View source for Quaternion

{{Short description|Four-dimensional number system}}
{{About|a number system}}
{| class="wikitable" align="right" style="text-align:center; margin-left:0.5em; max-width: 230px;"
|+ Quaternion multiplication table
|-
|width=15| {{nobr|↓ × →}}
!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]] of the [[quaternion group]] {{math|Q<sub>8</sub>}} 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 '''quaternions''' form a [[number system]] similar to the [[complex number]]s, with the usual [[arithmetical operations]] of [[addition]], [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]], but with four [[real number|real-number]] components instead of two. Unlike with the complex numbers, quaternion multiplication is not [[commutative]], meaning that the result of multiplying two quaternions depends on their order. Quaternions can be used to represent [[vector (mathematics and physics)|vector]]s in [[three-dimensional space]], which provides a definition of the quotient of two vectors.<ref>{{cite book |last=Hamilton |first=W.R., Sir |author-link=William Rowan Hamilton |year=1853 |title=Lectures on Quaternions |publisher=Hodges & Smith |place=Dublin, EI |page=[https://archive.org/details/bub_gb_TCwPAAAAIAAJ/page/n188 60] |quote=quaternion quotient lines tridimensional space time |url=https://archive.org/details/bub_gb_TCwPAAAAIAAJ }}</ref><ref>{{cite journal |last=Hardy |first=A.S. |author-link=Arthur Sherburne Hardy |year=1881 |title=Elements of Quaternions |journal=[[Science (journal)|Science]] |volume=2 |issue=75 |page=32 |publisher=Ginn, Heath, & Co. |doi=10.1126/science.os-2.75.564 |isbn=9781429701860 |pmid=17819877 |url=https://books.google.com/books?id=YNE2AAAAMAAJ&q=quotient+two+vectors+called+quaternion }}</ref>

Quaternions were first described by the Irish mathematician and physicist [[William Rowan Hamilton]] in 1843,<ref name=letter1843>Hamilton described quaternions in a letter to [[John T. Graves|John Graves]], 17 October 1843. It was published in {{harvp|Hamilton|1844}}.</ref><ref>{{cite book |first=Boris Abramovich |last=Rozenfelʹd |year=1988 |title=The history of non-euclidean geometry: Evolution of the concept of a geometric space |publisher=Springer |page=385 |isbn=9780387964584  |url=https://books.google.com/books?id=DRLpAFZM7uwC&pg=PA385 }}</ref> and in his honor the set of all quaternions is conventionally denoted by <math>\mathbb H</math> or {{math|'''H'''}}. A generic quaternion is usually 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 numbers, 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.

As an abstract mathematical structure, 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]]. Because of their non-commutative multiplication, they do not form a [[Field (mathematics)|field]]. The quaternions are also 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>

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 [[sedenion]]s, 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)
}}
}}
}}]]

== History ==
<!-- {{refimprove section|date=October 2015}} -->
{{main|History of quaternions}}

[[File:Inscription on Broom Bridge (Dublin) regarding the discovery of Quaternions multiplication by Sir William Rowan Hamilton.jpg|right|thumb|Quaternion plaque on [[Broom Bridge|Brougham (Broom) Bridge]], [[Dublin]], which reads:

{{bi|1= <poem>
     Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
     {{math|1= ''i''{{sup|2}} = ''j''{{sup|2}} = ''k''{{sup|2}} = ''i{{hairsp}}j{{hairsp}}k'' = −1}}
& cut it on a stone of this bridge
</poem> }} ]]

Quaternions were introduced by Hamilton in 1843.<ref name="SeeHazewinkel">See {{harvnb|Hazewinkel|Gubareni|Kirichenko|2004|p=[https://books.google.com/books?id=AibpdVNkFDYC&pg=PA12 12]}}</ref> Important precursors to this work included [[Euler's four-square identity]] (1748) and [[Olinde Rodrigues]]' [[Euler–Rodrigues parameters|parameterization of general rotations by four parameters]] (1840), but neither of these writers treated the four-parameter rotations as an algebra.<ref>{{harvnb|Conway|Smith|2003|p=[https://books.google.com/books?id=E_HCwwxMbfMC&pg=PA9 9]}}</ref><ref name=Bradley-Sandifer-2007>{{cite book |first1=Robert E. |last1=Bradley |first2=Charles Edward |last2=Sandifer |year=2007 |title=Leonhard Euler: Life, work and legacy |isbn=978-0-444-52728-8 |page=193 |publisher=Elsevier |url=https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA193 }}</ref>{{efn|
{{harvp|Bradley|Sandifer|2007|p=193}} mention [[Wilhelm Blaschke]]'s claim in 1959 that "the quaternions were first identified by L.&nbsp;Euler in a letter to Goldbach written on 4&nbsp;May 1748," and they comment that "it makes no sense whatsoever to say that Euler 'identified' the quaternions in this letter ... this claim is absurd."<ref name=Bradley-Sandifer-2007/>
}} [[Carl Friedrich Gauss|Gauss]] had discovered quaternions in 1819, but this work was not published until 1900.<ref>{{cite journal |last=Pujol |first=J. |year=2012 |title=Hamilton, Rodrigues, Gauss, quaternions, and rotations: A historical reassessment |journal=[[Communications in Mathematical Analysis]] |volume=13 |issue=2 |pages=1–14 |url=https://projecteuclid.org/journals/communications-in-mathematical-analysis/volume-13/issue-2/Hamilton-Rodrigues-Gauss-Quaternions-and-Rotations-a-Historical-Reassessment/cma/1349803591.full |via=projecteuclid.org }}</ref><ref>{{cite book |first=C.F. |last=Gauss |author-link=Carl Friedrich Gauss |date=c. 1819 |section=Mutationen des Raumes |trans-section=Transformations of space |others=Prof.&nbsp;Stäckel of Kiel, Germany (article edited by) |editor=Martin Brendel |title=Carl Friedrich Gauss Werke |trans-title=The works of Carl Friedrich Gauss |publication-date=1900 |volume=8 |pages=357–361 |location=Göttingen, DE |publisher=Königlichen Gesellschaft der Wissenschaften [Royal Society of Sciences] |url=https://books.google.com/books?id=aecGAAAAYAAJ&pg=PA357 |via=Google }}</ref>

Hamilton knew that the complex numbers could be interpreted as [[point (geometry)|points]] in a [[plane (mathematics)|plane]], and he was looking for a way to do the same for points in three-dimensional [[space]]. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact, [[Ferdinand Georg Frobenius]] later [[Frobenius theorem (real division algebras)|proved]] in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras: <math>\mathbb {R, C}</math> (complex numbers) and <math>\mathbb H</math> (quaternions) which have dimension 1, 2, and 4 respectively.{{citation needed|date=August 2025}}

The great breakthrough in quaternions finally came on Monday 16&nbsp;October 1843 in [[Dublin]], when Hamilton was on his way to the [[Royal Irish Academy]] to preside at a council meeting. As he walked along the towpath of the [[Royal Canal]] with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the defining formula for the quaternions into the stone of [[Broom Bridge|Brougham Bridge]] with his pocket knife:

<math display=block>\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i\; j\; k} = -1</math>

Although the carving has since faded away, there has been an annual pilgrimage since 1989, called the ''[[Hamilton Walk]]'', for scientists and mathematicians who process from the [[Dunsink Observatory]] to the Royal Canal bridge in remembrance of Hamilton's discovery.

On the following day, Hamilton wrote a letter to his friend and fellow mathematician, [[John T. Graves|J.T. Graves]], describing the train of thought that led to his discovery. The letter was later published in a letter to the ''[[Philosophical Magazine]]'';{{r|letter1843}} Hamilton states:
{{blockquote|And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.{{r|letter1843}} }}

Hamilton called a quadruple with these rules of multiplication a ''quaternion'', and he devoted most of the remainder of his life to studying and teaching them. [[Classical Hamiltonian quaternions|Hamilton's treatment]] is more [[Geometry|geometric]] than the modern approach, which emphasizes quaternions' [[algebra]]ic properties. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books, ''Elements of Quaternions'',<ref name=HamiltonElements>{{cite book |title=Elements of Quaternions |last=Hamilton |first=W.R., Sir |author-link=William Rowan Hamilton |editor-last=Hamilton |editor-first=W.E. |editor-link=William Edwin Hamilton |year=1866 |url=https://archive.org/details/elementsofquater00hamiuoft |location=London |publisher=Longmans, Green, & Co.}}</ref> was 800&nbsp;pages long; it was edited by [[William Edwin Hamilton|his son]] and published shortly after his death.{{Citation needed|date=September 2025}}

After Hamilton's death, the Scottish mathematical physicist [[Peter Guthrie Tait|Peter Tait]] became the chief exponent of quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as [[kinematics]] in space and [[Maxwell's equations]], were described entirely in terms of quaternions. There was even a professional research association, the [[Quaternion Association]], devoted to the study of quaternions and other [[hypercomplex number]] systems.<ref>[https://mathshistory.st-andrews.ac.uk/Societies/Quaternion/ The Quaternion Association] at [[MacTutor History of Mathematics archive]]</ref>

From the mid-1880s, quaternions began to be displaced by [[vector analysis]], which had been developed by [[Josiah Willard Gibbs]], [[Oliver Heaviside]], and [[Hermann von Helmholtz]]. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and [[physics]]. A side-effect of this transition is that [[classical Hamiltonian quaternions|Hamilton's work]] is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to follow.{{Citation needed|date=September 2025}}

However, quaternions have had a revival since the late 20th&nbsp;century, primarily due to their utility in [[Quaternions and spatial rotation|describing spatial rotations]]. The representations of rotations by quaternions are more compact and quicker to compute than the representations by [[matrix (mathematics)|matrices]]. In addition, unlike Euler angles, they are not susceptible to "[[gimbal lock]]". For this reason, quaternions are used in [[computer graphics]],<ref name=Shoemake>{{cite journal |first=Ken |last=Shoemake <!-- |author-link=Ken Shoemake --- n/a --> |year=1985 |title=Animating rotation with quaternion curves |journal=Computer Graphics |volume=19 |issue=3 |pages=245–254 |doi=10.1145/325165.325242 |url=https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf }} Presented at [[SIGGRAPH]] '85.</ref>{{efn|
''[[Tomb Raider (1996 video game)|Tomb Raider]]'' (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth three-dimensional rotations.<ref>{{cite magazine |first=Nick |last=Bobick |date=July 1998 |title=Rotating objects using quaternions |magazine=Game Developer |url=https://www.gamedeveloper.com/programming/rotating-objects-using-quaternions }}</ref>
}} [[computer vision]], [[robotics]],<ref>{{cite book |first=J.M. |last=McCarthy |year=1990 |title=An Introduction to Theoretical Kinematics |publisher=MIT Press |isbn=978-0-262-13252-7 |url=https://books.google.com/books?id=glOqQgAACAAJ }}</ref> [[nuclear magnetic resonance]] image sampling,<ref name=Mamone /> [[control theory]], [[signal processing]], [[Spacecraft attitude control|attitude control]], [[physics]], [[bioinformatics]], [[molecular dynamics]], [[computer simulation]]s, and [[orbital mechanics]]. For example, it is common for the [[Spacecraft attitude control|attitude control]] systems of spacecraft to be commanded in terms of quaternions. Quaternions also have contributed to [[number theory]], because of their relationships with the [[quadratic form]]s.<ref>{{cite book |last=Hurwitz |first=A. |year=1919
|title=Vorlesungen über die Zahlentheorie der Quaternionen |jfm=47.0106.01
|place=Berlin, DE |publisher=J. Springer }} — concerning [[Hurwitz quaternion]]s</ref>

=== Quaternions in physics ===
Hamilton had introduced [[biquaternion]]s in his ''Lectures on Quaternions'', and these were used by [[Ludwik Silberstein]] in 1914 to exhibit the [[Lorentz transformation]]s of [[special relativity]].<ref>[[Ludwik Silberstein]] (1914) [https://archive.org/details/theoryofrelativi00silbrich/page/n7/mode/2up ''The Theory of Relativity''] via [[Internet Archive]]</ref> This representation of Lorentz transformations was also used by [[Cornelius Lanczos]] in 1949.<ref>[[Cornelius Lanczos]] (1949) ''The Variational Principles of Mechanics'', [[University of Toronto Press]] {{ISBN|0-8020-1743-6}}</ref>

The finding of 1924 that in [[quantum mechanics]] the [[Spin (physics)|spin]] of an electron and other matter particles (known as [[spinors]]) can be described using quaternions (in the form of the famous [[Pauli matrices|Pauli spin matrices]]) furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the "[[Plate trick]]").<ref>{{cite web |last=Huerta |first=John |date=27 September 2010 |title=Introducing the quaternions |url=http://math.ucr.edu/~huerta/introquaternions.pdf |via=[[University of California, Riverside|U.C. Riverside]] (math.ucr.edu) |access-date=8 June 2018 |archive-url=https://web.archive.org/web/20141021093534/http://math.ucr.edu/~huerta/introquaternions.pdf |archive-date=2014-10-21 |url-status=live }}</ref><ref>{{cite web |first=Charlie |last=Wood |title=The strange numbers that birthed modern algebra |date=6 September 2018 |website=Abstractions |type=blog |publisher=Quanta Magazine |url=https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/}}</ref> {{As of|2018}}, their use has not yet overtaken [[orthogonal group|rotation groups]].{{efn|
A more personal view of quaternions was written by [[Joachim Lambek]] in 1995. He wrote in his essay ''If Hamilton had prevailed: Quaternions in physics'': "My own interest as a graduate student was raised by the inspiring book by Silberstein". He concluded by stating "I firmly believe that quaternions can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics."<ref>{{cite magazine |last=Lambek |first=J. |year=1995 |title=If Hamilton had prevailed: Quaternions in physics |magazine=[[The Mathematical Intelligencer]] |volume=17 |issue=4 |pages=7–15 |doi=10.1007/BF03024783}}</ref>
<!-- end footnote -->}}

[[W. K. Clifford]]<ref name="ref_cfifford_1878">{{cite journal |last1=Clifford |first1= W.K. |date=1878 |title=Applications of Grassmann's Extensive Algebra |url= |journal=American Journal of Mathematics |volume=1 |issue=4 |publisher=The Johns Hopkins University Press |pages=350–358 |doi=10.2307/2369379 |jstor=2369379 |access-date=}}/</ref> (1845 − 1879) introduced his algebras as a tensor product (”compound of algebras”) of quaternion algebras (and its even sub-algebra), a concept introduced by [[Benjamin Peirce|B. Peirce]]<ref>{{cite journal |last1=Peirce |first1=B. |date=1881 |title=Linear associative algebra |url= |journal=American Journal of Mathematics |volume=4 |issue=1 |publisher=Johns Hopkins University |pages=221–226 |doi=10.2307/2369153 |jstor=2369153 |access-date=}}</ref> (1809 − 1880). [[R. Lipschitz]]<ref>{{cite journal |last1=Lipschitz |first1=R. |date=1880 |title=Principes d'un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions |url= |journal=C. R. Acad. Sci. Paris |volume=91 |issue= |language=fr |pages=619–621, 660–664 |doi= |access-date=}}</ref> (1832 − 1903) rediscovered independently the even subalgebra. In 1922, [[Clarence Lemuel Elisha Moore|C. L. E. Moore]]<ref>{{cite journal |last1=Moore |first1=C.L.E. |date=1922 |title=Hyperquaternions |url= |journal=Journal of Mathematics and Physics |volume=1 |issue=2 |pages=63–77 |doi= 10.1002/sapm19221263|access-date=}}</ref> (1876 − 1931) was to call [[Rudolf Lipschitz|Lipschitz]]’ algebras ”hyperquaternions”. The term ”hyperquaternion” designates nowadays both the tensor product of <math>n</math> quaternion algebras <math>\mathbb{H}^{\otimes n}</math> and its even subalgebra <math>\mathbb{H}^{\otimes n-1} \otimes \mathbb{C} </math>.<ref>{{cite journal |last1=Girard |first1=P.R. |last2=Clarysse |first2=P. |last3=Pujol |first3=R. |last4=Goutte |first4=R. |last5=Delachartre |first5=P. |date=2018 |title=Hyperquaternions: a new tool for physics|url=https://doi.org/10.1007/s00006-018-0881-8 |journal=Advances in Applied Clifford Algebras |volume=28 |issue= 3|publisher= Springer |pages=1–14 |article-number=68 |jstor= |doi=10.1007/s00006-018-0881-8 |access-date=}}</ref>

Examples of hyperquaternions are: <math>\mathbb{H},\mathbb{H}^{\otimes 2}=\mathbb{H}\otimes_\mathbb{R}\mathbb{H} </math> (isomorphic to the Clifford algebra <math>Cl_{3,1}\mathbb{(R)}</math> and to <math>4\times 4</math> real matrices <math>M(4,\mathbb{R})</math>) leading to applications in [[special relativity]]. Its even subalgebra is <math>\mathbb{H}\otimes \mathbb{C} </math> ([[biquaternions]]).<ref>{{cite book |last1=Girard |first1=P. R. |title= Quaternions, Clifford algebras and relativistic physics |date=2007 |publisher=Springer Science & Business Media |location= |doi=10.1007/978-3-7643-7791-5 |isbn=978-3-7643-7790-8 |language= |url=https://link.springer.com/book/10.1007/978-3-7643-7791-5}}</ref><ref>{{cite book |last1=G'''ü'''rlebeck |first1=K. |last2=Spr'''ö'''ssig |first2=W. |title=Quaternionic and Clifford calculus for physicists and engineers |date=1997 |publisher=John Wiley & Sons |location=New York |language= |url=https://www.wiley.com/Quaternionic+and+Clifford+Calculus+for+Physicists+and+Engineers-p-9780471962007 }}</ref>

Another example is <math>\mathbb{H}^{\otimes 3}= M(4,\mathbb{H}) </math> yielding a [[quaternionic matrix]] and its even subalgebra <math>\mathbb{H}^{\otimes 2}\otimes_\mathbb{R}\mathbb{C} </math> ([[Dirac algebra]]).<ref>{{cite journal |last1=Girard |first1=P.R. |last2=Clarysse |first2=P. |last3=Pujol |first3=R. |last4=Delachartre |first4=P. |date=2025 |title=Hyperquaternionic unitary symplectic groups: A unifying tool for physics|url= https://hal.science/hal-05172172|journal=Advances in Applied Clifford Algebras |volume=35 |issue=40 |publisher= Springer |article-number=40 |jstor= |doi=10.1007/s00006-025-01402-w |access-date=}}</ref><ref>{{cite book |last1=G'''ü'''rlebeck |first1=K. |last2=Habetha |first2=K. |last3=Spr'''ö'''ssig |first3=W. |title=Application of holomorphic functions in two and higher dimensions |chapter= |date=2016 |pages= |publisher=Springer |location= |doi=10.1007/978-3-0348-0964-1 |isbn=978-3-0348-0962-7 |language= |url=https://doi.org/10.1007/978-3-0348-0964-1}}</ref><ref>{{cite book |last1=Spr'''ö'''ssig |first1=W. |editor-first1=Daniel |editor-first2=Michael Th. |editor-last1=Breaz |editor-last2=Rassias |title=Advancements in Complex Analysis: From Theory to Practice |chapter=Some new aspects in hypercomplex analysis |date=2020 |pages=497–518 |publisher=Springer |location= |doi=10.1007/978-3-030-40120-7 |arxiv=2006.16386 |isbn=978-3-030-40119-1 |language= |url=https://doi.org/10.1007/978-3-030-40120-7}}</ref>

== Definition ==
A ''quaternion'' is an [[expression (mathematics)|expression]] of the form

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

where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}}, are real numbers, and {{math|'''i'''}}, {{math|'''j'''}}, {{math|'''k'''}}, are [[symbol (mathematics)|symbols]] that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} is 0, the corresponding term is omitted; if {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are all zero, the quaternion is the ''zero quaternion'', denoted 0; if one of {{mvar|b}}, {{mvar|c}}, {{mvar|d}}  equals 1, the corresponding term is written simply {{math|'''i''', '''j'''}}, or {{math|'''k'''}}.

A quaternion <math>q = a + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}</math>, can be decomposed into its ''scalar part'' {{tmath|a}} (sometimes ''real part'') and its ''vector part'' <math> q-a=b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}</math> (sometimes ''imaginary part''). A quaternion that equals its real part (that is, its vector part is zero) is called a '''{{vanchor|scalar quaternion}}''' (sometimes ''real quaternion'' or simply ''scalar''), and is identified with the corresponding real number. That is, the real numbers are ''embedded'' in the quaternions.{{efn|More properly, the [[field (mathematics)|field]] of real numbers is isomorphic to a unique [[subring]] of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.<ref>{{harvtxt|Eves|1976|page=391}}</ref>}} A quaternion that equals its vector part is called a '''{{vanchor|vector quaternion}}''' (sometimes ''right quaternion'').

Quaternions form a 4-dimensional [[vector space]] over the real numbers, with <math>\left\{ 1, \mathbf i, \mathbf j, \mathbf k\right\}</math> as a [[basis (linear algebra)|basis]], by the component-wise addition

<math display=block>\begin{align}
& (a_1 + b_1\mathbf i + c_1\mathbf j + d_1\mathbf k) + (a_2 + b_2\mathbf i + c_2\mathbf j + d_2\mathbf k) \\[3mu]
& \qquad = (a_1 + a_2) + (b_1 + b_2)\mathbf i + (c_1 + c_2)\mathbf j + (d_1 + d_2)\mathbf k,
\end{align}</math>

and the component-wise scalar multiplication

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

A multiplicative group structure, called the ''Hamilton product'', denoted by juxtaposition, can be defined on the quaternions in the following way:
* The scalar quaternion {{math|1}} is the [[identity element]].
* The scalar quaternions commute with all other quaternions, that is {{math|''aq'' {{=}} ''qa''}} for every quaternion {{mvar|q}} and every scalar quaternion {{mvar|a}}. In algebraic terminology this is to say that the field of the scalar quaternions is the [[center (ring theory)|''center'']] of the quaternion algebra.
* The product is first given for the basis elements, and then extended to all quaternions by using the [[distributive property]] and the center property of the scalar quaternions (see below for details). The Hamilton product is not [[commutative property|commutative]], but is [[associative property|associative]], thus the quaternions form an [[associative algebra]] over the real numbers.
* Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product: <math display=block>(a + b\,\mathbf i + c\,\mathbf j + d \,\mathbf k)^{-1} = \frac{1}{a^2 + b^2 + c^2 + d^2}\,(a - b\,\mathbf i - c\,\mathbf j- d\,\mathbf k).</math>
Thus the quaternions form a [[division algebra]].

=== Multiplication of basis elements ===
{|class="wikitable" align="right" style="text-align:center"
|+ [[Multiplication table]]
|-
|width=15| {{nobr|↓ × →}}
!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}}
|style="background: #FF9999;"| {{math|'''k'''}}
|style="background: #9999FF;"| {{math|−'''j'''}}
|-
! {{math|'''j'''}}
| {{math|'''j'''}}
|style="background: #FF9999;"| {{math|−'''k'''}}
| {{math|−1}}
|style="background: #99FF99;"| {{math|'''i'''}}
|-
! {{math|'''k'''}}
| {{math|'''k'''}}
|style="background: #9999FF;"| {{math|'''j'''}}
|style="background: #99FF99;"| {{math|−'''i'''}}
| {{math|−1}}
|-
|colspan=5| Non commutativity is emphasized by colored squares 
|}
The multiplication with {{math|1}} of the basis elements {{math|'''i''', '''j'''}}, and {{math|'''k'''}} is defined by the fact that {{math|1}} is a [[multiplicative identity]], that is,

<math display=block>
\mathbf i \, 1 = 1 \, \mathbf i = \mathbf i, \qquad
\mathbf j \, 1 = 1 \, \mathbf j = \mathbf j, \qquad
\mathbf k \, 1 = 1 \, \mathbf k= \mathbf k .
</math>

The products of other basis elements are

<math display=block>\begin{align}
\mathbf i^2 &= \mathbf j^2 = \mathbf k^2 = -1, \\[5mu]
\mathbf{i\,j} &= - \mathbf{j\,i} = \mathbf k, \qquad
\mathbf{j\,k} = - \mathbf{k\,j} = \mathbf i, \qquad
\mathbf{k\,i} = - \mathbf{i\,k} = \mathbf j .
\end{align}</math>

Combining these rules,

<math display=block>\begin{align}
\mathbf{i\,j\,k}&=-1 .
\end{align}</math>

=== Center ===
The [[center (ring theory)|''center'']] of a [[noncommutative ring]] is the subring of elements {{mvar|c}} such that {{math|1=''cx'' = ''xc''}} for every {{mvar|x}}. The center of the quaternion algebra is the subfield of scalar quaternions. In fact, it is a part of the definition that the scalar quaternions belong to the center. Conversely, if {{math|''q'' {{=}} ''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} belongs to the center, then
<math display=block>0 = \mathbf i\,q - q\,\mathbf i = 2c\,\mathbf{i}\mathbf{j} + 2d\,\mathbf{i}\mathbf{k} = 2c\,\mathbf k - 2d\,\mathbf j,</math>

and {{math|''c'' {{=}} ''d'' {{=}} 0}}. A similar computation with {{math|'''j'''}} instead of {{math|'''i'''}} shows that one has also {{math|''b'' {{=}} 0}}. Thus {{math|''q'' {{=}} ''a''}} is a scalar quaternion.

The quaternions form a division algebra. This means that the non-commutativity of multiplication is the only property that makes quaternions different from a field. This non-commutativity has some unexpected consequences, among them that a [[polynomial equation]] over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation {{nowrap|{{math|''z''<sup>2</sup> + 1 {{=}} 0}},}} has infinitely many quaternion solutions, which are the quaternions {{math|''z'' {{=}} ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} such that {{math|''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> {{=}} 1}}. Thus these [[imaginary unit]]s form a [[unit sphere]] in the three-dimensional space of quaternion vectors.

=== Hamilton product ===
For two elements {{math|''a''<sub>1</sub> + ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}} and {{math|''a''<sub>2</sub> + ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}}, their product, called the  '''Hamilton product''' ({{math|''a''<sub>1</sub> + ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}}) ({{math|''a''<sub>2</sub> + ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}}), is determined by the products of the basis elements and the [[distributive law]]. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

<math display=block>\begin{alignat}{4}
        &a_1a_2  &&+ a_1b_2 \mathbf i   &&+ a_1c_2 \mathbf j   &&+ a_1d_2 \mathbf k\\
  {}+{} &b_1a_2 \mathbf i &&+ b_1b_2 \mathbf i^2 &&+ b_1c_2 \mathbf{ij} &&+ b_1d_2 \mathbf{ik}\\
  {}+{} &c_1a_2 \mathbf j &&+ c_1b_2 \mathbf{ji} &&+ c_1c_2 \mathbf j^2 &&+ c_1d_2 \mathbf{jk}\\
  {}+{} &d_1a_2 \mathbf k &&+ d_1b_2 \mathbf{ki} &&+ d_1c_2 \mathbf{kj}  &&+ d_1d_2 \mathbf k^2
\end{alignat}</math>

Now the basis elements can be multiplied using the rules given above to get:<ref name="SeeHazewinkel" />

<math display=block>\begin{alignat}{4}
          &a_1a_2 &&- b_1b_2 &&- c_1c_2 &&- d_1d_2\\
   {}+{} (&a_1b_2 &&+ b_1a_2 &&+ c_1d_2 &&- d_1c_2) \mathbf i\\
   {}+{} (&a_1c_2 &&- b_1d_2 &&+ c_1a_2 &&+ d_1b_2) \mathbf j\\
   {}+{} (&a_1d_2 &&+ b_1c_2 &&- c_1b_2 &&+ d_1a_2) \mathbf k
\end{alignat}</math>

=== Scalar and vector parts ===
A quaternion of the form {{math|''a'' + 0 '''i''' + 0 '''j''' + 0 '''k'''}}, where {{mvar|a}} is a real number, is called a ''scalar quaternion'' (sometimes ''real quaternion''), and a quaternion of the form {{math|0 + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}}, where {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are real numbers, and at least one of {{mvar|b}}, {{mvar|c}}, or {{mvar|d}} is nonzero, is called a ''vector quaternion'' (sometimes ''right quaternion''). For any quaternion {{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}}, {{mvar|a}} is called its ''scalar part'' and {{math|''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} is called its ''vector part''. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space <math>\mathbb R^3.</math>{{efn|
The vector part of a quaternion is a [[pseudovector]] or ''axial vector'', not an ordinary or [[polar vector]].<ref>{{cite book |first=Simon L. |last=Altmann |date=1986 |at=Ch.&nbsp;12 |title=Rotations, Quaternions, and Double Groups |publisher=Clarendon Press |isbn=0-19-855372-2 |lccn=85013615 }}</ref> A polar vector can be represented in calculations (for example, for rotation by a "quaternion similarity transform") by a vector quaternion, with no loss of information, but the two should not be confused. The axis of a "binary" (180°) rotation quaternion corresponds to the direction of the represented polar vector in such a case.
}}

Hamilton also called vector quaternions ''right quaternions''<ref>{{cite book |url=https://archive.org/details/bub_gb_fIRAAAAAIAAJ |title=Elements of Quaternions |publisher=Longmans, Green, & Company |article=Article 285 |page=[https://archive.org/details/bub_gb_fIRAAAAAIAAJ/page/n381 310] |last=Hamilton |first=W.R., Sir |author-link=William Rowan Hamilton |year=1866 }}</ref><ref>{{cite journal |last=Hardy |first=A.S. |author-link=Arthur Sherburne Hardy |year=1881 |title=Elements of Quaternions |journal=[[Science (journal)|Science]] |volume=2 |issue=75 |page=65 or 81 (?) |publisher=Ginn, Heath, & Co. |doi=10.1126/science.os-2.75.564 |isbn=((9781429701860)) |pmid=17819877 <!-- url is dead, no such DOI, and PMID leads to book, not ''Science'' (journal) |doi=10.1126/science.os-2.75.564 |url=http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;q1=right%20quaternion;rgn=full%20text;idno=05140001;didno=05140001;view=image;seq=81 |via=library.cornell.edu |journal=[[Science (journal)|Science]] |year=1881 |volume=2 |issue=75 |page=65 --> }}</ref> and real numbers (considered as quaternions with zero vector part) ''scalar quaternions''.

If a quaternion is divided up into its scalar part and its vector part, that is,

<math display=block>
\mathbf q = (r,\,\vec{v}),\ \mathbf q \in \mathbb{H},\ r \in \mathbb{R},\ \vec{v}\in \mathbb{R}^3,
</math>

then the formulas for addition, multiplication, and multiplicative inverse are

<math display=block>\begin{align}
(r_1,\,\vec{v}_1) + (r_2,\,\vec{v}_2)
&= (r_1 + r_2,\,\vec{v}_1 + \vec{v}_2), \\[5mu]
(r_1,\,\vec{v}_1) (r_2,\,\vec{v}_2)
&= (r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\,r_1\vec{v}_2+r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2), \\[5mu]
(r,\,\vec{v})^{-1}
&= \left(\frac{r}{r^2 + \vec{v}\cdot\vec{v}},\ \frac{-\vec{v}}{r^2 + \vec{v}\cdot\vec{v}}\right)
\end{align}</math>

where "<math>{}\cdot{}</math>" and "<math>\times</math>" denote respectively the [[dot product]] and the [[cross product]].

== Conjugation, the norm, and reciprocal ==
<!-- Should perhaps add an {{anchor|section name}} here if the section name changes -->

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let <math>q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k </math> be a quaternion. The '''[[Conjugate (algebra)|conjugate]]''' of {{mvar|q}} is the quaternion <math> q^* = a - b\,\mathbf i - c\,\mathbf j - d\,\mathbf k </math>. It is denoted by {{math|''q''<sup>∗</sup>}}, ''q<sup>t</sup>'', <math>\tilde q</math>, or {{overline|''q''}}.<ref name="SeeHazewinkel"/> Conjugation is an [[involution (mathematics)|involution]], meaning that it is its own [[inverse function|inverse]], so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates ''in the reverse order''. That is, if {{mvar|p}} and {{mvar|q}} are quaternions, then {{math|1=(''pq'')<sup>∗</sup> = ''q''<sup>∗</sup>''p''<sup>∗</sup>}}, not {{math|''p''<sup>∗</sup>''q''<sup>∗</sup>}}.

<!-- While the conjugation of a complex number is a special function, that requires, e.g., isolating the imaginary part, and cannot be performed by only applying addition, multiplication and the construction of the respective inverses, which make up the field of complex numbers (in fact, there is no holomorphic function doing the conjugation), – This should be in the section on complex numbers --> 
The conjugation of a quaternion, in contrast to the complex setting, can be expressed with multiplication and addition of quaternions:

<math display=block>
q^* = - \tfrac{1}{2} (q + \mathbf i \,q \,\mathbf i + \mathbf j \,q \,\mathbf j + \mathbf k \,q \,\mathbf k).
</math>

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of {{mvar|p}} is {{math|{{sfrac|1|2}}(''p'' + ''p''<sup>∗</sup>)}}, and the vector part of {{mvar|p}} is {{math|{{sfrac|1|2}}(''p'' − ''p''<sup>∗</sup>)}}.

{{Anchor|Norm}}The [[square root]] of the product of a quaternion with its conjugate is called its [[norm (mathematics)|''norm'']] and is denoted {{math|{{norm|''q''}}}} (Hamilton called this quantity the [[Tensor of a quaternion|''tensor'' of ''q'']], but this conflicts with the modern meaning of "[[tensor]]"). In formulas, this is expressed as follows:

<math display=block>\lVert q \rVert = \sqrt{qq^*} = \sqrt{q^*q} = \sqrt{a^2 + b^2 + c^2 + d^2}</math>

This is always a non-negative real number, and it is the same as the Euclidean norm on <math>\mathbb H</math> considered as the vector space <math>\mathbb R^4</math>. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if {{mvar|α}} is real, then

<math display=block>\lVert\alpha q\rVert = \left| \alpha\right|\,\lVert q\rVert.</math>

This is a special case of the fact that the norm is ''multiplicative'', meaning that

<math display=block>\lVert pq \rVert = \lVert p \rVert\,\lVert q \rVert</math>

for any two quaternions {{mvar|p}} and {{mvar|q}}. Multiplicativity is a consequence of the formula for the conjugate of a product.
Alternatively it follows from the identity

<math display=block> \det \begin{pmatrix}
a + i b & i d + c \\
 i d - c & a - i b
\end{pmatrix} = a^2 + b^2 + c^2 + d^2,</math>

(where {{mvar|i}} denotes the usual [[imaginary unit]]) and hence from the multiplicative property of [[determinant]]s of square matrices.

This norm makes it possible to define the '''distance''' {{math|''d''(''p'', ''q'')}} between {{mvar|p}} and {{mvar|q}} as the norm of their difference:

<math display=block>d(p, q) = \lVert p - q \rVert.</math>

This makes <math>\mathbb H</math> a [[metric space]].
Addition and multiplication are [[continuous function (topology)|continuous]] in regard to the associated [[Metric space#Open and closed sets, topology and convergence|metric topology]].
This follows with exactly the same proof as for the real numbers <math>\mathbb R</math> from the fact that <math>\mathbb H</math> is a normed algebra.

=== Unit quaternion ===
{{Main|Versor}}
A '''unit quaternion''' is a quaternion of norm one. Dividing a nonzero quaternion {{mvar|q}} by its norm produces a unit quaternion {{math|'''U'''''q''}} called the ''[[versor]]'' of {{mvar|q}}:

<math display=block>\mathbf{U}q = \frac{q}{\lVert q\rVert}.</math>

Every nonzero quaternion has a unique [[polar decomposition]] <math> q = \lVert q \rVert \cdot \mathbf{U} q, </math> while the zero quaternion can be formed from any unit quaternion.

Using conjugation and the norm makes it possible to define the [[Multiplicative inverse|reciprocal]] of a nonzero quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of <math>q</math> and <math>q^* / \left \Vert q \right \| ^2</math> is 1 (for either order of multiplication). So the ''[[reciprocal (mathematics)|reciprocal]]'' of {{mvar|q}} is defined to be

<math display=block>q^{-1} = \frac{q^*}{\lVert q\rVert^2}.</math>

Since the multiplication is non-commutative, the quotient quantities {{math|''p q''<sup>−1</sup>}} or {{math|''q''<sup>−1</sup>''p''}} are different (except if {{mvar|p}} and {{mvar|q}} have parallel vector parts): the notation {{math|{{sfrac|''p''|''q''}}}} is ambiguous and should not be used.

== Algebraic properties ==
[[File:Cayley graph Q8.svg|right|thumb|[[Cayley graph]] of {{math|Q<sub>8</sub>}}. The red arrows represent multiplication on the right by {{math|'''i'''}}, and the green arrows represent multiplication on the right by {{math|'''j'''}}.]]

The set <math>\mathbb H</math> of all quaternions is a vector space over the real numbers with [[dimension (vector space)|dimension]]&nbsp;4.{{efn|
For comparison, the [[natural number]]s, <math>\N,</math> [[rational number]]s, <math>\Q,</math> and [[real number]]s, <math>\R,</math> all have dimension&nbsp;1; the [[complex number]]s, <math>\C,</math> have dimension&nbsp;2; and the [[octonion]]s <math>\mathbb O,</math> have dimension&nbsp;8.
}} Multiplication of quaternions is [[associative]] and [[distributive property|distributive]] over [[vector addition]], but it is not [[commutative]]. Therefore, the quaternions, <math>\mathbb H,</math> are a non-commutative, associative algebra over the [[real number]]s. Even though <math>\mathbb H</math> embeds multiple ''copies'' of the complex numbers, it is not itself an ''[[associative algebra]]'' over the [[complex number]]s.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a [[field (mathematics)|field]] except that it admits non-commutative multiplication. ''Finite-dimensional [[associative division algebra]]s'' over the real numbers are very rare: The [[Frobenius theorem (real division algebras)|Frobenius theorem]] states that there are exactly three. Those are <math>\R,</math> <math>\C,</math> and <math>\mathbb H.</math> The [[norm (mathematics)|norm]] makes the quaternions into a [[composition algebra|''normed algebra'']], and normed division algebras over the real numbers are also very rare: [[Hurwitz's theorem (composition algebras)|Hurwitz's theorem]] says that there are only four: <math>\R,</math> <math>\C,</math> <math>\mathbb H,</math> and <math>\mathbb O</math> (the [[octonion]]s). The quaternions are also an example of a [[composition algebra]] and of a unital [[Banach algebra]].

[[File:Quaternion-multiplication-cayley-3d-with-legend.png|thumb|Three-dimensional graph of Q<sub>8</sub>. Red, green and blue arrows represent multiplication by {{math|'''i'''}}, {{math|'''j'''}}, and {{math|'''k'''}}, respectively. Multiplication by negative numbers is omitted for clarity.]]

Because the product of any two basis vectors is plus or minus another basis vector, the set {{math|{{mset|±1, ±'''i''', ±'''j''', ±'''k'''}}}} forms a group under multiplication. This [[non-abelian group]] is called the quaternion group and is denoted {{math|Q<sub>8</sub>}}.<ref>{{cite web |title=quaternion group |website=Wolframalpha.com |url=http://www.wolframalpha.com/input/?i=quaternion+group}}</ref> The real [[group ring]] of {{math|Q<sub>8</sub>}} is a ring <math>\mathbb R[\mathrm Q_8]</math> which is also an eight-dimensional vector space over <math>\mathbb R.</math> It has one basis vector for each element of <math>\mathrm Q_8.</math> The quaternions are isomorphic to the [[quotient ring]] of <math>\mathbb R[\mathrm Q_8]</math> by the [[ideal (ring theory)|ideal]] generated by the elements {{math|1 + (−1)}}, {{math|'''i''' + (−'''i''')}}, {{math|'''j''' + (−'''j''')}}, and {{math|'''k''' + (−'''k''')}}. Here the first term in each of the sums is one of the basis elements {{math|1, '''i''', '''j'''}}, and {{math|'''k'''}}, and the second term is one of basis elements {{math|−1, −'''i''', −'''j'''}}, and {{math|−'''k'''}}, not the additive inverses of {{math|1, '''i''', '''j'''}}, and {{math|'''k'''}}.

== Quaternions and three-dimensional geometry ==
The vector part of a quaternion can be interpreted as a coordinate vector in <math>\mathbb R^3;</math> therefore, the algebraic operations of the quaternions reflect the geometry of <math>\mathbb R^3.</math> Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics.<ref name=Shoemake/>

For the remainder of this section, {{math|'''i'''}}, {{math|'''j'''}}, and {{math|'''k'''}} will denote both the three imaginary<ref>{{cite book |url=https://archive.org/details/vectoranalysisa00wilsgoog |quote=right tensor dyadic |title=Vector Analysis |publisher=Yale University Press |last1=Gibbs |first1=J. Willard |last2=Wilson |first2= Edwin Bidwell |year=1901 |page=[https://archive.org/details/vectoranalysisa00wilsgoog/page/n452 428]}}</ref> basis vectors of <math>\mathbb H</math> and a basis for <math>\mathbb R^3.</math> Replacing {{math|'''i'''}} by {{math|−'''i'''}}, {{math|'''j'''}} by {{math|−'''j'''}}, and {{math|'''k'''}} by {{math|−'''k'''}} sends a vector to its [[additive inverse]], so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the ''spatial inverse''.

For two vector quaternions {{nowrap|{{math|''p'' {{=}} ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}} }} and {{nowrap|{{math|''q'' {{=}} ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}} }} their [[dot product]], by analogy to vectors in <math>\mathbb R^3,</math> is

<math display=block>p \cdot q = b_1 b_2 + c_1 c_2 + d_1 d_2.</math>

It can also be expressed in a component-free manner as

<math display=block>p \cdot q = \textstyle\frac{1}{2}(p^*q + q^*p) = \textstyle\frac{1}{2}(pq^* + qp^*).</math>

This is equal to the scalar parts of the products {{math|''pq''<sup>∗</sup>, ''qp''<sup>∗</sup>, ''p''<sup>∗</sup>''q'', and ''q''<sup>∗</sup>''p''}}. Note that their vector parts are different.

The [[cross product]] of {{mvar|p}} and {{mvar|q}} relative to the orientation determined by the ordered basis {{math|'''i''', '''j'''}}, and {{math|'''k'''}} is

<math display="block">p \times q = (c_1 d_2 - d_1 c_2)\mathbf i + (d_1 b_2 - b_1 d_2)\mathbf j + (b_1 c_2 - c_1 b_2)\mathbf k.</math>

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product {{math|''pq''}} (as quaternions), as well as the vector part of {{math|−''q''<sup>∗</sup>''p''<sup>∗</sup>}}. It also has the formula

<math display=block>p \times q = \textstyle\tfrac{1}{2}(pq - qp).</math>

For the [[Commutator#Ring theory|commutator]], {{math|[''p'', ''q''] {{=}} ''pq'' − ''qp''}}, of two vector quaternions one obtains

<math display=block>[p,q]= 2p \times q,</math>

{{anchor|commutation relationship}}which gives the commutation relationship

<math display=block>qp= pq - 2p \times q.</math>

In general, let {{mvar|p}} and {{mvar|q}} be quaternions and write

<math display=block>\begin{align}
p &= p_\text{s} + p_\text{v}, \\[5mu]
q &= q_\text{s} + q_\text{v},
\end{align}</math>

where {{math|''p''<sub>s</sub>}} and {{math|''q''<sub>s</sub>}} are the scalar parts, and {{math|''p''<sub>v</sub>}} and {{math|''q''<sub>v</sub>}} are the vector parts of {{mvar|p}} and {{mvar|q}}. Then we have the formula

<math display="block">pq = (pq)_\text{s} + (pq)_\text{v} = (p_\text{s}q_\text{s} - p_\text{v}\cdot q_\text{v}) + (p_\text{s} q_\text{v} + q_\text{s} p_\text{v}  + p_\text{v} \times q_\text{v}).</math>

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of vector quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear. Hamilton<ref name="Hamilton">{{cite journal |author-link=William Rowan Hamilton |first=W.R. |last=Hamilton |year=1844–1850 |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/ |title=On quaternions or a new system of imaginaries in algebra |journal=[[Philosophical Magazine]] |department=David R. Wilkins collection |publisher=[[Trinity College Dublin]]}}</ref> showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in [[Elliptic geometry]].

Unit quaternions can be identified with rotations in <math>\mathbb R^3</math> and were called [[versor]]s by Hamilton.<ref name="Hamilton" /> Also see [[Quaternions and spatial rotation]] for more information about modeling three-dimensional rotations using quaternions.

See [[Andrew J. Hanson|Hanson]] (2005)<ref>{{cite web |url=http://www.cs.indiana.edu/~hanson/quatvis/ |title=Visualizing Quaternions |publisher=Morgan-Kaufmann/Elsevier |year=2005 |access-date=2007-01-01 |archive-date=2007-12-27 |archive-url=https://web.archive.org/web/20071227231029/http://www.cs.indiana.edu/~hanson/quatvis/ |url-status=dead }}</ref> for visualization of quaternions.

== Matrix representations ==
Just as complex numbers can be [[Matrix representation of complex numbers|represented as matrices]], so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and [[matrix multiplication]]. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 [[real number|real]] matrices. In each case, the representation given is one of a family of linearly related representations. These are [[injective function|injective]] [[ring homomorphism|homomorphism]]s from <math>\mathbb H</math> to the [[matrix ring]]s {{math|M(2,'''C''')}} and {{math|M(4,'''R''')}}, respectively.

=== Representation as complex 2 × 2 matrices ===
The quaternion {{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} can be represented using a complex {{nobr|2 × 2 matrix}} as

<math display="block">\begin{bmatrix}
\phantom-a + bi & c + di \\
        -c + di & a - bi
\end{bmatrix}.</math>

This representation has the following properties:
* Constraining any two of {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} to zero produces a representation of complex numbers. For example, setting {{math|1=''c'' = ''d'' = 0}} produces a diagonal complex matrix representation of complex numbers, and setting {{math|1=''b'' = ''d'' = 0}} produces a real matrix representation.
* The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the [[determinant]] of the corresponding matrix.<ref>{{cite web |title={{grey|[no title]}} |website=Wolframalpha.com |url=http://www.wolframalpha.com/input/?i=det+%7B{a%2Bb*i%2C+c%2Bd*i}%2C+{-c%2Bd*i%2C+a-b*i}%7D }} — determinant evaluation via online [[computer algebra]] software.</ref>
* The scalar part of a quaternion is one half of the [[matrix trace]].
* The conjugate of a quaternion corresponds to the [[conjugate transpose]] of the matrix.
* By restriction this representation yields a [[group isomorphism]] between the subgroup of unit quaternions and their image [[SU(2)]]. Topologically, the [[unit quaternion]]s are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group {{math|SU(2)}} is important for describing [[Spin (physics)|spin]] in quantum mechanics; see [[Pauli matrices]].
* There is a strong relation between quaternions and [[Pauli matrices]]. The {{nobr|2 × 2 complex}} matrix above can be written as <math>a\,I + b\,i\,\sigma_3 + c\,i\,\sigma_2 + d\,i\,\sigma_1,</math> so in this representation the quaternion units {{math|{{mset|1, '''i''', '''j''', '''k'''}}}} correspond to <math>\left\{I,i\,\sigma_3,i\,\sigma_2,i\,\sigma_1\right\}</math> = <math>\left\{I,\sigma_1\,\sigma_2,\sigma_3\,\sigma_1,\sigma_2\,\sigma_3\right\}</math>. Multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for {{nobr|{{math|−1}} .}} One obtains −1 via {{nobr| {{math|1='''i'''<sup>2</sup> = '''j'''<sup>2</sup> = '''k'''<sup>2</sup> = '''i j k''' = −1}} ;}} e.g. the last equality is <math display=block>\mathbf{i\;j\;k} = \sigma_1\,\sigma_2\,\sigma_3\,\sigma_1\,\sigma_2\,\sigma_3 = -1.</math>
The representation in {{math|M(2,ℂ)}} is not unique: A different convention, that preserves the direction of cyclic ordering between the quaternions and the Pauli matrices, is to choose <math display="block">
1 \mapsto \mathbf{I}, \quad \mathbf{i} \mapsto - i\,\sigma_1 = - \sigma_2\,\sigma_3, \quad \mathbf{j} \mapsto - i\,\sigma_2  = - \sigma_3\,\sigma_1, \quad \mathbf{k} \mapsto - i\,\sigma_3 = - \sigma_1\,\sigma_2,
</math>{{pb}}This gives an alternative representation,{{sfnp|Altmann|1986|p=212, eqn&nbsp;5}}
<math display="block">
a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k
\mapsto \begin{bmatrix}
  a - d\,i & -c - b\,i \\
  c - b\,i & \phantom-a + d\,i \end{bmatrix}.
</math>

=== Representation as real 4 × 4 matrices ===
Using 4 × 4 real matrices, that same quaternion can be written as

<math display=block>\begin{align}
\left[ \begin{array}{rrrr}
 a & -b & -c & -d \\
 b & a  & -d & c  \\
 c & d  & a  & -b \\
 d & -c & b  & a
\end{array} \right]
&= a
\left[ \begin{array}{rrrr}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1
\end{array} \right]
+ b
\left[ \begin{array}{rrrr}
 0 & -1 & 0 & 0  \\
 1 & 0  & 0 & 0  \\
 0 & 0  & 0 & -1 \\
 0 & 0  & 1 & 0
\end{array} \right] \\[10mu]
&\qquad + c
\left[ \begin{array}{rrrr}
 0 & 0 & -1 & 0 \\
 0 & 0 & 0  & 1 \\
 1 & 0 & 0  & 0 \\
 0 & -1 & 0 & 0
\end{array} \right]
+ d
\left[ \begin{array}{rrrr}
 0 & 0 & 0  & -1 \\
 0 & 0 & -1 & 0  \\
 0 & 1 & 0  & 0  \\
 1 & 0 & 0  & 0
\end{array} \right].
\end{align}</math>

However, the representation of quaternions in {{math|M(4,ℝ)}} is not unique. For example, the same quaternion can also be represented as

<math display=block>\begin{align}
\left[ \begin{array}{rrrr}
 a  & d  & -b & -c \\
 -d & a  & c  & -b \\
 b  & -c & a  & -d \\
 c  & b  & d  & a
\end{array} \right]
&= a
\left[ \begin{array}{rrrr}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1
\end{array} \right]
+ b
\left[ \begin{array}{rrrr}
 0 & 0 & -1 & 0  \\
 0 & 0 & 0  & -1 \\
 1 & 0 & 0  & 0  \\
 0 & 1 & 0  & 0
\end{array} \right] \\[10mu]
&\qquad + c
\left[ \begin{array}{rrrr}
 0 & 0  & 0 & -1 \\
 0 & 0  & 1 & 0  \\
 0 & -1 & 0 & 0  \\
 1 & 0  & 0 & 0
\end{array} \right]
+ d
\left[ \begin{array}{rrrr}
 0  & 1 & 0 & 0  \\
 -1 & 0 & 0 & 0  \\
 0  & 0 & 0 & -1 \\
 0  & 0 & 1 & 0
\end{array} \right].
\end{align}</math>

There are 48&nbsp;distinct matrix representations of this form, in which one of the matrices represents the scalar part and the other three are all skew-symmetric. More precisely, there are 48&nbsp;sets of quadruples of matrices with these symmetry constraints, such that a function sending {{math|1, '''i''', '''j'''}}, and {{math|'''k'''}} to the matrices in the quadruple is a [[homomorphism]], that is, it sends sums and products of quaternions to sums and products of matrices.<ref name="MatRep">{{cite journal |last1=Farebrother |first1=Richard William |last2=Groß |first2=Jürgen |last3=Troschke |first3=Sven-Oliver |title=Matrix representation of quaternions |journal=Linear Algebra and Its Applications |date=2003 |volume=362 |pages=251–255 |doi=10.1016/s0024-3795(02)00535-9 | doi-access=free }}</ref> 
In this representation, the conjugate of a quaternion corresponds to the [[transpose]] of the matrix. The fourth power of the norm of a quaternion is the [[determinant]] of the corresponding matrix. The scalar part of a quaternion is one quarter of the matrix trace. As with the {{nobr|2 × 2 complex}} representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two {{nobr|2 × 2 blocks}} by setting {{nobr|&hairsp; {{math| ''c'' {{=}} ''d'' {{=}} 0}} .}}

Each {{nobr|4 × 4 matrix}} representation of quaternions corresponds to a multiplication table of unit quaternions. For example, the last matrix representation given above corresponds to the multiplication table
{|class="wikitable" style="text-align:center"
|-
|width=15| {{nobr|↓ × →}}
!width=15| {{mvar|a}}
!width=15| {{mvar|d}}
!width=15| {{mvar|−b}}
!width=15| {{mvar|−c}}
|-
! {{mvar|a}}
| {{mvar|a}}
| {{mvar|d}}
| {{mvar|−b}}
| {{mvar|−c}}
|-
! {{mvar|−d}}
| {{mvar|−d}}
| {{mvar|a}}
| {{mvar|c}}
| {{mvar|−b}}
|-
! {{mvar|b}}
| {{mvar|b}}
| {{mvar|−c}}
| {{mvar|a}}
| {{mvar|−d}}
|-
! {{mvar|c}}
| {{mvar|c}}
| {{mvar|b}}
| {{mvar|d}}
| {{mvar|a}}
|-
|}

which is isomorphic, through <math>\{a \mapsto 1,\ b \mapsto i,\ c \mapsto j,\ d \mapsto k\},</math> to

{|class="wikitable" style="text-align:center"
|-
|width=15| {{nobr|↓ × →}}
!width=15| {{math| 1 }}
!width=15| {{math|'''k'''}}
!width=15| {{math|−'''i'''}}
!width=15| {{math|−'''j'''}}
|-
! {{math| 1 }}
| {{math| 1 }}
| {{math|'''k'''}}
| {{math|−'''i'''}}
| {{math|−'''j'''}}
|-
! {{math|−'''k'''}}
| {{math|−'''k'''}}
| {{math| 1 }}
| {{math|'''j'''}}
| {{math|−'''i'''}}
|-
! {{math|'''i'''}}
| {{math|'''i'''}}
| {{math|−'''j'''}}
| {{math| 1 }}
| {{math|−'''k'''}}
|-
! {{math|'''j'''}}
| {{math|'''j'''}}
| {{math|'''i'''}}
| {{math|'''k'''}}
| {{math| 1 }}
|-
|}

Constraining any such multiplication table to have the identity in the first row and column and for the signs of the row headers to be opposite to those of the column headers, then there are 3&nbsp;possible choices for the second column (ignoring sign), 2&nbsp;possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6&nbsp;possibilities. Then, the second column can be chosen to be either positive or negative, the third column can be chosen to be positive or negative, and the fourth column can be chosen to be positive or negative, giving 8 possibilities for the sign. Multiplying the possibilities for the letter positions and for their signs yields 48. Then replacing {{math|1}} with {{mvar|a}}, {{math|'''i'''}} with {{mvar|b}}, {{math|'''j'''}} with {{mvar|c}}, and {{math|'''k'''}} with {{mvar|d}} and removing the row and column headers yields a matrix representation of {{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k''' }}.

== Lagrange's four-square theorem ==
{{Main|Lagrange's four-square theorem}}
Quaternions are also used in one of the proofs of Lagrange's four-square theorem in [[number theory]], which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as [[combinatorial design]] theory. The quaternion-based proof uses [[Hurwitz quaternion]]s, a subring of the ring of all quaternions for which there is an analog of the [[Euclidean algorithm]].

== Quaternions as pairs of complex numbers ==
{{Main|Cayley–Dickson construction}}
Quaternions can be represented as pairs of complex numbers. From this perspective, quaternions are the result of applying the [[Cayley–Dickson construction]] to the complex numbers. This is a generalization of the construction of the complex numbers as pairs of real numbers.

Let <math>\mathbb C^2</math> be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements {{math|1}} and {{math|'''j'''}}. A vector in <math>\C^2</math> can be written in terms of the basis elements {{math|1}} and {{math|'''j'''}} as

<math display=block>(a + b\,i) 1 + (c + d\,i) \mathbf j .</math>

If we define {{math|1='''j'''<sup>2</sup> = −1}} and {{math|1=''i'' '''j''' = −'''j''' ''i''}}, then we can multiply two vectors using the distributive law. Using {{math|'''k'''}} as an abbreviated notation for the product {{math|''i'' '''j'''}} leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion {{math|''a'' + ''b i'' + ''c'' '''j''' + ''d'' '''k'''}}. If we write the elements of <math>\mathbb C^2</math> as ordered pairs and quaternions as quadruples, then the correspondence is

<math display=block>(a + bi,\,c + di) \leftrightarrow (a,\,b,\,c,\,d).</math>

== Square roots ==

=== Square roots of −1 ===
In the complex numbers, <math>\C,</math> there are exactly two numbers, {{mvar|i}} and {{math|−''i''}}, that give −1 when squared. In <math>\mathbb H</math> there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit [[sphere]] in <math>\R^3.</math> To see this, let {{nowrap|{{math|''q'' {{=}} ''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} }} be a quaternion, and assume that its square is −1. In terms of {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and {{mvar|d}}, this means

<math display=block>\begin{align}
a^2 - b^2 - c^2 - d^2 &= -1, \vphantom{x^|} \\[3mu]
2ab &= 0, \\[3mu]
2ac &= 0, \\[3mu]
2ad &= 0.
\end{align}</math>

To satisfy the last three equations, either {{nobr|{{math|''a'' {{=}} 0}}}} or {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are all {{nobr|{{math|0}} .}} The latter is impossible because {{math|''a''}} is a real number and the first equation would imply that {{nobr|{{math|''a''<sup>2</sup> {{=}} −1}}.}} Therefore, {{nobr|{{math|''a'' {{=}} 0}}}} and {{nowrap|{{math|''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> {{=}} 1}}.}} In other words: A quaternion squares to {{math|−1}} if and only if it is a vector quaternion with {{nobr|norm {{math|1}} .}} By definition, the set of all such vectors forms the unit sphere.

Only negative real quaternions have infinitely many square roots. All others have just two (or one in the case of 0).{{citation needed|date=September 2017}}{{efn|The identification of the square roots of minus one in <math>\mathbb H</math> was given by Hamilton<ref>{{cite book |author=Hamilton, W.R. |year=2009 |orig-year=First published 1899 |title=Elements of Quaternions |edition=2nd |page=244 |publisher=Cambridge University Press |isbn=978-1-108-00171-7}}</ref> but was frequently omitted in other texts. By 1971 the sphere was included by Sam Perlis in his three-page exposition included in ''Historical Topics in Algebra'' published by the [[National Council of Teachers of Mathematics]].<ref>{{Cite book |last=Perlis |first=Sam |chapter=Capsule 77: Quaternions |title=Historical Topics in Algebra |chapter-url=https://archive.org/details/historicaltopics0000nati/page/38/mode/2up |chapter-url-access=registration |publisher=[[National Council of Teachers of Mathematics]] |location=Reston, VA |series=Historical Topics for the Mathematical Classroom |volume=31 |year=1971 |page=39 |isbn=9780873530583 |oclc=195566 }}</ref> More recently, the sphere of square roots of minus one is described in [[Ian R. Porteous]]'s book ''Clifford Algebras and the Classical Groups'' (Cambridge, 1995) in proposition 8.13.<ref>{{Cite book |last=Porteous |first=Ian R. |author-link=Ian R. Porteous |chapter=Chapter 8: Quaternions |url=https://www.maths.ed.ac.uk/~v1ranick/papers/porteous3.pdf |title=Clifford Algebras and the Classical Groups |series=Cambridge Studies in Advanced Mathematics |publisher=[[Cambridge University Press]] |location=Cambridge |volume=50 |pages=60 |year=1995 |doi=10.1017/CBO9780511470912.009 |isbn=9780521551779 |oclc=32348823 |mr=1369094 }}</ref>}}

==== As a union of complex planes ====
Each [[antipodal points|antipodal pair]] of square roots of {{math|−1}} creates a distinct copy of the complex numbers inside the quaternions. If {{nobr|{{math|''q''<sup>2</sup> {{=}} −1}} ,}} then the copy is the [[image (mathematics)|image]] of the function

<math display=block>a + bi \mapsto a + b q.</math>

This is an [[injective function|injective]] [[ring homomorphism]] from <math>\mathbb C</math> to <math>\mathbb H,</math> which defines a field [[isomorphism]] from <math>\Complex</math> onto its [[image (mathematics)|image]]. The images of the embeddings corresponding to {{mvar|+q}} and {{mvar|−q}} are identical.

Every non-real quaternion generates a [[subalgebra]] of the quaternions that is isomorphic to <math>\C</math>, and is thus a planar subspace of <math>\mathbb H</math>: write {{mvar|q}} as the sum of its scalar part and its vector part:
<math display=block>
q = q_s + \vec{q}_v.
</math>

Decompose the vector part further as the product of its norm and its [[versor]]:
<math display=block>
q = q_s + \lVert \vec{q}_v \rVert \cdot \mathbf{U}\,\vec{q}_v = q_s + \|\vec q_v\|\,\frac{ \vec q_v }{ \|\vec q_v\| } .
</math>

(This is not the same as <math>q_s + \lVert q\rVert\cdot\mathbf{U}q .</math>) The versor of the vector part of {{mvar|q}}, <math>\mathbf{U}\vec{q}_v,</math> is a right versor with –1 as its square. A straightforward verification shows that
<math display=block>a + bi \mapsto a + b\mathbf{U}\vec{q}_v</math>
defines an injective [[algebra homomorphism|homomorphism]] of [[normed algebra]]s from <math>\mathbb C</math> into the quaternions. Under this homomorphism, {{mvar|q}} is the image of the complex number <math>q_s + \lVert\vec{q}_v\rVert i .</math>

As <math>\mathbb H</math> is the [[union (set theory)#Arbitrary unions|union]] of the images of all these homomorphisms, one can view the quaternions as a [[pencil of planes]] intersecting on the [[real line]]. Each of these [[complex plane]]s contains exactly one pair of [[antipodal points]] of the sphere of square roots of minus one.

==== Commutative subrings ====
The relationship of quaternions to each other within the complex subplanes of <math>\mathbb H</math> can also be identified and expressed in terms of commutative [[subring]]s. Specifically, since two quaternions {{mvar|p}} and {{mvar|q}} commute (i.e., {{math|''p q'' {{=}} ''q p''}}) only if they lie in the same complex subplane of <math>\mathbb H,</math> the profile of <math>\mathbb H</math> as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion [[ring (mathematics)|ring]].

=== Square roots of arbitrary quaternions ===

Any quaternion <math>\mathbf q = (r,\, \vec{v})</math> (represented here in scalar–vector representation) has at least one square root <math>\sqrt{\mathbf q} = (x,\, \vec{y})</math> which solves the equation <math>(\sqrt{\mathbf q})^{2} = (x,\,\vec{y})^2 = \mathbf q .</math> Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions

<math display=block>
\sqrt{\mathbf q}
= \sqrt{(r,\, \vec{v})}
= \pm \left( \sqrt{ \tfrac{1}{2} \bigl( \|\mathbf q \| + r \bigr)},\ 
  \frac{ \vec{v} }{ \|\vec{v}\| }\sqrt{ \tfrac{1}{2} \bigl( \|\mathbf q\| - r \bigr)}\right),
</math>

where <math display="inline">\|\vec{v}\| = \sqrt{\vec{v}\cdot\vec{v}}</math> is the norm of <math>\vec{v}</math> and <math display="inline">\|\mathbf q\| = \sqrt{ \mathbf q^*\mathbf q} = \sqrt{ r^2 + \|\vec{v}\|^2}</math> is the norm of <math>\mathbf q .</math>  For any scalar quaternion <math>\mathbf q</math>, this equation provides the correct square roots if <math display="inline">\vec{v} / \|\vec{v}\|</math> is interpreted as an arbitrary unit vector.

Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions, have exactly two roots, while 0 has exactly one root (0), and negative scalar quaternions have infinitely many roots, which are the vector quaternions located on <math>\{ 0 \} \times S^2 \bigl( \sqrt{-r} \bigr)</math>, i.e., where the scalar part is zero and the vector part is located on the [[n-sphere|2-sphere]] with radius <math>\sqrt{-r} .</math>

==Functions of a quaternion variable==
{{Main|Quaternionic analysis}}
[[File:Quaternion Julia x=-0,75 y=-0,14.jpg|thumb|The Julia sets and Mandelbrot sets can be extended to the Quaternions, but they must use cross sections to be rendered visually in 3&nbsp;dimensions. This Julia set is cross sectioned at the {{mvar|x y}} plane.]]
Like functions of a [[complex variable]], functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the [[Mandelbrot set]] and [[Julia set]]s into 4-dimensional space.<ref>{{cite web |title=[no title cited] |website=bridgesmathart.org |series=archive |url=http://archive.bridgesmathart.org/2010/bridges2010-247.pdf |access-date=19 August 2018}}</ref>

===Exponential, logarithm, and power functions===
A function of a quaternion can be defined from a power series with real coefficients. For example, given a quaternion,

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

the exponential is computed as<ref name=Särkkä2007>{{cite web |url=http://www.lce.hut.fi/~ssarkka/pub/quat.pdf |website=Lce.hut.fi |title=Notes on Quaternions |first=Simo |last=Särkkä |date=June 28, 2007 |archive-url=https://web.archive.org/web/20170705123142/http://www.lce.hut.fi/~ssarkka/pub/quat.pdf |archive-date=5 July 2017}}</ref>

<math display=block>
\exp(q) = \sum_{n=0}^\infty \frac{q^n}{n!}
= e^{a} \biggl({\cos \|\mathbf{v}\|} + \frac{ \mathbf{v} }{ \|\mathbf{v}\| } \sin \|\mathbf{v}\| \biggr),
</math>

and the logarithm is<ref name=Särkkä2007/>

<math display=block>\ln(q) = \ln \|q\| + \frac{ \mathbf{v} }{ \|\mathbf{v}\| } \arccos \frac{ a }{ \|q\| }.</math>

It follows that the polar decomposition of a quaternion may be written

<math display=block>q = \|q\| e^{ \varphi \hat{n} } = \|q\| \bigl({\cos(\varphi) + \hat{n}\, \sin(\varphi)} \bigr),</math>

where the [[angle]] <math>\varphi</math>{{efn|name="θ"|Books on applied mathematics, such as Corke (2017)<ref>{{cite book |title=Robotics, Vision, and Control – Fundamental Algorithms in MATLAB |last=Corke |first=Peter |publisher=[[Springer Publishing|Springer]] |year=2017 |isbn=978-3-319-54413-7}}</ref> often use different notation with {{math|''φ'' :{{=}} {{sfrac|1|2}}''θ''}} — that is, [[change of variables|another variable]] {{nobr|{{math|''θ'' {{=}} 2''φ''}}}} .}}

<math display=block>a = \| q \|\, \cos( \varphi )</math>

and the unit vector <math>\hat{n}</math> is defined by:

<math display=block>\mathbf{v} = \hat{n} \|\mathbf{v}\| = \hat{n}\|q\|\,\sin(\varphi).</math>

Any unit quaternion may be expressed in polar form as:

<math display=block>q = \exp{(\hat{n}\varphi)}.</math>

The [[Power (mathematics)|power]] of a quaternion raised to an arbitrary (real) exponent {{mvar|x}} is given by:

<math display=block>q^x = \|q\|^x e^{\hat{n} x \varphi} = \|q\|^x \bigl(\cos(x\varphi) + \hat{n} \sin(x\varphi) \bigr).</math>

===Geodesic norm===
The [[Great-circle distance|geodesic distance]] {{nobr|{{math|''d''<sub>g</sub>(''p'', ''q'')}}}} between unit quaternions {{mvar|p}} and {{mvar|q}} is defined as:<ref name=Geodesic>{{cite journal |last1=Park |first1=F.C. |last2=Ravani |first2=Bahram |title=Smooth invariant interpolation of rotations |journal=ACM Transactions on Graphics |date=1997 |volume=16 |number=3 |pages=277–295 |doi=10.1145/256157.256160|s2cid=6192031 |doi-access=free }}</ref>

<math display=block>d_\text{g}(p, q) = \lVert \ln(p^{-1} q) \rVert .</math>

and amounts to the absolute value of half the angle subtended by {{mvar|p}} and {{mvar|q}} along a [[Arc (geometry)|great arc]] of the {{math|S<sup>3</sup>}} sphere.
This angle can also be computed from the quaternion [[dot product]] without the logarithm as:

<math display=block>d_\text{g}(p, q) = {\arccos}\bigl(2(p \cdot q)^2 - 1 \bigr).</math>

==Three-dimensional and four-dimensional rotation groups==
{{Main|Quaternions and spatial rotation|Rotation operator (vector space)}}
The word "[[conjugation (group theory)|conjugation]]", besides the meaning given above, can also mean taking an element {{mvar|a}} to {{math|''r a r''<sup>−1</sup>}} where {{mvar|r}} is some nonzero quaternion. All [[conjugacy class|elements that are conjugate to a given element]] (in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.)<ref>{{cite arXiv |last=Hanson |first=Jason |year=2011 |title=Rotations in three, four, and five dimensions |class=math.MG |eprint=1103.5263 }}</ref>

Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy of <math>\mathbb R^3</math> consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part {{math|cos(''φ'')}} is a rotation by an angle {{math|2''φ''}}, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:<ref>{{cite thesis |type=BS |last=Günaşti |first=Gökmen |year=2016 |title=Quaternions Algebra, Their Applications in Rotations and Beyond Quaternions |publisher=Linnaeus University |url=http://www.diva-portal.org/smash/get/diva2:535712/FULLTEXT02 }}</ref>

* Avoiding [[gimbal lock]], a problem with systems such as Euler angles.
* Faster and more compact than matrices.
* Nonsingular representation (compared with Euler angles for example).
* Pairs of unit quaternions represent a rotation in [[Four-dimensional space|4D]] space (see ''[[Rotations in 4-dimensional Euclidean space#Algebra of 4D rotations|Rotations in 4-dimensional Euclidean space: Algebra of 4D&nbsp;rotations]]'').

<!-- ''S''<sup>3</sup> itself has not a canonical group structure -->The set of all unit quaternions ([[versor]]s) forms a 3-sphere {{math|''S''<sup>3</sup>}} and a group (a [[Lie group]]) under multiplication, [[Covering space#Properties|double covering]] the group <math>\text{SO}(3,\mathbb{R})</math> of real orthogonal 3×3&nbsp;[[orthogonal matrix|matrices]] of [[determinant]]&nbsp;1 since ''two'' unit quaternions correspond to every rotation under the above correspondence. See [[plate trick]].

{{further|Point groups in three dimensions}}
The image of a subgroup of versors is a [[Point groups in three dimensions|point group]], and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same name, with the prefix '''binary'''. For instance, the preimage of the [[icosahedral group]] is the [[binary icosahedral group]].

The versors' group is isomorphic to {{math|SU(2)}}, the group of complex [[unitary matrix|unitary]] {{nobr|2 × 2 matrices}} of [[determinant]] 1.

{{further|Hurwitz quaternion}}
Let {{mvar|A}} be the set of quaternions of the form {{nobr|{{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} }} where {{mvar|a, b, c,}} and {{mvar|d}} are either all [[integer]]s or all [[half-integer]]s. The set {{mvar|A}} is a ring (in fact a [[domain (ring theory)|domain]]) and a [[Lattice (group)|lattice]] and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of a [[24-cell|regular 24&nbsp;cell]] with [[Schläfli symbol]] {{nobr|{{math|&thinsp;{3,4,3}&thinsp;}}.}} They correspond to the double cover of the rotational symmetry group of the regular [[tetrahedron]]. Similarly, the vertices of a [[600-cell|regular 600 cell]] with Schläfli symbol {{math|&thinsp;{3,3,5}&thinsp;}} can be taken as the unit [[icosian]]s, corresponding to the double cover of the rotational symmetry group of the [[regular icosahedron]]. The double cover of the rotational symmetry group of the regular [[octahedron]] corresponds to the quaternions that represent the vertices of the [[disphenoidal 288-cell]].<ref>{{cite web |title=Three-Dimensional Point Groups |url=https://www.classe.cornell.edu/~dms79/xrd/xtallography/Three-Dimensional%20Point%20Groups.htm |access-date=2022-12-09 |website=www.classe.cornell.edu}}</ref>

== Quaternion algebras ==
{{Main|Quaternion algebra}}

The Quaternions can be generalized into further algebras called ''quaternion algebras''. Take {{mvar|F}} to be any field with characteristic different from 2, and {{mvar|a}} and {{mvar|b}} to be elements of {{mvar|F}}; a four-dimensional unitary associative algebra can be defined over {{mvar|F}} with basis {{math| 1, '''i''', '''j''',}} and {{math|'''i j'''}}, where {{nowrap|{{math|'''i'''<sup>2</sup> {{=}} ''a''}}}}, {{nowrap|{{math|'''j'''<sup>2</sup> {{=}} ''b''}}}} and {{nobr|{{math|'''i j''' {{=}} −'''j i'''}}}} (so {{nobr|{{math|'''(i j)'''<sup>2</sup> {{=}} −''a b''}}}}).

Quaternion algebras are isomorphic to the algebra of 2×2 matrices over {{mvar|F}} or form division algebras over {{mvar|F}}, depending on the choice of {{mvar|a}} and {{mvar|b}}.

== Quaternions as the even part of {{math|Cl<sub>3,0</sub>(ℝ)}} ==
{{main|Spinor#Three dimensions}}
The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part <math>\operatorname{Cl}_{3,0}^+ (\R)</math> of the Clifford algebra <math>\operatorname{Cl}_{3,0}(\R).</math> This is an associative multivector algebra built up from fundamental basis elements {{math|[[Pauli matrices|''σ''<sub>1</sub>, ''σ''<sub>2</sub>, ''σ''<sub>3</sub>]]}} using the product rules

<math display=block>\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 1,</math>
<math display=block>\sigma_m \sigma_n = - \sigma_n \sigma_m \qquad (m \neq n).</math>

If these fundamental basis elements are taken to represent vectors in 3D space, then it turns out that the ''reflection'' of a vector {{mvar|r}} in a plane perpendicular to a unit vector {{mvar|w}} can be written:

<math display=block>r^{\prime} = - w\, r\, w.</math>

Two reflections make a rotation by an angle twice the angle between the two reflection planes, so

<math display=block>r^{\prime\prime} = \sigma_2 \sigma_1 \, r \, \sigma_1 \sigma_2</math>

corresponds to a rotation of 180° in the plane containing ''σ''<sub>1</sub> and ''σ''<sub>2</sub>. This is very similar to the corresponding quaternion formula,

<math display=block>r^{\prime\prime} = -\mathbf{k}\, r\, \mathbf{k} .</math>

Indeed, the two structures <math>\operatorname{Cl}_{3,0}^+(\mathbb R)</math> and <math>\mathbb H</math> are [[isomorphic]]. One natural identification is

<math display=block>1 \mapsto 1, \quad \mathbf{i} \mapsto - \sigma_2 \sigma_3, \quad \mathbf{j} \mapsto - \sigma_3 \sigma_1, \quad \mathbf{k} \mapsto - \sigma_1 \sigma_2,</math>

and it is straightforward to confirm that this preserves the Hamilton relations

<math display=block>\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i \,j \,k} = -1 .</math>

In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to [[bivector]]s – quantities with [[magnitude (mathematics)|magnitudes]] and [[orientation (mathematics)|orientations]] associated with particular 2D&nbsp;''planes'' rather than 1D&nbsp;''directions''. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions {{math|''σ''<sub>1</sub>}} and {{math|''σ''<sub>2</sub>}}, there is only one bivector basis element {{math|''σ''<sub>1</sub>''σ''<sub>2</sub>}}, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements {{math|''σ''<sub>2</sub>''σ''<sub>3</sub>}}, {{math|''σ''<sub>3</sub>''σ''<sub>1</sub>}}, {{math|''σ''<sub>1</sub>''σ''<sub>2</sub>}}, so three imaginaries.

This reasoning extends further. In the Clifford algebra <math>\operatorname{Cl}_{4,0}(\mathbb R),</math> there are six bivector basis elements, since with four different basic vector directions, six different pairs and therefore six different linearly independent planes can be defined. Rotations in such spaces using these generalisations of quaternions, called [[rotor (mathematics)|rotors]], can be very useful for applications involving [[homogeneous coordinates]]. But it is only in 3D that the number of basis bivectors equals the number of basis vectors, and each bivector can be identified as a [[pseudovector]].

There are several advantages for placing quaternions in this wider setting:<ref>{{cite web |url=http://www.geometricalgebra.net/quaternions.html |title=Quaternions and Geometric Algebra |website=geometricalgebra.net |access-date=2008-09-12 |archive-date=2008-09-08 |archive-url=https://web.archive.org/web/20080908073855/http://www.geometricalgebra.net/quaternions.html |url-status=dead }} See also: {{cite book |first1=Leo |last1=Dorst |first2=Daniel |last2=Fontijne |first3=Stephen |last3=Mann |year=2007 |url=http://www.geometricalgebra.net/index.html |title=Geometric Algebra for Computer Science |publisher=[[Morgan Kaufmann]] |isbn=978-0-12-369465-2 |archive-date=2008-09-08 |access-date=2008-09-12 |archive-url=https://web.archive.org/web/20080908073807/http://www.geometricalgebra.net/index.html |url-status=dead }}</ref>
* Rotors are a natural part of geometric algebra and easily understood as the encoding of a double reflection.
* In geometric algebra, a rotor and the objects it acts on live in the same space. This eliminates the need to change representations and to encode new data structures and methods, which is traditionally required when augmenting linear algebra with quaternions.
* Rotors are universally applicable to any element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on.
* In the [[Conformal geometric algebra|conformal model]] of Euclidean geometry, rotors allow the encoding of rotation, translation and scaling in a single element of the algebra, universally acting on any element. In particular, this means that rotors can represent rotations around an arbitrary axis, whereas quaternions are limited to an axis through the origin.
* Rotor-encoded transformations make interpolation particularly straightforward.
* Rotors carry over naturally to [[pseudo-Euclidean space]]s, for example, the [[Minkowski space]] of [[special relativity]]. In such spaces rotors can be used to efficiently represent [[Lorentz boost]]s, and to interpret formulas involving the [[gamma matrices]].{{citation needed|date=October 2024}}

For further detail about the geometrical uses of Clifford algebras, see [[Geometric algebra]].

== Brauer group ==
{{further|Brauer group}}

The quaternions are "essentially" the only (non-trivial) [[central simple algebra]] (CSA) over the real numbers, in the sense that every CSA over the real numbers is [[Brauer equivalent]] to either the real numbers or the quaternions. Explicitly, the [[Brauer group]] of the real numbers consists of two classes, represented by the real numbers 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. 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 real numbers.

CSAs – finite dimensional rings over a field, which are [[simple algebra]]s (have no non-trivial 2-sided ideals, just as with fields) whose center is exactly the field – are a noncommutative analog of [[extension field]]s, and are more restrictive than general ring extensions. The fact that the quaternions are the only non-trivial CSA over the real numbers (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial finite field extension of the real numbers.

== See also ==
* {{Annotated link|Conversion between quaternions and Euler angles}}
* {{Annotated link|Dual quaternion}}
* {{Annotated link|Dual-complex number}}
* {{Annotated link|Biquaternion}}
* {{Annotated link|Biquaternion functions}}
* {{Annotated link|Exterior algebra}}
* {{Annotated link|Hurwitz quaternion order}}
* {{Annotated link|Hyperbolic quaternion}}
* {{Annotated link|Lénárt sphere}}
* {{Annotated link|Pauli matrices}}
* {{Annotated link|Quaternionic manifold}}
* {{Annotated link|Quaternionic matrix}}
* {{Annotated link|Quaternionic polytope}}
* {{Annotated link|Quaternionic projective space}}
* {{Annotated link|Rotations in 4-dimensional Euclidean space}}
* {{Annotated link|Slerp}}
* {{Annotated link|Split-quaternion}}
* {{Annotated link|Tesseract}}
* {{Annotated link|Versor}}

==Notes==
{{notelist}}

==References==
{{reflist|25em}}

==Further reading==
===Books and publications===
{{refbegin|25em|small=yes}}
* {{cite book |last=Adler |first=Stephen L. |title=Quaternionic quantum mechanics and quantum fields |publisher=Oxford University Press |year=1995 |isbn=0-19-506643-X |lccn=94006306 |series=International series of monographs on physics  |volume=88|ref=none }}
* {{cite journal |first=Simon L. |last=Altmann |title=Hamilton, Rodrigues, and the Quaternion Scandal |journal=Mathematics Magazine |volume=62 |issue=5 |pages=291–308 |year=1989 |doi=10.1080/0025570X.1989.11977459 |ref=none }}
* {{cite book |last1=Binz |first1=Ernst |first2=Sonja |last2=Pods |chapter=1. The Skew Field of Quaternions |title=Geometry of Heisenberg Groups |publisher=[[American Mathematical Society]] |year=2008 |isbn=978-0-8218-4495-3 |ref=none }}
* {{cite EB1911|wstitle=Algebra|ref=none }} (''See section on quaternions.'')
* {{cite book |last=Clerk Maxwell |first=James |author-link=James Clerk Maxwell |year=1873 |title=[[A Treatise on Electricity and Magnetism]] |publisher=Clarendon Press |location=Oxford |ref=none}}
* {{cite book |last1=Conway |first1=John Horton |author-link=John Horton Conway |last2=Smith |first2=Derek A. |title=On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry |publisher=A.K. Peters |year=2003 |isbn=1-56881-134-9 }} ([http://nugae.wordpress.com/2007/04/25/on-quaternions-and-octonions/ review]).
* {{cite book |last=Crowe |first=Michael J. |year=1967 |title=[[A History of Vector Analysis]]: The Evolution of the Idea of a Vectorial System |publisher=University of Notre Dame Press |ref=none}} Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
* {{cite book |last1=Doran |first1=Chris J.L. |first2=Anthony N. |last2=Lasenby |title=Geometric Algebra for Physicists |year=2003 |publisher=Cambridge University Press |isbn=978-0-521-48022-2 |author1-link=Chris J. L. Doran|ref=none }}
* {{cite book |last=Du Val |first=Patrick |author-link=Patrick du Val |title=Homographies, quaternions, and rotations |publisher=Clarendon Press |series=Oxford mathematical monographs |year=1964 |lccn=64056979|ref=none }}
* {{cite journal |last=Evans |first=D.J. |title=On the Representation of Orientation Space |journal=Mol. Phys. |volume=34 |issue=2 |pages=317–325 |year=1977 |doi=10.1080/00268977700101751 |bibcode=1977MolPh..34..317E |ref=none }} For molecules that can be regarded as classical rigid bodies, [[molecular dynamics]] computer simulation employs quaternions.
* {{ cite book |last1=Eves |first1=Howard |title=An Introduction to the History of Mathematics |edition=4th | location=New York |publisher=Holt, Rinehart and Winston |year=1976 |isbn=0-03-089539-1 }}
* {{cite journal |last1=Finkelstein |first1=David |first2=Josef M. |last2=Jauch |first3=Samuel |last3=Schiminovich |first4=David |last4=Speiser |title=Foundations of quaternion quantum mechanics |journal=J. Math. Phys. |volume=3 |pages=207–220 |year=1962 |issue=2 |doi=10.1063/1.1703794 |bibcode=1962JMP.....3..207F |s2cid=121453456 |url=https://archive-ouverte.unige.ch/unige:162173 |ref=none }}
* {{cite journal |first=Fuzhen |last=Zhang |title=Quaternions and Matrices of Quaternions |journal=Linear Algebra and Its Applications |volume=251 |pages=21–57 |year=1997 |doi=10.1016/0024-3795(95)00543-9 |doi-access=free |ref=none }}
* {{cite book |last=Goldman |first=Ron |title=Rethinking Quaternions: Theory and Computation|year=2010|publisher=Morgan & Claypool |isbn=978-1-60845-420-4|ref=none }}
* {{cite book |last1=Gürlebeck |first1=Klaus |last2=Sprössig |first2=Wolfgang |title=Quaternionic and Clifford calculus for physicists and engineers |publisher=Wiley |year=1997 |isbn=0-471-96200-7 |series=Mathematical methods in practice |volume=1 |lccn=98169958|ref=none }}
* {{cite journal |last=Hamilton |first=William Rowan |author-link=William Rowan Hamilton |year=1844 |journal=[[Philosophical Magazine]] |series=Ser. 3 |volume='''25''' supplement |number=169 |pages=489–495 |title=On Quaternions; or on a new System of Imaginaries in Algebra |url=https://archive.org/details/s3philosophicalmag25londuoft/page/489/ |doi=10.1080/14786444408645047 }}
* [[William Rowan Hamilton|Hamilton, William Rowan]] (1853), "''[https://web.archive.org/web/20140808040037/http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 Lectures on Quaternions]''". Royal Irish Academy.
* [[William Rowan Hamilton|Hamilton]] (1866) ''[https://archive.org/details/bub_gb_fIRAAAAAIAAJ Elements of Quaternions]'' [[University of Dublin]] Press. Edited by William Edwin Hamilton, son of the deceased author.
* [[William Rowan Hamilton|Hamilton]] (1899) ''Elements of Quaternions'' volume I, (1901) volume II. Edited by [[Charles Jasper Joly]]; published by [[Longmans, Green & Co.]]
* {{cite book |last=Hanson |first=Andrew J. |title=Visualizing Quaternions |publisher=Elsevier |year=2006 |isbn=0-12-088400-3 |url=http://www.cs.indiana.edu/~hanson/quatvis/ |ref=none |archive-date=2007-12-27 |access-date=2007-01-01 |archive-url=https://web.archive.org/web/20071227231029/http://www.cs.indiana.edu/~hanson/quatvis/ |url-status=dead }}
* {{cite book |last1=Hazewinkel |first1=Michiel |author-link=Michiel Hazewinkel |first2=Nadiya |last2=Gubareni |first3=Vladimir V. |last3=Kirichenko |title=Algebras, rings and modules |publisher=Springer |year=2004 |isbn=1-4020-2690-0 |volume=1 |url=https://books.google.com/books?id=AibpdVNkFDYC }}
* {{cite arXiv |last=Jack |first=P.M. |title=Physical space as a quaternion structure, I: Maxwell equations. A brief Note |date=2003 |eprint=math-ph/0307038|ref=none }} 
* {{cite book |last=Joly |first=Charles Jasper |title=A manual of quaternions |publisher=Macmillan |year=1905 |lccn=05036137|ref=none }}
* {{cite book |last1=Kantor |first1=I.L. |last2=Solodnikov |first2=A.S. |title=Hypercomplex numbers, an elementary introduction to algebras |publisher=Springer-Verlag |year=1989 |isbn=0-387-96980-2 |ref=none }}
* {{cite book |last=Kravchenko |first=Vladislav |title=Applied Quaternionic Analysis |publisher=Heldermann Verlag |year=2003 |isbn=3-88538-228-8 |ref=none }}
* {{cite book |last=Kuipers |first=Jack |title=Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality |publisher=[[Princeton University Press]] |year=2002 |isbn=0-691-10298-8 |ref=none }}
* {{cite book |last=Macfarlane |first=Alexander |author-link=Alexander Macfarlane |title=Vector analysis and quaternions |publisher=Wiley |edition=4th |year=1906 |lccn=16000048|ref=none }}
* {{cite journal |last=Pujol |first=Jose |title=On Hamilton's Nearly-Forgotten Early Work on the Relation between Rotations and Quaternions and on the Composition of Rotations |journal=The American Mathematical Monthly |volume=121 |issue=6 |pages=515–522 |year=2014 |doi=10.4169/amer.math.monthly.121.06.515 |s2cid=1543951 |ref=none }}
* {{cite book |last=Tait |first=Peter Guthrie |author-link=Peter Guthrie Tait |year=1873|title=An elementary treatise on quaternions |edition=2nd |location=Cambridge |publisher=The University Press |ref=none}}
* {{cite book |last=Vince |first=John A. |title=Geometric Algebra for Computer Graphics |publisher=Springer |year=2008 |isbn=978-1-84628-996-5 |ref=none }}
* {{cite book |last=Voight |first=John |title=Quaternion Algebras |series=Graduate Texts in Mathematics |publisher=Springer |year=2021 |volume=288 |isbn=978-3-030-57467-3 |doi=10.1007/978-3-030-56694-4 |doi-access=free |ref=none }}
* {{cite book |last=Ward |first=J.P. |title=Quaternions and Cayley Numbers: Algebra and Applications |publisher=Kluwer Academic |year=1997 |isbn=0-7923-4513-4 |ref=none }}
{{refend}}

===Links and monographs===
{{refbegin|25em|small=yes}}
* {{cite web |title=Quaternion notices |url=https://quaternionnews.commons.gc.cuny.edu/ }} Notices and materials related to Quaternion conference presentations
* {{springer|title=Quaternion|id=p/q076770}}
* {{cite web |department=Frequently Asked Questions |title=Matrix and quaternion  |id=1.21 |url=http://www.j3d.org/matrix_faq/matrfaq_latest.html|ref=none}}
* {{cite web |first=Doug |last=Sweetser |title=Doing physics with quaternions |url=http://world.std.com/~sweetser/quaternions/qindex/qindex.html |ref=none }}
* {{cite web |title=Quaternions for computer graphics and mechanics |first=Gernot |last=Hoffman |website=fho-emden.de  |url=http://www.fho-emden.de/~hoffmann/quater12012002.pdf |url-status=dead <!-- presumed --> |archive-url=https://web.archive.org/web/20050408193941/http://www.fho-emden.de/~hoffmann/quater12012002.pdf |archive-date=2005-04-08 }}
* {{cite arXiv |title=The Physical Heritage of Sir W. R. Hamilton |eprint=math-ph/0201058|last1=Gsponer|first1=Andre|last2=Hurni|first2=Jean-Pierre|year=2002|ref=none }}
* {{cite web |first=D.R. |last=Wilkins |title=Hamilton's research on quaternions |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html |ref=none }}
* {{cite web |first=David J. |last=Grossman |title=Quaternion Julia fractals |url=http://www.unpronounceable.com/julia/ |ref=none }} 3D Raytraced Quaternion [[Julia set|Julia Fractals]]
* {{cite web |title=Quaternion math and conversions |url=http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/index.htm|ref=none }} Great page explaining basic math with links to straight forward rotation conversion formulae.
* {{cite web |first=John H. |last=Mathews |title=Bibliography for quaternions |url=http://math.fullerton.edu/mathews/c2003/QuaternionBib/Links/QuaternionBib_lnk_3.html|archive-url=https://web.archive.org/web/20060902200454/http://math.fullerton.edu/mathews/c2003/QuaternionBib/Links/QuaternionBib_lnk_3.html |archive-date=2006-09-02 |ref=none }} 
* {{cite web |title=Quaternion powers |website=GameDev.net |url=https://www.gamedev.net/articles/programming/math-and-physics/quaternion-powers-r1095/|ref=none }}
* {{cite web |first=Andrew |last=Hanson |title=Visualizing quaternions |type=publisher page |url=http://books.elsevier.com/companions/0120884003/vq/index.html |archive-url=https://web.archive.org/web/20061105174313/http://books.elsevier.com/companions/0120884003/vq/index.html |archive-date=2006-11-05 |ref=none }}
* {{cite journal |first=Charles F.F. |last=Karney |title=Quaternions in molecular modeling |journal=J. Mol. Graph. Mod. |volume=25 |issue=5 |pages=595–604 |date=January 2007 |doi=10.1016/j.jmgm.2006.04.002 |pmid=16777449 |arxiv=physics/0506177|bibcode=2007JMGM...25..595K |s2cid=6690718 |ref=none }}
* {{cite arXiv |first=Johan E. |last=Mebius |title=A matrix-based proof of the quaternion representation theorem for four-dimensional rotations |year=2005 |eprint=math/0501249|ref=none }}
* {{cite arXiv |first=Johan E. |last=Mebius |title=Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations |year=2007 |eprint=math/0701759|ref=none }}  
* {{cite web |title=Hamilton Walk |publisher=Department of Mathematics, [[NUI Maynooth]] |url=http://www.maths.nuim.ie/links/hamilton.shtml|ref=none }}
* {{cite web |title=Using Quaternions to represent rotation |work=OpenGL:Tutorials |url=http://gpwiki.org/index.php/OpenGL:Tutorials:Using_Quaternions_to_represent_rotation |archive-url=https://web.archive.org/web/20071215235040/http://gpwiki.org/index.php/OpenGL:Tutorials:Using_Quaternions_to_represent_rotation |archive-date=2007-12-15 |ref=none }}
* David Erickson, [[Defence Research and Development Canada]] (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper. 
* {{cite web |first=Alberto |last=Martinez |title=Negative math: How mathematical rules can be positively bent |publisher=University of Texas |series=Department of History |url=https://webspace.utexas.edu/aam829/1/m/NegativeMath.html|archive-url=https://web.archive.org/web/20110924161347/https://webspace.utexas.edu/aam829/1/m/NegativeMath.html |archive-date=2011-09-24 |ref=none }}
* {{cite web |first=D. |last=Stahlke |title=Quaternions in classical mechanics |url=http://www.stahlke.org/dan/phys-papers/quaternion-paper.pdf|ref=none }}
* {{cite arXiv |last1=Morier-Genoud |first1=Sophie |first2=Valentin |last2=Ovsienko |title=Well, Papa, can you multiply triplets? |year=2008 |class=math.AC |eprint=0810.5562|ref=none }} describes how the quaternions can be made into a skew-commutative algebra graded by {{nobr|'''Z'''/2 × '''Z'''/2 × '''Z'''/2}}.
* {{cite web |first=Helen |last=Joyce |title=Curious Quaternions |date=November 2004 |publisher=hosted by [[John Baez]] |url=http://plus.maths.org/content/os/issue32/features/baez/index |ref=none }}
* {{cite web |first=Luis |last=Ibanez |title=Part&nbsp;I |department=Tutorial on quaternions |website=itk.org |type=course software |url=http://www.itk.org/CourseWare/Training/QuaternionsI.pdf |access-date=2011-12-05 |archive-url=https://web.archive.org/web/20120204055438/http://www.itk.org/CourseWare/Training/QuaternionsI.pdf |archive-date=2012-02-04 |url-status=dead |ref=none }} (uses Hamilton's terminology, which differs from the modern usage)
* {{cite web |first=Luis |last=Ibanez |title=Part&nbsp;II |department=Tutorial on quaternions |website=itk.org |type=course software |url=http://www.itk.org/CourseWare/Training/QuaternionsII.pdf |url-status=dead <!-- presumed --> |archive-url=https://web.archive.org/web/20121005003247/http://www.itk.org/CourseWare/Training/QuaternionsII.pdf |archive-date=2012-10-05 }} (uses Hamilton's terminology, which differs from the modern usage)
* {{cite journal |first1=R. |last1=Ghiloni |first2=V. |last2=Moretti |first3=A. |last3=Perotti |title=Continuous slice functional calculus in quaternionic Hilbert spaces |journal=Rev. Math. Phys. |volume=25 |pages=1350006–126 |year=2013 |issue=4 |doi=10.1142/S0129055X13500062 |arxiv=1207.0666|bibcode=2013RvMaP..2550006G |s2cid=119651315 |ref=none }}<br />{{cite journal |first1=R. |last1=Ghiloni |first2=V. |last2=Moretti |first3=A. |last3=Perotti |title=Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures |journal=Rev. Math. Phys. |volume=29 |pages=1750034 |year=2017 |doi=10.1142/S0129055X17500349 |arxiv=1602.02661|s2cid=124709652 |ref=none }} two expository papers about continuous functional calculus and spectral theory in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.
* {{cite web |last=Moritz |first=Will |title=Quaternions |type=[[Android (operating system)|Android]] software |website=play.google.com/store/apps |url=https://play.google.com/store/apps/details?id=com.MoritzWillProduction.Quaternions }} the Android app shows the quaternion corresponding to the orientation of the device.
* {{cite web |url=https://www.gamedeveloper.com/programming/rotating-objects-using-quaternions |title=Rotating objects using quaternions |website=gamedeveloper.com/programming}} article describing to the use of Quaternions for rotation in video games/computer graphics.
{{refend}}

==External links==
{{wikibooks|Associative Composition Algebra|Quaternions}}
{{Wikiquote}}
{{Wiktionary}}
*{{Commons category-inline}}
* [[Lawrence Paulson|Paulson, Lawrence C.]] [https://www.isa-afp.org/entries/Quaternions.html Quaternions (Formal proof development in Isabelle/HOL, Archive of Formal Proofs) ]
* [https://quaternions.online/ Quaternions – Visualisation]

{{Number systems|state=collapsed}}
{{Dimension topics|state=collapsed}}
{{Authority control|state=collapsed}}

[[Category:Composition algebras]]
[[Category:Quaternions| ]]
[[Category:William Rowan Hamilton]]