Normed division algebras with R: introducing the onion package

where multiplication is denoted by juxtaposition; note the absence of associativity. A division algebra is a nontrivial algebra in which division is possible: for any a ∈ V and any nonzero b ∈ V , there exists precisely one element x with a = bx and precisely one element y with a = yb. A normed division algebra is a division algebra with a norm ||·|| satisfying ||xy|| = ||x|| ||y||. There are precisely four normed division algebras: the reals themselves (R), the complex numbers (C), the quaternions (H) and the octonions (O); the generic term is“onion”, although the term includes other algebras such as the sedenions. The R computer language (R Development Core Team 2008) is well-equipped to deal with the first two: here, I introduce the onion package that provides some functionality for the quaternions and the octonions, and illustrate these interesting algebras using numerical examples.