In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface or blackboard bold . There are three lower dimensional algebras: the real numbers , the complex numbers , and the quaternions . The Higher-dimensional algebras include the sedenions . 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.