Quaternion

The quaternions (singular: the - but sometimes: the - quaternion, from Latin quaternio " Tetrad " ) are a range of numbers, which extends the speed range of real numbers - similar to the complex numbers and beyond. Described ( and systematically developed further ) they were in 1843 by Sir William Rowan Hamilton; they are also called Hamiltonian quaternions or Hamiltonian numbers. Olinde Rodrigues discovered in 1840 independently by Hamilton. Nevertheless, the amount of quaternions is often denoted by.

Quaternions allow in many cases an elegant mathematical description of three-dimensional Euclidean space and other spaces, particularly in the context of rotations. Therefore, they are used among other things in computation and representation algorithms for simulations. They are also available as an independent mathematical object of interest and serve as for example in the proof of the four -square theorem.

6.1 exponential, logarithm
6.2 continuations of complex functions
6.3 Analysis

7.1 Minkowski scalar product
7.2 Vector Analysis
7.3 rotations in the three-dimensional space 7.3.1 with respect to orthogonal matrices
7.3.2 with respect to Euler angles
7.3.3 Universal superposition of the rotation group; spin group

Construction

The quaternions arise from the real numbers by adding ( adjunction ) three new numbers, which based on the complex - imaginary unit names, and be given. Thus, a four-dimensional number system results ( mathematically: a vector space ) with a real part, which consists of a real component and an imaginary part of three components, also called the vector part.

Every quaternion can be uniquely in the form

With real numbers, write. So that the elements are a base, on the standard basis of quaternions. The addition is componentwise and is inherited from the vector space. Multiplicative be the new figures, according to the Hamilton - rules

Linked. The scalar multiplication that is also inherited by the vector space and in which the scalars be regarded as interchangeable with each item, together with the addition and the Hamilton rules make it possible to increase the multiplication of the base to all quaternions. Since so every scalar is as embedded in, can be understood as a body of.

The so- defined multiplication is associative and satisfies also the distributive, ie makes the quaternions a ring. However, it is not commutative, ie for two quaternions and the two products and usually different ( see below). The center of, ie the set of elements that commute with all elements is, exactly.

The quaternions form a skew field (division ring ), as there are at any quaternion inverse quaternion with

Because of the lack commutativity are notations with fraction line, such as avoided.

In sum, the quaternions are a four-dimensional division algebra over - and up to isomorphism the only one. Historically, the quaternions were the first example of a division ring, the second link is not commutative.

Spelling

In the further text the following conventions are used:

Is a quaternion, then their real components are denoted by and these are assigned as follows

Occasionally, a vector notation is needed. The components are, for example, combined to form a three -dimensional vector, so that one can identify with the 4-dimensional vector.

Analogous arrangements should apply to other characters like etc..

In older literature, quaternions were referred to fracture with large letters and the imaginary units as unit vectors with small fracture in, for example, like this:

With.

Basic arithmetic

The construction of the quaternions is the analog of the complex numbers, however, is not only a new number added, but three, with, and be referred to.

The linear combinations

Stretched across the base with real components on the 4-dimensional vector space of quaternions. ( The base element, which also represents the neutral element of multiplication and which embeds the real numbers is injective, usually omitted in the linear combination. ) The addition and subtraction is done component-wise as in any vector space. The vector space and the scalar multiplication is applied, that is, the left and right multiplication by a real number, which is multiplied to each component distributive.

This scalar multiplication is a limitation of the Hamilton - multiplication, which is defined on all of. The Hamilton - multiplication of the basis elements to each other or anything comprehensively within the amount

Done according to the Hamilton - rules

The full board for a link constitute together with the commutativity of the other with each item that turns out to be associative and makes them a group - the quaternion group.

Assuming the rule ( and the group axioms ) are the other two and in which is expressed, inter alia, the cyclical and anti - cyclical behavior of the three non - real quaternion units, equivalent to the short form

Use these replacement rules, the associative law and ( left as right ) distributive goes on to quite the multiplication. The can be treated as anti- commuting variables. If products of two of them on, we may replace it by the Hamilton rules.

The elaborated formulas for the two links of two quaternions

Read

This required for a ring 2 threads are defined. It is easily recalculated that all ring axioms are satisfied.

The additive inverse is (as in any vector space ) the product with the scalar -1. The subtraction is the addition of this inverse.

The Division required for a division ring needs to be replaced because of the lack commutativity by a multiplication by the ( multiplicative ) inverse (see inverse and division).

Basic concepts

Scalar and vector part

Because of the special position of the component of a quaternion

One calls it - like with the complex numbers - as the real part or scalar

While the components along the imaginary or vector part

Form. Often we identify the vector part also with the vector.

Conjugation

For each quaternion

Is defined as the conjugate quaternion

Because here, unlike in the complex, the imaginary part remains linked to its unit vectors and the real part is clearly embedded as a real number in the quaternions, the simple relations arising

And

From which immediately

And

Calculates.

Is a quaternion equal to its conjugate, it is real, that is, the vector part is zero. Is a quaternion equal to the negative of its conjugate, so it is a pure quaternion, that is, the scalar part is zero.

Other important properties of conjugation are:

The conjugation is an involution.

And for real numbers

The conjugation is an involutive Antiautomorphismus.

The conjugation can be represented " with arithmetic means."

Scalar product

The scalar product of two quaternions considered as vectors, is defined by:

It is a positive definite symmetric bilinear form on the standard and amount can be defined and can be determined with the angle and orthogonality.

Furthermore, one can thus isolate the individual components of a quaternion:

In the following, the dot product, both the 4 - as the 3-dimensional - as in physics usual - listed to the center point.

Cross product

The cross product of two quaternions is the cross product ( vector product ) of their vector parts and up to a factor of 2 their commutator. If and, then

Quaternion as dot and cross product

If we identify quaternions

With pairs of a scalar and a vector

With,

This is how the multiplication using the ( three-dimensional ) inner product and cross product describe:

Two quaternions are accordingly if and only commute if their cross product is 0, ie, when their vector parts as real vectors are linearly dependent on each other (see a embedding of complex numbers ).

Standard and amount

The scalar product of a quaternion with itself, which is equal to the Quaternionenprodukt with the conjugate is called standard:

In particular, this value is real and non-negative.

The square root thereof

Is amount or length of the quaternion called and agrees with amount or Euclidean length of the vector. It fulfills the important property

The multiplicativity of the amount. With the amount the quaternions to a real Banach algebra are.

Inverse and Division

In a non-commutative multiplication must equations

Differ. If the inverse exists, then

Respectively solutions, which only coincide if and commute, especially if the divisor is real. In such a case, the writing can be used - in general it would not be clear divisions.

In addition, if exists, the formula is

Because

Is the norm

Real and positive. The quaternion

Then satisfies the conditions of the right -

And the left- inverse

And can therefore be referred to as simply the inverse of.

Pure quaternion

A quaternion, whose vector part is 0 is identified with the corresponding scalar their real number.

A quaternion whose real part is 0, it is called a pure quaternion (also: pure imaginary or vectorial). Pure quaternions can be characterized whose square is real and not positive as those quaternions. For the set of pure quaternions to write

She is a three-dimensional real vector space with basis.

For pure quaternion multiplication takes on a particularly simple form:

Unit quaternion

A unit quaternion (also: normalized quaternion, quaternion of length 1 ) is a quaternion, whose amount is equal to 1. For them (analogous to the complex numbers )

For any quaternion

A unit quaternion, which are sometimes referred to as the sign of.

The product of two Einheitsquaternionen and the inverse of a unit quaternion are Einheitsquaternionen again. So the Einheitsquaternionen form a group.

Geometric can be considered the unit 3- sphere in four-dimensional Euclidean space and thus as a Lie group interpret the amount of Einheitsquaternionen, with the space of pure quaternions as the associated Lie algebra. The representation of a complex matrix illustrates the reversible one correspondence with the Einheitsquaternionen the special unitary group.

The only real Einheitsquaternionen are. They also make up the center of.

Pure unit quaternion

Einheitsquaternionen that are also pure quaternions, can be characterized as those quaternions give rise to their squares:

You lie on the boundary and in the equatorial hyperplane of the 3 - sphere and make the unit 2-sphere of three-dimensional space from.

Embedding of complex numbers

Each quaternion with square defines a Einbettungsisomorphismus of complex numbers in the quaternions

With and as an imaginary unit of complex numbers. The image sets and the corresponding embeddings are identical.

Each such quaternion may be called a perpendicular to it and their product. Each non-real quaternion is in exactly one such embedding. Two quaternions are then exactly interchangeable, if there is a common embedding.

Two different images have the real axis to the average.

In this sense, the quaternions are an association complex levels.

Polar representation

Each unit quaternion can uniquely in the form of

Are shown.

With the generalized exponential function can also write this because as

With the pure quaternion. So if you want a pure quaternion exponentiieren, it is and to form the pure unit quaternion, and it results in the unit quaternion

The case can be supplemented continuously. Thus, the exponential map is surjective - and bijective with restriction on, for it is for infinitely many with. It is continuous, because of the non- commutativity of multiplication but not a homomorphism.

General can be any non-real quaternion uniquely in the form

. Write Is by definition, so that points in the same direction as the vector portion.

Every non -negative real quaternion writes clearly as

With a pure quaternion with.

These representations are the polar form of complex numbers

Analog ( with the imaginary unit ). For the functional equation

However, must commute.

Function theory

Exponential, logarithm

The exponential of a non- real quaternion is:

With.

The (natural ) logarithm of a non- real quaternion is:

For non- real they are inverse functions of each other

And, optionally,

For non- real, commuting with the functional equations are

And

For the latter with a sufficiently small imaginary part.

Continuations of complex functions

As can be understood as a union of embeddings of complex levels ( see section # embedding of complex numbers ), you can try to lift functions, using these Einbettungsisomorphismen from the quaternionic complexes. This is to request that the functions thus obtained provide with the overlaps of the domains the same result, so that the combined function on the union of virtue than can be formed in a well defined manner.

Let be a complex-valued function of a complex variable with real and real. Embeddability: if and only embeddable in the quaternions, if an even and an odd function of is.

Proof: If any non-real quaternion, then with a pure and normalized quaternion. And are further, both are real. Both as a Einbettungsisomorphismus for the image. In the former case is the archetype of, in the second case we have because of the archetype; respectively as the imaginary unit of. The archetypes are different, but the image that will serve as an argument in the function is to be formed, both times. The " lifts" is called by the embedding of the function values

And

Completed ( see diagram). Now, by assumption,

So that

Results and does not depend on the choice of Einbettungsisomorphismus.

The condition is also necessary. Because vice versa: the function to be embedded in the quaternions, so we have to each a suitable pure unit quaternion with and

Now, the conjugate embedding the same image as, therefore, the same definition as quantity. The function value

So must be the same for all previous. ■

The embedded function is true on all subsets consistent with, and can therefore be viewed as a continuation of and if confusion is to be feared, and the function name is retained.

Is an embeddable function so is because of the oddness of the second variable, ie and for. Thus it follows from the embeddability, that the restriction on Real is real. This class of complex functions include standard and amount, but also all Laurent series with real coefficients, the exponential and logarithm.

For example, does not belong to this class, the function in which not is odd in. Nevertheless, it is a well-defined function and a continuation of, because there is agreement on the subset.

Analysis

It is more difficult to develop a general quaternionic analysis with differential and / or integral calculus is. A problem immediately jumps to the eye: the concept of the difference quotient, which is so successful in the real as complex analysis, needs to be defined because of the non- commutativity as a left and right version. If you put then on the same strict standards as for the complex differentiability, then it turns out that there are just linear functions, namely left and right, are differentiable. Always define but can be a directional derivative and the Gâteaux differential.

Based on the Cauchy- Riemann equations and the set of Morera Regularitätsbegriff following was found: A quaternionic function is regular at the point when their integral vanishes over every sufficiently small enclosing hypersurface.

Description of other constructs using quaternions

Minkowski scalar product

The Minkowski scalar product of two quaternions, considered as vectors in Minkowski space, the scalar part of:

Vector Analysis

The following vectors in three-dimensional space with pure quaternions, so the usual coordinates with the components to be identified. Defines the nabla operator is used ( as Hamilton) as

And applies it to a scalar function as a (formal ) scalar multiplication, we obtain the gradient

Applying on a vector field

As a ( formal ) scalar product yields the divergence

The application to a vector field as a ( formal ) cross product gives the rotation

The application to a vector field as a ( formal ) product of two pure quaternions results

With the scalar and vector part of the quaternion as.

Two-fold application to a function yields the Laplace operator

That is, acts like a Dirac operator as a (formal ) " square root " of the (negative) Laplacian.

Turns in three dimensional space

Einheitsquaternionen can be used for an elegant description of rotations in three-dimensional space: For a fixed unit quaternion is the mapping

A rotation. ( Here, as below, the speech that leave to set the origin, that is, is only of rotations whose axis of rotation passing through the origin. )

The polar representation, the unit quaternion by an angle and a pure unit quaternion clearly constitutes a

Then a rotation of the axis of rotation.

Define for each unit quaternion and the same rotation; particular match and both the identity map ( rotation with rotation angle 0). In contrast to the description of rotations by orthogonal matrices so there is no 1:1 correspondence to each rotation there are exactly two Einheitsquaternionen with.

The sequential execution of rotations corresponds to the multiplication of quaternions, that is,

Reversing the direction of rotation corresponding to the inverse of:

This is the picture

A homomorphism of the group of Einheitsquaternionen in the rotation group. She is a superposition of, and, as a picture element has exactly two preimages, two-leaved, so the homomorphism is also called 2:1 overlay ( shomomorphismus ). Furthermore, it is universal, there simply is contiguous.

Respect to orthogonal matrices

Explicitly corresponds to the unit quaternion,

And with the rotation matrix

It forms a pure quaternion onto.

Conversely, if the rotation matrix

Where and is the track

Then accomplished the quaternion

Rotation, as it is for each pure quaternion.

If you take the homogeneous formulated version of the input matrix, produces the solution shown with the quaternion. Because of the homogeneity can be maintained in the set.

The how the dimension of the 3 components of 9 so can not all be freely selected. Since each matrix corresponds to a quaternion from the rotation matrices cover the whole. When is. So if really, the unit quaternion is too.

Considerations for the numerical stability of the problem can be found in en: rotation matrix # conversions.

Respect to Euler angles

For Euler angles there are different conventions; The following discussion relates to the turning, which is obtained when one rotates about the first axis by an angle, then around the new axis by an angle, and finally the new - axis by the angle, i.e., the so-called "X- Convention " (z, x ', z ''), all angles in duplicate. The individual rotations correspond to the Einheitsquaternionen

And there is respectively rotated about the axes jointly rotated, the sequence of the composition is reversed. The total rotation corresponds therefore

For other conventions, similar formulas arise.

The Euler angles to a given quaternion can be read at the corresponding rotation matrix.

Universal covering of the rotation group; spin group

As shown in section Einheitsquaternionen, there is a brokered by the Hamiltonian numbers isomorphism between the group of Einheitsquaternionen and the special unitary group. These two groups are isomorphic to the group Spin ( physics: see spinning ).

So the 2:1 overlay provides a homomorphism of the group in the spin rotation group. This superposition is two-leaved and universal, as opposed to simply being contiguous. The natural action of on is a so-called spinor representation.

The well-known from quantum mechanics so-called Pauli matrices are simply related to the three generators of the. This is particularly evident in the presentation as complex matrices:

Here, the imaginary unit of complex numbers.

The Pauli matrices -1 for the determinant ( ie are not quaternions ) are traceless and Hermitian and are therefore in quantum mechanics as measurable quantities in question, which has been found for the applications (see mathematical structure of quantum mechanics ) as important. Details are presented in the article SU (2).

Orthogonal images of the four-dimensional space

Analogous to the three-dimensional case can be any orientation-preserving orthogonal map of into itself in the form

Describe for Einheitsquaternionen. It is

This construction provides an overlay

With core.

The finite subgroups

The 2-1 Überlagerungshomomorphismus

Of a unit quaternion, the 3D rotation

Assigns a finite group of quaternions must be converted into a finite group, which is then a finite rotation group. One finds cyclic groups and Polyedergruppen, ie the dihedral groups (counting the n- corner), the tetrahedral group, the octahedral group and the icosahedral group.

The generators of the cyclic groups are embeddings of unit roots. The archetypes of, among are considered, called so for a Polyedergruppe.

The finite groups of quaternions are therefore:

With

The convex hulls are (except for the cases in which one requires only 2 dimensions ) of 4- polytopes and have since all group elements of length 1 are the unit 3- sphere as order -3- sphere. The edges of this polytope 4, so the cells are collections of tetrahedra - except for the case where it is octahedron. In the regular among the convex hulls, it is clear that the cells are also regular and congruent to each other and there is an in -3- sphere that touches all the cells ( at its center ). The other, namely, and, so-called perfect tense 4- polytopes on. Here, the tetragonal Disphenoide cells, all of which are congruent to one another and also to be touched at its center point from the three - in- sphere.

Automorphisms

Each ring is an inner automorphism of, that is, there is a quaternion so. It follows:

The center remains fixed, ie for all.
It may be limited to the Einheitsquaternionen.
An automorphism does not change the scalar product, that is.
The automorphisms are exactly the angle and length-preserving rotations of the section of rotations in three-dimensional space.
Because of the length of loyalty which automorphisms are continuous, thus additionally topologically.
The Centre. Consequently, the automorphism group.

Conjugation as reflection on the real axis is antihomomorph in the multiplication, i.e., and is referred to as involutional Antiautomorphismus because it is also an involution.

Other constructions

Matrix representations

Complex matrices

The ring of the complex 2 × 2 matrix is formed from the elements of the

Generated subring, where the imaginary unit of the complex numbers is done as indicated. A matrix

With real and complex, the determinant is 0 only if. Thus, all of the zero matrix is invertible matrices different - and the ring is a division ring.

The thus constructed skew field turns out to be isomorphic to the quaternions. Because the picture with the assignments

Is homomorphic in the shortcuts addition and multiplication, the latter of matrix multiplication is to be assigned. The conjugate quaternion going to the adjoint matrix and the norm on the determinant. In addition, the map is injective and continuous, ie topologically.

There are various options for embedding, all of which are conjugate to each other and homeomorphic.

Real matrices

Analogous can be the quaternion as a real matrix of 4 × 4

. Write The conjugation of the quaternion corresponding to the transposition of the matrix and the amount of the fourth root of the determinant.

The model of real matrices is advantageous, for example, if you have a software for linear algebra with weaknesses in the complex numbers.

Quotient algebra

An elegant, but at the same time abstract construction provides the way on the quotient of the polynomial ring in three non-commutative indeterminates, whose images are, modulo the ideal generated by the Hamiltonian rules. Alternatively, one comes out with only two unknowns. In this way, the quaternion algebra is measured as the Clifford algebra of the two-dimensional Euclidean plane with producers. In the context of three-dimensional rotations, the interpretation as the even part of the Clifford algebra of the three-dimensional Euclidean space is important. Producers are then identified.

The quaternion algebra as

There are up to isomorphism exactly four finite dimensional algebras whose multiplication without zero divisors, namely the field of real numbers themselves, the field of complex numbers, the skew field of quaternions, and the alternative body of Cayley octaves.

The center of is; the quaternions are therefore a central simple algebra over. Reduced norm and trace are by

Given.

At the base change of the algebraic degree quaternions to a matrix algebra are:

The complex conjugation on the factor of the tensor product corresponds to an involution of the matrix algebra. The invariants of, d see the left by fixed elements, to form an isomorphic algebra. For the above matrix representation of the quaternions as complex matrices fit the involution

The fact that the Brauer group consists of only two elements, is also reflected in that

Is.

Generally referred to any four-dimensional central simple algebra over a field as a quaternion algebra.

The quaternions are the Clifford algebra to the room with a negative - definite symmetric bilinear form.

Other body

Quaternions over the rational numbers

In all the above forms of construction, the entirety of the coefficient supply does not matter. Therefore, one can ( instead of the real numbers on ) by other primitives, such as the rational numbers, go out via Gaussian numbers with quaternions with rational coefficients

To arrive - with the same formal calculation rules. Thereafter, if at all necessary, the completion of the metric sum are carried out with a final isomorphic.

To that extent, by many statements can be replaced by and by.

Since there is no finite field with non- commutative multiplication by the theorem of Wedderburn and the dimension of the vector space over its prime field and center with minimal heard as countable set to the "smallest" skew fields with non- commutative multiplication - definitely contains no smaller.

The body has a so-called wholeness ring, that is a subset of numbers, called Hurwitzquaternionen, which form a ring and have the quotient body - similar to the way it behaves with the integers and its quotient field. In such a ring can be, for example, Approximationsfragen, Teilbarkeitsfragen and the like. investigate.

Other basic body

Even bodies are suitable as starting point for the formation of non- commutative extension field to the type of quaternions. It is important that disappears in the sum of four squares only. Then there is no by and is a real square extension defining a conjugation. These conditions are all formally real bodies met, for example.

But also in articles which can not be arranged, the above condition relating to the sum can be met from 4 squares, for example, in the body of the 2- adic number. The quaternion so formed is isomorphic to the completion of the ( above ) body of quaternions with rational coefficients for the following ( nichtarchimedische discrete) valuation, the 2- exponent of the standard

With. The prime number is the only one for which the quaternion algebra over zero divisors and a skew field.

Applications

Euler four- square theorem

The identity of the two sums from the product of four squares

Makes a sum that is universal - including any variations caused by game and sign permutation, - in any polynomial ring over a commutative unitary ring and can be viewed in retrospect as a "waste product " of the multiplicativity of the quaternionic amount. Their discovery in 1748, long before the Quaternionenzeit, but goes back to Leonhard Euler, who with their help could significantly simplify the first time in 1770 provided evidence of Joseph Louis Lagrange for the long sought four- square theorem.

Engineering

The representation of rotations using quaternions is now used in the field of interactive computer graphics, especially in computer games, as well as in the management and control of satellites. When using quaternions instead of rotation matrices slightly less arithmetic operations are required. Particularly if many twists combined ( multiplied ) are the processing speed increases. Furthermore, quaternions, in addition to the Euler angles used for the programming of industrial robots (eg ABB).

Physics

Through the use of quaternions can be dispensed with in many cases to separate equations for the calculation of time and space. This offers advantages in physics, including in the fields of mechanics, wave equations, Special relativity and gravitation, electromagnetism and quantum mechanics.

As discussed in vector analysis vectors are identified in three-dimensional space with pure quaternions.

Electromagnetism

The Maxwell equations for the description of electromagnetism are the most well-known use case for quaternions. Maxwell's equations are defined by a set of commutators and anticommutators the differential operator, the electric field E and the magnetic field B in a vacuum. Essentially, these are the homogeneous Maxwell equation and Gauss's law.

Below modified commutators or anticommutators be used:

And

With the (formal) quaternions and various formal products.

The homogeneous Maxwell equation is defined by:

This means that no magnetic monopoles exist. is Faraday's law of induction.

The Gauss's law is defined by the reverse:

This results in Gauss's law and the corrected Maxwell Ampere's law Ampère.

Electromagnetic four-potential

The electric and magnetic fields are widely used as an electromagnetic four-potential (i.e. a 4-functional vector ) is expressed. This vector can be reformulated as a quaternion.

The electric field E is the anticommutator of the conjugated differentiated Four potential. The magnetic field B used the commutator. Through this display, you can use directly in the Maxwell equations:

As well as

Here, the expressions, and the two source fields, which are formed by the difference between the two commutators and two anticommutators.

The induction law and Ampere's law can be formed by the sum of the two nested commutators and anticommutators.

Lorentz force

The Lorentz force is derived in a similar manner from the Maxwell equations. However, the sign must be corrected.

Conservation law

The law of conservation of electric charge is formed by the application of the conjugate difference operator to the sources of the Maxwell equation. With here is called the real or scalar part of the quaternion. In the examples is a Quaternionenprodukt.

This equation shows that the inner product of the electric field plus the cross product of the magnetic field on the one hand, and the power density plus the frequency of the charge density on the other hand, is the same. This means that the charge is maintained during the forming process.

Poyntings conservation of energy is derived in the same way, with the difference that instead of the differential, the conjugated electric field is used.

Using the vector identities

Can this equation for

Forming, which corresponds to the Poynting equation. The term here corresponds to the Poynting vector.

History

William Rowan Hamilton in 1835 had given the construction of the complex numbers as pairs of numbers. Thus motivated, he has long sought a corresponding structure on the space of number triples; today we know that no such structure exists. In 1843 he came to the conclusion that it is possible to construct a multiplication on the set of 4- tuples, if one is willing to abandon the commutativity. In a letter to his son he gives the date of 16 October 1843 reported that he had spontaneously get carried to the multiplication rules in a stone at the Brougham Bridge (now Broombridge Road) to scratch in Dublin; later a plaque was placed there. The calculation rules for quaternions were earlier known in batches, so we find the formula for the four- square theorem already at Leonhard Euler ( 1748). Other, more general multiplication rules were examined by Graßmann (1855).

Shortly after the discovery of quaternions Hamilton was the representation of rotations of the space using quaternions and thus a first confirmation of the importance of the new structure; Arthur Cayley discovered in 1855 the corresponding statements about orthogonal images of the four-dimensional space. The mere parameterization of - rotation matrices, however, was already known to Euler. Cayley was in 1858 in the work in which he introduced matrices, the possibility of representation of quaternions by complex matrices to.

Hamilton devoted himself exclusively to the study of quaternions; they were a private exam specialist in Dublin. In following a " World Federation for the promotion of quaternions " was founded in 1895 even. The German mathematician Felix Klein writes retrospectively about this initial euphoria:

"As I have already indicated, Hamilton joined a school that even surpassed their master of rigidity and intolerance. [ ... ] The quaternions are good and useful in its place; but they will not reach their significance to the ordinary complex numbers. [ ... ] The ease and elegance with which results in the most far-reaching theorems here is surprising, in fact, and it can probably understand from here is anything but negative enthusiasm of quaternionists for their system, which [ ... ] will soon be over reasonable boundaries grew, in a neither of mathematics as a whole nor the Quaternionentheorie even beneficial manner. [ ... ] The persecution of the specified path - who wants to be new, although he actually only a meticulous transmission of long known thoughts on a single new object, so thoroughly does not mean a brilliant concept - leads to all kinds of extensions of the known records that in their generality lose the main characteristic and will become obsolete at best to special features which may provide a certain pleasure. "