Octonion

The octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system usually written with the boldface letter O. They have eight dimensions, twice the dimensions of the quaternions, of which they are an extension built through the Cayley-Dickson construction, the same construction that in turn builds the sedenions from the octonions. Octonions are noncommutative and, unlike the quaternions, nonassociative, though they satisfy weaker forms of associativity called alternative and power associative. They are much less widely studied than the quaternions and complex numbers, but connect to several exceptional structures in mathematics, among them the exceptional Lie groups, and have applications in string theory, special relativity and quantum logic.