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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space^[1] or equivalently as the quotient of two vectors.^[2] Quaternions can also be represented as the sum of a scalar and a vector.

Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and epipolar geometry, although they have been superseded in many applications by vectors and matrices.

In modern language, quaternions form a four-dimensional normed division algebra over the real numbers. In fact, the quaternions were the first noncommutative division algebra to be discovered.^[3] The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by (Unicode ℍ). It can also be given by the Clifford algebra classifications Cℓ_0,2(R) = Cℓ⁰_3,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers.

Contents 1 History 2 Definition 2.1 Multiplication of basis elements 2.2 Hamilton product 2.3 Ordered list form 2.4 Remarks 2.4.1 Scalar and vector parts 2.4.2 Noncommutative 2.4.3 Historical impact on physics 3 Conjugation, the norm, and division 4 Algebraic properties 5 Quaternions and the geometry of R3 6 Matrix representations 7 Quaternions as pairs of complex numbers 8 Square roots of −1 8.1 H as a union of complex planes 8.2 Commutative subrings 9 Functions of a quaternion variable 9.1 Exponentiation, logarithm 10 Three-dimensional and four-dimensional rotation groups 11 Generalizations 12 Quaternions as the even part of Cℓ3,0(R) 13 Quotes 14 See also 15 Notes 16 External articles and resources 16.1 Books and publications 16.2 Links and monographs 16.3 Software

[ History

Quaternionic algebra was introduced by Irish mathematician Sir William Rowan Hamilton in 1843^[4]. However the use of quaternions to describe rigid body orientations has a much older history dating back to at least 1776 in the work^[5] of L. Euler.

Hamilton knew that the complex numbers could be viewed as points in a plane, and he was looking for a way to do the same for points in space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years Hamilton had known how to add and multiply triples of numbers. But he had been stuck on the problem of division: He did not know how to take the quotient of two points in space.

The breakthrough 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. While walking along the tow path of the Royal Canal with his wife, the concept behind quaternions was taking shape in his mind. Hamilton could not resist the impulse to carve the formulae for the quaternions

i² = j² = k² = ijk = − 1.

into the stone of Brougham Bridge as he passed by it.

On the following day, he 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 subsequently published in the London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, vol. xxv (1844), pp 489-95. On the letter, Hamilton states,

And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the remainder of his life to studying and teaching them. He founded a school of "quaternionists" and popularized them in several books. The last and longest, Elements of Quaternions, had 800 pages and was published shortly after his death.

After Hamilton's death, his pupil 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, exclusively devoted to the study of quaternions.

From the mid 1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs and Oliver Heaviside. Vector analysis described the same phenomena as quaternions, so it borrowed ideas and terms liberally from the classical quaternion literature. 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 because Hamilton's original definitions are unfamiliar and his writing style is prolix and opaque.

However, quaternions have had a revival in the late 20th century, primarily due to their utility in describing spatial rotations. Representations of rotations by quaternions are more compact and faster to compute than representations by matrices. For this reason, quaternions are used in computer graphics,^[6] computer vision, robotics, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics computer simulation and orbital mechanics. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relation to quadratic forms.

Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth has organized a pilgrimage, where scientists (including physicists Murray Gell-Mann in 2002, Steven Weinberg in 2005, and mathematician Andrew Wiles in 2003) take a walk from Dunsink Observatory to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.

[ Definition

As a set, the quaternions H are equal to R⁴, a four-dimensional vector space over the real numbers. H has three operations: Addition, scalar multiplication, and quaternion multiplication. The sum of two elements of H is defined to be their sum as elements of R⁴. Similarly the product of an element of H by a real number is defined to be the same as the product in R⁴. To define multiplication in H requires a choice of basis for R⁴. The elements of this basis are customarily denoted as 1, i, j, and k. Every element of H can be uniquely written as a linear combination of these basis elements, that is, as a1 + bi + cj + dk, where a, b, c, and d are real numbers. The basis element 1 will be the identity element of H, meaning that multiplication by 1 does nothing, and for this reason, elements of H are usually written a + bi + cj + dk, suppressing the basis element 1. Given this basis, quaternion multiplication is defined by first defining the products of basis elements and then defining all other products using the distributive law.

[ Multiplication of basis elements

The equations

where i, j, and k are basis elements of H, determine all the possible products of i, j, and k. For example, since

right-multiplying both sides by k gives

All the other possible products can be determined by similar methods, resulting in

from which we have the following table:

Quaternion multiplication

×	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

[ Hamilton product

For two elements a₁ + b₁i + c₁j + d₁k and a₂ + b₂i + c₂j + d₂k, their 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:

a₁a₂ + a₁b₂i + a₁c₂j + a₁d₂k

+ b₁a₂i + b₁b₂i² + b₁c₂ij + b₁d₂ik

+ c₁a₂j + c₁b₂ji + c₁c₂j² + c₁d₂jk

+ d₁a₂k + d₁b₂ki + d₁c₂kj + d₁d₂k².

Now the basis elements can be multiplied using the rules given above to get^[7]:

a₁a₂ − b₁b₂ − c₁c₂ − d₁d₂

+ (a₁b₂ + b₁a₂ + c₁d₂ − d₁c₂)i

+ (a₁c₂ − b₁d₂ + c₁a₂ + d₁b₂)j

+ (a₁d₂ + b₁c₂ − c₁b₂ + d₁a₂)k.

[ Ordered list form

Using the basis 1, i, j, k of H makes it possible to write H as a set of quadruples:

Then the basis elements are:

1 = (1,0,0,0),

i = (0,1,0,0),

j = (0,0,1,0),

k = (0,0,0,1),

and the formulas for addition and multiplication are:

(a₁,b₁,c₁,d₁) + (a₂,b₂,c₂,d₂) =

(a₁ + a₂,b₁ + b₂,c₁ + c₂,d₁ + d₂).

(a₁,b₁,c₁,d₁)(a₂,b₂,c₂,d₂) =

(a₁a₂ − b₁b₂ − c₁c₂ − d₁d₂,

a₁b₂ + b₁a₂ + c₁d₂ − d₁c₂,

a₁c₂ − b₁d₂ + c₁a₂ + d₁b₂,

a₁d₂ + b₁c₂ − c₁b₂ + d₁a₂).

[ Remarks

[ Scalar and vector parts

A number of the form a + 0i + 0j + 0k, where a is a real number, is called real, and a number of the form 0 + bi + cj + dk, where b, c, and d are real numbers, is called pure imaginary. If a + bi + cj + dk is any quaternion, then a is called its scalar part and bi + cj + dk is called its vector part. The scalar part of a quaternion is always real, and the vector part is always pure imaginary. Even though every quaternion is a vector in a four-dimensional vector space, it is common to define a vector to mean a pure imaginary quaternion. With this convention, a vector is the same as an element of the vector space R³. This identification leads to many applications^[citation needed].

Hamilton called pure imaginary quaternions right quaternions^[8][9] and real numbers (considered as quaternions with zero vector part) scalar quaternions.

[ Noncommutative

Unlike multiplication of real or complex numbers, multiplication of quaternions is not commutative: For example, ij = k, while ji = − k. The noncommutativity of multiplication has some unexpected consequences, among them that polynomial equations over the quaternions can have more distinct solutions than the degree of the polynomial. The equation z² + 1 = 0, for instance, has infinitely many quaternion solutions z = bi + cj + dk with b² + c² + d² = 1, so that these solutions form a two-dimensional sphere centered on zero in the three-dimensional pure imaginary subspace of quaternions. This sphere intersects the complex plane at the two poles i and  − i.

[ Historical impact on physics

In 1984 the European Journal of Physics (5:25–32) published P.R. Girard’s essay "The quaternion group in modern physics". It "shows how various physical covariance groups: SO(3), the Lorentz group, the general relativity group, the Clifford algebra SU(2) and the conformal group can be easily related to the quaternion group". Girard begins by discussing group representations and by representing some space groups of crystallography. He proceeds to kinematics of rigid body motion. Next he uses complex quaternions (biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He cites five authors, beginning with Ludwik Silberstein that use a potential function of one quaternion variable to express Maxwell’s equations in a single differential equation. Concerning general relativity, he expresses the Runge-Lenz vector. He mentions the Clifford biquaternions (split-biquaternions) as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard describes conformal maps on spacetime. Among the fifty references he includes Alexander Macfarlane and his Bulletin of the Quaternion Society.

[ Conjugation, the norm, and division

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 +bi +cj + dk be a quaternion. The conjugate of q is the quaternion a − bi − cj − dk. It is denoted by q^*, ^[10], q^t, or . Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq)^* = q^*p^*, not p^*q^*.

Unlike the situation in the complex plane, the conjugation of a quaternion can be expressed entirely with multiplication and addition as shown in the article functions of a quaternion variable.

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 its norm and is denoted ||q||. (Hamilton called this quantity the tensor of q, but this conflicts with modern usage. See tensor.) It has the formula

This is always a non-negative real number, and it is same as the Euclidean norm on H considered as the vector space R⁴. 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. 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 into a metric space. Addition and multiplication are continuous in the metric topology.

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion q by its norm produces a unit quaternion Uq called the versor of q:

Every quaternion has a polar decomposition q = ||q|| Uq.

Using conjugation and the norm makes it possible to define the reciprocal of a quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of q and (in either order) is 1. So the reciprocal of q is defined to be

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

[ Algebraic properties

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.) The quaternions have a multiplication that is associative and that distributes over vector addition, but which 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 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, ±j, ±k} forms a group under multiplication. This group is called the quaternion group and is denoted Q₈^[11]. The real group ring of Q₈ is a ring RQ₈ 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 RQ₈ by the ideal generated by the elements 1 + (−1), i + (−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.

[ Quaternions and the geometry of R³

Because the vector part of a quaternion is a vector in R³, the geometry of R³ is reflected in the algebraic structure of the quaternions. Many operations on vectors can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. For instance, this is true in electrodynamics and 3D computer graphics.

For the remainder of this section, i, j, and k will denote both imaginary^[12] basis vectors of H and a basis for R³. Notice that replacing i by −i, j by −j, and k by −k sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse.

Choose two imaginary quaternions p = b₁i + c₁j + d₁k and q = b₂i + c₂j + d₂k. Their dot product is

This is equal to the scalar parts of p^*q, qp^*, pq^*, and q^*p. (Note that the vector parts of these four products are different.) It also has the formulas

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

In general, let p and q be quaternions (possibly non-imaginary), and write

where p_s and q_s are the scalar parts of p and q and and are the vector parts of p and q. Then we have the formula

This shows that noncommutative quaternion multiplication comes from the multiplication of pure imaginary quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear.

[ 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 the terminology of abstract algebra, these are injective homomorphisms from H to the matrix rings M₂(C) and M₄(R), respectively.

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

This representation has the following properties:

• Complex numbers (c = d = 0) correspond to diagonal matrices.
• The norm of a quaternion (the square root of a product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix^[13].
• The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
• Restricted to unit quaternions, this representation provides an isomorphism between S³ and SU(2). The latter group is important for describing spin in quantum mechanics; see Pauli matrices.

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

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. Complex numbers are block diagonal matrices with two 2×2 blocks.

[ 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 C² be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and j. A vector in C² 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 k in place of the product ij leads to the same rules for multiplication as the usual quaternions. Therefore the above vector of complex numbers corresponds to the quaternion a + bi + cj + dk. If we write the elements of C² as ordered pairs and quaternions as quadruples, then the correspondence is

[ Square roots of −1

In the complex numbers, 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 includes every point on the surface of the unit sphere in 3-space. 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

a² − b² − c² − d² = − 1,

2ab = 0,

2ac = 0,

2ad = 0.

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 (that is, pure imaginary) with norm 1. By definition, the set of all such vectors forms the unit sphere.

The identification of the square roots of minus one in H was given by Hamilton^[14] 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. Porteus's book Clifford Algebras and the Classical Groups (Cambridge, 1995) in proposition 8.13 on page 60. Also in Conway (2003) On Quaternions and Octonions we read on page 40: "any imaginary unit may be called i, and perpendicular one j, and their product k", another statement of the sphere.

[ H as a union of complex planes

Each square root 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, this is an injective ring homomorphism from C to H.

Every non-real quaternion lies in a unique copy of 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:

(Note that this is not the same as .) The versor of the vector part of q, , is a pure imaginary unit quaternion, so its square is −1. Therefore it determines a copy of the complex numbers by the function

Under this function, q is the image of the complex number . Thus H is the union of complex planes intersecting in a common real line, where the union is taken over the sphere of square roots of minus one.

[ Commutative subrings

The relationship of quaternions to each other within the complex subplanes of H can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions p and q commute (p q = q p) 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 coquaternions and 2 × 2 real matrices.

[ Functions of a quaternion variable

Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. However the complications of the quaternion variable still challenge investigators. Consider for example the function

which expresses quaternion conjugation.

[ Exponentiation, logarithm

The exponential and logarithm of a quaternion are relatively inexpensive to compute, particularly compared with the cost of those operations for other charts on SO(3) such as rotation matrices, which require computing thee matrix exponential and matrix logarithm respectively^{[citation needed]}.

Given a quaternion,

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