Octonion
Introduction
The octonion is a type of non-associative algebra over the real numbers. It is an extension of the quaternions, and like them, it is a normed division algebra. However, unlike the quaternions, the octonions do not form an associative algebra. They were discovered by John T. Graves in 1843 and independently by Arthur Cayley in 1845.
Definition
An octonion is a linear combination of the unit octonions 1, e1, e2, e3, e4, e5, e6, and e7. The multiplication of these basis elements is defined by a multiplication table, which can be visualized as a Fano plane. The multiplication is neither commutative nor associative, but it is alternative and satisfies the Moufang identities.
Properties
The octonions have a number of interesting properties. They are a normed division algebra, which means that every non-zero octonion has a multiplicative inverse. However, they are not associative, which means that the order in which multiplication is performed matters. They are also not commutative, which means that the order of the operands in a multiplication operation matters.
The octonions also have a norm, which is a function that assigns a positive length or size to each octonion. The norm of an octonion is always a non-negative real number, and it satisfies the property that the norm of a product of two octonions is the product of their norms.
Applications
The octonions have found applications in various areas of mathematics and physics. In mathematics, they are used in the study of exceptional Lie groups, Jordan algebras, and projective planes. In physics, they have been used in the study of quantum mechanics, string theory, and M-theory.