Quaternions means: plural of quaternion. Below you'll find 20 example sentences showing how to use Quaternions in practice.
Quaternions meaning
plural of quaternion
Example types with quaternions
Below, the same example sentences are grouped by length and sentence type:
After Hamilton's death, his student Peter Tait continued promoting quaternions. (11 words)
Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. (15 words)
He founded a school of "quaternionists", and he tried to popularize quaternions in several books. (15 words)
For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. (41 words)
Explicitly, the Brauer group of the reals consists of two classes, represented by the reals and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another. (41 words)
A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. (38 words)
Example sentences (20)
For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions.
Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions.
Quaternions as pairs of complex numbers main Quaternions can be represented as pairs of complex numbers.
Quaternions Quaternion Plaque on Broom Bridge The other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843.
The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.
Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature of quaternions.
A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions.
After Hamilton's death, his student Peter Tait continued promoting quaternions.
By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the reals.
Cayley–Dickson construction details All of the Clifford algebras Cℓ p,q (R) apart from the real numbers, complex numbers and the quaternions contain non-real elements that square to +1; and so cannot be division algebras.
Explicitly, the Brauer group of the reals consists of two classes, represented by the reals and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another.
For example: * Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.
For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude.
From this perspective, quaternions are the result of applying the Cayley–Dickson construction to the complex numbers.
Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research.
Hamilton retained his faculties unimpaired to the very last, and steadily continued the task of finishing the Elements of Quaternions which had occupied the last six years of his life.
Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties.
He failed to find a useful 3-dimensional system (in modern terminology, he failed to find a real, three-dimensional skew-field ), but in working with four dimensions he created quaternions.
He founded a school of "quaternionists", and he tried to popularize quaternions in several books.
Common combinations with quaternions
These word pairs occur most frequently in English texts:
- the quaternions 13×
- of quaternions 10×
- quaternions are 6×
- quaternions in 4×
- as quaternions 3×
- quaternions and 3×
- quaternions to 3×
- imaginary quaternions 2×
- quaternions can 2×
- like quaternions 2×