Cayley Number

There are two completely different definitions of Cayley numbers. The first type Cayley numbers is one of the eight elements in a Cayley Algebra, also known as an Octonion. A typical Cayley number is of the form

$$a+bi_0+ci_1+di_2+ei_3+fi_4+gi_5+hi_6,$$

where each of the triples $(i_0, i_1, i_3)$, $(i_1, i_2, i_4)$, $(i_2, i_3, i_5)$, $(i_3, i_4, i_6)$, $(i_4, i_5, i_0)$, $(i_5, i_6, i_1)$, $(i_6, i_0, i_2)$ behaves like the Quaternions $(i,j,k)$. Cayley numbers are not Associative. They have been used in the study of 7- and 8-D space, and a general rotation in 8-D space can be written

$$x'\to ((((((xc_1)c_2)c_3)c_4)c_5)c_6)c_7.$$

The second type of Cayley number is a quantity which describes a Del Pezzo Surface.

See also Complex Number, Del Pezzo Surface, Quaternion, Real Number

References

Conway, J. H. and Guy, R. K. ``Cayley Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 234-235, 1996.

Okubo, S. Introduction to Octonion and Other Non-Associative Algebras in Physics. New York: Cambridge University Press, 1995.