Octonions

The octonions are an eight-dimensional number system that extends the quaternions in the same way the quaternions extend the complex numbers, produced from the quaternions by the Cayley-Dickson construction, a doubling procedure that keeps producing larger number systems at the cost of losing some familiar algebraic properties at each step. The octonions were discovered in 1843 by the Irish mathematician John T. Graves, shortly after William Rowan Hamilton's discovery of the quaternions, and independently by the English mathematician Arthur Cayley in 1845, whose name is more often attached to them today. Octonion multiplication is noncommutative, meaning the order of multiplying two octonions matters, and also nonassociative, meaning the grouping of three or more factors being multiplied can change the result, a further loss of structure beyond what the quaternions already give up, though the octonions still form a normed division algebra, one of only four such algebras that exist over the real numbers. Despite their unusual algebra, octonions have found genuine application in areas such as string theory and the study of exceptional Lie groups, where their eight-dimensional structure connects to some of the most exotic symmetry groups known in mathematics.