Editing Quaternion (section)

{{Short description|Noncommutative extension of the complex numbers}}
{{About|quaternions in mathematics}}
{|class="wikitable" align="right" style="text-align:center; margin-left:0.5em; max-width: 230px;"

|+Quaternion multiplication table
|-
!width=15 nowrap|&darr;&thinsp;&times;&thinsp;&rarr;
!width=15|{{math|1}}
!width=15|{{math|'''i'''}}
!width=15|{{math|'''j'''}}
!width=15|{{math|'''k'''}}
|-
!{{math|1}}
|{{math|1}}
|{{math|'''i'''}}
|{{math|'''j'''}}
|{{math|'''k'''}}
|-
!{{math|'''i'''}}
|{{math|'''i'''}}
|{{math|−1}}
|{{math|'''k'''}}
|{{math|−'''j'''}}
|-
!{{math|'''j'''}}
|{{math|'''j'''}}
|{{math|−'''k'''}}
|{{math|−1}}
|{{math|'''i'''}}
|-
!{{math|'''k'''}}
|{{math|'''k'''}}
|{{math|'''j'''}}
|{{math|−'''i'''}}
|{{math|−1}}
|-
|colspan=5 |Left column shows the left factor, top row shows the right factor. Also, <math>a\mathbf{b}=\mathbf{b}a</math> and <math>-\mathbf{b} = (-1)\mathbf{b}</math> for <math>a\in \mathbb{R} </math>, <math>\mathbf{b}  = \mathbf{i}, \mathbf{j}, \mathbf{k}  </math>.
|}
[[File:Cayley_Q8_multiplication_graph.svg|thumb|[[Cayley graph|Cayley Q8 graph]] showing the six cycles of multiplication by {{red|'''i'''}}, {{green|'''j'''}} and {{blue|'''k'''}}. (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 number]]s. Quaternions were first described by the Irish mathematician [[William Rowan Hamilton]] in 1843<ref>{{cite journal |title=On Quaternions; or on a new System of Imaginaries in Algebra |journal=Letter to John T. Graves |date=17 October 1843}}</ref><ref>{{cite book |url=https://books.google.com/books?id=DRLpAFZM7uwC&pg=PA385 |title=The history of non-euclidean geometry: Evolution of the concept of a geometric space |year=1988 |publisher=Springer |first=Boris Abramovich |last=Rozenfelʹd |page=385 |isbn=9780387964584}}</ref> and applied to [[mechanics]] in [[three-dimensional space]]. The algebra of quaternions is often denoted by {{math|'''H'''}} (for ''Hamilton''), or in [[blackboard bold]] by <math>\mathbb H.</math> Quaternions are not a [[Field (mathematics)|field]], because multiplication of quaternions is not, in general, [[commutative]]. Quaternions provide a definition of the quotient of two [[vector (mathematics and physics)|vector]]s in a three-dimensional space.<ref>{{cite book |url=https://archive.org/details/bub_gb_TCwPAAAAIAAJ |quote=quaternion quotient lines tridimensional space time |title=Hamilton |page=[https://archive.org/details/bub_gb_TCwPAAAAIAAJ/page/n188 60] |year=1853 |publisher=Hodges and Smith}}</ref><ref>{{cite book |url=https://books.google.com/books?id=YNE2AAAAMAAJ&q=quotient+two+vectors+called+quaternion |title=Hardy 1881 |page=32 |year=1881 |publisher=Ginn, Heath, & co.|isbn=9781429701860 }}</ref> Quaternions are generally represented in the form

<math display=block>a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k,</math>

where the coefficients {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are [[real number]]s, and {{math|1, '''i''', '''j'''}}, {{math|'''k'''}} are the ''basis vectors'' or ''basis elements''.<ref>{{ citation | last = Curtis | first = Morton L. | title = Matrix Groups | edition = 2nd | location = New York | publisher = [[Springer-Verlag]] | year = 1984 | isbn = 978-0-387-96074-6 | page=10 }}</ref>

Quaternions are used in [[pure mathematics]], but also have practical uses in [[applied mathematics]], particularly for [[quaternions and spatial rotation|calculations involving three-dimensional rotations]], such as in [[3D computer graphics|three-dimensional computer graphics]], [[computer vision]], robotics, [[magnetic resonance imaging]]<ref name=Mamone>{{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }}</ref> and [[Texture (crystalline)|crystallographic texture]] analysis.<ref name=Lunze>{{cite journal |first1=Karsten |last1=Kunze |first2=Helmut |last2=Schaeben |title=The Bingham distribution of quaternions and its spherical radon transform in texture analysis |journal=Mathematical Geology |date=November 2004 |volume=36 |issue=8 |pages=917–943 |doi=10.1023/B:MATG.0000048799.56445.59|bibcode=2004MatGe..36..917K |s2cid=55009081 }}</ref> They can be used alongside other methods of rotation, such as [[Euler angles]] and [[rotation matrix|rotation matrices]], or as an alternative to them, depending on the application.

In modern terms, quaternions form a four-dimensional [[associative algebra|associative]] [[composition algebra|normed]] [[division algebra]] over the real numbers, and therefore a ring, also a [[division ring]] and a [[domain (ring theory)|domain]]. It is a special case of a [[Clifford algebra]], [[Classification of Clifford algebras|classified]] as <math>\operatorname{Cl}_{0,2}(\mathbb R)\cong \operatorname{Cl}_{3,0}^+(\mathbb R).</math> It was the first noncommutative division algebra to be discovered.

According to the [[Frobenius theorem (real division algebras)|Frobenius theorem]], the algebra <math>\mathbb H</math> is one of only two finite-dimensional [[division ring]]s containing a proper [[subring]] [[isomorphism|isomorphic]] to the real numbers; the other being the complex numbers. These rings are also [[Hurwitz's theorem (composition algebras)|Euclidean Hurwitz algebras]], of which the quaternions are the largest [[associative algebra]] (and hence the largest ring). Further extending the quaternions yields the [[Non-associative algebra|non-associative]] [[octonion]]s, which is the last [[normed division algebra]] over the real numbers. The next extension gives the [[sedenions]], which have [[zero divisor]]s and so cannot be a normed division algebra.<ref>{{cite web |url=http://www.tony5m17h.net/sedenion.html |title=Why not sedenion? |first=Frank (Tony) |last=Smith |access-date=8 June 2018 |archive-date=14 January 2024 |archive-url=https://web.archive.org/web/20240114134612/https://www.tony5m17h.net/sedenion.html |url-status=dead }}</ref>

The [[unit quaternion]]s give a [[group (mathematics)|group]] structure on the [[3-sphere]] {{math|S<sup>3</sup>}} isomorphic to the groups [[Spin(3)]] and [[SU(2)]], i.e. the [[universal cover]] group of [[SO(3)]]. The positive and negative basis vectors form the eight-element [[quaternion group]].

[[File:Quaternion 2.svg|thumb|right|Graphical representation of products of quaternion units as 90° rotations in the planes of 4-dimensional space spanned by two of {{math|{1, '''i''', '''j''', '''k'''}.}} The left factor can be viewed as being rotated by the right factor to arrive at the product. Visually {{Math|1={{font color|blue|'''i{{nbsp}}⋅{{nbsp}}j'''}} = −{{font color|red|('''j{{nbsp}}⋅{{nbsp}}i''')}}}}.
{{bulleted list |
{{font color|blue|In ''blue'':
{{bulleted list |{{Math|1=1{{nbsp}}⋅{{nbsp}}'''i'''{{nbsp}}{{=}}{{nbsp}}'''i'''}} (1/'''i''' plane) | {{Math|1='''i'''{{nbsp}}⋅{{nbsp}}'''j'''{{nbsp}}{{=}}{{nbsp}}'''k'''}} ('''i'''/'''k''' plane)
}} }} |
{{font color|red|In ''red'':
{{bulleted list |{{Math|1=1{{nbsp}}⋅{{nbsp}}'''j'''{{nbsp}}{{=}}{{nbsp}}'''j'''}} (1/'''j''' plane) | {{Math|1='''j'''{{nbsp}}⋅{{nbsp}}'''i'''{{nbsp}}{{=}}{{nbsp}}−'''k'''}} ('''j'''/'''k''' plane)
}}
}}
}}]]