Split-complex number: Difference between revisions

Revision as of 23:31, 17 December 2022

Revision as of 01:32, 17 January 2023

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A '''split-complex number''' is an ordered pair of real numbers, written in the form

<math display="block">z = x + jy</math>

where {{mvar|x}} and {{mvar|y}} are [[real number]]s and the quantity'''hyperbolic unit'''<ref>Vladimir V. Kisil (2012) ''Geometry of Mobius Transformations: Elliptic, Parabolic, and Hyperbolic actions of SL(2,R)'', pages 2, 161, Imperial College Press {{ISBN|978-1-84816-858-9}}</ref> {{mvar|j}} satisfies

<math display="block">j^2 = +1</math>

ChoosingIn <math>j^2the =field -1</math>of results[[complex number]]s in the [[complex number|compleximaginary numbersunit]]. Iti issatisfies this<math>i^2 sign= -1 .</math> The change whichof sign distinguishes the split-complex numbers from the ordinary complex ones. The quantityhyperbolic unit {{mvar|j}} here is ''not'' a real number but an independent quantity.

The collection of all such {{mvar|z}} is called the '''split-complex plane'''. [[Addition]] and [[multiplication]] of split-complex numbers are defined by