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In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the [[nine-point hyperbola]] of a triangle inscribed in {{math|1=''zz''{{sup|∗}} = 1}}.<ref>Allen, E.F. (1941) "On a Triangle Inscribed in a Rectangular Hyperbola", [[American Mathematical Monthly]] 48(10): 675–681</ref>
In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in ''Bulletin de l’Académie polonaise des sciences'' (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra.<ref>M. Warmus (1956) [http://www.cs.utep.edu/interval-comp/warmus.pdf "Calculus of Approximations"] {{Webarchive|url=https://web.archive.org/web/20120309164421/http://www.cs.utep.edu/interval-comp/warmus.pdf |date=2012-03-09 }}, ''Bulletin de l'Académie polonaise des sciences'', Vol. 4, No. 5, pp. 253–257, {{MR|id=0081372}}</ref> [[D. H. Lehmer]] reviewed the article in [[Mathematical Reviews]] and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article.
In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.
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