Sedenion

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In abstract algebra, sedenions form a 16-dimensional algebra over the reals. The set of sedenions is denoted as $\mathbb{S}$ . Two types are currently known:

Sedenions obtained by applying the Cayley-Dickson construction
Conic sedenions ("16-dimensional M-algebra") after Charles Musès, part of his hypernumber concept.

1 Cayley-Dickson Sedenions
- 1.1 Arithmetic
- 1.2 Further reading
2 Conic sedenions / "16-dim. M-algebra"
- 2.1 Arithmetic
3 See also

[edit] Cayley-Dickson Sedenions

[edit] Arithmetic

Like (Cayley-Dickson) octonions, multiplication of Cayley-Dickson sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of power associativity.

Every sedenion is a real linear combination of the unit sedenions 1, e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀, e₁₁, e₁₂, e₁₃, e₁₄ and e₁₅, which form a basis of the vector space of sedenions.

The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. This is because they have zero divisors; this means that two non-zero numbers can be multiplied to obtain a zero result: a trivial example is (e₃ + e₁₀)*(e₆ - e₁₅). All hypercomplex number systems based on the Cayley-Dickson construction from sedenions on contain zero divisors.

The multiplication table of these unit sedenions follows:

×	1	e₁	e₂	e₃	e₄	e₅	e₆	e₇	e₈	e₉	e₁₀	e₁₁	e₁₂	e₁₃	e₁₄	e₁₅
1	1	e₁	e₂	e₃	e₄	e₅	e₆	e₇	e₈	e₉	e₁₀	e₁₁	e₁₂	e₁₃	e₁₄	e₁₅
e₁	e₁	-1	e₃	-e₂	e₅	-e₄	-e₇	e₆	e₉	-e₈	-e₁₁	e₁₀	-e₁₃	e₁₂	e₁₅	-e₁₄
e₂	e₂	-e₃	-1	e₁	e₆	e₇	-e₄	-e₅	e₁₀	e₁₁	-e₈	-e₉	-e₁₄	-e₁₅	e₁₂	e₁₃
e₃	e₃	e₂	-e₁	-1	e₇	-e₆	e₅	-e₄	e₁₁	-e₁₀	e₉	-e₈	-e₁₅	e₁₄	-e₁₃	e₁₂
e₄	e₄	-e₅	-e₆	-e₇	-1	e₁	e₂	e₃	e₁₂	e₁₃	e₁₄	e₁₅	-e₈	-e₉	-e₁₀	-e₁₁
e₅	e₅	e₄	-e₇	e₆	-e₁	-1	-e₃	e₂	e₁₃	-e₁₂	e₁₅	-e₁₄	e₉	-e₈	e₁₁	-e₁₀
e₆	e₆	e₇	e₄	-e₅	-e₂	e₃	-1	-e₁	e₁₄	-e₁₅	-e₁₂	e₁₃	e₁₀	-e₁₁	-e₈	e₉
e₇	e₇	-e₆	e₅	e₄	-e₃	-e₂	e₁	-1	e₁₅	e₁₄	-e₁₃	-e₁₂	e₁₁	e₁₀	-e₉	-e₈
e₈	e₈	-e₉	-e₁₀	-e₁₁	-e₁₂	-e₁₃	-e₁₄	-e₁₅	-1	e₁	e₂	e₃	e₄	e₅	e₆	e₇
e₉	e₉	e₈	-e₁₁	e₁₀	-e₁₃	e₁₂	e₁₅	-e₁₄	-e₁	-1	-e₃	e₂	-e₅	e₄	e₇	-e₆
e₁₀	e₁₀	e₁₁	e₈	-e₉	-e₁₄	-e₁₅	e₁₂	e₁₃	-e₂	e₃	-1	-e₁	-e₆	-e₇	e₄	e₅
e₁₁	e₁₁	-e₁₀	e₉	e₈	-e₁₅	e₁₄	-e₁₃	e₁₂	-e₃	-e₂	e₁	-1	-e₇	e₆	-e₅	e₄
e₁₂	e₁₂	e₁₃	e₁₄	e₁₅	e₈	-e₉	-e₁₀	-e₁₁	-e₄	e₅	e₆	e₇	-1	-e₁	-e₂	-e₃
e₁₃	e₁₃	-e₁₂	e₁₅	-e₁₄	e₉	e₈	e₁₁	-e₁₀	-e₅	-e₄	e₇	-e₆	e₁	-1	e₃	-e₂
e₁₄	e₁₄	-e₁₅	-e₁₂	e₁₃	e₁₀	-e₁₁	e₈	e₉	-e₆	-e₇	-e₄	e₅	e₂	-e₃	-1	e₁
e₁₅	e₁₅	e₁₄	-e₁₃	-e₁₂	e₁₁	e₁₀	-e₉	e₈	-e₇	e₆	-e₅	-e₄	e₃	e₂	-e₁	-1

[edit] Further reading

Imaeda, K., Imaeda, M.: Sedenions: algebra and analysis, Applied Mathematics and Computation, 115:77-88 (2000)

Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: C-loops: Extensions and constructions, Journal of Algebra and its Applications 6 (2007), no. 1, 1-20. [1]

[edit] Conic sedenions / "16-dim. M-algebra"

[edit] Arithmetic

In contrast to Cayley-Dickson sedenions, which are built on one and 15 roots of negative one, conic sedenions are built on 8 square roots each of positive and negative one. They share non-commutativity and non-associativity with Cayley-Dickson sedenion ("circular sedenion") arithmetic, however, conic sedenions are modular, alternative, and flexible. With the exception of its nilpotents, zero divisors, and zero itself, the arithmetic is closed with respect to the power-of and logarithm operations. Conic sedenions are not power-associative.

For detailed information and isomorphic subalgebras see Musean hypernumber.

[edit] See also

v • d • e

Number systems

Basic		Complex extensions		Other extensions

Natural numbers $\mathbb{N}$ Negative numbers Integers $\mathbb{Z}$ Rational numbers $\mathbb{Q}$ Irrational numbers	Real numbers $\mathbb{R}$ Imaginary numbers $\mathbb{I}$ Complex numbers $\mathbb{C}$ Algebraic numbers $\mathbb{A}$ Transcendental numbers	Quaternions $\mathbb{H}$ Octonions $\mathbb{O}$ Sedenions $\mathbb{S}$ Cayley-Dickson construction Split-complex numbers $\mathbb{R}^{1,1}$	Bicomplex numbers Biquaternions Coquaternions Tessarines Hypercomplex numbers	Musean hypernumbers Superreal numbers Hyperreal numbers Surreal numbers Dual numbers Transfinite numbers