Octonion

In mathematics, the octonion number system extends the complex numbers into eight dimensions. It is represented using the symbol $\mathbb {O}$ . The 16-dimensional sedenions come after the octonions.

History

The octonions were first described by Irish mathematician John T. Graves in 1843, who originally called them "octaves".^[1] They were independently described by Arthur Cayley in 1845.

Introduction

The octonions take on the following form, with 8 total elements. $e 0$ is a real number, and the others are imaginary units belonging to 7 different dimensions.

x = e 0 + e 1 + e 2 + e 3 + e 4 + e 5 + e 6 + e 7

Multiplication

The Fano plane is a diagram that shows how another octonion element is obtained when two octonion elements are multiplied with each other.

The two examples below illustrate how a positive product is obtained when moving along with directions of the arrows in the Fano plane.^[2]

e 5 e 3 = e 6

e 7 e 6 = e 1

The two examples below illustrate how a negative product is obtained when moving against the directions of the arrows in the Fano plane.^[2]

e 2 e 4 = -e 6

e 7 e 5 = -e 2

Both quaternions and octonions are non-commutative, but octonions are also non-associative. However, quaternions are associative. The example below shows how the results of multiplying $e 5$ , $e 2$ , $e 4$ change when they are grouped differently (in order words, when the order of operations differ).^[2]

(e 5 e 3)e 1 = (e 6)e 1 = e 7

e 5 (e 3 e 1) = e 5 (e 2) = -e 7

References

↑ Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. S2CID 586512.
↑ ^2.0 ^2.1 ^2.2 Wolchover, Natalie (2018-07-20). "The Peculiar Math That Could Underlie the Laws of Nature". Quanta Magazine. Retrieved 2024-10-09.

Other websites

Media related to Octonions at Wikimedia Commons

[1] Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. S2CID 586512.

[Quanta-2] 2.0 ^2.1 ^2.2 Wolchover, Natalie (2018-07-20). "The Peculiar Math That Could Underlie the Laws of Nature". Quanta Magazine. Retrieved 2024-10-09.

[1]

[2]

v t e Number systems
Commonly used systems	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ )
Hypercomplex numbers	Quaternions ( $\mathbb {H}$ ) Octonions ( $\mathbb {O}$ ) Sedenions ( $\mathbb {S}$ ) Trigintaduonions ( $\mathbb {T}$ )

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