John Baez, of This Week’s Finds in Mathematical Physics fame, has a new article The Octonions.
The octonions are a mysterious example in mathematics that have been drawing attention in physics. Initially discovered by Graves in 1843, the octonions provided the first example of a number system with nonassociative multiplication: (ab)c is not equal to a(bc). Lots of examples of nonassociative multiplication are known, but the octonions remain the most interesting. For a more elementary introduction to the subject, there’s the article on Wikipedia.