What are quaternions and octonions?
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. They are also power associative.
Are octonions Clifford algebra?
but the octonions are not a Clifford algebra, since they are nonassociative. we can describe them with reference to their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups.
Who discovered octonions?
John Graves
John Graves, a lawyer friend of Hamilton’s, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space. John Graves, the Irish lawyer and mathematician who discovered the octonions in 1843.
Are the octonions a field?
The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. Octonion algebras are always nonassociative. They are, however, alternative algebras, alternativity being a weaker form of associativity.
How are octonions related to the quaternions?
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic.
How are octonions represented in a division algebra?
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.
How is an octonion related to a real number?
The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions: where e 0 is the scalar or real element; it may be identified with the real number 1. That is, every octonion x can be written in the form. with real coefficients {x i}.
Which is the only generalization of the quaternions?
When one wishes to retain the multiplicativity of the norm, N(XY) = N(X)N(Y) , there is only one possible generalization of the quaternions (over the reals): the octaves or octonions, which have 8 instead of 4 components (Hurwitz’s theorem, 1898; cf. Cayley numbers ). The centre of the skew-field of quaternions is the field of real numbers.