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 ${\displaystyle \mathbb {S} ={\mathcal {CD}}(\mathbb {O} ,1)}$. 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.

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).

Every sedenion is a linear combination of the unit sedenions ${\displaystyle e_{0}}$, ${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$, ${\displaystyle e_{3}}$, ..., ${\displaystyle e_{15}}$, which form a basis of the vector space of sedenions. Every sedenion can be represented in the form

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.

Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra they were constructed from. So they contain the octonions (generated by ${\displaystyle e_{0}}$ to ${\displaystyle e_{7}}$ in the table below), and therefore also the quaternions (generated by ${\displaystyle e_{0}}$ to ${\displaystyle e_{3}}$), complex numbers (generated by ${\displaystyle e_{0}}$ and ${\displaystyle e_{1}}$) and real numbers (generated by ${\displaystyle e_{0}}$).

Like octonions, multiplication of sedenions is neither commutative nor associative. However, 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, which can be stated as that, for any element ${\displaystyle x}$ of ${\displaystyle \mathbb {S} }$, the power ${\displaystyle x^{n}}$ is well defined. They are also flexible.

The sedenions have a multiplicative identity element ${\displaystyle e_{0}}$ and multiplicative inverses, but they are not a division algebra because they have zero divisors: two nonzero sedenions can be multiplied to obtain zero, for example ${\displaystyle (e_{3}+e_{10})(e_{6}-e_{15})}$. All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.