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The '''dual-complex numbers''' make up a four-dimensional [[Algebra over a field|algebra]] over the [[real number]]s.<ref>{{Citation|last=Matsuda|first=Genki|title=Anti-commutative Dual Complex Numbers and 2D Rigid Transformation|date=2014|url=https://doi.org/10.1007/978-4-431-55007-5_17|work=Mathematical Progress in Expressive Image Synthesis I: Extended and Selected Results from the Symposium MEIS2013|pages=131–138|editor-last=Anjyo|editor-first=Ken|series=Mathematics for Industry|publisher=Springer Japan|language=en|doi=10.1007/978-4-431-55007-5_17|isbn=9784431550075|access-date=2019-09-14|last2=Kaji|first2=Shizuo|last3=Ochiai|first3=Hiroyuki|arxiv=1601.01754}}</ref><ref>Gunn C. (2011) On the Homogeneous Model of Euclidean Geometry. In: Dorst L., Lasenby J. (eds) Guide to Geometric Algebra in Practice. Springer, London</ref> Their primary application is in representing [[rigid body motion|rigid body motions]] in 2D space.
== Definition ==
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