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In mathematics, the quaternions are a non-commutative extension of the complex numbers. They were first described by William Rowan Hamilton of Ireland in 1843. At first, the quaternions were regarded as pathological, because they disobeyed the commutative law ab = ba. However, today they find many uses in both theoretical and applied mathematics.
| · | 1 | i | j | k |
| 1 | 1 | i | j | k |
| i | i | −1 | k | −j |
| j | j | −k | −1 | i |
| k | k | j | −i | −1 |
While the complex numbers are obtained by adding the element i to the real numbers which satisfies i2 = −1, the quaternions are obtained by adding the elements i, j and k to the real numbers which satisfy the following relations.
Every quaternion is a real linear combination of the unit quaternions 1, i, j, and k, i.e. every quaternion is uniquely expressible in the form a + bi + cj + dk. In other words, as a vector space over the real numbers, the quaternions have dimension 4, whereas the complex numbers have dimension 2.Addition of quaternions is accomplished by adding corresponding coefficients, as with the complex numbers. By linearity, multiplication of quaternions is completely determined by the multiplication table for the unit quaternions; this table is given at the right. Under this multiplication, the unit quaternions form the quaternion group of order 8, Q8.
Let
Then
Unlike real or complex numbers, multiplication of quaternions is not commutative: e.g. ij = k, ji = −k, jk = i, kj = −i, ki = j, ik = −j. The quaternions are an example of a division ring, an algebraic structure similar to a fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil except for commutativity of multiplication. In particular, multiplication is still associative and every non-zero element has a unique inverse.
Quaternions form a 4-dimensional associative algebraAlgebra In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras. Definition An associative algebra A over over the reals (in fact a division algebraRing theory In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. Definitions Formally, we start with an algebra D over a field, and assume that D does not just) and contain the complex numbers, but they do not form an associative algebra over the complex numbers. The quaternions, along with the complex and real numbers, are the only finite-dimensional associative division algebras over the field of real numbers. The non-commutativity of multiplication has some unexpected consequences, among them that polynomialIn mathematics polynomial functions or polynomials are an important class of simple and smooth functions. Simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i. they have derivatives o equations over the quaternions can have more distinct solutions than the degree of the polynomial.
The equation z2 + 1 = 0, for instance, has the infinitely many quaternion solutions z = bi + cj + dk with b2 + c2 + d2 = 1. The conjugate of the quaternion z = a + bi + cj + dk is defined as
and the absolute value of z is the non-negative real number defined by
Note that (wz)*= z*w*, which is not in general equal to w*z*. The multiplicative inverse of the non-zero quaternion z can be conveniently computed as z−1 = z* / |z|2.
By using the distance function d(z, w) = |z − w|, the quaternions form a metric spaceIn mathematics, a metric space is a set (or "space") where a distance between points is defined. History Maurice Frechet introduced metric spaces in his work Sur quelques points du calcul fonctionnel Rendic. Palermo 22(1906) 1-74. Formal definition Formal (isometric to the usual Euclidean metric on R4) and the arithmetic operations are continuous. We also have |zw| = |z| |w| for all quaternions z and w. Using the absolute value as norm, the quaternions form a real Banach algebraIn functional analysis, a Banach algebra named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related.
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