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{{Inline|date=July 2022}}
In [[algebra]], a '''split complex number''' (or '''hyperbolic number''', also '''perplex number''', '''double number''') has two [[real number]] components {{mvar|x}} and {{mvar|y}}, and is written {{math|1=''z'' = ''x'' + ''y{{tsp}}j''}}, where {{math|1=''j''{{i sup|2}} = 1 }}. The ''conjugate'' of ''z'' is {{math|1=''z''<sup>∗</sup> = ''x'' − ''y j''}}. Since {{math|1=''j''{{i sup|2}} = 1 }}, the product of a number {{mvar|z}} with its conjugate is {{math|1=''N''(''z'') := 'zz''<sup>∗</sup> = ''x''{{i sup|2}} − ''y''{{i sup|2}}}}, an [[isotropic quadratic form]]
The collection {{mvar|D}} of all split complex numbers {{math|1=''z'' = ''x'' + ''y{{tsp}}j''}} for {{math|''x'', ''y'' ∈ '''R'''}} forms an [[algebra over a field|algebra over the field of real numbers]]. Two split-complex numbers ''w'' and ''z'' have a product {{math|''wz''}} that satisfies {{math|1=''N''(''wz'') = ''N''(''w'')''N''(''z'')}}. This composition of {{mvar|N}} over the algebra product makes {{math|(''D'', +, ×, *)}} a [[composition algebra]].
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