Sedenions

Definition

Sedenions are an extension of the number system created through the Cayley-Dickson construction, which are 16-dimensional algebras over the real numbers. They are non-associative and exhibit unique properties, such as the existence of zero divisors and the lack of a normed division algebra structure, making them significantly different from their predecessors, such as quaternions and octonions. Understanding sedenions is essential in the study of non-associative rings and their properties, as they serve as an example of how algebraic structures can be extended while losing certain desirable characteristics.

5 Must Know Facts For Your Next Test

1. Sedenions are formed by applying the Cayley-Dickson construction to octonions, effectively doubling their dimensions to create a 16-dimensional algebra.
2. They contain zero divisors, meaning that there exist non-zero elements whose product is zero, which is not true for quaternions or octonions.
3. Sedenions do not have a normed division algebra structure, making them fundamentally different from their lower-dimensional counterparts.
4. The lack of associativity in sedenions means that multiplication is not always well-defined in terms of grouping, leading to more complex behaviors in calculations.
5. Sedenions can be represented as pairs of octonions, and their structure allows for various interesting mathematical properties and applications in theoretical physics.

Review Questions

• How do sedenions illustrate the concept of non-associative rings and what are the implications of their properties?
  • Sedenions serve as a prime example of non-associative rings because they highlight how extending number systems can lead to the loss of associativity. Unlike quaternions and octonions, sedenions exhibit both zero divisors and a lack of a division structure. This illustrates important implications for algebraic structures in mathematics, showing that while you can create new algebras through extensions like the Cayley-Dickson construction, those algebras may lose vital properties that make them useful or manageable.
• Discuss the significance of zero divisors in sedenions compared to lower-dimensional algebras like quaternions and octonions.
  • The presence of zero divisors in sedenions is a significant characteristic that differentiates them from quaternions and octonions. In quaternions and octonions, the product of two non-zero elements will always yield a non-zero result. However, in sedenions, it's possible for two distinct non-zero elements to multiply to zero. This property complicates their use in applications where division is needed and raises questions about how these algebras can be applied within theoretical frameworks where division is essential.
• Analyze how sedenions could be used in theoretical physics compared to other algebras like quaternions and octonions.
  • Sedenions could potentially offer unique insights in theoretical physics due to their high dimensionality and complex algebraic properties. While quaternions and octonions have established roles in modeling rotations and certain physical phenomena, sedenions' lack of division may suggest alternative frameworks for exploring concepts such as field theories or higher-dimensional spaces. However, the complications arising from their non-associativity and existence of zero divisors necessitate careful treatment when applying them in physics. This invites researchers to consider novel applications or challenges in formulating physical theories using these more complex algebras.

Related terms

Cayley-Dickson Construction: A method for generating new algebras from existing ones by doubling their dimensions, leading to structures like complex numbers, quaternions, octonions, and sedenions.

Quaternions: A number system that extends complex numbers and can be represented as a four-dimensional vector space over the reals; they are non-commutative but associative.

Octonions: An eight-dimensional extension of the real numbers that forms a non-associative algebra, exhibiting both non-commutativity and non-associativity.