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{{About|quaternions in mathematics}}
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In [[mathematics]], the '''quaternions''' are a [[number system]] that extends the [[complex number]]s. They were first described by Irish mathematician [[William Rowan Hamilton]] in 1843<ref>{{cite conference |title=On Quaternions; or on a new System of Imaginaries in Algebra (letter to John T. Graves, dated 17&nbsp;October 1843) |year=1843}}</ref><ref>{{cite book |url=https://books.google.com/?id=DRLpAFZM7uwC&pg=PA385 |title=The history of non-euclidean geometry: Evolution of the concept of a geometric space |year=1988 |publisher=Springer |author=Boris Abramovich Rozenfelʹd |page=385 |isbn=9780387964584}}</ref>  and applied to [[mechanics]] in [[three-dimensional space]]. A feature of quaternions is that multiplication of two quaternions is [[noncommutative]]. Hamilton defined a quaternion as the [[quotient]] of two directed lines in a three-dimensional space<ref>{{cite book |url=https://books.google.com/?id=TCwPAAAAIAAJ&printsec=frontcover&dq=quaternion+quotient+lines+tridimensional+space+time#PPA60,M1 |title=Hamilton |page=60 |year=1853 | publisher=Hodges and Smith}}</ref> or equivalently as the quotient of two [[vector (geometry)|vector]]s.<ref>{{cite book |url=https://books.google.com/?id=YNE2AAAAMAAJ&printsec=frontcover&dq=quotient+two+vectors+called+quaternion#PPA32,M1 |title=Hardy 1881 pg. 32 |year=1881 | publisher=Ginn, Heath, & co.}}</ref>

Quaternions are generally represented in the form:
:<math>a + b\ \mathbf{i} + c\ \mathbf{j} +d\ \mathbf{k}</math>

where ''a'', ''b'', ''c'', and ''d'' are real numbers, and  '''''i''''', '''''j''''', and '''''k''''' are the fundamental ''quaternion units''.

Quaternions are used in [[pure mathematics]], and also have practical uses in [[applied mathematics]] — in particular for [[quaternions and spatial rotation|calculations involving three-dimensional rotations]] such as in [[3D computer graphics|three-dimensional computer graphics]], [[computer vision]], and [[Texture (crystalline)|crystallographic texture]] analysis.<ref name=Lunze>{{cite journal |authors=Karsten Kunze & Helmut Schaeben |title=The Bingham distribution of quaternions and its spherical radon transform in texture analysis |journal=Mathematical Geology |date=November 2004 |volume=8 |issue=8 |pages=917–943 |doi=10.1023/B:MATG.0000048799.56445.59}}</ref> In practical applications, they can be used alongside other methods, such as [[Euler angles]] and [[rotation matrix|rotation matrices]], or as an alternative to them, depending on the application.

In modern mathematical language, quaternions form a four-[[dimension (vector space)|dimensional]] associative [[composition algebra|normed division algebra]] over the [[real number]]s, and therefore also a [[domain (ring theory)|domain]]. In fact, the quaternions were the first [[division algebra|noncommutative division algebra]] to be discovered. The algebra of quaternions is often denoted by '''H''' (for ''Hamilton''), or in [[blackboard bold]] by <math>\mathbb H</math> ([[Unicode]] U+210D, ℍ). It can also be given by the [[Clifford algebra]] [[Classification of Clifford algebras|classifications]] {{nowrap|''C''ℓ<sub>0,2</sub>('''ℝ''') ≅ ''C''ℓ<sup>0</sup><sub>3,0</sub>('''ℝ''')}}. The algebra '''ℍ''' holds a special place in analysis since, according to the [[Frobenius theorem (real division algebras)|Frobenius theorem]], it is one of only two finite-dimensional [[division ring]]s containing the [[real numbers]] as a proper [[subring]], the other being the complex numbers. These rings are also [[Hurwitz's theorem (composition algebras)|Euclidean Hurwitz algebras]], of which quaternions are the largest [[associative algebra]]. Further extending the quaternions yields the [[Non-associative algebra|non-associative]] [[octonion]]s, which is the last [[normed division algebra]] over the reals (the extension of the octonions, [[sedenions]], has [[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? |author=Frank (Tony) Smith |access-date=8 June 2018}}</ref>

The [[unit quaternion]]s can be thought of as a choice of a group structure on the [[3-sphere]] S<sup>3</sup> that gives the group [[Spin(3)]], which is isomorphic to [[SU(2)]] and also to the [[universal cover]] of [[SO(3)]].
[[File:Quaternion 2.svg|thumb|300px|right|Graphical representation of products of quaternion units as 90°-rotations in the planes of 4D-space, spanned by two of {1, ''i'', ''j'', ''k''}. The left factor can be taken as rotated by the right. For example:<br>
in blue: 1/''i''-plane: 1&nbsp;⋅&nbsp;''i''&nbsp;=&nbsp;''i'', ''i''/''k''-plane: ''i''&nbsp;⋅&nbsp;''j''&nbsp;=&nbsp;''k''<br>
in ''red'': 1/''j''-plane: 1&nbsp;⋅&nbsp;''j''&nbsp;=&nbsp;''j'', ''j''/''k''-plane: ''j''&nbsp;⋅&nbsp;''i''&nbsp;=&nbsp;-''k'']]

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

[[File:Inscription on Broom Bridge (Dublin) regarding the discovery of Quaternions multiplication by Sir William Rowan Hamilton.jpg|right|thumb|Quaternion plaque on [[Broom Bridge|Brougham (Broom) Bridge]], [[Dublin]], which says: <br><br><center>Here as he walked by<br>on the 16th of October 1843<br>Sir William Rowan Hamilton<br>in a flash of genius discovered<br>the fundamental formula for<br>quaternion multiplication<br> {{math|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = −1}}<br>& cut it on a stone of this bridge]]

Quaternions were introduced by Hamilton in 1843.<ref name="SeeHazewinkel">See Hazewinkel et al. (2004), p. 12.</ref> Important precursors to this work included [[Euler's four-square identity]] (1748) and [[Olinde Rodrigues]]' [[Euler–Rodrigues parameters|parameterization of general rotations by four parameters]] (1840), but neither of these writers treated the four-parameter rotations as an algebra.<ref>{{cite book |first1=John Horton |last1=Conway |first2=Derek Alan |last2=Smith |title=On quaternions and octonions: Their geometry, arithmetic, and symmetry |year=2003 |isbn=1-56881-134-9 |url=https://books.google.com/books?id=E_HCwwxMbfMC&pg=PA9 |page=9}}</ref><ref>{{cite book |authors=Robert E. Bradley, Charles Edward Sandifer |title=Leonhard Euler: life, work and legacy |year=2007 |isbn=0-444-52728-1 |url=https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA193 |page=193}}. They mention [[Wilhelm Blaschke]]'s claim in 1959 that "the quaternions were first identified by L.&nbsp;Euler in a letter to Goldbach written on 4&nbsp;May 1748," and they comment that "it makes no sense whatsoever to say that Euler "identified" the quaternions in this letter ... this claim is absurd."</ref> [[Carl Friedrich Gauss]] had also discovered quaternions in 1819, but this work was not published until 1900.<ref>{{Cite journal |author=Simon L. Altmann |journal=Mathematics Magazine |volume=62 |title=Hamilton, Rodrigues, and the quaternion scandal |issue=5 |date=December 1989 |page=306 |jstor=2689481 |doi=10.2307/2689481}}</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#v=onepage&q&f=false}}</ref>

Hamilton knew that the [[complex number]]s could be interpreted as [[point (geometry)|points]] in a [[plane (mathematics)|plane]], and he was looking for a way to do the same for points in three-dimensional [[space]]. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the [[quotient]] of the coordinates of two points in space.

The great breakthrough in quaternions finally came on Monday 16&nbsp;October 1843 in [[Dublin]], when Hamilton was on his way to the [[Royal Irish Academy]] where he was going 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>\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{ijk} = -1</math>

into the stone of [[Broom Bridge|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 a science magazine;<ref name=letter1844>{{cite magazine |author=Hamilton, 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:
{{quote|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]]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. |author-link=William Rowan Hamilton |editor=Hamilton, W.E. |editor-link=William Edwin Hamilton |year=1866 |url=https://archive.org/details/elementsofquater00hamiuoft |location=London, UK |publisher=Longmans, Green, & Co.}}</ref> was 800&nbsp;pages long; it was edited by [[William Edwin Hamilton|his son]] and published shortly after his death.

After Hamilton's death, his student [[Peter Guthrie Tait|Peter Tait]] continued promoting quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as [[kinematics]] in space and [[Maxwell's equations]], were described entirely in terms of quaternions. There was even a professional research association, the [[Quaternion Society]], devoted to the study of quaternions and other [[hypercomplex number]] systems.

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

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 matrices.  In addition, unlike [[Euler angles]], they are not susceptible to “[[gimbal lock]]”. For this reason, quaternions are used in [[computer graphics]],<ref name="Shoemake">{{cite journal| doi=10.1145/325165.325242 |author=Ken Shoemake |author-link=Ken 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>''[[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]], [[robotics]],<ref>J. M. McCarthy, 1990, [https://books.google.com/books?id=glOqQgAACAAJ&dq=introduction+to+theoretical+kinematics&hl=en&sa=X&ved=0ahUKEwjgoIOmnMncAhUPUK0KHT3cB5gQ6AEIJzAA ''Introduction to Theoretical Kinematics''], MIT Press</ref> [[control theory]], [[signal processing]], [[attitude control]], [[physics]], [[bioinformatics]],<ref>{{cite journal |title=Pairwise alignment of the DNA sequence using hypercomplex number representation |journal=Bulletin of Mathematical Biology |date=2004 |first1=Jian-Jun |last1=Shu |first2=L.S. |last2=Ouw |volume=66 |issue=5 |pages=1423–1438 |doi=10.1016/j.bulm.2004.01.005 |url=https://arxiv.org/pdf/1403.2658 |arxiv=1403.2658}}</ref><ref>{{cite journal |title=Hypercomplex cross-correlation of DNA sequences |journal=Journal of Biological Systems |date=2010 |first1=Jian-Jun |last1=Shu |first2=Y. |last2=Li |volume=18 |issue=4 |pages=711–725 |doi=10.1142/S0218339010003470 |url=https://arxiv.org/pdf/1402.5341 |arxiv=1402.5341}}</ref> [[molecular dynamics]], [[computer simulation]]s, and [[orbital mechanics]]. For example, it is common for the [[attitude control]] systems of spacecraft to be commanded in terms of quaternions.  Quaternions have received another boost from [[number theory]] because of their relationships with the [[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 |author=Girard, P. R. |title=The quaternion group and modern physics |year=1984 |journal=European Journal of Physics |volume=5 |pages=25–32 |doi=10.1088/0143-0807/5/1/007|bibcode=1984EJPh....5...25G }}</ref> discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely SO(3), the Lorentz group, the general theory of relativity group, the Clifford algebra SU(2) and the conformal group, can easily be related to the [[quaternion group]] in [[Abstract algebra|modern algebra]]. Girard began by discussing [[group representation]]s and by representing some [[space group]]s of [[crystallography]]. He proceeded to [[kinematics]] of [[rigid body]] motion. Next he used complex quaternions ([[biquaternion]]s) to represent the [[Lorentz group]] of special relativity, including the [[Thomas precession]]. He cited five authors, beginning with [[Ludwik Silberstein]], who used a [[potential]] function of one [[quaternion variable]] to express [[Maxwell's equations]] in a single [[differential equation]]. Concerning general relativity, he expressed the [[Runge–Lenz vector]]. He mentioned the Clifford biquaternions ([[split-biquaternion]]s) as an instance of [[Clifford algebra]]. Finally, invoking the reciprocal of a biquaternion, Girard described [[conformal map]]s on [[spacetime]]. Among the fifty references, Girard included [[Alexander Macfarlane (mathematician)|Alexander Macfarlane]] and his ''Bulletin'' of the [[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}}</ref> 

The finding of 1924 that in [[quantum mechanics]] the [[spin (physics)|spin]] of an electron and other matter particles (known as [[spinors]]) can be described using quaternions 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 |accessdate=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>https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/ Retrieved September 9, 2018</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 [[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>a + b\ \mathbf{i} +  c\ \mathbf{j} + d\ \mathbf{k}\ ,</math>
where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}}, are [[real number]]s, and ''{{mvar|i}}'', ''{{mvar|j}}'', ''{{mvar|k}}'', are [[symbol (mathematics)|symbols]] that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} is 0, the corresponding term is omitted; if {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are all zero, the quaternion is the ''zero quaternion'', denoted 0; if one of {{mvar|b}}, {{mvar|c}}, {{mvar|d}}  equals 1, the corresponding term is written simply ''{{mvar|i}}'', ''{{mvar|j}}'', or ''{{mvar|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'', and is identified with the corresponding real number. That is, the real numbers are a subset of the quaternions. A quaternion that equals its vector part is called a ''vector quaternion''.

The set of quaternions is made a 4&nbsp;dimensional [[vector space]] over the real numbers, with <math>\left\{ 1, \mathbf{i}, \mathbf{j}, \mathbf{k}\right\}</math> as a [[basis (linear algebra)|basis]], by the componentwise addition
:<math>(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}) = (a_1 + a_2) + (b_1 + b_2)\ \mathbf{i} + (c_1 + c_2)\ \mathbf{j} + (d_1 + d_2)\ \mathbf{k}</math>
and the componentwise scalar multiplication
:<math>\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]].
*The real quaternions commute with all other quaternions, that is {{math|1=''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]] and the center property of the real quaternions. The Hamilton product is not [[commutative property|commutative]], but [[associative property|associative]], thus the quaternions form an [[associative algebra]] over the reals.
*Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product:
:<math>(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 ===
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The basis elements {{math|''i''}}, {{math|''j''}}, and {{math|''k''}} commute with the real quaternion 1, that is
:<math>i\cdot 1 = 1\cdot i =i, \qquad j\cdot 1 = 1\cdot j = j, \qquad k\cdot 1 = 1\cdot k=k.</math>

The other products of basis elements are defined by
:<math>i^2=j^2=k^2=-1,</math>
and
:<math>\begin{alignat}{2}
ij & = k, & \qquad ji & = -k, \\
jk & = i, & kj & = -i, \\
ki & = j, & ik & = -j. 
\end{alignat}</math>

These multiplication formulas are equivalent to
:<math>i^2 = j^2 = k^2 = ijk = -1~.</math>
In fact, the equality {{math|1= ''ijk'' = –1}} results from
:<math>(ij)k = k^2 = -1.</math>
The converse implication results from manipulations similar to the following. By right-multiplying both sides of {{math|1=−1 = ''ijk''}} by {{math|–''k''}}, one gets
:<math>k = (i j k)(-k) = (i j) (-k^2) = i j.</math>
All other products can be determined by similar methods.

=== Center ===

The [[center (ring theory)|center]] of a [[noncommutative ring]] is the subring of elements {{mvar|c}} such that {{math|1=''cx'' = ''xc''}} for every {{mvar|x}}. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if {{mvar|1=''q'' = ''a'' + ''bi'' + ''cj'' + ''dk''}} belongs to the center, then
:<math>0 = iq-qi = 2cij + 2dik = 2ck-2dj,</math>
and {{math|1= ''c'' = ''d'' = 0}}. A similar computation with {{mvar|j}} instead of {{mvar|i}} shows that one has also {{math|1=''b'' = 0}}. Thus {{math|1=''q'' = ''a''}} is a real quaternion.

The quaternion form a [[division algebra]]. This means that the noncommutativity of multiplication is the only property that makes quaternions different from a [[field (mathematics)|field]]. This noncommutativity 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 {{math|1=''z''<sup>2</sup> + 1 = 0}}, has infinitely many quaternion solutions, which are the quaternions {{math|1=''z'' = ''bi'' + ''cj'' + ''dk''}} such that {{math|1=''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> = 1}}. Thus these "roots of –1" form a [[unit sphere]] in the three-dimensional space of vector quaternions.

=== Hamilton product ===
For two elements {{nowrap|''a''<sub>1</sub> + ''b''<sub>1</sub>''i'' + ''c''<sub>1</sub>''j'' + ''d''<sub>1</sub>''k''}} and {{nowrap|''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''' ({{nowrap|''a''<sub>1</sub> + ''b''<sub>1</sub>''i'' + ''c''<sub>1</sub>''j'' + ''d''<sub>1</sub>''k''}}) ({{nowrap|''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>a_1a_2 + a_1b_2i + a_1c_2j + a_1d_2k</math>
:<math>{}+ b_1a_2i + b_1b_2i^2 + b_1c_2ij + b_1d_2ik</math>
:<math>{}+ c_1a_2j + c_1b_2ji + c_1c_2j^2 + c_1d_2jk</math>
:<math>{}+ d_1a_2k + d_1b_2ki + d_1c_2kj + d_1d_2k^2.</math>
Now the basis elements can be multiplied using the rules given above to get:<ref name="SeeHazewinkel" />
:<math>a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2</math>
:<math>{}+ (a_1b_2 + b_1a_2 + c_1d_2 - d_1c_2)i</math>
:<math>{}+ (a_1c_2 - b_1d_2 + c_1a_2 + d_1b_2)j</math>
:<math>{}+ (a_1d_2 + b_1c_2 - c_1b_2 + d_1a_2)k.</math>

The product of two [[Quaternions and spatial rotation#Using quaternion 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 q1 followed by a rotation of q2 is equivalent to a single rotation of 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|''a''<sub>2</sub> + ''b''<sub>2</sub>''i'' + ''c''<sub>2</sub>''j'' + ''d''<sub>2</sub>''k''}} followed by the rotation  {{nowrap|''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 {{nowrap|''a'' + 0''i'' + 0''j'' + 0''k''}}, where ''a'' is a real number, is called '''scalar''', and a quaternion of the form {{nowrap|0 + ''bi'' + ''cj'' + ''dk''}}, where ''b'', ''c'', and ''d'' are real numbers, and at least one of ''b'', ''c'' or ''d'' is nonzero, is called a '''vector quaternion'''.  If {{nowrap|''a'' + ''bi'' + ''cj'' + ''dk''}} is any quaternion, then ''a'' is called its '''scalar part''' and ''bi'' + ''cj'' + ''dk'' 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 '''R'''<sup>3</sup>.<ref group=lower-alpha>It is important to note that the vector part of a quaternion is, in truth, an "axial" vector or "[[pseudovector]]", ''not'' an ordinary or "polar" vector, as was formally proven by S.L. Altmann in Ch. 12 of his 1986 book, ''Rotations, Quaternions and Double Groups''. 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.</ref>

Hamilton also called vector quaternions '''right quaternions'''<ref>{{cite book |url=https://books.google.com/?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion#PPA310,M1 |title=Elements of Quaternions |article=Article 285 |page=310 |author=Hamilton, Sir William Rowan |year=1866}}</ref><ref>{{cite book |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 |page=65 |publisher=library.cornell.edu}}</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, i.e.
:<math>q = (r,\ \vec{v}),\ q\in\mathbf{H},\ r\in\mathbf{R},\ \vec{v}\in\mathbf{R}^3</math>

then the formulas for addition and multiplication are:
:<math> (r_1,\ \vec{v}_1) + (r_2,\ \vec{v}_2) = (r_1 + r_2,\ \vec{v}_1+\vec{v}_2)</math>
:<math>(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)</math>

where "'''·'''" is the [[dot product]] and "'''×'''" is the [[cross product]].

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

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of [[Clifford algebra]]s. To define it, let <math>q = a + bi + c j + d k </math> be a quaternion. The '''[[Conjugate (algebra)|conjugate]]''' of ''q'' is the quaternion <math> q^* = a - bi - c j - d k </math>. It is denoted by ''q''<sup>∗</sup>, {{overline|''q''}},<ref name="SeeHazewinkel" /> ''q<sup>t</sup>'', or <math>\tilde q</math>. Conjugation is an [[involution (mathematics)|involution]], meaning that it is its own 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 ''p'' and ''q'' are quaternions, then {{nowrap|1=(''pq'')<sup>∗</sup> = ''q''<sup>∗</sup>''p''<sup>∗</sup>}}, not ''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>q^* = - \frac 1 2 (q + iqi + jqj + kqk).</math>

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

The square root of the product of a quaternion with its conjugate is called its '''[[norm (mathematics)|norm]]''' and is denoted ||''q''|| (Hamilton called this quantity the [[Tensor of a quaternion|''tensor'' of ''q'']], but this conflicts with modern meaning of "[[tensor]]"). In formula, this is expressed as follows:
:<math>\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 '''H''' considered as the vector space '''R'''<sup>4</sup>. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then
:<math>\lVert\alpha q\rVert = \lVert \alpha\lVert\lVert q\rVert.</math>
This is a special case of the fact that the norm is ''multiplicative'', meaning that
:<math>\lVert pq \rVert = \lVert p \rVert\lVert q \rVert.</math>
for any two quaternions ''p'' and ''q''. Multiplicativity is a consequence of the formula for the conjugate of a product.
Alternatively it follows from the identity
:<math>  \det \Bigl(\begin{array}{cc} a+ib & id+c \\ id-c & a-ib \end{array}\Bigr) = a^2 + b^2 + c^2 + d^2,</math>

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

This norm makes it possible to define the '''distance''' {{nowrap|''d''(''p'', ''q'')}} between ''p'' and ''q'' as the norm of their difference:
:<math>d(p, q) = \lVert p - q \rVert.</math>
This makes '''H''' a [[metric space]]. Addition and multiplication are continuous in the metric topology. Indeed, for any scalar, positive ''a'' it holds
:<math>\lVert (p+a p_1 + q+a q_1)-(p+q)\rVert = a\lVert p_1+q_1\rVert.</math>
Continuity follows from taking ''a'' to zero in the limit. Continuity for multiplication holds similarly.

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

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

Every quaternion has a [[polar decomposition]] <math> q = \lVert q \rVert \cdot \mathbf{U} q </math>.

Using conjugation and the norm makes it possible to define the [[Multiplicative inverse|reciprocal]] of a non-zero 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^\ast / \left \Vert q \right \| ^2</math> (in either order) is 1. So the '''reciprocal''' of ''q'' is defined to be

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

This makes it possible to divide two quaternions ''p'' and ''q'' in two different ways (when q is non-zero). That is, their quotient can be either ''p q''<sup>−1</sup> or ''q''<sup>−1</sup>''p''. The notation {{sfrac|p|q}} is ambiguous because it does not specify whether ''q'' divides on the left or the right.

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

The set '''H''' of all quaternions is a [[vector space]] over the [[real number]]s with [[Hamel dimension|dimension]] 4. (In comparison, the real numbers have dimension 1, the complex numbers have dimension 2, and the [[octonion]]s have dimension 8.) Multiplication of quaternions is associative and distributes over vector addition, but it is not commutative. Therefore, the quaternions '''H''' are a non-commutative [[associative algebra]] over the real numbers. Even though '''H''' 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 (mathematics)|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: '''R''', '''C''', and '''H'''. The norm makes the quaternions into a [[composition algebra|normed algebra]], and normed division algebras over the reals are also very rare: [[Hurwitz's theorem (composition algebras)|Hurwitz's theorem]] says that there are only four: '''R''', '''C''', '''H''', and '''O''' (the [[octonions]]). The quaternions are also an example of a [[composition algebra]] and of a unital [[Banach algebra]].

[[File:Quaternion-multiplication-cayley-3d-with-legend.png|thumb|Three dimensional graph of Q<sub>8</sub>.  Red, green and blue arrows represent multiplication by i, j and 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 {{nowrap|{±1, ±''i'', ±''j'', ±''k''} }} forms a [[group (mathematics)|group]] under multiplication. This non-[[Abelian Group]] is called the [[quaternion group]] and is denoted 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]] of Q<sub>8</sub> is a ring '''R'''[Q<sub>8</sub>] which is also an eight-dimensional vector space over '''R'''. It has one basis vector for each element of Q<sub>8</sub>.  The quaternions are the [[quotient ring]] of '''R'''[Q<sub>8</sub>] by the [[ideal (ring theory)|ideal]] generated by the elements {{nowrap|1 + (−1)}}, {{nowrap|''i'' + (−''i'')}}, {{nowrap|''j'' + (−''j'')}}, and {{nowrap|''k'' + (−''k'')}}. Here the first term in each of the differences is one of the basis elements 1, ''i'', ''j'', and ''k'', and the second term is one of basis elements −1, −''i'', −''j'', and −''k'', not the additive inverses of 1, ''i'', ''j'', and ''k''.

== Quaternions and the geometry of R<sup>3</sup> ==
The vector part of a quaternion can be interpreted as a coordinate vector in '''R'''<sup>3</sup>; therefore, the algebraic operations of the quaternions reflect the geometry of '''R'''<sup>3</sup>.  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, ''i'', ''j'', and ''k'' will denote both the three imaginary<ref>{{Cite book|url=https://books.google.com/?id=RC8PAAAAIAAJ&printsec=frontcover&dq=right+tensor+dyadic#PPA428,M1|title= Vector Analysis |publisher=Yale University Press|last1 = Gibbs |first1= J. Willard|last2= Wilson |first2= Edwin Bidwell|year=1901 |page= 428}}</ref> basis vectors of '''H''' and a basis for '''R'''<sup>3</sup>. Replacing ''i'' by −''i'', ''j'' by −''j'', and ''k'' by −''k'' sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the ''spatial inverse''.

For two vector quaternions {{nowrap|1=''p'' = ''b''<sub>1</sub>''i'' + ''c''<sub>1</sub>''j'' + ''d''<sub>1</sub>''k''}} and {{nowrap|1=''q'' = ''b''<sub>2</sub>''i'' + ''c''<sub>2</sub>''j'' + ''d''<sub>2</sub>''k''}} their [[dot product]], by analogy to vectors in '''R'''<sup>3</sup>, is
:<math>p \cdot q = b_1b_2 + c_1c_2 + d_1d_2.</math>
It can also be expressed in a component-free manner as
:<math>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 ''pq''<sup>∗</sup>, ''qp''<sup>∗</sup>, ''p''<sup>∗</sup>''q'', and ''q''<sup>∗</sup>''p''. Note that their vector parts are different.

The [[cross product]] of ''p'' and ''q'' relative to the orientation determined by the ordered basis ''i'', ''j'', and ''k'' is
:<math>p \times q = (c_1d_2 - d_1c_2)i + (d_1b_2 - b_1d_2)j + (b_1c_2 - c_1b_2)k.</math>
(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product ''pq'' (as quaternions), as well as the vector part of −''q''<sup>∗</sup>''p''<sup>∗</sup>. It also has the formula
:<math>p \times q = \textstyle\frac{1}{2}(pq - q^*p^*).</math>
For the commutator, [''p'', ''q''] = ''pq'' − ''qp'', of two vector quaternions one obtains
:<math>[p,q]= 2p \times q.</math>

In general, let ''p'' and ''q'' be quaternions and write
:<math>p = p_s + p_v,</math>
:<math>q = q_s + q_v,</math>
where ''p<sub>s</sub>'' and ''q<sub>s</sub>'' are the scalar parts, and <math>p_v</math> and <math>q_v</math> are the vector parts of ''p'' and ''q''. Then we have the formula
:<math>pq = (pq)_s + (pq)_v = (p_sq_s - p_v\cdot q_v) + (p_s q_v + q_s p_v  + p_v \times q_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">[[William Rowan Hamilton]] (1844 to 1850) [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/ On quaternions or a new system of imaginaries in algebra], [[Philosophical Magazine]], link to David R. Wilkins collection at [[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 '''R'''<sup>3</sup> and were called [[versor]]s by Hamilton.<ref name="Hamilton"/>  Also see [[Quaternions and spatial rotation]] for more information about modeling three-dimensional rotations using quaternions.

See [[Andrew J. Hanson]].<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 [[matrix (mathematics)|matrices]] in such a way that quaternion addition and multiplication correspond to matrix addition and [[matrix multiplication]]. One is to use 2&thinsp;×&thinsp;2 [[complex number|complex]] matrices, and the other is to use 4&thinsp;×&thinsp;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]], these are [[injective function|injective]] [[homomorphism]]s from '''H''' to the [[matrix ring]]s {{nowrap|M(2, '''C''')}} and {{nowrap|M(4, '''R''')}}, respectively.

Using 2&thinsp;×&thinsp;2 complex matrices, the quaternion {{nowrap|''a'' + ''bi'' + ''cj'' + ''dk''}} can be represented as

: <math>\begin{bmatrix}a+bi & c+di \\ -c+di & a-bi \end{bmatrix}.</math>

This representation has the following properties:
* Constraining any two of ''b'', ''c'' and ''d'' to zero produces a representation of [[complex number]]s.  For example, setting {{nowrap|1=''c'' = ''d'' = 0}} produces a diagonal complex matrix representation of complex numbers, and setting {{nowrap|1=''b'' = ''d'' = 0}} produces a real matrix representation.
* The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the [[determinant]] of the corresponding matrix.<ref>[http://www.wolframalpha.com/input/?i=det+{{a%2Bb*i%2C+c%2Bd*i}%2C+{-c%2Bd*i%2C+a-b*i}} Wolframalpha.com]</ref>
* The conjugate of a quaternion corresponds to the [[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 quaternion]]s are the [[3-sphere]], so the underlying space of SU(2) is also a 3-sphere. The group SU(2) is important for describing [[spin (physics)|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 ''a'', ''b'', ''c'' and ''d'', set three of them at zero and the fourth at 1 or &minus;1. Multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for &minus;1. One obtains &minus;1 via ''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = &minus;1. E.g. the last equality is
:<math>\begin{align}
ijk = \sigma_1 \sigma_2 \sigma_3 \sigma_1 \sigma_2 \sigma_3 = -1
\end{align}</math>

Using 4&thinsp;×&thinsp;4 real matrices, that same quaternion can be written as
:<math>\begin{bmatrix}
 a & -b & -c & -d \\ 
 b & a & -d & c \\
 c & d & a & -b \\
 d & -c & b & a 
\end{bmatrix}= a
\begin{bmatrix}
 1 & 0 & 0 & 0 \\ 
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1 
\end{bmatrix}
+ b
\begin{bmatrix}
 0 & -1 & 0 & 0 \\ 
 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & -1 \\
 0 & 0 & 1 & 0 
\end{bmatrix}
+ c
\begin{bmatrix}
 0 & 0 & -1 & 0 \\ 
 0 & 0 & 0 & 1 \\
 1 & 0 & 0 & 0 \\
 0 & -1 & 0 & 0 
\end{bmatrix}
+ d
\begin{bmatrix}
 0 & 0 & 0 & -1 \\ 
 0 & 0 & -1 & 0 \\
 0 & 1 & 0 & 0 \\
 1 & 0 & 0 & 0 
\end{bmatrix}.
</math>
However, the representation of quaternions in M(4,ℝ) is not unique.  For example, the same quaternion can also be represented as
:<math>\begin{bmatrix}
 a & d & -b & -c \\ 
 -d & a & c & -b \\
 b & -c & a & -d \\
 c & b & d & a 
\end{bmatrix}= a
\begin{bmatrix}
 1 & 0 & 0 & 0 \\ 
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1 
\end{bmatrix}
+ b
\begin{bmatrix}
 0 & 0 & -1 & 0 \\ 
 0 & 0 & 0 & -1 \\
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 
\end{bmatrix}
+ c
\begin{bmatrix}
 0 & 0 & 0 & -1 \\ 
 0 & 0 & 1 & 0 \\
 0 & -1 & 0 & 0 \\
 1 & 0 & 0 & 0 
\end{bmatrix}
+ d
\begin{bmatrix}
 0 & 1 & 0 & 0 \\ 
 -1 & 0 & 0 & 0 \\
 0 & 0 & 0 & -1 \\
 0 & 0 & 1 & 0 
\end{bmatrix}.
</math>
In fact, there exist 48 distinct representations of this form.  More precisely, there are 48 sets of quadruples of matrices such that a function sending 1, ''i'', ''j'', and ''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}}</ref> 
In this representation, the conjugate of a quaternion corresponds to the [[transpose]] of the matrix. The fourth power of the norm of a quaternion is the [[determinant]] of the corresponding matrix. As with the 2&thinsp;×&thinsp;2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2&thinsp;×&thinsp;2 blocks by setting {{nowrap|1=''c'' = ''d'' = 0}}.

Each 4x4 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|&minus;''b''
!width=15|&minus;''c''
|-
!''a''
|''a''
|''d''
|''&minus;b''
|''&minus;c''
|-
!''&minus;d''
|''&minus;d''
|''a''
|''c''
|''&minus;b''
|-
!''b''
|''b''
| &minus;''c''
|''a''
|&minus;''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|&minus;''i''
!width=15|&minus;''j''
|-
!''1''
|''1''
|''k''
|''&minus;i''
|''&minus;j''
|-
!''&minus;k''
|''&minus;k''
|''1''
|''j''
|''&minus;i''
|-
!''i''
|''i''
| &minus;''j''
|''1''
|&minus;''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 possible choices for the second column (ignoring sign), 2 possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6 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 1 with ''a'', ''i'' with ''b'', ''j'' with ''c'', and ''k'' with ''d'' and removing the row and column headers yields a matrix representation of ''a + bi + cj + dk''.

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

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

Let '''C'''<sup>2</sup> be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and ''j''.  A vector in '''C'''<sup>2</sup> can be written in terms of the basis elements 1 and ''j'' as
:<math>(a + bi)1 + (c + di)j.\ </math>
If we define {{nowrap|1=''j''<sup>2</sup> = −1}} and {{nowrap|1=''ij'' = −''ji''}}, then we can multiply two vectors using the distributive law. Writing ''k'' in place of the product ''ij'' leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion {{nowrap|''a'' + ''bi'' + ''cj'' + ''dk''}}. If we write the elements of '''C'''<sup>2</sup> as ordered pairs and quaternions as quadruples, then the correspondence is

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

== Square roots of &minus;1 ==
In the complex numbers, '''C''', there are just two numbers, ''i'' and −''i'', whose square is &minus;1&nbsp;. In '''H''' there are infinitely many square roots of minus one: the quaternion solution for the square root of &minus;1 is the unit [[sphere]] in '''R'''<sup>3</sup>.  To see this, let {{nowrap|1=''q'' = ''a'' + ''bi'' + ''cj'' + ''dk''}} be a quaternion, and assume that its square is −1.  In terms of ''a'', ''b'', ''c'', and ''d'', this means
:<math>a^2 - b^2 - c^2 - d^2 = -1,</math>
:<math>2ab = 0,</math>
:<math>2ac = 0,</math>
:<math>2ad = 0.</math>
To satisfy the last three equations, either {{nowrap|''a'' {{=}} 0}} or ''b'', ''c'', and ''d'' are all 0.  The latter is impossible because ''a'' is a real number and the first equation would imply that {{nowrap| ''a''<sup>2</sup> {{=}} &minus;1}}.  Therefore, {{nowrap| ''a'' {{=}} 0}} and {{nowrap| ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> {{=}} 1}}. In other words: a quaternion squares to &minus;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}}

The identification of the square roots of minus one in '''H''' was given by Hamilton<ref>{{cite book|author=Hamilton |year=1899|title=Elements of Quaternions |edition= 2nd |page= 244|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'' (page&nbsp;39) published by the [[National Council of Teachers of Mathematics]]. More recently, the sphere of square roots of minus one is described in [[Ian R. Porteous]]'s book ''Clifford Algebras and the Classical Groups'' (Cambridge, 1995) in proposition 8.13 on page&nbsp;60.

=== H as a union of complex planes ===
Each pair of square roots of &minus;1 creates a distinct copy of the complex numbers inside the quaternions.  If {{nowrap| ''q''<sup>2</sup> {{=}} &minus;1}}, then the copy is determined by the function

:<math>a + b\sqrt{-1} \mapsto a + bq.</math>

In the language of [[abstract algebra]], each is an [[injective function|injective]] [[ring (abstract algebra)|ring]] [[homomorphism]] from '''C''' to '''H'''.  The images of the embeddings corresponding to ''q'' and &minus;''q'' are identical.

Every non-real quaternion determines a planar subspace in '''H''' that is isomorphic to '''C''':  Write ''q'' as the sum of its scalar part and its vector part:
:<math>q = q_s + \vec{q}_v.</math>
Decompose the vector part further as the product of its norm and its [[versor]]:
:<math>q = q_s + \lVert\vec{q}_v\rVert\cdot\mathbf{U}\vec{q}_v.</math>
(Note that this is not the same as <math>q_s + \lVert q\rVert\cdot\mathbf{U}q</math>.) The versor of the vector part of ''q'', <math>\mathbf{U}\vec{q}_v</math>, is a right versor with –1 as its square.  Therefore, it determines a copy of the complex numbers by the function
:<math>a + b\sqrt{-1} \mapsto a + b\mathbf{U}\vec{q}_v.</math>
Under this function, {{math|''q''}} is the image of the complex number <math>q_s + \lVert\vec{q}_v\rVert i</math>. Thus '''H''' is the [[union (set theory)#Arbitrary unions|union]] of complex planes intersecting in a common [[real line]], where the union is taken over the sphere of square roots of minus one, bearing in mind that the same plane is associated with any pair of [[antipodal points]] on the sphere of right versors.

=== Commutative subrings ===
The relationship of quaternions to each other within the complex subplanes of '''H''' can also be identified and expressed in terms of [[commutative]] [[subring]]s. Specifically, since two quaternions ''p'' and ''q'' commute (i.e., {{nowrap|''pq'' {{=}} ''qp''}}) only if they lie in the same complex subplane of '''H''', the profile of '''H''' as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion [[ring (mathematics)|ring]]. This method of commutative subrings is also used to profile the [[split-quaternion]]s, which as an algebra over the reals are isomorphic to [[2 × 2 real matrices]].

==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 dimensions. This Julia set is cross sectioned at the xy 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>http://archive.bridgesmathart.org/2010/bridges2010-247.pdf retrieved August 19, 2018</ref>

===Exponential, logarithm, and power functions===
Given a quaternion,
:<math> q= a+ bi + cj + dk = a + \mathbf{v} </math>
the exponential is computed as
:<math>\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>

:<math>\ln(q) = \ln \|q\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \arccos \frac{a}{\|q\|}</math>.<ref>{{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|archiveurl = http://web.archive.org/web/20170705123142/http://www.lce.hut.fi/~ssarkka/pub/quat.pdf |archivedate=5 July 2017 }}</ref>

It follows that the polar decomposition of a quaternion may be written
:<math>q=\|q\|e^{\hat{n}\varphi} = \|q\| \left(\cos(\varphi) + \hat{n} \sin(\varphi)\right),</math>

where the [[angle]] <math>\varphi</math><ref name="θ">Books on applied mathematics, such as {{Cite book |title=Robotics, Vision and Control Fundamental Algorithms In MATLAB® |last=Corke |first=Peter |publisher=[[Springer Publishing|Springer]] |year=2017 |isbn=978-3-319-54413-7}}, often use different notation with {{math|1=''φ'' := ''θ''/2}}—that is, [[change of variables|another variable]] {{math|1=''θ'' = 2''φ''}}.</ref> and the unit vector <math>\hat{n}</math> are defined by:
:<math>a=\|q\|\cos(\varphi)</math>
and
:<math>\mathbf{v}=\hat{n} \|\mathbf{v}\|=\hat{n}\|q\|\sin(\varphi).</math>
Any unit quaternion may be expressed in polar form as <math>e^{\hat{n}\varphi}</math>.

The [[Power (mathematics)|power]] of a quaternion raised to an arbitrary (real) exponent <math>\alpha</math> is given by:
:<math>q^\alpha=\|q\|^\alpha e^{\hat{n}\alpha\varphi} = \|q\|^\alpha \left(\cos(\alpha\varphi) + \hat{n} \sin(\alpha\varphi)\right).</math>

===Geodesic norm===
The [[Great-circle distance|geodesic distance]] {{nowrap|''d''<sub>g</sub> (''p'', ''q'')}} between unit quaternions ''p'' and ''q'' is defined as:
:<math>d_g(p, q) = \lVert \ln(p^{-1} q) \rVert.</math><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}}</ref>
and amounts to the absolute value of half the angle subtended by ''p'' and ''q'' along a [[Arc (geometry)|great arc]] of the S<sup>3</sup> sphere.
This angle can also be computed from the quaternion [[dot product]] without the logarithm as:
:<math>\arccos(2(p \cdot q)^2 - 1).</math>

==Three-dimensional and four-dimensional rotation groups==
{{unreferenced section|date=October 2018}}
{{Main|Quaternions and spatial rotation|Rotation operator (vector space)}}
The word "[[conjugation (group theory)|conjugation]]", besides the meaning given above, can also mean taking an element ''a'' to ''r a r''<sup>−1</sup> where ''r'' is some non-zero 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.)

Thus the multiplicative group of non-zero quaternions acts by conjugation on the copy of '''R'''<sup>3</sup> consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(''φ'') is a rotation by an angle 2''φ'', the axis of the rotation being the direction of the vector part. The advantages of quaternions are:

* Avoiding [[gimbal lock]], a problem with systems such as [[Euler angles]].
* Faster and more compact than [[matrix (mathematics)|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 rotations]]'').

The set of all unit quaternions ([[versor]]s) forms a [[3-sphere]] ''S''<sup>3</sup> and a [[group (mathematics)|group]] (a [[Lie group]]) under multiplication, [[Covering space#Elementary properties|double covering]] the group {{nowrap|SO(3, '''R''')}} of real orthogonal 3×3 [[orthogonal matrix|matrices]] of [[determinant]] 1 since ''two'' unit quaternions correspond to every rotation under the above correspondence. See the [[plate trick]].
{{details|Point groups in three dimensions}}
The image of a subgroup of versors <!-- ''S''³ itself has not a canonical group structure --> 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 SU(2), the group of complex [[unitary matrix|unitary]] 2×2 matrices of [[determinant]] 1.

Let ''A'' be the set of quaternions of the form {{nowrap|''a'' + ''bi'' + ''cj'' + ''dk''}} where ''a'', ''b'', ''c'', and ''d'' are either all [[integer]]s or all [[rational number]]s with odd numerator and denominator 2. The set ''A'' is a [[ring (mathematics)|ring]] (in fact a [[domain (ring theory)|domain]]) and a [[Lattice (group)|lattice]] and is called the ring of [[Hurwitz quaternion]]s. There are 24 unit quaternions in this ring, and they are the vertices of a [[24-cell|regular 24-cell]] with [[Schläfli symbol]] {3,4,3}. They correspond to the double cover of the rotational symmetry group of the regular [[tetrahedron]]. Similarly, the vertices of a [[600-cell|regular 600-cell]] with Schläfli symbol {3,3,5} can be taken as the unit [[icosian]]s, corresponding to the double cover of the rotational symmetry group of the [[regular icosahedron]]. The double cover of the rotational symmetry group of the regular [[octahedron]] corresponds to the quaternions that represent the vertices of the [[disphenoidal 288-cell]].

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

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

Quaternion algebras are isomorphic to the algebra of 2×2 [[matrix (mathematics)|matrices]] over ''F'' or form [[division algebra]]s over ''F'', depending on 
the choice of ''a'' and ''b''.

== Quaternions as the even part of Cℓ<sub>3,0</sub>(R) ==
{{main|Spinor#Three_dimensions}}
The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part Cℓ<sup>+</sup><sub>3,0</sub>('''R''') of the [[Clifford algebra]] Cℓ<sub>3,0</sub>('''R''').  This is an associative multivector algebra built up from fundamental basis elements [[Pauli matrices|σ<sub>1</sub>, σ<sub>2</sub>, σ<sub>3</sub>]] using the product rules
:<math>\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 1,</math>
:<math>\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 ''r'' in a plane perpendicular to a unit vector ''w'' can be written:
:<math>r^{\prime} = - w\, r\, w.</math>
Two reflections make a rotation by an angle twice the angle between the two reflection planes, so
:<math>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>r^{\prime\prime} = -\mathbf{k}\, r\, \mathbf{k}. </math>

In fact, the two are identical, if we make the identification
:<math>\mathbf{k} = \sigma_2 \sigma_1, \mathbf{i} = \sigma_3 \sigma_2, \mathbf{j} = \sigma_1 \sigma_3</math>
and it is straightforward to confirm that this preserves the Hamilton relations
:<math>\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i} \mathbf{j} \mathbf{k} = -1.</math>

In this picture, quaternions correspond not to vectors but to [[bivector]]s – quantities with magnitude and orientations associated with particular 2D ''planes'' rather than 1D ''directions''.   The relation to [[complex number]]s becomes clearer, too: in 2D, with two vector directions σ<sub>1</sub> and σ<sub>2</sub>, there is only one bivector basis element 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 σ<sub>1</sub>σ<sub>2</sub>, σ<sub>2</sub>σ<sub>3</sub>, σ<sub>3</sub>σ<sub>1</sub>, so three imaginaries.

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

There are several advantages for placing quaternions in this wider setting:<ref>[http://www.geometricalgebra.net/quaternions.html Quaternions and Geometric Algebra]. Accessed 2008-09-12.  See also: Leo Dorst, Daniel Fontijne, Stephen Mann, (2007), ''[http://www.geometricalgebra.net/index.html Geometric Algebra For Computer Science]'', [[Morgan Kaufmann]].  {{isbn|0-12-369465-5}}</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 geometry|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.

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

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

The quaternions are "essentially" the only (non-trivial) [[central simple algebra]] (CSA) over the real numbers, in the sense that every CSA over the reals is [[Brauer equivalent]] to either the reals or the quaternions. Explicitly, the [[Brauer group]] of the reals consists of two classes, represented by the reals and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a [[matrix ring]] over another. By the [[Artin–Wedderburn theorem]] (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the reals.

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

== Quotations ==
* "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 x, y, z, etc." &mdash; William Rowan Hamilton [Quoted in a letter from Tait to Cayley].
* "Time is said to have only one dimension, and space to have three dimensions. […] 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. ''And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be''." &mdash; William Rowan Hamilton [Quoted in R.P. Graves, ''Life of Sir William Rowan Hamilton''].
* "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 [[James Clerk Maxwell|Clerk Maxwell]]." &mdash; [[Lord Kelvin|W. Thompson, Lord Kelvin]]. (1892).
* "I came later to see that, as 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." &mdash; [[Oliver Heaviside]]. (1893). ''Electromagnetic Theory'' volume&nbsp;I, pp.&nbsp;134&ndash;135. London: The Electrician Printing and Publishing Company.
* "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 every-day 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." &mdash; [[Ludwik Silberstein]]. (1924). Notes on preparing the second edition of his [[List of important publications in physics#The Theory of Relativity|Theory of Relativity]].
* "... 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." &mdash; [[Simon L. Altmann]]. (1986).

==See also==
{{Div col|colwidth=20em}}
* [[3-sphere]]
* [[Associative algebra]]
* [[Biquaternion]]
* [[Clifford algebra]]
* [[Complex number]]
* [[Conversion between quaternions and Euler angles]]
* [[Division algebra]]
* [[Dual quaternion]]
* [[Dual complex number]]
* [[Euler angles]]
* [[Exterior algebra]]
* [[Geometric algebra]]
* [[Hurwitz quaternion]]
* [[Hurwitz quaternion order]]
* [[Hyperbolic quaternion]]
* [[Hypercomplex number]]
* [[Lénárt sphere]]
* [[Octonion]]
* [[Pauli matrices]]
* [[Quaternion group]]
* [[Quaternion variable]]
* [[Quaternionic matrix]]
* [[Quaternionic polytope]]
* [[Quaternionic projective space]]
* [[Quaternions and spatial rotation]]
* [[Rotation operator (vector space)]]
* [[Rotations in 4-dimensional Euclidean space]]
* [[Slerp]]
* [[Split-quaternion]]
* [[Tesseract]]
{{Div col end}}

==Notes==
{{notelist}}

==References==
{{Reflist}}

==External articles and resources==
{{Wiktionary}}

===Books and publications===
{{wikibooks|Associative Composition Algebra|Quaternions}}
*[[William Rowan Hamilton|Hamilton, William Rowan]]. [https://web.archive.org/web/20140808040037/http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 On quaternions, or on a new system of imaginaries in algebra]. Philosophical Magazine. Vol. 25, n 3. p.&nbsp;489–495. 1844.
*[[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://books.google.com/books?id=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 [[Charles Jasper Joly]]; published by [[Longmans, Green & Co.]].
*[[Peter Guthrie Tait|Tait, Peter Guthrie]] (1873), "''An elementary treatise on quaternions''". 2d ed., Cambridge, [Eng.] : The University Press.
*Maxwell, James Clerk (1873), "''[[A Treatise on Electricity and Magnetism]]''". Clarendon Press, Oxford.
*[[Peter Guthrie Tait|Tait, Peter Guthrie]] (1886), "''{{cite web|url=http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 |title=Archived copy |accessdate=June 26, 2005 |url-status=unfit |archiveurl=https://web.archive.org/web/20140808040037/http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 |archivedate=August 8, 2014 }}''". M.A. Sec. R.S.E. [[Encyclopædia Britannica]], Ninth Edition, 1886, Vol. XX, pp.&nbsp;160&ndash;164. (bzipped [[PostScript]] file)
*Joly, Charles Jasper (1905), "''A manual of quaternions''". London, Macmillan and co., limited; New York, The Macmillan company. LCCN 05036137 //r84
*[[Alexander Macfarlane|Macfarlane, Alexander]] (1906), "''Vector analysis and quaternions''", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048
*{{cite EB1911|wstitle=Algebra}} (''See section on quaternions.'')
*Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "''Foundations of quaternion quantum mechanics''". J. Math. Phys. 3, pp.&nbsp;207&ndash;220, MathSciNet.
*[[Patrick du Val|Du Val, Patrick]] (1964), "''Homographies, quaternions, and rotations''". Oxford, Clarendon Press (Oxford mathematical monographs). LCCN 64056979 //r81
*Crowe, Michael J. (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).
*Altmann, Simon L. (1986), "''Rotations, quaternions, and double groups''". Oxford [Oxfordshire] : Clarendon Press ; New York : Oxford University Press. LCCN 85013615 {{isbn|0-19-855372-2}}
*Altmann, Simon L. (1989), "''Hamilton, Rodrigues, and the Quaternion Scandal''". Mathematics Magazine. Vol. 62, No. 5. p.&nbsp;291–308, December  1989.
*Adler, Stephen L. (1995), "''Quaternionic quantum mechanics and quantum fields''". New York : Oxford University Press. International series of monographs on physics (Oxford, England) 88. LCCN 94006306 {{isbn|0-19-506643-X}}
*[https://web.archive.org/web/20110805044706/http://members.cox.net/vtrifonov/ Trifonov, Vladimir] (1995), "''A Linear Solution of the Four-Dimensionality Problem''", Europhysics Letters, '''32 (8)''' 621&ndash;626, {{doi|10.1209/0295-5075/32/8/001}}
*Ward, J. P. (1997), "''Quaternions and Cayley Numbers: Algebra and Applications''", Kluwer Academic Publishers. {{isbn|0-7923-4513-4}}
*Kantor, I. L. and Solodnikov, A. S. (1989), "''Hypercomplex numbers, an elementary introduction to algebras''", Springer-Verlag, New York,  {{isbn|0-387-96980-2}}
*Gürlebeck, Klaus and Sprössig, Wolfgang (1997), "''Quaternionic and Clifford calculus for physicists and engineers''". Chichester ; New York : Wiley (Mathematical methods in practice; v. 1). LCCN 98169958 {{isbn|0-471-96200-7}}
*Kuipers, Jack (2002), "''Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality''" (reprint edition), [[Princeton University Press]]. {{isbn|0-691-10298-8}}
*[[John Horton Conway|Conway, John Horton]], and Smith, Derek A. (2003), "''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry''", A. K. Peters, Ltd. {{isbn|1-56881-134-9}} ([http://nugae.wordpress.com/2007/04/25/on-quaternions-and-octonions/ review]).
* Jack, P. M. (2003).  ''Physical space as a quaternion structure, I: Maxwell equations. A brief Note.''  {{arXiv|math-ph/0307038}}
*[[Vladislav Kravchenko|Kravchenko, Vladislav]] (2003),  "''Applied Quaternionic Analysis''", Heldermann Verlag {{isbn|3-88538-228-8}}.
*[[Michiel Hazewinkel]], Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004. {{isbn|1-4020-2690-0}}
*[http://www.cs.indiana.edu/~hanson/quatvis/ Hanson, Andrew J.] (2006), "''Visualizing Quaternions''", Elsevier: Morgan Kaufmann; San Francisco. {{isbn|0-12-088400-3}}
*[https://web.archive.org/web/20110805044706/http://members.cox.net/vtrifonov/ Trifonov, Vladimir] (2007), "''Natural Geometry of Nonzero Quaternions''", International Journal of Theoretical Physics, '''46 (2)''' 251&ndash;257, {{doi|10.1007/s10773-006-9234-9}}
*Ernst Binz & Sonja Pods (2008) ''Geometry of Heisenberg Groups'' [[American Mathematical Society]], Chapter 1: "The Skew Field of Quaternions" (23 pages) {{isbn|978-0-8218-4495-3}}.
* {{cite book|author1=[[Chris J. L. Doran]]|author2=Anthony N. Lasenby|title=Geometric Algebra for Physicists|year=2003|publisher=Cambridge University Press|isbn=978-0-521-48022-2}}
*Vince, John A. (2008), ''Geometric Algebra for Computer Graphics'',  Springer, {{isbn|978-1-84628-996-5}}.
*For molecules that can be regarded as classical rigid bodies [[molecular dynamics]] computer simulation employs quaternions.  They were first introduced for this purpose by D.J. Evans,  (1977), "On the Representation of Orientation Space", Mol. Phys., vol 34, p 317.
*Zhang, Fuzhen (1997), "Quaternions and Matrices of Quaternions", Linear Algebra and its Applications, Vol. 251, pp.&nbsp;21–57.
* {{cite book|author=Ron Goldman|title=Rethinking Quaternions: Theory and Computation|year=2010|publisher=Morgan & Claypool Publishers|isbn=978-1-60845-420-4}}

===Links and monographs===
* Notices and materials related to Quaternion conference presentations, [https://quaternionnews.commons.gc.cuny.edu/ Quaternion Notices]
* {{springer|title=Quaternion|id=p/q076770}}
* [http://www.j3d.org/matrix_faq/matrfaq_latest.html Matrix and Quaternion FAQ v1.21] Frequently Asked Questions
* Doug Sweetser, [http://world.std.com/~sweetser/quaternions/qindex/qindex.html Doing Physics with Quaternions]
* [https://web.archive.org/web/20050408193941/http://www.fho-emden.de/~hoffmann/quater12012002.pdf Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)]
* [https://arxiv.org/pdf/math-ph/0201058 The Physical Heritage of Sir W. R. Hamilton] (PDF)
* D. R. Wilkins, [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html Hamilton's Research on Quaternions]
* [http://www.unpronounceable.com/julia/ Quaternion Julia Fractals] 3D Raytraced Quaternion [[Julia set|Julia Fractals]] by David J. Grossman
* [http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/index.htm Quaternion Math and Conversions] Great page explaining basic math with links to straight forward rotation conversion formulae.
* John H. Mathews, [https://web.archive.org/web/20060902200454/http://math.fullerton.edu/mathews/c2003/QuaternionBib/Links/QuaternionBib_lnk_3.html Bibliography for Quaternions].
* [https://www.gamedev.net/articles/programming/math-and-physics/quaternion-powers-r1095/ Quaternion powers on GameDev.net]
* Andrew Hanson, [https://web.archive.org/web/20061105174313/http://books.elsevier.com/companions/0120884003/vq/index.html Visualizing Quaternions home page].
* Charles F. F. Karney, ''Quaternions in molecular modeling'', J. Mol. Graph. Mod. '''25'''(5), 595–604 (January  2007); {{doi|10.1016/j.jmgm.2006.04.002}}; E-print [https://arxiv.org/abs/physics/0506177 arxiv:0506177].
* Johan E. Mebius, [https://arxiv.org/abs/math/0501249 A matrix-based proof of the quaternion representation theorem for four-dimensional rotations.], ''arXiv General Mathematics'' 2005.
* Johan E. Mebius, [https://arxiv.org/abs/math/0701759  Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations.], ''arXiv General Mathematics'' 2007.
* [[NUI Maynooth]] [http://www.maths.nuim.ie/links/hamilton.shtml Department of Mathematics, Hamilton Walk].
* [https://web.archive.org/web/20071215235040/http://gpwiki.org/index.php/OpenGL:Tutorials:Using_Quaternions_to_represent_rotation OpenGL:Tutorials:Using Quaternions to represent rotation]
* David Erickson, [[Defence Research and Development Canada]] (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper. 
* Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent",[https://web.archive.org/web/20110924161347/https://webspace.utexas.edu/aam829/1/m/NegativeMath.html Utexas.edu]
* D. Stahlke, Quaternions in Classical Mechanics [http://www.stahlke.org/dan/phys-papers/quaternion-paper.pdf Stahlke.org] (PDF)
* Morier-Genoud, Sophie, and Valentin Ovsienko. "Well, Papa, can you multiply triplets?", [https://arxiv.org/abs/0810.5562 arxiv.org]  describes how the quaternions can be made into a skew-commutative algebra graded by {{nowrap|'''Z'''/2 × '''Z'''/2 × '''Z'''/2}}.
* [http://plus.maths.org/content/os/issue32/features/baez/index Curious Quaternions] by Helen Joyce hosted by [[John Baez]].
* Luis Ibanez "Tutorial on Quaternions" [http://www.itk.org/CourseWare/Training/QuaternionsI.pdf Part I] [http://www.itk.org/CourseWare/Training/QuaternionsII.pdf Part II] (PDF; using Hamilton's terminology, which differs from the modern usage)
* R. Ghiloni, V. Moretti, A. Perotti  "[https://arxiv.org/pdf/1207.0666.pdf Continuous slice functional calculus in quaternionic Hilbert spaces,]" Rev.Math.Phys.25(2013)1350006 and "[https://arxiv.org/pdf/1602.02661v2.pdf Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures,]" Rev. Math.Phys.29(2017)1750034 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.

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