Sedenion - Citizendia

In abstract algebra, sedenions form a 16-dimensional algebra over the reals. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, the real numbers may be described informally in several different ways 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. Charles A Muses (1919&ndash2000 a figure who wrote articles and books under various pseudonyms (including Musès Musaios Kyril Demys Arthur Fontaine Kenneth Demarest and Carl Musean hypernumbers are an algebraic concept envisioned by Charles A

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

Cayley-Dickson Sedenions

Arithmetic

Like (Cayley-Dickson) octonions, multiplication of Cayley-Dickson sedenions is neither commutative nor associative. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have But in contrast to the octonions, the sedenions do not even have the property of being alternative. In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only alternative. They do, however, have the property of power associativity. In Abstract algebra, power associativity is a weak form of 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. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible 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). In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 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

Conic sedenions / "16-dim. M-algebra"

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. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 Conic sedenions are not power-associative.

For detailed information and isomorphic subalgebras see Musean hypernumber. Musean hypernumbers are an algebraic concept envisioned by Charles A

Contents

Cayley-Dickson Sedenions

Arithmetic

Further reading

Conic sedenions / "16-dim. M-algebra"

Arithmetic

See also