In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold . There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative.