Sedenion

Sedenions
Symbol	${\displaystyle \mathbb {S} }$
Type	Hypercomplex algebra
Units	e0, ..., e15
Multiplicative identity	e0
Main properties	Power associativity Distributivity
Common systems
${\displaystyle \mathbb {N} }$ Natural numbers ${\displaystyle \mathbb {Z} }$ Integers ${\displaystyle \mathbb {Q} }$ Rational numbers ${\displaystyle \mathbb {R} }$ Real numbers ${\displaystyle \mathbb {C} }$ Complex numbers ${\displaystyle \mathbb {H} }$ Quaternions Less common systems ${\displaystyle \mathbb {O} }$ Octonions ${\displaystyle \mathbb {S} }$ Sedenions ${\displaystyle \mathbb {T} }$ Trigintaduonions

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface S or blackboard bold ${\displaystyle \mathbb {S} }$

The sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mathematically expressed as ${\displaystyle \mathbb {S} ={\mathcal {CD}}(\mathbb {O} ,1)}$.^[1] As such, the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, called the trigintaduonions or sometimes the 32-nions.^[2] It is possible to continue applying the Cayley–Dickson construction arbitrarily many times.

The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matrices over the real numbers, or that studied by Smith (1995).