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!)

  • $\begingroup$ incidentally, I'm not familiar with the tags here, feel free to fix mine as appropriate. Thanks in advance for any guidance. $\endgroup$ Commented Nov 23, 2016 at 3:38
  • $\begingroup$ Paracomplex numbers were discussed at length, with that name (in French as nombre paracomplexe) by Paulette Libermann in her memoir Sur le problème d’équivalence de certaines structures infinitésimales, Ann. Mat. Pura Appl. (4) 36 (1954), 27–120. (this is cited in the paper of Cruceanu, Fortuny and Gadea). Whether the use is older still, I don't know. $\endgroup$
    – Dan Fox
    Commented Dec 8, 2016 at 21:56

1 Answer 1


According to the text Bicomplex Holomorphic Funtions by Luna-Elizarraras, et al, as far as we know Cockle was the first to introduce and study these objects, and their introduction was almost contemporary to Hamilton's introduction of the quaternions and certainly stimulated by it. From this text (using a sightly different notation - $k$ in place of $j$) :

...it is not unreasonable to consider whether a four-dimensional algebra, containing $\mathbb C$ as a subalgebra, can be introduced in a way that perserves commutativity. Not surprisingly, this can be done by simply considering two imaginary units $i, j$, introducing $k = ij$ (as in the quaternionic case), but now imposing that $ij = ji$. This turns $k$ into what is known as a hyperbolic imaginary unit, i.e., and element such that $k^2 = 1$. As far as we know, the first time these objects were introduced was almost contemporary with Hamilton's construction, and in fact J. Cockle wrote, in 1848, a series of papers in which he introduced a new algebra that he called the algebra of tessarines. Cockle's work was certainly stimulated by Hamilton's and he was the first to use tessarines to isolate the hyperboic trigonometric series as components of the exponential series. Not surprisingly, Cockle immediately realised that there was a price to pay for commutativity in four dimensions, and the price was the existence of zero-divisors. This discovery led him to call such numbers impossibles, and the theory has no further significant development for a while.

It was only in 1892 that the mathematician Corrado Segre, inspired by the work of Hamilton and Clifford, introduced what he called bicomplex numbers ....

So Luna-Elizarraras history marks Segre's 1892 work as the next instance of the development of hyperbolic numbers, rather than that of Clifford, while noting Clifford and Hamilton's inspiration.

Cockle's original papers are available on the Biodiversity Heritage Library site as :

On Certain Functions Resembling Quaternions ... (1848)

On a New Imaginary in Algebra (1849)

On the Symbols of Algebra and the Theory of Tessarines (1849)

On the True Amplitude of Tessarines (1850)

On Impossible Equations, on Impossible Quantities, and on Tessarines (1850)

  • $\begingroup$ Thanks! I guess this doesn't rule out something between Segre and Cockle. I can't help but wonder about Euler... always Euler... but, your links are great this really helps with a paper I'm writing at the moment. $\endgroup$ Commented Nov 25, 2016 at 21:17
  • $\begingroup$ I think the nmain disadvantage is that it is quite trivial, being isomorphic to $\mathbb C ^2$. But why do so many people consider zero divisors a disadvantage? $\endgroup$
    – Anixx
    Commented Oct 31, 2022 at 11:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.