View source for Quaternion

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

|+Quaternion multiplication table
|-
!width=15 nowrap|&darr;&thinsp;&times;&thinsp;&rarr;
!width=15|{{math|1}}
!width=15|{{math|'''i'''}}
!width=15|{{math|'''j'''}}
!width=15|{{math|'''k'''}}
|-
!{{math|1}}
|{{math|1}}
|{{math|'''i'''}}
|{{math|'''j'''}}
|{{math|'''k'''}}
|-
!{{math|'''i'''}}
|{{math|'''i'''}}
|{{math|−1}}
|{{math|'''k'''}}
|{{math|−'''j'''}}
|-
!{{math|'''j'''}}
|{{math|'''j'''}}
|{{math|−'''k'''}}
|{{math|−1}}
|{{math|'''i'''}}
|-
!{{math|'''k'''}}
|{{math|'''k'''}}
|{{math|'''j'''}}
|{{math|−'''i'''}}
|{{math|−1}}
|-
|colspan=5 |Left column shows  premultiplier, top row shows post-multiplier. 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_quaternion_multiplication_graph.svg|thumb|[[Cayley graph|Cayley Q8 graph]] showing the six cycles of multiplication by {{red|'''i'''}}, {{green|'''j'''}} and {{blue|'''k'''}}. (If the image is opened in the [[Wikimedia Commons|Wikimedia Commons]] by clicking twice on it, cycles can be highlighted by hovering over or clicking on them.)]]

In [[Mathematics|mathematics]], the '''quaternion''' number system extends the [[Complex number|complex number]]s. Quaternions were first described by the Irish mathematician [[Biography:William Rowan Hamilton|William Rowan Hamilton]] in 1843<ref>{{cite journal |title=On Quaternions; or on a new System of Imaginaries in Algebra |journal=Letter to John T. Graves |date=17 October 1843}}</ref><ref>{{cite book |url=https://books.google.com/books?id=DRLpAFZM7uwC&pg=PA385 |title=The history of non-euclidean geometry: Evolution of the concept of a geometric space |year=1988 |publisher=Springer |first=Boris Abramovich |last=Rozenfelʹd |page=385 |isbn=9780387964584}}</ref> and applied to [[Physics:Mechanics|mechanics]] in [[Three-dimensional space|three-dimensional space]]. The algebra of quaternions is often denoted by {{math|'''H'''}} (for ''Hamilton''), or in [[Blackboard bold|blackboard bold]] by <math>\mathbb H.</math> Although multiplication of quaternions is noncommutative, it gives a definition of the quotient of two vectors in a three-dimensional space.<ref>{{cite book |url=https://archive.org/details/bub_gb_TCwPAAAAIAAJ |quote=quaternion quotient lines tridimensional space time |title=Hamilton |page=[https://archive.org/details/bub_gb_TCwPAAAAIAAJ/page/n188 60] |year=1853 |publisher=Hodges and Smith}}</ref><ref>{{cite book |url=https://books.google.com/books?id=YNE2AAAAMAAJ&q=quotient+two+vectors+called+quaternion |title=Hardy 1881 |page=32 |year=1881 |publisher=Ginn, Heath, & co.|isbn=9781429701860 }}</ref> Quaternions are generally represented in the form

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

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

Quaternions are used in [[Pure mathematics|pure mathematics]], but also have practical uses in [[Applied mathematics|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|computer vision]], and [[Physics: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|s2cid=55009081 }}</ref> They can be used alongside other methods of rotation, such as [[Euler angles]] and [[Rotation matrix|rotation matrices]], or as an alternative to them, depending on the application.

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

According to the [[Frobenius theorem (real division algebras)|Frobenius theorem]], the algebra <math>\mathbb H</math> is one of only two finite-dimensional [[Division ring|division ring]]s containing a proper [[Subring|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|associative algebra]] (and hence the largest ring). Further extending the quaternions yields the [[Non-associative algebra|non-associative]] [[Octonion|octonion]]s, which is the last normed division algebra over the real numbers. The next extension gives the sedenions, which have [[Zero divisor|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}}</ref>

The unit quaternions give a [[Group (mathematics)|group]] structure on the [[3-sphere|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|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 ==
<!-- -->
{{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 Brougham (Broom) Bridge, [[Place:Dublin|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</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''i{{thinsp}}j{{thinsp}}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 [[Biography:Olinde Rodrigues|Olinde Rodrigues]]' 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>{{cite book |first1=Robert E. |last1=Bradley |first2=Charles Edward |last2=Sandifer |title=Leonhard Euler: life, work and legacy |year=2007 |isbn=978-0-444-52728-8 |url=https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA193 |page=193|publisher=Elsevier }} They mention [[Biography:Wilhelm Blaschke|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> [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]] had also discovered quaternions in 1819, but this work was not published until 1900.<ref>Pujol, J., "[https://projecteuclid.org/journals/communications-in-mathematical-analysis/volume-13/issue-2/Hamilton-Rodrigues-Gauss-Quaternions-and-Rotations-a-Historical-Reassessment/cma/1349803591.full Hamilton, Rodrigues, Gauss, Quaternions, and Rotations: A Historical Reassessment]"  ''Communications in Mathematical Analysis'' (2012), 13(2), 1–14</ref><ref>{{cite book |first=C.F. |last=Gauss |article=Mutationen des Raumes [Transformations of space] (c.&nbsp;1819) |others=article edited by Prof.&nbsp;Stäckel of Kiel, Germany |editor=Martin Brendel |title=Carl Friedrich Gauss Werke |trans-title=The works of Carl Friedrich Gauss |year=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}}</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|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, [[Biography:Ferdinand Georg Frobenius|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.

The great breakthrough in quaternions finally came on Monday 16&nbsp;October 1843 in [[Place:Dublin|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 formula for the quaternions,

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

into the stone of Brougham Bridge as he paused on it. Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from 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, describing the train of thought that led to his discovery. This letter was later published in a letter to the ''London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'';<ref name=letter1844>{{cite magazine |last=Hamilton |first=W.R. |title=Letter |magazine=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=xxv |year=1844 |pages=489–495}}</ref> 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.<ref name=letter1844/>}}

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|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 |author=Hamilton, Sir W.R. |editor=Hamilton, W.E. |year=1866 |url=https://archive.org/details/elementsofquater00hamiuoft |location=London |publisher=Longmans, Green, & Co.}}</ref> was 800&nbsp;pages long; it was edited by his son and published shortly after his death.

After Hamilton's death, the Scottish mathematical physicist  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 [[Physics:Kinematics|kinematics]] in space and [[Physics:Maxwell's equations|Maxwell's equations]], were described entirely in terms of quaternions. There was even a professional research association, the [[Organization:Quaternion Society|Quaternion Society]], devoted to the study of quaternions and other [[Hypercomplex number|hypercomplex number]] systems.

From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by [[Biography:Josiah Willard Gibbs|Josiah Willard Gibbs]], Oliver Heaviside, and [[Biography:Hermann von Helmholtz|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:Physics|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.

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|gimbal lock]]". For this reason, quaternions are used in [[Computer graphics|computer graphics]],<ref name="Shoemake">{{cite journal |doi=10.1145/325165.325242 |first=Ken |last=Shoemake |year=1985 |url=https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf |title=Animating Rotation with Quaternion Curves |journal=Computer Graphics |volume=19 |issue=3 |pages=245–254}} Presented at SIGGRAPH '85.</ref><ref>''[[Software: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. See, for example {{cite magazine |author=Nick Bobick |url=http://www.gamasutra.com/view/feature/3278/rotating_objects_using_quaternions.php |title=Rotating objects using quaternions |magazine=Game Developer |date=July 1998}}</ref> [[Computer vision|computer vision]], robotics,<ref>{{cite book |first=J.M. |last=McCarthy |title=An Introduction to Theoretical Kinematics |url=https://books.google.com/books?id=glOqQgAACAAJ |year=1990 |publisher=MIT Press |isbn=978-0-262-13252-7}}</ref> [[Control theory|control theory]], [[Signal processing|signal processing]], [[Astronomy:Spacecraft attitude control|attitude control]], [[Physics:Physics|physics]], [[Biology:Bioinformatics|bioinformatics]], [[Physics:Molecular dynamics|molecular dynamics]], [[Computer simulation|computer simulation]]s, and [[Astronomy:Orbital mechanics|orbital mechanics]]. For example, it is common for the [[Astronomy:Spacecraft attitude control|attitude control]] systems of spacecraft to be commanded in terms of quaternions.  Quaternions have received another boost from [[Number theory|number theory]] because of their relationships with the [[Quadratic form|quadratic form]]s.<ref>{{citation |last=Hurwitz |first=A.
|title=Vorlesungen über die Zahlentheorie der Quaternionen |jfm=47.0106.01
|place=Berlin |publisher=J. Springer |year=1919}}, concerning [[Hurwitz quaternion]]s</ref>

=== Quaternions in physics ===
P.R. Girard's 1984 essay ''The quaternion group and modern physics''<ref>{{cite journal |last=Girard |first=P.R. |title=The quaternion group and modern physics |year=1984 |journal=European Journal of Physics |volume=5 |issue=1 |pages=25–32 |doi=10.1088/0143-0807/5/1/007|bibcode=1984EJPh....5...25G |s2cid=250775753 }}</ref> discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely {{math|SO(3)}}, the Lorentz group, the general theory of relativity group, the Clifford algebra {{math|SU(2)}} and the conformal group, can easily be related to the [[Quaternion group|quaternion group]] in [[Abstract algebra|modern algebra]]. Girard began by discussing [[Group representation|group representation]]s and by representing some [[Space group|space group]]s of [[Physics:Crystallography|crystallography]]. He proceeded to [[Physics:Kinematics|kinematics]] of [[Rigid body|rigid body]] motion. Next he used complex quaternions ([[Biquaternion|biquaternion]]s) to represent the [[Lorentz group]] of special relativity, including the [[Physics:Thomas precession|Thomas precession]]. He cited five authors, beginning with [[Biography:Ludwik Silberstein|Ludwik Silberstein]], who used a [[Physics:Potential|potential]] function of one quaternion variable to express [[Physics:Maxwell's equations|Maxwell's equations]] in a single [[Differential equation|differential equation]]. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions ([[Split-biquaternion|split-biquaternion]]s) as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described [[Conformal map|conformal map]]s on [[Physics:Spacetime|spacetime]]. Among the fifty references, Girard included Alexander Macfarlane and his ''Bulletin'' of the [[Organization:Quaternion Society|Quaternion Society]]. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions.<ref>{{cite journal |url=http://clifford-algebras.org/v9/v92/GIRAR92.pdf |title=Einstein's equations and Clifford algebra |archive-url=https://web.archive.org/web/20101217224618/http://clifford-algebras.org/v9/v92/GIRAR92.pdf |archive-date=17 December 2010 |journal=Advances in Applied Clifford Algebras |volume=9 |issue=2 |pages=225–230 |year=1999|doi=10.1007/BF03042377 |last1=Girard |first1=Patrick R. |s2cid=122211720 }}</ref>

The finding of 1924 that in [[Physics:Quantum mechanics|quantum mechanics]] the [[Physics:Spin|spin]] of an electron and other matter particles (known as [[Physics:Spinors|spinors]]) can be described using quaternions (in the form of the famous 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 |last1=Huerta |first1=John |title=Introducing The Quaternions |url=http://math.ucr.edu/~huerta/introquaternions.pdf |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 |date=27 September 2010}}</ref><ref>{{cite web |first=Charlie |last=Wood |title=The Strange Numbers That Birthed Modern Algebra |date=6 September 2018 |work=Abstractions 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 overtaken [[Orthogonal group|rotation groups]].<ref group=lower-alpha>A more personal view of quaternions was written by [[Biography:Joachim Lambek|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." {{cite magazine |author=Lambek, J. |title=If Hamilton had prevailed: Quaternions in physics |year=1995 |magazine=Math. Intelligencer |volume=17 |issue=4 |pages=7–15 |doi=10.1007/BF03024783}}</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 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'''}}.

Hamilton describes a quaternion <math>q = a + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}</math>, as consisting of a [[Scalar (mathematics)|scalar]] part and a vector part. The quaternion <math> b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}</math> is called the ''vector part'' (sometimes ''imaginary part'') of {{mvar|q}}, and {{mvar|a}} is the ''scalar part'' (sometimes ''real part'') of {{mvar|q}}. A quaternion that equals its real part (that is, its vector part is zero) is called a ''scalar'' or ''real quaternion'', and is identified with the corresponding real number. That is, the real numbers are ''embedded'' in the quaternions. (More properly, the [[Field (mathematics)|field]] of real numbers is isomorphic to a subset 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 ''vector quaternion''.

The set of quaternions is a 4-dimensional [[Vector space|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 real quaternion {{math|'''1'''}} is the [[Identity element|identity element]].
*The '''real''' quaternions commute with all other quaternions, that is {{math|''aq'' {{=}} ''qa''}} for every quaternion {{mvar|q}} and every real quaternion {{mvar|a}}. In algebraic terminology this is to say that the field of real quaternions are the [[Center (ring theory)|''center'']] of this quaternion algebra.
*The product is first given for the basis elements (see next subsection), and then extended to all quaternions by using the [[Distributive property|distributive property]] and the center property of the real quaternions. 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|×
!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}}
|-
|+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|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 real quaternions. In fact, it is a part of the definition that the real 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{ij} + 2d\,\mathbf{ik} = 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 ''real'' 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 "roots of –1" form a [[Unit sphere|unit sphere]] in the three-dimensional space of vector quaternions.

=== 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>

The product of two [[Quaternions and spatial rotation#Using quaternions as rotations|rotation quaternions]]<ref>{{cite web |title=Maths – Transformations using Quaternions |url=http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/transforms/index.htm |website=EuclideanSpace |quote=A rotation of {{math|q1}} followed by a rotation of {{math|q2}} is equivalent to a single rotation of {{math|q2 q1}}. Note the reversal of order, that is, we put the first rotation on the right hand side of the multiplication.}}</ref> will be equivalent to the rotation {{nowrap|{{math|''a''<sub>2</sub> + ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}}}} followed by the rotation  {{nowrap|{{math|''a''<sub>1</sub> + ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}}.}}

=== 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 '''scalar''', 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'''.  If {{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} is any quaternion, then {{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|It is important to note that the vector part of a quaternion is, in truth, an "axial" vector or "[[Pseudovector|pseudovector]]", ''not'' an ordinary or "polar" vector, as was formally proven by Altmann (1986).<ref>{{cite book |first=S.L. |last=Altmann |at=Ch.&nbsp;12 |title=Rotations, Quaternions, and Double Groups}}</ref> A polar vector can be represented in calculations (for example, for rotation by a quaternion "similarity transform") by a pure imaginary 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] |author=Hamilton, Sir William Rowan |year=1866}}</ref><ref>{{cite journal |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 |author=Hardy |title=Elements of Quaternions |journal=Science |year=1881 |volume=2 |issue=75 |page=65 |publisher=library.cornell.edu|doi=10.1126/science.os-2.75.564 |pmid=17819877 }}</ref> and real numbers (considered as quaternions with zero vector part) '''scalar quaternions'''.

If a quaternion is divided up into a scalar part and a 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|dot product]] and the [[Cross product|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''' 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 stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:

<math display=block>
q^* = - \frac{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|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 ''q'', but this conflicts with the modern meaning of "[[Tensor|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|imaginary unit]]) and hence from the multiplicative property of [[Determinant|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|metric space]].
Addition and multiplication are 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|versor]]'' of {{mvar|q}}:

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

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

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'' 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''}}&nbsp; are different (except if {{mvar|p}} and {{mvar|q}} are scalar multiples of each other or if one is a scalar): the notation {{math|{{sfrac|''p''|''q''}}}} is ambiguous and should not be used.

== Algebraic properties ==
[[File:Cayley graph Q8.svg|right|thumb|[[Cayley graph|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|In comparison, the real numbers <math>\mathbb R</math> have dimension&nbsp;1, the complex numbers <math>\mathbb C</math> have dimension&nbsp;2, and the octonions <math>\mathbb O</math> have dimension&nbsp;8.}} Multiplication of quaternions is associative and distributes over vector addition, but with the exception of the scalar subset, it is not commutative. Therefore, the quaternions <math>\mathbb H</math> are a non-commutative, associative algebra over the real numbers. Even though <math>\mathbb H</math> contains copies of the complex numbers, it is not an associative algebra over the complex numbers.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the non-commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The [[Frobenius theorem (real division algebras)|Frobenius theorem]] states that there are exactly three: <math>\mathbb R</math>, <math>\mathbb C</math>, and <math>\mathbb H</math>. The 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>\mathbb R</math>, <math>\mathbb C</math>, <math>\mathbb H</math>, and <math>\mathbb O</math> (the octonions). The quaternions are also an example of a [[Composition algebra|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 are 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|abelian group]] is called the quaternion group and is denoted {{math|Q<sub>8</sub>}}.<ref>{{cite web |url=http://www.wolframalpha.com/input/?i=quaternion+group |title=quaternion group |work=Wolframalpha.com}}</ref> The real [[Group ring|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|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 differences 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|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|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|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>

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 |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=[[Physics:Philosophical Magazine|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|versor]]s by Hamilton.<ref name="Hamilton" /> Also see [[Quaternions and spatial rotation]] for more information about modeling three-dimensional rotations using quaternions.

See [[Biography: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}}</ref> for visualization of quaternions.

== Matrix representations ==
Just as complex numbers can be [[Complex number#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|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. In the terminology of [[Abstract algebra|abstract algebra]], these are [[Injective function|injective]] [[Homomorphism|homomorphism]]s from <math>\mathbb H</math> to the [[Matrix ring|matrix ring]]s {{math|M(2,'''C''')}} and {{math|M(4,'''R''')}}, respectively.

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

: <math> \left[ \begin{array}{rr}  a+bi & c+di \\ -c + d i & a - b i \end{array} \right].</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|determinant]] of the corresponding matrix.<ref>{{cite web |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 |title=[no title cited; determinant evaluation] |website=Wolframalpha.com}}</ref>
* The conjugate of a quaternion corresponds to the [[Conjugate transpose|conjugate transpose]] of the matrix.
* By restriction this representation yields an [[Group isomorphism|isomorphism]] between the subgroup of unit quaternions and their image SU(2). Topologically, the unit quaternions 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 [[Physics:Spin|spin]] in quantum mechanics; see [[Pauli matrices]].
* There is a strong relation between quaternion units and Pauli matrices. Obtain the eight quaternion unit matrices by taking {{mvar|a}}, {{mvar|b}}, {{mvar|c}} and {{mvar|d}}, set three of them at zero and the fourth at 1 or −1. Multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for −1. One obtains −1 via {{nowrap|{{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>ijk = \sigma_1 \sigma_2 \sigma_3 \sigma_1 \sigma_2 \sigma_3 = -1.</math>

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 exist 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|transpose]] of the matrix. The fourth power of the norm of a quaternion is the [[Determinant|determinant]] of the corresponding matrix. As with the 2 × 2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2 × 2 blocks by setting {{math|1=''c'' = ''d'' = 0}}.

Each 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|×
!width=15|''a''
!width=15|''d''
!width=15|−''b''
!width=15|−''c''
|-
!''a''
|''a''
|''d''
|''−b''
|''−c''
|-
!''−d''
|''−d''
|''a''
|''c''
|''−b''
|-
!''b''
|''b''
| −''c''
|''a''
|−''d''
|-
!''c''
|''c''
|''b''
|''d''
|''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|×
!width=15| 1 
!width=15|'''k'''
!width=15|−'''i'''
!width=15|−'''j'''
|-
!1 
|1 
|'''k'''
|−'''i'''
|−'''j'''
|-
!−'''k'''
|−'''k'''
|1
|'''j'''
|−'''i'''
|-
!'''i'''
|'''i'''
|−'''j'''
|1 
|−'''k'''
|-
!'''j'''
|'''j'''
|'''i'''
|'''k'''
|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|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|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>\mathbb 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>\mathbb 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|sphere]] in <math>\mathbb 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 {{nowrap|{{math|''a'' {{=}} 0}}}} or {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are all 0. The latter is impossible because ''a'' is a real number and the first equation would imply that {{nowrap|{{math|''a''<sup>2</sup> {{=}} −1}}.}} Therefore, {{nowrap|{{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 −1 if and only if it is a vector quaternion with norm 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=1899 |title=Elements of Quaternions |edition=2nd |page=244 |publisher=Cambridge University Press |isbn=1-108-00171-8}}</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 [[Organization:National Council of Teachers of Mathematics|National Council of Teachers of Mathematics]].<ref>{{Cite book |last=Perlis |first=Sam |chapter=Capsule 77: Quaternions |title=Historical Topics in Algebra |url=https://archive.org/details/historicaltopics0000nati/page/38/mode/2up |url-access=registration |publisher=[[Organization:National Council of Teachers of Mathematics|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 [[Biography:Ian R. Porteous|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. |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={{wipe|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 pair of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. If {{nowrap|{{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|ring homomorphism]] from <math>\mathbb C</math> to <math>\mathbb H,</math> which defines a field [[Isomorphism|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|subalgebra]] of the quaternions that is isomorphic to <math>\mathbb C,</math> and is thus a planar subspace of <math>\mathbb H\colon</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|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|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|real line]]. Each of these [[Complex plane|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|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{\frac{\|\mathbf q\|+r}{2}},\ \frac{\vec{v}}{\|\vec{v}\|}\sqrt{\frac{\|\mathbf q\|-r}{2}}\right),
</math>

where <math display="inline">\|\vec{v}\| = \sqrt{\vec{v}\cdot\vec{v}}=\sqrt{-\vec v^2}</math> is the norm of <math>\vec{v}</math> and <math display="inline">\|\mathbf q\| = \sqrt{\mathbf q^*\mathbf q} = 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">\frac{\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(\sqrt{-r})</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===
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} \left(\cos \|\mathbf{v}\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \sin \|\mathbf{v}\|\right),
</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^{\hat{n}\varphi} = \|q\| \left(\cos(\varphi) + \hat{n} \sin(\varphi)\right),</math>

where the [[Angle|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=[[Company: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]] {{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 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 \left(\cos(x\varphi) + \hat{n}\,\sin(x\varphi)\right)~.</math>

===Geodesic norm===
The [[Great-circle distance|geodesic distance]] {{nowrap|{{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|dot product]] without the logarithm as:

<math display=block>\arccos(2(p \cdot q)^2 - 1).</math>

==Three-dimensional and four-dimensional rotation groups==
The word "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|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''³ itself has not a canonical group structure -->The set of all unit quaternions ([[Versor|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|determinant]]&nbsp;1 since ''two'' unit quaternions correspond to every rotation under the above correspondence. See [[Plate trick|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]] 2×2&nbsp;matrices of [[Determinant|determinant]] 1.

Let {{mvar|A}} be the set of quaternions of the form {{nowrap|{{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} }} where {{mvar|a, b, c,}} and {{mvar|d}} are either all [[Integer|integer]]s or all [[Half-integer|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 cell]] with [[Schläfli symbol]] {{math|{3,4,3}.}} They correspond to the double cover of the rotational symmetry group of the regular [[Tetrahedron|tetrahedron]]. Similarly, the vertices of a [[600-cell|regular 600 cell]] with Schläfli symbol {{math|{3,3,5}}} can be taken as the unit [[Icosian|icosian]]s, corresponding to the double cover of the rotational symmetry group of the [[Regular icosahedron|regular icosahedron]]. The double cover of the rotational symmetry group of the regular [[Octahedron|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 {{nowrap|{{math|'''i j''' {{=}} −'''j i'''}}}} (so {{nowrap|{{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)}} ==
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}^+(\mathbb R)</math> of the Clifford algebra <math>\operatorname{Cl}_{3,0}(\mathbb 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_i \sigma_j = - \sigma_j \sigma_i \qquad (j \neq i).</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{k} \mapsto \sigma_2 \sigma_1\,, \quad \mathbf{i} \mapsto \sigma_3 \sigma_2\,, \quad \mathbf{j} \mapsto \sigma_1 \sigma_3\,,</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|bivector]]s – quantities with magnitude and 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>1</sub>''σ''<sub>2</sub>}}, {{math|''σ''<sub>2</sub>''σ''<sub>3</sub>}}, {{math|''σ''<sub>3</sub>''σ''<sub>1</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|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|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}} 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}}</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|pseudo-Euclidean space]]s, for example, the [[Physics:Minkowski space|Minkowski space]] of [[Physics:Special relativity|special relativity]]. In such spaces rotors can be used to efficiently represent Lorentz boosts, and to interpret formulas involving the [[Gamma matrices|gamma matrices]].

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|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|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|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 fields, 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.

== Quotations ==
{{blockquote|text=I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to {{math|x, y, z,}} etc.|author=William Rowan Hamilton (circa 1848)<ref>{{cite book |last1=Hamilton |first1=William Rowan |title=Lectures on quaternions |date=1853 |publisher=Dublin: Hodges and Smith |page=522 |url=https://archive.org/details/lecturesonquater00hami/page/522/mode/2up}}</ref>}}

{{blockquote|text=Time is said to have only one dimension, and space to have three dimensions.&nbsp;... The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions.&nbsp;... ''And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be''.|author=William Rowan Hamilton (circa 1853)<ref>{{cite book |first=R.P. |last=Graves |title=Life of Sir William Rowan Hamilton|url=https://archive.org/details/lifeofsirwilliam03gravuoft/page/634/mode/2up|pages=635–636 |publisher=Dublin Hodges, Figgis }}</ref>}}

{{blockquote|text=Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including [[Biography:James Clerk Maxwell|Clerk Maxwell]].|author=[[Biography:Lord Kelvin|W. Thompson, Lord Kelvin]] (1892)<ref>{{cite book |last1=Thompson |first1=Silvanus Phillips |title=The life of William Thomson (Vol. 2) |date=1910 |publisher=London, Macmillan |page=1138 |url=https://archive.org/details/lifeofwillthom02thomrich/page/n581/mode/2up}}</ref>}}

{{blockquote|text=There was a time, indeed, when I, although recognizing the appropriateness of vector analysis in electromagnetic theory (and in mathematical physics generally), did think it was harder to understand and to work than the Cartesian analysis. But that was before I had thrown off the quaternionic old-man-of-the-sea who fastened himself about my shoulders when reading the only accessible treatise on the subject – Prof. Tait's Quaternions. But I came later to see that, so far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work. There is not a ghost of a quaternion in any of my papers (except in one, for a special purpose). The vector analysis I use may be described either as a convenient and systematic abbreviation of Cartesian analysis; or else, as Quaternions without the quaternions, .... "Quaternion" was, I think, defined by an American schoolgirl to be "an ancient religious ceremony". This was, however, a complete mistake. The ancients – unlike Prof. Tait – knew not, and did not worship Quaternions. |author=Oliver Heaviside (1893)<ref>{{cite book |first=Oliver |last=Heaviside |year=1893 |title=Electromagnetic Theory |volume=I |pages=134–135 |place=London, UK |publisher=The Electrician Printing and Publishing Company |url=https://archive.org/details/electromagnetict01heavrich/page/134/ }}</ref>}}

{{blockquote|text=Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity. Moreover, in science as well as in everyday life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols.|author=[[Biography:Ludwik Silberstein|Ludwik Silberstein]] (1924)<ref>Ludwik Silberstein (1924). [https://archive.org/details/in.ernet.dli.2015.212395/page/n5/mode/2up?view=theater Preface to second edition] of ''The Theory of Relativity''</ref>}}

{{blockquote|text=...&nbsp;quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.|author=Simon L. Altmann (1986)<ref>{{cite book |first=Simon L. |last=Altmann |title=Rotations, quaternions, and double groups |publisher=Clarendon Press |year=1986 |isbn=0-19-855372-2 |lccn=85013615 }}</ref>}}

== See also ==
* {{Annotated link|Conversion between quaternions and Euler angles}}
* {{Annotated link|Dual quaternion}}
* {{Annotated link|Dual-complex number}}
* {{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}}

==Notes==
{{notelist}}

==References==
{{Reflist|25em}}

==Further reading==
===Books and publications===
{{refbegin|30em}}
*{{cite journal |first=William Rowan |last=Hamilton |title=On quaternions, or on a new system of imaginaries in algebra |journal=Philosophical Magazine |volume=25 |issue=3 |pages=489–495 |year=1844 |doi=10.1080/14786444408645047 |url=https://zenodo.org/record/1431043 }}
*[[Biography: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.
*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.
*Hamilton (1899) ''Elements of Quaternions'' volume I, (1901) volume II. Edited by [[Biography:Charles Jasper Joly|Charles Jasper Joly]]; published by Longmans, Green & Co.
*Tait, Peter Guthrie (1873), "''An elementary treatise on quaternions''". 2d ed., Cambridge, [Eng.] : The University Press.
*Maxwell, James Clerk (1873), "''[[Physics:A Treatise on Electricity and Magnetism|A Treatise on Electricity and Magnetism]]''". Clarendon Press, Oxford.
*Tait, Peter Guthrie (1886), "''{{cite web|url=http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 |title=Archived copy |access-date=June 26, 2005 |url-status=unfit |archive-url=https://web.archive.org/web/20140808040037/http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 |archive-date=August 8, 2014 }}''". M.A. Sec. R.S.E. [[Encyclopædia Britannica]], Ninth Edition, 1886, Vol. XX, pp.&nbsp;160&ndash;164. (bzipped [[PostScript]] file)
*{{cite book |first=Charles Jasper |last=Joly |title=A manual of quaternions |publisher=Macmillan |year=1905 |lccn=05036137}}
*{{cite book |first=Alexander |last=Macfarlane |title=Vector analysis and quaternions |publisher=Wiley |edition=4th |year=1906 |lccn=16000048}}
* (''See section on quaternions.'')
*{{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 }}
*{{cite book |first=Patrick |last=Du Val |title=Homographies, quaternions, and rotations |publisher=Clarendon Press |series=Oxford mathematical monographs |year=1964 |lccn=64056979}}
*Michael J. Crowe (1967), [[A History of Vector Analysis]]: ''The Evolution of the Idea of a Vectorial System'', University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
*{{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 }}
*{{cite journal |first=Jose |last=Pujol |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 }}
*{{cite book |first=Stephen L. |last=Adler |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}}
*{{cite book |first=J.P. |last=Ward |title=Quaternions and Cayley Numbers: Algebra and Applications |publisher=Kluwer Academic |year=1997 |isbn=0-7923-4513-4 }}
*{{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 }}
*{{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}}
*{{cite book |first=Jack |last=Kuipers |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 }}
*{{cite book |last1=Conway |first1=John Horton |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 arXiv |first=P.M. |last=Jack |title=Physical space as a quaternion structure, I: Maxwell equations. A brief Note |date=2003 |eprint=math-ph/0307038}} 
*{{cite book |first=Vladislav |last=Kravchenko |title=Applied Quaternionic Analysis |publisher=Heldermann Verlag |year=2003 |isbn=3-88538-228-8 }}
*{{cite book |first1=Michiel |last1=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 book |first=Andrew J. |last=Hanson |title=Visualizing Quaternions |publisher=Elsevier |year=2006 |isbn=0-12-088400-3 |url=http://www.cs.indiana.edu/~hanson/quatvis/}}
*{{cite book |first1=Ernst |last1=Binz |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 }}
*{{cite book |first1=Chris J.L. |last1=Doran |first2=Anthony N. |last2=Lasenby |title=Geometric Algebra for Physicists |year=2003 |publisher=Cambridge University Press |isbn=978-0-521-48022-2 |}}
*{{cite book |first=John A. |last=Vince |title=Geometric Algebra for Computer Graphics |publisher=Springer |year=2008 |isbn=978-1-84628-996-5 }}
*For molecules that can be regarded as classical rigid bodies [[Physics:Molecular dynamics|molecular dynamics]] computer simulation employs quaternions.  They were first introduced for this purpose by {{cite journal |first=D.J. |last=Evans |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 }}
*{{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 }}
* {{cite book|author=Ron Goldman|title=Rethinking Quaternions: Theory and Computation|year=2010|publisher=Morgan & Claypool |isbn=978-1-60845-420-4}}
* {{ citation | 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 book |first=John |last=Voight |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 }}
{{refend}}

===Links and monographs===
{{refbegin|30em}}
* {{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 |title=Frequently Asked Questions |work=Matrix and Quaternion  |id=1.21 |url=http://www.j3d.org/matrix_faq/matrfaq_latest.html}}
* {{cite web |first=Doug |last=Sweetser |title=Doing Physics with Quaternions |url=http://world.std.com/~sweetser/quaternions/qindex/qindex.html }}
* [https://web.archive.org/web/20050408193941/http://www.fho-emden.de/~hoffmann/quater12012002.pdf Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)]
* {{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}}
* {{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 }}
* {{cite web |first=David J. |last=Grossman |title=Quaternion Julia Fractals |url=http://www.unpronounceable.com/julia/ }} 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}} 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 }} 
* {{cite web |title=Quaternion powers |publisher=GameDev.net |url=https://www.gamedev.net/articles/programming/math-and-physics/quaternion-powers-r1095/}}
* {{cite web |first=Andrew |last=Hanson |title=Visualizing Quaternions home 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 }}
* {{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|s2cid=6690718 }}
* {{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}}
* {{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}}  
* {{cite web |title=Hamilton Walk |publisher=Department of Mathematics, NUI Maynooth |url=http://www.maths.nuim.ie/links/hamilton.shtml}}
* {{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 }}
* 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=Department of History, University of Texas |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 }}
* {{cite web |first=D. |last=Stahlke |title=Quaternions in Classical Mechanics |url=http://www.stahlke.org/dan/phys-papers/quaternion-paper.pdf}}
* {{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}} describes how the quaternions can be made into a skew-commutative algebra graded by {{nowrap|'''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}}
* {{cite web |first=Luis |last=Ibanez |title=Tutorial on Quaternions. Part I |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 }} [https://web.archive.org/web/20121005003247/http://www.itk.org/CourseWare/Training/QuaternionsII.pdf Part II] (PDF; using 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 }}<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 }} two expository papers about continuous functional calculus and spectral theory in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.
* [https://play.google.com/store/apps/details?id=com.MoritzWillProduction.Quaternions Quaternions] the Android app shows the quaternion corresponding to the orientation of the device.
* [https://www.gamasutra.com/view/feature/131686/rotating_objects_using_quaternions.php Rotating Objects Using Quaternions] article speaking to the use of Quaternions for rotation in video games/computer graphics.
{{refend}}

==External links==
* [[Biography: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}}


[[Category:Composition algebras]]
[[Category:Quaternions| ]]

{{Sourceattribution|Quaternion}}