In

^{t}'', $\backslash tilde\; q$, or . Conjugation is an 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 and are quaternions, then , not .
The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:
:$q^*\; =\; -\; \backslash frac\; (q\; +\; \backslash ,\backslash mathbf\; i\; \backslash ,q\; \backslash ,\backslash mathbf\; i\; +\; \backslash ,\backslash mathbf\; j\; \backslash ,q\; \backslash ,\backslash mathbf\; j\; +\; \backslash ,\backslash mathbf\; k\; \backslash ,q\; \backslash ,\backslash mathbf\; k)~.$
Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of is , and the vector part of is .
The square root of the product of a quaternion with its conjugate is called its ''norm'' and is denoted (Hamilton called this quantity the ''tensor'' of ''q'', but this conflicts with the modern meaning of "

versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...

s) forms a 3-sphere and a group (a

_{1} and ''σ''_{2}. This is very similar to the corresponding quaternion formula,
:$r^\; =\; -\backslash mathbf\backslash ,\; r\backslash ,\; \backslash mathbf.$
In fact, the two are identical, if we make the identification
:$\backslash mathbf\; =\; \backslash sigma\_2\; \backslash sigma\_1\backslash ,,\; \backslash quad\; \backslash mathbf\; =\; \backslash sigma\_3\; \backslash sigma\_2\backslash ,,\; \backslash quad\; \backslash mathbf\; =\; \backslash sigma\_1\; \backslash sigma\_3\backslash ,,$
and it is straightforward to confirm that this preserves the Hamilton relations
:$\backslash mathbf^2\; =\; \backslash mathbf^2\; =\; \backslash mathbf^2\; =\; \backslash mathbf\; =\; -1~.$
In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to bivectors – quantities with magnitude and orientations associated with particular 2D ''planes'' rather than 1D ''directions''. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions and , there is only one bivector basis element , so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements , , , so three imaginaries.
This reasoning extends further. In the Clifford algebra $\backslash operatorname\_(\backslash mathbb\; R),$ 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

Lectures on Quaternions

'". Royal Irish Academy. *Hamilton (1866)

Elements of Quaternions

'

review

. * * * * * * * *For molecules that can be regarded as classical rigid bodies

Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)

* * * 3D Raytraced Quaternion Julia Fractals * Great page explaining basic math with links to straight forward rotation conversion formulae. * * * * * * * * * David Erickson,

Part II

(PDF; using Hamilton's terminology, which differs from the modern usage) *

two expository papers about continuous functional calculus and spectral theory in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.

Quaternions

the Android app shows the quaternion corresponding to the orientation of the device.

Rotating Objects Using Quaternions

article speaking to the use of Quaternions for rotation in video games/computer graphics.

Quaternions – Visualisation

{{Authority control Composition algebras William Rowan Hamilton

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the quaternion number system extends the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. Quaternions were first described by the Irish mathematician William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ire ...

in 1843 and applied to mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...

in three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...

. Hamilton defined a quaternion as the quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...

s. Multiplication of quaternions is noncommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

.
Quaternions are generally represented in the form
:$a\; +\; b\backslash \; \backslash mathbf\; i\; +\; c\backslash \; \backslash mathbf\; j\; +d\backslash \; \backslash mathbf\; k$
where , and are real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s; and , and are the ''basic quaternions''.
Quaternions are used in pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...

, but also have practical uses in applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...

, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the huma ...

, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref>
Th ...

and rotation matrices, or as an alternative to them, depending on the application.
In modern mathematical language, quaternions form a four- dimensional associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

normed division algebra over the real numbers, and therefore a ring, being both a division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

and a domain. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pro ...

by $\backslash mathbb\; H.$ It can also be given by the Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...

classifications Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood.
Classification is the grouping of related facts into classes.
It may also refer to:
Business, organizat ...

$\backslash operatorname\_(\backslash mathbb\; R)\backslash cong\; \backslash operatorname\_^+(\backslash mathbb\; R).$ In fact, it was the first noncommutative division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fiel ...

to be discovered.
According to the Frobenius theorem, the algebra $\backslash mathbb\; H$ is one of only two finite-dimensional division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

s containing a proper subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...

isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...

s, which is the last normed division algebra over the real numbers. (The sedenions, the extension of the octonions, have zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...

s and so cannot be a normed division algebra.)
The unit quaternions can be thought of as a choice of a group structure on the 3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...

that gives the group Spin(3), which is isomorphic to SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the spec ...

and also to the universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete sp ...

of SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a ...

.
History

Quaternions were introduced by Hamilton in 1843.See Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

had also discovered quaternions in 1819, but this work was not published until 1900.
Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...

. 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
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of the coordinates of two points in space. In fact, Ferdinand Georg Frobenius later 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: $\backslash mathbb$ (complex numbers) and $\backslash mathbb\; H$ (quaternions) which have dimension 1, 2, and 4 respectively.
The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin
Dublin (; , or ) is the capital and largest city of Ireland. On a bay at the mouth of the River Liffey, it is in the province of Leinster, bordered on the south by the Dublin Mountains, a part of the Wicklow Mountains range. At the 2016 cen ...

, 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
The Royal Canal ( ga, An Chanáil Ríoga) is a canal originally built for freight and passenger transportation from Dublin to Longford in Ireland. It is one of two canals from Dublin to the River Shannon and was built in direct competition ...

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,
:$\backslash mathbf^2\; =\; \backslash mathbf^2\; =\; \backslash mathbf^2\; =\; \backslash mathbf\; =\; -1$
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''; Hamilton states:
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. Hamilton's treatment is more geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

than the modern approach, which emphasizes quaternions' algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...

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'', was 800 pages long; it was edited by his son and published shortly after his death.
After Hamilton's death, the Scottish mathematical physicist Peter Tait Peter Tait may refer to:
* Peter Tait (physicist) (1831–1901), Scottish mathematical physicist
* Peter Tait (footballer) (1936–1990), English professional footballer
* Peter Tait (mayor) (1915–1996), New Zealand politician
* Peter Tait (r ...

became the chief exponent of quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...

in space and Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...

, 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
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...

, which had been developed by Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...

, Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ve ...

, and Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...

. 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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

. A side-effect of this transition is that 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 century, primarily due to their utility in 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
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...

, Presented at SIGGRAPH '85. computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the huma ...

, robotics
Robotics is an interdisciplinarity, interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist human ...

, control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to ...

, signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, d ...

, attitude control
Attitude control is the process of controlling the orientation of an aerospace vehicle with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, etc.
Controlling vehicle ...

, physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

, bioinformatics
Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combine ...

, molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...

, computer simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...

s, and orbital mechanics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws ...

. For example, it is common for the attitude control
Attitude control is the process of controlling the orientation of an aerospace vehicle with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, etc.
Controlling vehicle ...

systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

because of their relationships with the quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...

s.
Quaternions in physics

P.R. Girard's 1984 essay ''The quaternion group and modern physics'' discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely , the Lorentz group, the general theory of relativity group, the Clifford algebra and the conformal group, can easily be related to the quaternion group in modern algebra. Girard began by discussinggroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to r ...

s and by representing some space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unc ...

s of crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

. He proceeded to kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...

of rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...

motion. Next he used complex quaternions (biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions c ...

s) to represent the Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physi ...

of special relativity, including the Thomas precession. He cited five authors, beginning with Ludwik Silberstein, who used a potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...

function of one quaternion variable to express Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...

in a single differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions ( split-biquaternions) as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differe ...

. Among the fifty references, Girard included Alexander Macfarlane
Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician.
Life
Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowire) and Ann Small. He s ...

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.
The finding of 1924 that in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

the spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...

of an electron and other matter particles (known as spinors
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

) 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"). , their use has not overtaken rotation groups.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."
Definition

A ''quaternion'' is anexpression
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphorical expression, a particular word, phrase, ...

of the form
:$a\; +\; b\backslash ,\backslash mathbf\; +\; c\backslash ,\backslash mathbf\; +\; d\backslash ,\backslash mathbf\backslash \; ,$
where , , , , are real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s, and , , , are symbols
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different con ...

that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of , , , is 0, the corresponding term is omitted; if , , , are all zero, the quaternion is the ''zero quaternion'', denoted 0; if one of , , equals 1, the corresponding term is written simply , or .
Hamilton describes a quaternion $q\; =\; a\; +\; b\backslash ,\backslash mathbf\; +\; c\backslash ,\backslash mathbf\; +\; d\backslash ,\backslash mathbf$, as consisting of a scalar part and a vector part. The quaternion $b\backslash ,\backslash mathbf\; +\; c\backslash ,\backslash mathbf\; +\; d\backslash ,\backslash mathbf$ is called the ''vector part'' (sometimes ''imaginary part'') of , and is the ''scalar part'' (sometimes ''real part'') of . 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 of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.) A quaternion that equals its vector part is called a ''vector quaternion''.
The set of quaternions is made a 4-dimensional vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...

over the real numbers, with $\backslash left\backslash $ as a basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting o ...

, by the componentwise addition
:$(a\_1+b\_1\backslash ,\backslash mathbf\; i\; +\; c\_1\backslash ,\backslash mathbf\; j\; +\; d\_1\backslash ,\backslash mathbf\; k)\; +\; (a\_2\; +\; b\_2\backslash ,\backslash mathbf\; i\; +\; c\_2\backslash ,\backslash mathbf\; j\; +\; d\_2\backslash ,\backslash mathbf\; k)\; =\; (a\_1\; +\; a\_2)\; +\; (b\_1\; +\; b\_2)\backslash ,\backslash mathbf\; i\; +\; (c\_1\; +\; c\_2)\backslash ,\backslash mathbf\; j\; +\; (d\_1\; +\; d\_2)\backslash ,\backslash mathbf\; k\backslash ,,$
and the componentwise scalar multiplication
:$\backslash lambda(a\; +\; b\backslash ,\backslash mathbf\; i\; +\; c\backslash ,\backslash mathbf\; j\; +\; d\backslash ,\backslash mathbf\; k)\; =\; \backslash lambda\; a\; +\; (\backslash lambda\; b)\backslash ,\backslash mathbf\; i\; +\; (\backslash lambda\; c)\backslash ,\backslash mathbf\; j\; +\; (\backslash lambda\; d)\backslash ,\backslash mathbf\; k.$
A multiplicative group structure, called the ''Hamilton product'', denoted by juxtaposition, can be defined on the quaternions in the following way:
*The real quaternion is the identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

.
*The real quaternions commute with all other quaternions, that is for every quaternion and every real quaternion . In algebraic terminology this is to say that the field of real quaternions are the ''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
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic ...

and the center property of the real quaternions. The Hamilton product is not commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

, but is associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

, thus the quaternions form an associative algebra over the real numbers.
*Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product:
:$(a\; +\; b\backslash ,\backslash mathbf\; i\; +\; c\backslash ,\backslash mathbf\; j\; +\; d\; \backslash ,\backslash mathbf\; k)^\; =\; \backslash frac\backslash ,(a\; -\; b\backslash ,\backslash mathbf\; i\; -\; c\backslash ,\backslash mathbf\; j-\; d\backslash ,\backslash mathbf\; k).$
Thus the quaternions form a division algebra.
Multiplication of basis elements

The multiplication with of the basis elements , and is defined by the fact that is a multiplicative identity, that is, :$\backslash mathbf\; i\; \backslash ,\; 1\; =\; 1\; \backslash ,\; \backslash mathbf\; i\; =\; \backslash mathbf\; i,\; \backslash qquad\; \backslash mathbf\; j\; \backslash ,\; 1\; =\; 1\; \backslash ,\; \backslash mathbf\; j\; =\; \backslash mathbf\; j,\; \backslash qquad\; \backslash mathbf\; k\; \backslash ,\; 1\; =\; 1\; \backslash ,\; \backslash mathbf\; k=\; \backslash mathbf\; k\; \backslash ,.$ The products of basis elements are derived from the product rules for $\backslash mathbf\; i$ and $\backslash mathbf\; j\backslash ,\backslash colon$ :$\backslash mathbf\; i^2\; =\; \backslash mathbf\; j^2\; =\; -1$ and :$\backslash begin\; \backslash mathbf\; \&=\; \backslash mathbf\; k\backslash ,,\; \backslash quad\; \&\backslash mathbf\; \&=\; -\backslash mathbf\; k\backslash ,.\; \backslash end$ The remaining product rules are obtained by multiplying both sides of these latter rules by $\backslash mathbf\; j$ or $\backslash mathbf\; i$ from the left or right and applyingassociativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...

which gives
:$\backslash begin\; \backslash mathbf\; \&=\; \backslash mathbf\; i\backslash ,,\backslash quad\; \&\; \backslash mathbf\&\; =\; -\backslash mathbf\; i\backslash ,,\backslash \backslash \; \backslash mathbf\; \&=\; \backslash mathbf\; j\backslash ,,\backslash quad\; \&\; \backslash mathbf\; \&\; =\; -\backslash mathbf\; j\; \backslash ,,\backslash \backslash \; \backslash mathbf\&=-1\backslash ,,\backslash quad\; \&\; \backslash mathbf\; k^2\; \&=-1\backslash ,\backslash \backslash \; \backslash end$
Center

The ''center'' of anoncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...

is the subring of elements such that for every . 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 belongs to the center, then
:$0\; =\; \backslash mathbf\; i\backslash ,q\; -\; q\backslash ,\backslash mathbf\; i\; =\; 2c\backslash ,\backslash mathbf\; +\; 2d\backslash ,\backslash mathbf\; =\; 2c\backslash ,\backslash mathbf\; k\; -\; 2d\backslash ,\backslash mathbf\; j\backslash ,,$
and . A similar computation with instead of shows that one has also . Thus 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
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...

over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation has infinitely many quaternion solutions, which are the quaternions such that . Thus these "roots of –1" form a unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...

in the three-dimensional space of vector quaternions.
Hamilton product

For two elements and , their product, called the Hamilton product () (), is determined by the products of the basis elements and thedistributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic ...

. 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:
:$\backslash begin\; \&a\_1a\_2\; \&\&+\; a\_1b\_2\; \backslash mathbf\; i\; \&\&+\; a\_1c\_2\; \backslash mathbf\; j\; \&\&+\; a\_1d\_2\; \backslash mathbf\; k\backslash \backslash \; +\; \&b\_1a\_2\; \backslash mathbf\; i\; \&\&+\; b\_1b\_2\; \backslash mathbf\; i^2\; \&\&+\; b\_1c\_2\; \backslash mathbf\; \&\&+\; b\_1d\_2\; \backslash mathbf\backslash \backslash \; +\; \&c\_1a\_2\; \backslash mathbf\; j\; \&\&+\; c\_1b\_2\; \backslash mathbf\; \&\&+\; c\_1c\_2\; \backslash mathbf\; j^2\; \&\&+\; c\_1d\_2\; \backslash mathbf\backslash \backslash \; +\; \&d\_1a\_2\; \backslash mathbf\; k\; \&\&+\; d\_1b\_2\; \backslash mathbf\; \&\&+\; d\_1c\_2\; \backslash mathbf\; \&\&+\; d\_1d\_2\; \backslash mathbf\; k^2\; \backslash end$
Now the basis elements can be multiplied using the rules given above to get:
:$\backslash begin\; \&a\_1a\_2\; \&\&-\; b\_1b\_2\; \&\&-\; c\_1c\_2\; \&\&-\; d\_1d\_2\backslash \backslash \; +\; (\&a\_1b\_2\; \&\&+\; b\_1a\_2\; \&\&+\; c\_1d\_2\; \&\&-\; d\_1c\_2)\; \backslash mathbf\; i\backslash \backslash \; +\; (\&a\_1c\_2\; \&\&-\; b\_1d\_2\; \&\&+\; c\_1a\_2\; \&\&+\; d\_1b\_2)\; \backslash mathbf\; j\backslash \backslash \; +\; (\&a\_1d\_2\; \&\&+\; b\_1c\_2\; \&\&-\; c\_1b\_2\; \&\&+\; d\_1a\_2)\; \backslash mathbf\; k\; \backslash end$
The product of two rotation quaternions will be equivalent to the rotation followed by the rotation
Scalar and vector parts

A quaternion of the form , where is a real number, is called scalar, and a quaternion of the form , where , and are real numbers, and at least one of or is nonzero, is called a vector quaternion. If is any quaternion, then is called its scalar part and 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 $\backslash mathbb\; R^3.$ Hamilton also called vector quaternions right quaternions 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, :$\backslash mathbf\; q\; =\; (r,\backslash \; \backslash vec),~~\; \backslash mathbf\; q\; \backslash in\; \backslash mathbb,~~\; r\; \backslash in\; \backslash mathbb,~~\; \backslash vec\backslash in\; \backslash mathbb^3,$ then the formulas for addition and multiplication are :$(r\_1,\backslash \; \backslash vec\_1)\; +\; (r\_2,\backslash \; \backslash vec\_2)\; =\; (r\_1\; +\; r\_2,\backslash \; \backslash vec\_1+\backslash vec\_2),$ :$(r\_1,\backslash \; \backslash vec\_1)\; (r\_2,\backslash \; \backslash vec\_2)\; =\; (r\_1\; r\_2\; -\; \backslash vec\_1\backslash cdot\backslash vec\_2,\backslash \; r\_1\backslash vec\_2+r\_2\backslash vec\_1\; +\; \backslash vec\_1\backslash times\backslash vec\_2),$ where "$\backslash cdot$" and "$\backslash times$" denote respectively thedot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...

and the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

.
Conjugation, the norm, and reciprocal

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 $q\; =\; a\; +\; b\backslash ,\backslash mathbf\; i\; +\; c\backslash ,\backslash mathbf\; j\; +\; d\backslash ,\backslash mathbf\; k$ be a quaternion. The conjugate of is the quaternion $q^*\; =\; a\; -\; b\backslash ,\backslash mathbf\; i\; -\; c\backslash ,\backslash mathbf\; j\; -\; d\backslash ,\backslash mathbf\; k$. It is denoted by , ''qtensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

"). In formulas, this is expressed as follows:
:$\backslash lVert\; q\; \backslash rVert\; =\; \backslash sqrt\; =\; \backslash sqrt\; =\; \backslash sqrt$
This is always a non-negative real number, and it is the same as the Euclidean norm on $\backslash mathbb\; H$ considered as the vector space $\backslash mathbb\; R^4$. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if is real, then
:$\backslash lVert\backslash alpha\; q\backslash rVert\; =\; \backslash left,\; \backslash alpha\backslash \backslash ,\backslash lVert\; q\backslash rVert~.$
This is a special case of the fact that the norm is ''multiplicative'', meaning that
:$\backslash lVert\; pq\; \backslash rVert\; =\; \backslash lVert\; p\; \backslash rVert\backslash ,\backslash lVert\; q\; \backslash rVert$
for any two quaternions and . Multiplicativity is a consequence of the formula for the conjugate of a product.
Alternatively it follows from the identity
:$\backslash det\; \backslash begin\; a\; +\; i\; b\; \&\; i\; d\; +\; c\; \backslash \backslash \; i\; d\; -\; c\; \&\; a\; -\; i\; b\; \backslash end\; =\; a^2\; +\; b^2\; +\; c^2\; +\; d^2,$
(where denotes the usual imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...

) and hence from the multiplicative property of determinants of square matrices.
This norm makes it possible to define the distance between and as the norm of their difference:
:$d(p,\; q)\; =\; \backslash lVert\; p\; -\; q\; \backslash rVert~.$
This makes $\backslash mathbb\; H$ a metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

.
Addition and multiplication are continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...

in regard to the associated metric topology.
This follows with exactly the same proof as for the real numbers $\backslash mathbb\; R$ from the fact that $\backslash mathbb\; H$ is a normed algebra.
Unit quaternion

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion by its norm produces a unit quaternion called the ''versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...

'' of :
:$\backslash mathbfq\; =\; \backslash frac.$
Every quaternion has a polar decomposition $q\; =\; \backslash lVert\; q\; \backslash rVert\; \backslash cdot\; \backslash mathbf\; q$.
Using conjugation and the norm makes it possible to define the 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 $q$ and $q^*\; /\; \backslash left\; \backslash Vert\; q\; \backslash right\; \backslash ,\; ^2$ is 1 (for either order of multiplication). So the '' reciprocal'' of is defined to be
:$q^\; =\; \backslash frac.$
This makes it possible to divide two quaternions and in two different ways (when is non-zero). That is, their quotient can be either or ; in general, those products are different, depending on the order of multiplication, except for the special case that and are scalar multiples of each other (which includes the case where ). Hence, the notation is ambiguous because it does not specify whether divides on the left or the right (whether multiplies on its left or its right).
Algebraic properties

The set $\backslash mathbb\; H$ of all quaternions is avector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...

over the real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s with dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...

4. 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 $\backslash mathbb\; H$ are a non-commutative, associative algebra over the real numbers. Even though $\backslash mathbb\; H$ 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 except for the non-commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The Frobenius theorem states that there are exactly three: $\backslash mathbb\; R$, $\backslash mathbb\; C$, and $\backslash mathbb\; H$. The norm makes the quaternions into a normed algebra, and normed division algebras over the real numbers are also very rare: Hurwitz's theorem says that there are only four: $\backslash mathbb\; R$, $\backslash mathbb\; C$, $\backslash mathbb\; H$, and $\backslash mathbb\; O$ (the octonions). The quaternions are also an example of a composition algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involutio ...

and of a unital Banach algebra.
Because the product of any two basis vectors is plus or minus another basis vector, the set forms a group under multiplication. This non-abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

is called the quaternion group and is denoted . The real group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...

of is a ring $\backslash mathbb\; R;\; href="/html/ALL/s/mathrm\_Q\_8.html"\; ;"title="mathrm\; Q\_8">mathrm\; Q\_8$quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...

of $\backslash mathbb\; R;\; href="/html/ALL/s/mathrm\_Q\_8.html"\; ;"title="mathrm\; Q\_8">mathrm\; Q\_8$ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring consider ...

generated by the elements , , , and . Here the first term in each of the differences is one of the basis elements , and , and the second term is one of basis elements , and , not the additive inverses of , and .
Quaternions and the space geometry

The vector part of a quaternion can be interpreted as a coordinate vector in $\backslash mathbb\; R^3;$ therefore, the algebraic operations of the quaternions reflect the geometry of $\backslash mathbb\; R^3.$ 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. For the remainder of this section, , , and will denote both the three imaginary basis vectors of $\backslash mathbb\; H$ and a basis for $\backslash mathbb\; R^3.$ Replacing by , by , and by 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 and theirdot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...

, by analogy to vectors in $\backslash mathbb\; R^3,$ is
:$p\; \backslash cdot\; q\; =\; b\_1\; b\_2\; +\; c\_1\; c\_2\; +\; d\_1\; d\_2~.$
It can also be expressed in a component-free manner as
:$p\; \backslash cdot\; q\; =\; \backslash textstyle\backslash frac(p^*q\; +\; q^*p)\; =\; \backslash textstyle\backslash frac(pq^*\; +\; qp^*).$
This is equal to the scalar parts of the products . Note that their vector parts are different.
The cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

of and relative to the orientation determined by the ordered basis , and is
:$p\; \backslash times\; q\; =\; (c\_1\; d\_2\; -\; d\_1\; c\_2)\backslash mathbf\; i\; +\; (d\_1\; b\_2\; -\; b\_1\; d\_2)\backslash mathbf\; j\; +\; (b\_1\; c\_2\; -\; c\_1\; b\_2)\backslash mathbf\; k\backslash ,.$
(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product (as quaternions), as well as the vector part of . It also has the formula
:$p\; \backslash times\; q\; =\; \backslash textstyle\backslash tfrac(pq\; -\; qp).$
For the commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...

, , of two vector quaternions one obtains
:$;\; href="/html/ALL/s/,q.html"\; ;"title=",q">,q$
In general, let and be quaternions and write
:$p\; =\; p\_\backslash text\; +\; p\_\backslash text,$
:$q\; =\; q\_\backslash text\; +\; q\_\backslash text,$
where and are the scalar parts, and and are the vector parts of and . Then we have the formula
:$pq\; =\; (pq)\_\backslash text\; +\; (pq)\_\backslash text\; =\; (p\_\backslash textq\_\backslash text\; -\; p\_\backslash text\backslash cdot\; q\_\backslash text)\; +\; (p\_\backslash text\; q\_\backslash text\; +\; q\_\backslash text\; p\_\backslash text\; +\; p\_\backslash text\; \backslash times\; q\_\backslash text).$
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 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 $\backslash mathbb\; R^3$ and were called versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...

s by Hamilton. Also see Quaternions and spatial rotation Unit quaternions, known as ''versors'', provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation ab ...

for more information about modeling three-dimensional rotations using quaternions.
See Hanson (2005) for visualization of quaternions.
Matrix representations

Just as complex numbers can be 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 andmatrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

. One is to use 2 × 2 complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...

matrices, and the other is to use 4 × 4 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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sam ...

s from $\backslash mathbb\; H$ to the matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...

s and , respectively.
Using 2 × 2 complex matrices, the quaternion can be represented as
: $\backslash begina+bi\; \&\; c+di\; \backslash \backslash \; -c\; +\; d\; i\; \&\; a\; -\; b\; i\; \backslash end.$
Note that the "i" of the complex numbers is distinct from the "i" of the quaternions.
This representation has the following properties:
* Constraining any two of , and to zero produces a representation of complex numbers. For example, setting produces a diagonal complex matrix representation of complex numbers, and setting 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.
* The conjugate of a quaternion corresponds to the conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...

of the matrix.
* By restriction this representation yields an isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

between the subgroup of unit quaternions and their image SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the spec ...

. Topologically, the unit quaternions are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group is important for describing spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...

in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

; see Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...

.
* There is a strong relation between quaternion units and Pauli matrices. Obtain the eight quaternion unit matrices by taking and , 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 ; e.g. the last equality is
:$ijk\; =\; \backslash sigma\_1\; \backslash sigma\_2\; \backslash sigma\_3\; \backslash sigma\_1\; \backslash sigma\_2\; \backslash sigma\_3\; =\; -1$
Using 4 × 4 real matrices, that same quaternion can be written as
:$\backslash begin\; a\; \&\; -b\; \&\; -c\; \&\; -d\; \backslash \backslash \; b\; \&\; a\; \&\; -d\; \&\; c\; \backslash \backslash \; c\; \&\; d\; \&\; a\; \&\; -b\; \backslash \backslash \; d\; \&\; -c\; \&\; b\; \&\; a\; \backslash end=\; a\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash end\; +\; b\; \backslash begin\; 0\; \&\; -1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; -1\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\; \backslash end\; +\; c\; \backslash begin\; 0\; \&\; 0\; \&\; -1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; -1\; \&\; 0\; \&\; 0\; \backslash end\; +\; d\; \backslash begin\; 0\; \&\; 0\; \&\; 0\; \&\; -1\; \backslash \backslash \; 0\; \&\; 0\; \&\; -1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash end.$
However, the representation of quaternions in is not unique. For example, the same quaternion can also be represented as
:$\backslash begin\; a\; \&\; d\; \&\; -b\; \&\; -c\; \backslash \backslash \; -d\; \&\; a\; \&\; c\; \&\; -b\; \backslash \backslash \; b\; \&\; -c\; \&\; a\; \&\; -d\; \backslash \backslash \; c\; \&\; b\; \&\; d\; \&\; a\; \backslash end=\; a\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash end\; +\; b\; \backslash begin\; 0\; \&\; 0\; \&\; -1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; -1\; \backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash end\; +\; c\; \backslash begin\; 0\; \&\; 0\; \&\; 0\; \&\; -1\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; -1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash end\; +\; d\; \backslash begin\; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; -1\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; -1\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\; \backslash end.$
There exist 48 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 sets of quadruples of matrices with these symmetry constraints such that a function sending , and to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices.
In this representation, the conjugate of a quaternion corresponds to the transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The t ...

of the matrix. The fourth power of the norm of a quaternion is the 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 .
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
which is isomorphic — through $\backslash $ — to
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 with , with , with , and with and removing the row and column headers yields a matrix representation of .
Lagrange’s four-square theorem

Quaternions are also used in one of the proofs of Lagrange's four-square theorem innumber theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, 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 quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an e ...

.
Quaternions as pairs of complex numbers

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 $\backslash mathbb\; C^2$ be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements and . A vector in $\backslash mathbb\; C^2$ can be written in terms of the basis elements and as :$(a\; +\; b\; i)1\; +\; (c\; +\; d\; i)\backslash mathbf\; j\backslash ,.$ If we define and , then we can multiply two vectors using the distributive law. Using as an abbreviated notation for the product leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion . If we write the elements of $\backslash mathbb\; C^2$ as ordered pairs and quaternions as quadruples, then the correspondence is :$(a\; +\; bi,\backslash \; c\; +\; di)\; \backslash leftrightarrow\; (a,\; b,\; c,\; d).$Square roots

Square roots of −1

In the complex numbers, $\backslash mathbb\; C,$ there are just two numbers, ''i'' and −''i'', whose square is −1 . In $\backslash mathbb\; H$ there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unitsphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

in $\backslash mathbb\; R^3.$ To see this, let be a quaternion, and assume that its square is −1. In terms of , and , this means
:$a^2\; -\; b^2\; -\; c^2\; -\; d^2\; =\; -1,$
:$2ab\; =\; 0,$
:$2ac\; =\; 0,$
:$2ad\; =\; 0.$
To satisfy the last three equations, either or , and are all 0. The latter is impossible because ''a'' is a real number and the first equation would imply that Therefore, and 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).
As a union of complex planes

Each pair of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. If then the copy is determined by the function :$a\; +\; b\backslash sqrt\; \backslash mapsto\; a\; +\; b\; q\backslash ,.$ This is aninjective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preserv ...

from $\backslash mathbb\; C$ to $\backslash mathbb\; H,$ which defines a field isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

from $\backslash Complex$ onto its image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...

. The images of the embeddings corresponding to and − are identical.
Every non-real quaternion generates a subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...

of the quaternions that is isomorphic to $\backslash mathbb\; C,$ and is thus a planar subspace of $\backslash mathbb\; H\backslash colon$ write as the sum of its scalar part and its vector part:
:$q\; =\; q\_s\; +\; \backslash vec\_v.$
Decompose the vector part further as the product of its norm and its versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...

:
:$q\; =\; q\_s\; +\; \backslash lVert\backslash vec\_v\backslash rVert\backslash cdot\backslash mathbf\backslash vec\_v=q\_s+\backslash frac.$
(Note that this is not the same as $q\_s\; +\; \backslash lVert\; q\backslash rVert\backslash cdot\backslash mathbfq$.) The versor of the vector part of , $\backslash mathbf\backslash vec\_v$, is a right versor with –1 as its square. A straightforward verification shows that
:$a\; +\; b\backslash sqrt\; \backslash mapsto\; a\; +\; b\backslash mathbf\backslash vec\_v$
defines an injective homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sam ...

of normed algebras from $\backslash mathbb\; C$ into the quaternions. Under this homomorphism, is the image of the complex number $q\_s\; +\; \backslash lVert\backslash vec\_v\backslash rVert\; i$.
As $\backslash mathbb\; H$ is the union of the images of all these homomorphisms, this allows viewing the quaternions as a union of complex planes intersecting on the real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...

. Each of these complex planes 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 $\backslash mathbb\; H$ can also be identified and expressed in terms ofcommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...

s. Specifically, since two quaternions and commute (i.e., ) only if they lie in the same complex subplane of $\backslash mathbb\; H$, the profile of $\backslash mathbb\; H$ as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring.
Square roots of arbitrary quaternions

Any quaternion $\backslash mathbf\; q\; =\; (r,\backslash ,\; \backslash vec)$ (represented here in scalar–vector representation) has at least one square root $\backslash sqrt\; =\; (x,\backslash ,\; \backslash vec)$ which solves the equation $\backslash sqrt^2\; =\; (x,\backslash ,\; \backslash vec)^2\; =\; \backslash mathbf\; q$. Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions $$\backslash sqrt\; =\; \backslash sqrt\; =\; \backslash pm\backslash left(\backslash sqrt,\backslash \; \backslash frac\backslash sqrt\backslash right),$$ where $\backslash ,\; \backslash vec\backslash ,\; =\; \backslash sqrt=\backslash sqrt$ is the norm of $\backslash vec$ and $\backslash ,\; \backslash mathbf\; q\backslash ,\; =\; \backslash sqrt\; =\; r^2\; +\; \backslash ,\; \backslash vec\backslash ,\; ^2$ is the norm of $\backslash mathbf\; q$. For any scalar quaternion $\backslash mathbf\; q$, this equation provides the correct square roots if $\backslash frac$ is interpreted as an arbitrary unit vector. Therefore, non-zero, 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 $\backslash \; \backslash times\; S^2(\backslash sqrt)$, i.e., where the scalar part is zero and the vector part is located on the2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

with radius $\backslash sqrt$.
Functions of a quaternion variable

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 theMandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...

and Julia sets into 4-dimensional space.
Exponential, logarithm, and power functions

Given a quaternion, :$q\; =\; a\; +\; b\backslash mathbf\; i\; +\; c\backslash mathbf\; j\; +\; d\backslash mathbf\; k\; =\; a\; +\; \backslash mathbf$ the exponential is computed as :$\backslash exp(q)\; =\; \backslash sum\_^\backslash infty\; \backslash frac=e^\; \backslash left(\backslash cos\; \backslash ,\; \backslash mathbf\backslash ,\; +\; \backslash frac\; \backslash sin\; \backslash ,\; \backslash mathbf\backslash ,\; \backslash right)~~$ and the logarithm is :$\backslash ln(q)\; =\; \backslash ln\; \backslash ,\; q\backslash ,\; +\; \backslash frac\; \backslash arccos\; \backslash frac~~$ It follows that the polar decomposition of a quaternion may be written :$q=\backslash ,\; q\backslash ,\; e^\; =\; \backslash ,\; q\backslash ,\; \backslash left(\backslash cos(\backslash varphi)\; +\; \backslash hat\; \backslash sin(\backslash varphi)\backslash right),$ where theangle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...

$\backslash varphi$
:$a\; =\; \backslash ,\; q\; \backslash ,\; \backslash cos(\; \backslash varphi\; )$
and the unit vector $\backslash hat$ is defined by:
:$\backslash mathbf\; =\; \backslash hat\; \backslash ,\; \backslash mathbf\backslash ,\; =\; \backslash hat\backslash ,\; q\backslash ,\; \backslash sin(\backslash varphi)\backslash ,.$
Any unit quaternion may be expressed in polar form as:
:$q=\backslash exp$.
The power of a quaternion raised to an arbitrary (real) exponent is given by:
:$q^x\; =\; \backslash ,\; q\backslash ,\; ^x\; e^\; =\; \backslash ,\; q\backslash ,\; ^x\; \backslash left(\backslash cos(x\backslash varphi)\; +\; \backslash hat\backslash ,\backslash sin(x\backslash varphi)\backslash right)~.$
Geodesic norm

The geodesic distance between unit quaternions and is defined as: :$d\_\backslash text(p,\; q)\; =\; \backslash lVert\; \backslash ln(p^\; q)\; \backslash rVert.$ and amounts to the absolute value of half the angle subtended by and along a great arc of the sphere. This angle can also be computed from the quaterniondot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...

without the logarithm as:
:$\backslash arccos(2(p\; \backslash cdot\; q)^2\; -\; 1).$
Three-dimensional and four-dimensional rotation groups

The word "conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
*Complex conjugation, the change ...

", besides the meaning given above, can also mean taking an element to where is some non-zero quaternion. All 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 $\backslash mathbb\; R^3$ consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part is a rotation by an angle , 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 matrices.
* Nonsingular representation (compared with Euler angles for example).
* Pairs of unit quaternions represent a rotation in 4D space (see '' Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations'').
The set of all unit quaternions (Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...

) under multiplication, double covering the group of real orthogonal 3×3 matrices of determinant 1 since ''two'' unit quaternions correspond to every rotation under the above correspondence. See plate trick.
The image of a subgroup of versors is a point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...

, 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
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of th ...

is the binary icosahedral group.
The versors' group is isomorphic to , the group of complex unitary 2×2 matrices of determinant 1.
Let be the set of quaternions of the form where and are either all integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s or all half-integer
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...

s. The set is a ring (in fact a domain) and a lattice and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of a regular 24 cell with Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

They correspond to the double cover of the rotational symmetry group of the regular tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

. Similarly, the vertices of a regular 600 cell with Schläfli symbol can be taken as the unit icosian
In mathematics, the icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. The term can be used to refer to two related, but distinct, concepts:
* The icosian group: a multiplicative group of 120 quaterni ...

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
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...

corresponds to the quaternions that represent the vertices of the disphenoidal 288-cell.
Quaternion algebras

The Quaternions can be generalized into further algebras called ''quaternion algebras''. Take to be any field with characteristic different from 2, and and to be elements of ; a four-dimensional unitary associative algebra can be defined over with basis and , where , and (so ). Quaternion algebras are isomorphic to the algebra of 2×2 matrices over or form division algebras over , depending on the choice of and .Quaternions as the even part of

The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part $\backslash operatorname\_^+(\backslash mathbb\; R)$ of the Clifford algebra $\backslash operatorname\_(\backslash mathbb\; R).$ This is an associative multivector algebra built up from fundamental basis elements using the product rules :$\backslash sigma\_1^2\; =\; \backslash sigma\_2^2\; =\; \backslash sigma\_3^2\; =\; 1,$ :$\backslash sigma\_i\; \backslash sigma\_j\; =\; -\; \backslash sigma\_j\; \backslash sigma\_i\; \backslash qquad\; (j\; \backslash neq\; i).$ If these fundamental basis elements are taken to represent vectors in 3D space, then it turns out that the ''reflection'' of a vector in a plane perpendicular to a unit vector can be written: :$r^\; =\; -\; w\backslash ,\; r\backslash ,\; w.$ Two reflections make a rotation by an angle twice the angle between the two reflection planes, so :$r^\; =\; \backslash sigma\_2\; \backslash sigma\_1\; \backslash ,\; r\; \backslash ,\; \backslash sigma\_1\; \backslash sigma\_2$ corresponds to a rotation of 180° in the plane containing ''σ''rotors
Rotor may refer to:
Science and technology
Engineering
*Rotor (electric), the non-stationary part of an alternator or electric motor, operating with a stationary element so called the stator
*Helicopter rotor, the rotary wing(s) of a rotorcraft ...

, can be very useful for applications involving homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...

. 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
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...

.
There are several advantages for placing quaternions in this wider setting: See also:
* 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 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 spaces, for example, the Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...

of special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...

. In such spaces rotors can be used to efficiently represent Lorentz boosts, and to interpret formulas involving the gamma matrices.
For further detail about the geometrical uses of Clifford algebras, see Geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...

.
Brauer group

The quaternions are "essentially" the only (non-trivial)central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simpl ...

(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
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...

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 In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a si ...

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

See also

* Conversion between quaternions and Euler angles * Dual quaternion * Dual-complex number * Exterior algebra * Hurwitz quaternion order * Hyperbolic quaternion * Lénárt sphere *Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...

* Quaternionic manifold
* Quaternionic matrix
* Quaternionic polytope
* Quaternionic projective space
* Rotations in 4-dimensional Euclidean space
* Slerp
In computer graphics, Slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation. It refers to constant-speed motion along a unit-radius gr ...

* Split-quaternion
* Tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of ei ...

Notes

References

Further reading

Books and publications

* * Hamilton, William Rowan (1853), "Lectures on Quaternions

'". Royal Irish Academy. *Hamilton (1866)

Elements of Quaternions

'

University of Dublin
The University of Dublin ( ga, Ollscoil Átha Cliath), corporately designated the Chancellor, Doctors and Masters of the University of Dublin, is a university located in Dublin, Ireland. It is the degree-awarding body for Trinity College Dubl ...

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.
* Tait, Peter Guthrie (1873), "''An elementary treatise on quaternions''". 2d ed., Cambridge, ng.: The University Press.
*Maxwell, James Clerk (1873), "''A Treatise on Electricity and Magnetism
''A Treatise on Electricity and Magnetism'' is a two-volume treatise on electromagnetism written by James Clerk Maxwell in 1873. Maxwell was revising the ''Treatise'' for a second edition when he died in 1879. The revision was completed by Wi ...

''". Clarendon Press, Oxford.
* Tait, Peter Guthrie (1886), "'". M.A. Sec. R.S.E. Encyclopædia Britannica
The (Latin for "British Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various ti ...

, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped PostScript
PostScript (PS) is a page description language in the electronic publishing and desktop publishing realm. It is a dynamically typed, concatenative programming language. It was created at Adobe Systems by John Warnock, Charles Geschke, Doug B ...

file)
*
*
* (''See section on quaternions.'')
*
*
*Crowe, Michael J. (1967), A History of Vector Analysis
''A History of Vector Analysis'' (1967) is a book on the history of vector analysis by Michael J. Crowe, originally published by the University of Notre Dame Press.
As a scholarly treatment of a reformation in technical communication, the text ...

: ''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).
*
*
*
*
*
*
*review

. * * * * * * * *For molecules that can be regarded as classical rigid bodies

molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...

computer simulation employs quaternions. They were first introduced for this purpose by
*
*
*
Links and monographs

* Notices and materials related to Quaternion conference presentations * * *Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)

* * * 3D Raytraced Quaternion Julia Fractals * Great page explaining basic math with links to straight forward rotation conversion formulae. * * * * * * * * * David Erickson,

Defence Research and Development Canada
Defence Research and Development Canada (DRDC; french: Recherche et développement pour la défense Canada, ''RDDC'') is a special operating agency of the Department of National Defence (DND), whose purpose is to provide the Canadian Armed Forces ...

(DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper.
*
*
* describes how the quaternions can be made into a skew-commutative algebra graded by .
*
*Part II

(PDF; using Hamilton's terminology, which differs from the modern usage) *

two expository papers about continuous functional calculus and spectral theory in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.

Quaternions

the Android app shows the quaternion corresponding to the orientation of the device.

Rotating Objects Using Quaternions

article speaking to the use of Quaternions for rotation in video games/computer graphics.

External links

* * Paulson, Lawrence C.br>Quaternions (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)Quaternions – Visualisation

{{Authority control Composition algebras William Rowan Hamilton