Hyperbolic numbers have the form $a+bj$ where $a,b \in \mathbb{R}$ and $j^2=1$. These are also known as a the Lorentz numbers or double numbers by some authors. I suppose these are a particular type of the Tessarines introduced by James Cockle in 1848, see this wikipedia article on the Tessarines for details. However, the hyperbolic numbers themselves are not so complicated and I wonder if they were known to other mathematicians before Cockle's Tessarines.
Question: was Cockle the first to introduce hyperbolic numbers ?
I happened to discover some of the hyperbolic numbers interesting features working with some students for fun back in 2012. We found, $$ e^{j\theta} = \cosh (\theta) + j \sinh (\theta) $$ and a few other fun things before we stumbled upon the Wikipedia article which had much of what we found. We're not alone in our experience. The current article on split-complex numbers lists the following synonyms:
- (real) tessarines, James Cockle (1848)
- (algebraic) motors, W.K. Clifford (1882)
- hyperbolic complex numbers, J.C. Vignaux (1935)
- bireal numbers, U. Bencivenga (1946)
- approximate numbers, Warmus (1956), for use in interval analysis
- countercomplex or hyperbolic numbers from Musean hypernumbers
- double numbers, I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014)
- anormal-complex numbers, W. Benz (1973)
- perplex numbers, P. Fjelstad (1986) and Poodiack & LeClair (2009)
- Lorentz numbers, F.R. Harvey (1990)
- hyperbolic numbers, G. Sobczyk (1995)
- paracomplex numbers, Cruceanu, Fortuny & Gadea (1996)
- semi-complex numbers, F. Antonuccio (1994)
- split binarions, K. McCrimmon (2004)
- split-complex numbers, B. Rosenfeld (1997)[13]
- spacetime numbers, N. Borota (2000)
- Study numbers, P. Lounesto (2001)
- twocomplex numbers, S. Olariu (2002)
The multitude of names reflects the fact that for the most part each author starts anew without much guidance from the previous generation. This seems to be the one uniting theme in the majority of the literature on hypercomplex analysis (if I could call it that, but, I really need to call it by many names, see how horrible the situation is even for the particular example of the hyperbolic numbers!)