quaternion
Sign in to save{| class="wikitable" align="right" style="text-align:center; margin-left:0.5em; max-width: 230px;" |+ Quaternion multiplication table |- |width=15| !width=15| !width=15| !width=15| !width=15| |- ! | | | | |- ! | | | | |- ! | | | | |- ! | | | | |- |colspan=5| Left column shows the left factor, top row shows the right factor. Also, a\mathbf{b}=\mathbf{b}a and -\mathbf{b} = (-1)\mathbf{b} for a\in \mathbb{R} , \mathbf{b} = \mathbf{i}, \mathbf{j}, \mathbf{k} . |} thumb|Cayley graph of the [[quaternion group showing the six cycles of multiplication by , and . (If the image
AI overview
I cannot write an accurate overview based solely on this context. The provided material is only a quaternion multiplication table with technical notation—it shows *how* quaternions multiply but contains no information about what quaternions are, their definition, or why they matter. To write a responsible overview for a general reader, I would need contextual material that explains quaternions' nature and significance.
AI-generated from the Wikipedia summary — may contain errors.
Article
36 sectionsContents
- History
- Quaternions in physics
- Definition
- Multiplication of basis elements
- Center
- Hamilton product
- Scalar and vector parts
- Conjugation, the norm, and reciprocal
- Unit quaternion
- Algebraic properties
- Quaternions and three-dimensional geometry
- Matrix representations
- Representation as complex 2 × 2 matrices
- Representation as real 4 × 4 matrices
- Lagrange's four-square theorem
- Quaternions as pairs of complex numbers
- Square roots
- Square roots of −1
- As a union of complex planes
- Commutative subrings
- Square roots of arbitrary quaternions
- Functions of a quaternion variable
- Exponential, logarithm, and power functions
- Geodesic norm
- Three-dimensional and four-dimensional rotation groups
- Quaternion algebras
- Quaternions as the even part of {{math|Cl<sub>3,0</sub>(ℝ)}}
- Brauer group
- Quotations
- See also
- Notes
- References
- Further reading
- Books and publications
- Links and monographs
- External links
{| class="wikitable" align="right" style="text-align:center; margin-left:0.5em; max-width: 230px;" |+ Quaternion multiplication table |- |width=15| !width=15| !width=15| !width=15| !width=15| |- ! | | | | |- ! | | | | |- ! | | | | |- ! | | | | |- |colspan=5| Left column shows the left factor, top row shows the right factor. Also, a\mathbf{b}=\mathbf{b}a and -\mathbf{b} = (-1)\mathbf{b} for a\in \mathbb{R} , \mathbf{b} = \mathbf{i}, \mathbf{j}, \mathbf{k} . |} thumb|Cayley graph of the [[quaternion group showing the six cycles of multiplication by , and . (If the image is opened in the Wikimedia Commons by clicking twice on it, cycles can be highlighted by hovering over or clicking on them.)]]
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The set of all quaternions is conventionally denoted by \ \mathbb H\ ('H' for Hamilton) or by Quaternions are not a field because multiplication of quaternions is not commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form