Get to know Octonions better with 10+ real example sentences, the meaning.
Octonions meaning
plural of octonion
Using Octonions
- The main meaning on this page is: plural of octonion
- In the example corpus, octonions often appears in combinations such as: octonions is, the octonions, integral octonions.
Context around Octonions
- Average sentence length in these examples: 20.4 words
- Position in the sentence: 10 start, 10 middle, 0 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis for Octonions
- In this selection, "octonions" usually appears near the start of the sentence. The average example has 20.4 words, and this corpus slice is mostly made up of statements.
- Around the word, integral, two, irreducible and usually stand out and add context to how "octonions" is used.
- Recognizable usage signals include addition of octonions is also and contain the octonions e 0. That gives this page its own corpus information beyond isolated example sentences.
- By corpus frequency, "octonions" sits close to words such as aaditya, aardman and abbess, which helps place it inside the broader word index.
Example types with octonions
The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:
Split-octonions are non-associative and contain nilpotents. (8 words)
The octonions were discovered independently by Cayley citation. (8 words)
Addition of octonions is also associative, but multiplication of octonions is non-associative. (13 words)
So, they contain the octonions ( e 0 to e 7 in the table below), and therefore also the quaternions ( e 0 to e 3 ), complex numbers ( e 0 and e 1 ) and reals ( e 0 ). (35 words)
Hamilton invented the word associative so that he could say that octonions were not associative. citation Definition The octonions can be thought of as octets (or 8-tuples) of real numbers. (31 words)
It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively. (28 words)
Example sentences (20)
Addition of octonions is also associative, but multiplication of octonions is non-associative.
Hamilton invented the word associative so that he could say that octonions were not associative. citation Definition The octonions can be thought of as octets (or 8-tuples) of real numbers.
Integral octonions There are several natural ways to choose an integral form of the octonions.
The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions.
A convenient mnemonic for remembering the products of unit octonions is given by the diagram at the right, which represents the multiplication table of Cayley and Graves.
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions.
Cayley Dickson construction A more systematic way of defining the octonions is via the Cayley Dickson construction.
Commutator and cross product The commutator of two octonions x and y is given by : This is antisymmetric and imaginary.
It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively.
Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions.
Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the product of all the terms, again like quaternions.
Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used.
So, they contain the octonions ( e 0 to e 7 in the table below), and therefore also the quaternions ( e 0 to e 3 ), complex numbers ( e 0 and e 1 ) and reals ( e 0 ).
Split-octonions are non-associative and contain nilpotents.
The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system.
The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the E 8 lattice.
The isotopy group of the octonions is the group Spin 8 (R), with a, b, and c acting as the three 8-dimensional representations.
The octonions were discovered independently by Cayley citation.
The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders.
Common combinations with octonions
These word pairs occur most frequently in English texts:
- octonions is 7×
- the octonions 7×
- integral octonions 4×
- of octonions 3×
- octonions can 3×
- octonions were 2×
- octonions are 2×
- two octonions 2×
- octonions and 2×
- and octonions 2×