{i + x \varepsilon j + y \varepsilon k \mid x \in \mathbb R, y \in \mathbb R\}</math>.
An element <math>v = i + x \varepsilon j + y \varepsilon k</math> on <math>\Pi</math> represents the point on the [[Euclidean plane]] with [[cartesian coordinate]] <math>(x,y)</math>.
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