In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843^[3][4] and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space^[5] or equivalently as the quotient of two vectors.^[6]

Quaternions are generally represented in the form:

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

Quaternions find uses in both pure and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis.^[7] In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

The unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).

Quaternions were introduced by Hamilton in 1843.^[9] 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.^[10][11] Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900.^[12][13]

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. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space.

The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions,

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 a science magazine;^[14] 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' algebraic 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,^[15] was 800 pages long; it was edited by his son and published shortly after his death.

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

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

However, quaternions have had a revival since the late 20th 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,^[16][17] computer vision, robotics,^[18] control theory, signal processing, attitude control, physics, bioinformatics,^[19][20] molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms.^[21]

P.R.

Girard's 1984 essay The quaternion group and modern physics ^[22] discusses some roles of quaternions in physics. The essay

in modern algebra. Girard began by discussing group representations and by representing some space groups of crystallography. He proceeded to kinematics 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, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions (split-biquaternions) as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. 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.^[23]

The finding of 1924 that in quantum mechanics the spin of an electron and other matter particles (known as spinors) can be described using quaternions furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the “Plate trick”).^[24][25] As of 2018, their use has not overtaken rotation groups.^[1]

A quaternion is an expression of the form

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

and the componentwise scalar multiplication

A multiplicative group structure, called the Hamilton product, denoted by juxtaposition, can be defined on the quaternions in the following way:

Thus the quaternions form a division algebra.

The basis elements i, j, and k commute with the real quaternion 1, that is

The other products of basis elements are defined by

and

These multiplication formulas are equivalent to

In fact, the equality ijk = –1 results from

The converse implication results from manipulations similar to the following.

By right-multiplying both sides of −1 = ijk by –k, one gets

All other products can be determined by similar methods.

The center of a noncommutative ring is the subring of elements c such that cx = xc for every x. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if q = a + bi + cj + dk belongs to the center, then

and c = d = 0. A similar computation with j instead of i shows that one has also b = 0. Thus q = a is a real quaternion.

The quaternion form a division algebra. This means that the noncommutativity of multiplication is the only property that makes quaternions different from a field. This noncommutativity has some unexpected consequences, among them that a polynomial equation over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation z2 + 1 = 0, has infinitely many quaternion solutions, which are the quaternions z = bi + cj + dk such that b2 + c2 + d2 = 1. Thus these "roots of –1" form a unit sphere in the three-dimensional space of vector quaternions.

For two elements a1 + b1i + c1j + d1k and a2 + b2i + c2j + d2k, their product, called the Hamilton product (a1 + b1i + c1j + d1k) (a2 + b2i + c2j + d2k), 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:

Now the basis elements can be multiplied using the rules given above to get:^[9]

The product of two rotation quaternions^[26] will be equivalent to the rotation a2 + b2i + c2j + d2k followed by the rotation a1 + b1i + c1j + d1k.

A quaternion of the form a + 0i + 0j + 0k, where a is a real number, is called scalar, and a quaternion of the form 0 + bi + cj + dk, where b, c, and d are real numbers, and at least one of b, c or d is nonzero, is called a vector quaternion. If a + bi + cj + dk is any quaternion, then a is called its scalar part and bi + cj + dk is called its vector part. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space R 3.^[2]

Hamilton also called vector quaternions right quaternions ^[27][28] and real numbers (considered as quaternions with zero vector part) scalar quaternions.

If a quaternion is divided up into a scalar part and a vector part, i.e.

then the formulas for addition and multiplication are:

where "·" is the dot product and "×" is the cross product.

The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:

Conjugation can be used to extract the scalar and vector parts of a quaternion.

The scalar part of p is (p + p∗) / 2, and the vector part of p is (p − p∗) / 2.

The square root of the product of a quaternion with its conjugate is called itsnorm and is denoted ||qtensor* of q g of "tensor"). In formula, this is expressed as follows:

This is always a non-negative real number, and it is the same as the Euclidean norm on H considered as the vector space R 4. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then

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

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

(where i denotes the usual imaginary unit) and hence from the multiplicative property of determinants of square matrices.

This norm makes it possible to define the distance d (p, q) between p and q as the norm of their difference:

This makes H a metric space. Addition and multiplication are continuous in the metric topology. Indeed, for any scalar, positive a it holds

Continuity follows from taking a to zero in the limit. Continuity for multiplication holds similarly.

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion q by its norm produces a unit quaternion U qcalled the versor

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

The set H of all quaternions is a vector space over the real numbers with dimension 4. (In comparison, the real numbers have dimension 1, the complex numbers have dimension 2, and the octonions have dimension 8.) Multiplication of quaternions is associative and distributes over vector addition, but it is not commutative. Therefore, the quaternions H are a non-commutative associative algebra over the real numbers. Even though H contains copies of the complex numbers, it is not an associative algebra over the complex numbers.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field 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: R, C, and H. The norm makes the quaternions into a normed algebra, and normed division algebras over the reals are also very rare: Hurwitz's theorem says that there are only four: R, C, H, and O (the octonions). The quaternions are also an example of a composition algebra and of a unital Banach algebra.

Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±i, ±, ±k*} forms a group under multiplication. This non-Abelian Group is called the quaternion group and is denoted Q8.^[29] The real group ring of Q8 is a ring R [Q8] which is also an eight-dimensional vector space over R. It has one basis vector for each element of Q8. The quaternions are the quotient ring of R [Q8] by the ideal generated by the elements 1 + (−1), i+ (−), j*+ (−j), and* k*+ (−k). Here the first term in each of the differences is one of the basis elements 1,* i*,* j*, and* k*, and the second term is one of basis elements −1, −i, −j, and −k, not the additive inverses of 1,* i*,* j*, and* k*.

The vector part of a quaternion can be interpreted as a coordinate vector in R 3; therefore, the algebraic operations of the quaternions reflect the geometry of 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.^[16]

For the remainder of this section, i, j, and kl denote both the three imaginary^[30] basis vectors of H and a basis for R 3. Replacing i by −*, jby −j, and* kby −ksends 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 p = b1i + c1j + d1k and q = b2i + c2j + d2k their dot product, by analogy to vectors in R 3, is

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

This is equal to the scalar parts of the products pq∗, qp∗, p∗, and q∗p*. Note that their vector parts are different.

The cross product of p and q relative to the orientation determined by the ordered basis i, j, and k is

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product pq (as quaternions), as well as the vector part of −q∗p∗. It also has the formula

For the commutator, [p,* q*] =* pq− qp*, of two vector quaternions one obtains

In general, let p and q be quaternions and write

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^[31] showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in Elliptic geometry.

Unit quaternions can be identified with rotations in R 3 and were called versors by Hamilton.^[31] Also see Quaternions and spatial rotation for more information about modeling three-dimensional rotations using quaternions.

See Andrew J. Hanson.^[32] for visualization of quaternions.

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, these are injective homomorphisms from H to the matrix rings M(2, C) and M(4, R), respectively.

Using 2 × 2 complex matrices, the quaternion a + bi + cj + dk can be represented as

This representation has the following properties:

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

However, the representation of quaternions in M(4,ℝ) is not unique.

For example, the same quaternion can also be represented as

In fact, there exist 48 distinct representations of this form.

More precisely, there are 48 sets of quadruples of matrices such that a function sending 1, i, j, and k to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices.^[34] In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. As with the 2 × 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 c = d = 0.

Each 4x4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions.

For example, the last matrix representation given above corresponds to the multiplication table

Constraining any such multiplication table to have the identity in the first row and column and for the signs of the row headers to be opposite to those of the column headers, then there are 3 possible choices for the second column (ignoring sign), 2 possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6 possibilities.

Then, the second column can be chosen to be either positive or negative, the third column can be chosen to be positive or negative, and the fourth column can be chosen to be positive or negative, giving 8 possibilities for the sign.

Multiplying the possibilities for the letter positions and for their signs yields 48.

Then replacing 1 with a, i with b, j with c, and k with d and removing the row and column headers yields a matrix representation of a + bi + cj + dk.

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

Quaternions can be represented as pairs of complex numbers.

From this perspective, quaternions are the result of applying the Cayley–Dickson construction to the complex numbers. This is a generalization of the construction of the complex numbers as pairs of real numbers.

Let C 2 be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and j. A vector in C 2 can be written in terms of the basis elements 1 and j as

If we define j2 = −1 and ij = −ji, then we can multiply two vectors using the distributive law. Writing* kin place of the product ijleads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion a*+* bi*+* cj*+* dk*. If we write the elements of* C 2 as ordered pairs and quaternions as quadruples, then the correspondence is

In the complex numbers, C, there are just two numbers, i and −i, whose square is −1. In* H there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in R 3. To see this, let q = a + bi + cj + dk be a quaternion, and assume that its square is −1. In terms of a, b, c, and d, this means

To satisfy the last three equations, either a = 0 or b, c, and d are all 0. The latter is impossible because a is a real number and the first equation would imply that a2 = −1. Therefore, a = 0 and b2 + c2 + d2 = 1. In other words: a quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere.

Only negative real quaternions have infinitely many square roots.

All others have just two (or one in the case of 0).

The identification of the square roots of minus one in H was given by Hamilton^[35] but was frequently omitted in other texts. By 1971 the sphere was included by Sam Perlis in his three-page exposition included in Historical Topics in Algebra (page 39) published by the National Council of Teachers of Mathematics. More recently, the sphere of square roots of minus one is described in Ian R. Porteous's book Clifford Algebras and the Classical Groups (Cambridge, 1995) in proposition 8.13 on page 60.

Each pair of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions.

If q2 = −1, then the copy is determined by the function

In the language of abstract algebra, each is an injective ring homomorphism from C to H. The images of the embeddings corresponding to q and −q are identical.

Every non-real quaternion determines a planar subspace in H that is isomorphic to C: Write q as the sum of its scalar part and its vector part:

Decompose the vector part further as the product of its norm and its versor:

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

Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the Mandelbrot set and Julia sets into 4 dimensional space.^[36]

Given a quaternion,

the exponential is computed as

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

and

The geodesic distance dg (p, q) between unit quaternions p and q is defined as:

and amounts to the absolute value of half the angle subtended by p and q along a great arc of the S3 sphere. This angle can also be computed from the quaternion dot product without the logarithm as:

The word "conjugation", besides the meaning given above, can also mean taking an element a to r a r−1 where r 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 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 cos(φ) is a rotation by an angle 2φ, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:

The set of all unit quaternions (versors) forms a 3-sphere S3 and a group (a Lie group) under multiplication, double covering the group SO(3, R) of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence. See the 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 SU(2), the group of complex unitary 2×2 matrices of determinant 1.

Let A be the set of quaternions of the form a + bi + cj + dk where a, b, c, and d are either all integers or all rational numbers with odd numerator and denominator 2. The set A 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 {3,4,3}. They correspond to the double cover of the rotational symmetry group of the regular tetrahedron. Similarly, the vertices of a regular 600-cell with Schläfli symbol {3,3,5} 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.

The Quaternions can be generalized into further algebras called quaternion algebras. Take F to be any field with characteristic different from 2, and a and b to be elements of F; a four-dimensional unitary associative algebra can be defined over F with basis 1, i, j, and ij, where i2 = a, j2 = b and ij = −ji (so* (ij)2 = −ab*).

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

The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part Cℓ+3,0(R) of the Clifford algebra Cℓ3,0(R). This is an associative multivector algebra built up from fundamental basis elements σ1, σ2, σ3 using the product rules

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

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

corresponds to a rotation of 180° in the plane containing σ1 and σ2.

This is very similar to the corresponding quaternion formula,

In fact, the two are identical, if we make the identification

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

In this picture, 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 σ1 and σ2, there is only one bivector basis element σ1σ2, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ1σ2, σ2σ3, σ3σ1, so three imaginaries.

This reasoning extends further.

In the Clifford algebra Cℓ4,0(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. But it is only in 3D that the number of basis bivectors equals the number of basis vectors, and each bivector can be identified as a pseudovector.

There are several advantages for placing quaternions in this wider setting:^[40]

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

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

CSAs – rings over a field, which are simple 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 reals (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial field extension of the reals.