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Concept
Octonion
Summary
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. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
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. Applying the Cayley–Dickson construction to the octonions produces the sedenions.
The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions. Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 26 December 1843. He first published his result slightly later than Arthur Cayley's article. 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's discovery.
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 e0 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 xi.
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions.
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