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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 fo ...
s. Quaternions were first described by the Irish mathematician
William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
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 objec ...
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 informa ...
. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. 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 o ...
. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \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, 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 mathemati ...
, 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 human ...
, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application. In modern mathematical language, quaternions form a four-
dimensional 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 ...
associative normed division algebra over the real numbers, and therefore a ring, being both a division ring and a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. It can also be given by the Clifford algebra classifications \operatorname_(\mathbb R)\cong \operatorname_^+(\mathbb R). In fact, it was the first noncommutative division algebra to be discovered. According to the Frobenius theorem, the algebra \mathbb H is one of only two finite-dimensional division rings 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 hav ...
s, which is the last normed division algebra over the real numbers. (The
sedenions In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...
, 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 ze ...
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 that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).


History

Quaternions were introduced by Hamilton in 1843.See Important precursors to this work included
Euler's four-square identity In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. Algebraic identity For any pair of quadruples from a commutative ring, the following expre ...
(1748) and
Olinde Rodrigues Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer. In mathematics Rodrigues is remembered for Rodrigues' rotation formula for vectors ...
' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss 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 Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * 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 consi ...
. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact,
Ferdinand Georg Frobenius Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famou ...
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: \mathbb (complex numbers) and \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 ...
, when Hamilton was on his way to the
Royal Irish Academy The Royal Irish Academy (RIA; ga, Acadamh Ríoga na hÉireann), based in Dublin, is an academic body that promotes study in the sciences, humanities and social sciences. It is Ireland's premier learned society and one its leading cultural ...
where he was going to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, :\mathbf^2 = \mathbf^2 = \mathbf^2 = \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 The Hamilton Walk from Dunsink Observatory to Broom Bridge on the Royal Canal in Dublin takes place on 16 October each year. This is the anniversary of the day in 1843 when William Rowan Hamilton discovered the non-commutative algebraic system k ...
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 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 ...
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 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 fiel ...
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, 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 develope ...
, 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 Associat ...
. 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 ...
. 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 de ...
, 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 human ...
,
robotics Robotics is an 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 humans. Robotics integrat ...
,
control theory Control theory is a field of mathematics that deals with the control system, 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 ...
,
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, ...
, attitude control,
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 ...
,
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 combi ...
,
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 th ...
,
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 deter ...
s, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from
number theory 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, "Ma ...
because of their relationships with the quadratic forms.


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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
. Girard began by discussing
group 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 ...
s and by representing some space groups of crystallography. 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 fiel ...
of rigid body motion. Next he used complex quaternions ( biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He cited five authors, beginning with
Ludwik Silberstein Ludwik Silberstein (1872 – 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmillan in 1914 with a ...
, who used a potential function of one
quaternion variable In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. ...
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, ...
. 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 differ ...
. Among the fifty references, Girard included Alexander Macfarlane and his ''Bulletin'' of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions. 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 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 sli ...
) 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 Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus ...
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 an expression of the form :a + b\,\mathbf + c\,\mathbf + d\,\mathbf\ , 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 co ...
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\,\mathbf + c\,\mathbf + d\,\mathbf, as consisting of a scalar part and a vector part. The quaternion b\,\mathbf + c\,\mathbf + d\,\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 can ...
over the real numbers, with \left\ as a basis, by the componentwise addition :(a_1+b_1\,\mathbf i + c_1\,\mathbf j + d_1\,\mathbf k) + (a_2 + b_2\,\mathbf i + c_2\,\mathbf j + d_2\,\mathbf k) = (a_1 + a_2) + (b_1 + b_2)\,\mathbf i + (c_1 + c_2)\,\mathbf j + (d_1 + d_2)\,\mathbf k\,, and the componentwise scalar multiplication :\lambda(a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k) = \lambda a + (\lambda b)\,\mathbf i + (\lambda c)\,\mathbf j + (\lambda d)\,\mathbf k. 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. *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 and the center property of the real quaternions. The Hamilton product is not commutative, but is associative, thus the quaternions form an associative algebra over the real numbers. *Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product: :(a + b\,\mathbf i + c\,\mathbf j + d \,\mathbf k)^ = \frac\,(a - b\,\mathbf i - c\,\mathbf j- d\,\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, :\mathbf i \, 1 = 1 \, \mathbf i = \mathbf i, \qquad \mathbf j \, 1 = 1 \, \mathbf j = \mathbf j, \qquad \mathbf k \, 1 = 1 \, \mathbf k= \mathbf k \,. The products of basis elements are derived from the product rules for \mathbf i and \mathbf j\,\colon :\mathbf i^2 = \mathbf j^2 = -1 and :\begin \mathbf &= \mathbf k\,, \quad &\mathbf &= -\mathbf k\,. \end The remaining product rules are obtained by multiplying both sides of these latter rules by \mathbf j or \mathbf i from the left or right and applying associativity which gives :\begin \mathbf &= \mathbf i\,,\quad & \mathbf& = -\mathbf i\,,\\ \mathbf &= \mathbf j\,,\quad & \mathbf & = -\mathbf j \,,\\ \mathbf&=-1\,,\quad & \mathbf k^2 &=-1\,\\ \end


Center

The ''center'' of a noncommutative ring 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 = \mathbf i\,q - q\,\mathbf i = 2c\,\mathbf + 2d\,\mathbf = 2c\,\mathbf k - 2d\,\mathbf j\,, 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 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 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 the distributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression: :\begin &a_1a_2 &&+ a_1b_2 \mathbf i &&+ a_1c_2 \mathbf j &&+ a_1d_2 \mathbf k\\ + &b_1a_2 \mathbf i &&+ b_1b_2 \mathbf i^2 &&+ b_1c_2 \mathbf &&+ b_1d_2 \mathbf\\ + &c_1a_2 \mathbf j &&+ c_1b_2 \mathbf &&+ c_1c_2 \mathbf j^2 &&+ c_1d_2 \mathbf\\ + &d_1a_2 \mathbf k &&+ d_1b_2 \mathbf &&+ d_1c_2 \mathbf &&+ d_1d_2 \mathbf k^2 \end Now the basis elements can be multiplied using the rules given above to get: :\begin &a_1a_2 &&- b_1b_2 &&- c_1c_2 &&- d_1d_2\\ + (&a_1b_2 &&+ b_1a_2 &&+ c_1d_2 &&- d_1c_2) \mathbf i\\ + (&a_1c_2 &&- b_1d_2 &&+ c_1a_2 &&+ d_1b_2) \mathbf j\\ + (&a_1d_2 &&+ b_1c_2 &&- c_1b_2 &&+ d_1a_2) \mathbf k \end 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 \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, : \mathbf q = (r,\ \vec),~~ \mathbf q \in \mathbb,~~ r \in \mathbb,~~ \vec\in \mathbb^3, then the formulas for addition and multiplication are : (r_1,\ \vec_1) + (r_2,\ \vec_2) = (r_1 + r_2,\ \vec_1+\vec_2), :(r_1,\ \vec_1) (r_2,\ \vec_2) = (r_1 r_2 - \vec_1\cdot\vec_2,\ r_1\vec_2+r_2\vec_1 + \vec_1\times\vec_2), where "\cdot" and "\times" denote respectively the
dot 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 alg ...
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 ...
.


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\,\mathbf i + c\,\mathbf j + d\,\mathbf k be a quaternion. The conjugate of is the quaternion q^* = a - b\,\mathbf i - c\,\mathbf j - d\,\mathbf k . It is denoted by , ''qt'', \tilde q, or . Conjugation is an involution, meaning that it is its own
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
, 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^* = - \frac (q + \,\mathbf i \,q \,\mathbf i + \,\mathbf j \,q \,\mathbf j + \,\mathbf k \,q \,\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 In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
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 "
tensor 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 tensor ...
"). In formulas, this is expressed as follows: :\lVert q \rVert = \sqrt = \sqrt = \sqrt This is always a non-negative real number, and it is the same as the Euclidean norm on \mathbb H considered as the vector space \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 :\lVert\alpha q\rVert = \left, \alpha\\,\lVert q\rVert~. This is a special case of the fact that the norm is ''multiplicative'', meaning that :\lVert pq \rVert = \lVert p \rVert\,\lVert q \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 : \det \begin a + i b & i d + c \\ i d - c & a - i b \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
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
s of square matrices. This norm makes it possible to define the distance between and as the norm of their difference: :d(p, q) = \lVert p - q \rVert~. This makes \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 g ...
in regard to the associated metric topology. This follows with exactly the same proof as for the real numbers \mathbb R from the fact that \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'' of : :\mathbfq = \frac. Every quaternion has a polar decomposition q = \lVert q \rVert \cdot \mathbf q . Using conjugation and the norm makes it possible to define the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
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^* / \left \Vert q \right \, ^2 is 1 (for either order of multiplication). So the ''
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
'' of is defined to be :q^ = \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 \mathbb H of all quaternions is a
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 can ...
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 coord ...
 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 \mathbb H are a non-commutative, associative algebra over the real numbers. Even though \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: \mathbb R, \mathbb C, and \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: \mathbb R, \mathbb C, \mathbb H, and \mathbb O (the octonions). The quaternions are also an example of a composition algebra 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 of is a ring \mathbb R mathrm Q_8/math> which is also an eight-dimensional vector space over \mathbb R. It has one basis vector for each element of \mathrm Q_8. The quaternions are isomorphic to the quotient ring of \mathbb R mathrm Q_8/math> by the
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 considered ...
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 \mathbb R^3; therefore, the algebraic operations of the quaternions reflect the geometry of \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 \mathbb H and a basis for \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 their
dot 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 alg ...
, by analogy to vectors in \mathbb R^3, is :p \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 \cdot q = \textstyle\frac(p^*q + q^*p) = \textstyle\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 ...
of and relative to the orientation determined by the ordered basis , and is :p \times q = (c_1 d_2 - d_1 c_2)\mathbf i + (d_1 b_2 - b_1 d_2)\mathbf j + (b_1 c_2 - c_1 b_2)\mathbf k\,. (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 \times q = \textstyle\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, ...
, , of two vector quaternions one obtains : ,q 2p \times q. In general, let and be quaternions and write :p = p_\text + p_\text, :q = q_\text + q_\text, where and are the scalar parts, and and are the vector parts of and . Then we have the formula :pq = (pq)_\text + (pq)_\text = (p_\textq_\text - p_\text\cdot q_\text) + (p_\text q_\text + q_\text p_\text + p_\text \times q_\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 \mathbb R^3 and were called versors by Hamilton. Also see Quaternions and spatial rotation for more information about modeling three-dimensional rotations using quaternions. See
Hanson Hanson or Hansson may refer to: People * Hanson (surname) * Hansson (surname) * Hanson (wrestler), ringname of an American professional wrestler Musical groups * Hanson (band), an American pop rock band * Hanson (UK band), an English rock ...
(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 and matrix multiplication. One is to use 2 × 2 complex 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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, these are injective
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s from \mathbb H to the matrix rings and , respectively. Using 2 × 2 complex matrices, the quaternion can be represented as : \begina+bi & c+di \\ -c + d i & a - b i \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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the corresponding matrix. * The conjugate of a quaternion corresponds to the conjugate transpose 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). 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 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. * 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 = \sigma_1 \sigma_2 \sigma_3 \sigma_1 \sigma_2 \sigma_3 = -1 Using 4 × 4 real matrices, that same quaternion can be written as :\begin a & -b & -c & -d \\ b & a & -d & c \\ c & d & a & -b \\ d & -c & b & a \end= a \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end + b \begin 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end + c \begin 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end + d \begin 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end. However, the representation of quaternions in is not unique. For example, the same quaternion can also be represented as :\begin a & d & -b & -c \\ -d & a & c & -b \\ b & -c & a & -d \\ c & b & d & a \end= a \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end + b \begin 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end + c \begin 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end + d \begin 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \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 of the matrix. The fourth power of the norm of a quaternion is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 \ — 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 in
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, "Ma ...
, 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 \mathbb C^2 be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements and . A vector in \mathbb C^2 can be written in terms of the basis elements and as :(a + b i)1 + (c + d i)\mathbf j\,. 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 \mathbb C^2 as ordered pairs and quaternions as quadruples, then the correspondence is :(a + bi,\ c + di) \leftrightarrow (a, b, c, d).


Square roots


Square roots of −1

In the complex numbers, \mathbb C, there are just two numbers, ''i'' and −''i'', whose square is −1 . In \mathbb H there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit
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 c ...
in \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\sqrt \mapsto a + b q\,. This is an injective ring homomorphism from \mathbb C to \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 \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-dimensio ...
. The images of the embeddings corresponding to and − are identical. Every non-real quaternion generates a subalgebra of the quaternions that is isomorphic to \mathbb C, and is thus a planar subspace of \mathbb H\colon write as the sum of its scalar part and its vector part: :q = q_s + \vec_v. Decompose the vector part further as the product of its norm and its versor: :q = q_s + \lVert\vec_v\rVert\cdot\mathbf\vec_v=q_s+\frac. (Note that this is not the same as q_s + \lVert q\rVert\cdot\mathbfq.) The versor of the vector part of , \mathbf\vec_v, is a right versor with –1 as its square. A straightforward verification shows that :a + b\sqrt \mapsto a + b\mathbf\vec_v defines an injective
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of normed algebras from \mathbb C into the quaternions. Under this homomorphism, is the image of the complex number q_s + \lVert\vec_v\rVert i. As \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. Each of these complex planes contains exactly one pair of
antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
of the sphere of square roots of minus one.


Commutative subrings

The relationship of quaternions to each other within the complex subplanes of \mathbb H can also be identified and expressed in terms of commutative
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 \mathbb H, the profile of \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 \mathbf q = (r,\, \vec) (represented here in scalar–vector representation) has at least one square root \sqrt = (x,\, \vec) which solves the equation \sqrt^2 = (x,\, \vec)^2 = \mathbf q. Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions \sqrt = \sqrt = \pm\left(\sqrt,\ \frac\sqrt\right), where \, \vec\, = \sqrt=\sqrt is the norm of \vec and \, \mathbf q\, = \sqrt = r^2 + \, \vec\, ^2 is the norm of \mathbf q. For any scalar quaternion \mathbf q, this equation provides the correct square roots if \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 \ \times S^2(\sqrt), i.e., where the scalar part is zero and the vector part is located on the 2-sphere with radius \sqrt.


Functions of a quaternion variable

Like functions of a
complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the Mandelbrot set and Julia sets into 4-dimensional space.


Exponential, logarithm, and power functions

Given a quaternion, : q = a + b\mathbf i + c\mathbf j + d\mathbf k = a + \mathbf the exponential is computed as :\exp(q) = \sum_^\infty \frac=e^ \left(\cos \, \mathbf\, + \frac \sin \, \mathbf\, \right)~~ and the logarithm is :\ln(q) = \ln \, q\, + \frac \arccos \frac~~ It follows that the polar decomposition of a quaternion may be written :q=\, q\, e^ = \, q\, \left(\cos(\varphi) + \hat \sin(\varphi)\right), where the
angle 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 ...
\varphi :a = \, q \, \cos( \varphi ) and the unit vector \hat is defined by: :\mathbf = \hat \, \mathbf\, = \hat\, q\, \sin(\varphi)\,. Any unit quaternion may be expressed in polar form as: :q=\exp. The power of a quaternion raised to an arbitrary (real) exponent is given by: :q^x = \, q\, ^x e^ = \, q\, ^x \left(\cos(x\varphi) + \hat\,\sin(x\varphi)\right)~.


Geodesic norm

The geodesic distance between unit quaternions and is defined as: :d_\text(p, q) = \lVert \ln(p^ q) \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 quaternion
dot 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 alg ...
without the logarithm as: :\arccos(2(p \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 \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 ( versors) forms a 3-sphere and a group (a
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 addi ...
) under multiplication, double covering the group of real orthogonal 3×3  matrices of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
 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, and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same name, with the prefix binary. For instance, the preimage of the icosahedral group is the binary icosahedral group. The versors' group is isomorphic to , the group of complex
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...
2×2 matrices of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 languag ...
s or all half-integers. The set is a ring (in fact a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
) 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 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 ...
. Similarly, the vertices of a regular 600 cell with Schläfli symbol can be taken as the unit icosians, corresponding to the double cover of the rotational symmetry group of the regular icosahedron. The double cover of the rotational symmetry group of the regular octahedron corresponds to the quaternions that represent the vertices of the disphenoidal 288-cell.


Quaternion algebras

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 \operatorname_^+(\mathbb R) of the Clifford algebra \operatorname_(\mathbb R). This is an associative multivector algebra built up from fundamental basis elements using the product rules :\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 1, :\sigma_i \sigma_j = - \sigma_j \sigma_i \qquad (j \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\, r\, w. Two reflections make a rotation by an angle twice the angle between the two reflection planes, so :r^ = \sigma_2 \sigma_1 \, r \, \sigma_1 \sigma_2 corresponds to a rotation of 180° in the plane containing ''σ''1 and ''σ''2. This is very similar to the corresponding quaternion formula, :r^ = -\mathbf\, r\, \mathbf. In fact, the two are identical, if we make the identification :\mathbf = \sigma_2 \sigma_1\,, \quad \mathbf = \sigma_3 \sigma_2\,, \quad \mathbf = \sigma_1 \sigma_3\,, and it is straightforward to confirm that this preserves the Hamilton relations :\mathbf^2 = \mathbf^2 = \mathbf^2 = \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 \operatorname_(\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 rotors, 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 geometr ...
. 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 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 law ...
. In such spaces rotors can be used to efficiently represent
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
s, and to interpret formulas involving the gamma matrices. For further detail about the geometrical uses of Clifford algebras, see Geometric algebra.


Brauer group

The quaternions are "essentially" the only (non-trivial) central simple algebra (CSA) over the real numbers, in the sense that every CSA over the real numbers is Brauer equivalent to either the real numbers or the quaternions. Explicitly, the
Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...
of the real numbers consists of two classes, represented by the real numbers and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another. By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the real numbers. CSAs – finite dimensional rings over a field, which are simple algebras (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 Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler ...
* Dual quaternion * Dual-complex number * Exterior algebra * Hurwitz quaternion order *
Hyperbolic quaternion In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form :q = a + bi + cj + dk, \quad a,b,c,d \in \mathbb \! where the squares of i, j, and k are +1 and distinct eleme ...
*
Lénárt sphere A Lénárt sphere is a educational manipulative and writing surface for exploring spherical geometry, invented by Hungarian István Lénárt as a modern replacement for a spherical blackboard. It can be used for visualizing the geometry of poi ...
* Pauli matrices *
Quaternionic manifold In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part ...
* Quaternionic matrix * Quaternionic polytope *
Quaternionic projective space In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions \mathbb. Quaternionic projective space of dimension ''n'' ...
* 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 g ...
* Split-quaternion * Tesseract


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 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 Charles Jasper Joly (27 June 1864 – 4 January 1906) was an Irish mathematician and astronomer who became Royal Astronomer of Ireland.Obituary, New York Times, 5 January 1906 Life He was born at St Catherine's Rectory, Hop Hill, Tullamor ...
; 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 t ...
, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped PostScript 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 th ...
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