Cayley Numbers or Alternative Division Algebras

Abstract

With the creation by Hamilton of a “system of hypercomplex numbers” a process of rethinking began to take place. Mathematicians began to realize that, by abandoning the vague principle of permanence, it was possible to create “out of nothing” new number systems which were still further removed from the real and complex numbers than were the quaternions. In December 1843 for example, only two months after Hamilton’s discovery, Graves discovered the eight-dimensional division algebra of octo-nions (octaves) which—as Hamilton observed—is no longer associative. Graves communicated his results about octonions to Hamilton in a letter dated 4th January 1844, but they were not published until 1848 (Note by Professor Sir W.R. Hamilton, respecting the researches of John T. Graves, esq. Trans. R. Irish Acad., 1848, Science 338-341). Octonions were rediscovered by Cayley in 1845 and published as an appendix in a work on elliptic functions (Math. Papers 1, p. 127) and have since then been called Cayley numbers.

It is possible to form an analogous theory with seven imaginary roots of (−1) (A. Cayley 1845).