Los octoniones son la extensión no asociativa de los cuaterniones. Fueron descubiertos por John T. Graves en 1843, e independientemente por Arthur Cayley, quien lo publicó por primera vez en 1845. Son llamados, a veces números de Cayley. Los octoniones forman un álgebra 8-dimensional sobre los números reales y pueden ser comprendidos como un octeto ordenado de números reales. Cada octonión forma una combinación lineal de la base: 1, e1, e2, e3, e4, e5, e6, e7.La forma de multiplicar octoniones está dada en la tabla siguiente: Este producto no es conmutativo ni asociativo. A causa de esta no asociatividad, los octoniones, a diferencia de los cuaterniones, no admiten una representación matricial.

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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. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this, they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic. The octonions were discovered in 1843 by John T. Graves, inspired by his friend W. R. Hamilton's discovery of quaternions. Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December 1843, but his first publication of his result in () was slightly later than Arthur Cayley's article on them. The octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves' discovery. Hamilton invented the word associative so that he could say that octonions were not associative.

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