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## QuaternionThequaternions are an extension of the real numbers, similar to the complex numbers, except: they have dimension 4 rather than 2 over the real numbers, and the multiplication of quaternions is not commutative.
## DefinitionA quaternion then is a number of the forma + bi + cj + dk, where a, b, c, and d are real numbers uniquely determined by the quaternion. The multiplication of quaternions could be deduced from the following multiplication table:
These products form the quaternion group of order 8, ## Properties
Unlike real or complex numbers, multiplication of quaternions is not commutative:
The non-commutativity of multiplication has some unexpected consequences, among them that polynomial equations over the quaternions can have more solutions than the polynomial's degree indicates. The equation
The
By using the distance function
As is explained in more detail in quaternions and spatial rotation, the multiplicative group of non-zero quaternions acts by conjugation on the copy of
The set of all unit quaternions forms a 3-dimensional sphere
Let ## Representing quaternions by matricesThere 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 2x2 complex matrices, and the other is to use 4x4 real matrices.
In the first way, the quaternion
- All complex numbers (
*c*=*d*= 0) correspond to matrices with only real entries. - The square of the absolute value of a quaternion is the same as the determinant of the corresponding matrix.
- The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
- Restricted to unit quaternions, this representation provides the isomorphism between
*S*^{3}and SU(2). The latter group is important in quantum mechanics when dealing with spin; see all Pauli matrices.
a + bi + cj + dk is represented as:
## History
Quaternions were discovered by William Rowan Hamilton of Ireland in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. He could not do so for 3-dimensions, but 4-dimensions produce quaternions. According to a story he told, he was out walking one day with his wife when the solution in the form of equation This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices were still in the future. Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the 'scalar' part, and the remaining three as the 'vector' part. If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the negative of the dot product of the vector parts, while the vector part of the product is the cross product. But the significance of these was still to be discovered.
Hamilton proceeded to popularize quaternions with several books, the last of which, Even by this time there was controversy about the use of quaternions. Some of Hamilton's supporters vociferously opposed the growing fields of vector algebra and vector calculus (developed by Oliver Heaviside and Willard Gibbs among others), maintaining that quaternions provided a superior notation. While this is debatable in three dimensions, quaternions cannot be used in other dimensions (though extensions like octonions and Clifford algebras may be more applicable). In any case, vector notation had nearly universally replaced quaternions in science and engineering by the mid-20th century. Today, quaternions see use in computer graphics, control theory, signal processing and orbital mechanics, mainly for representing rotations/orientations in three dimensions. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations. ## Generalizations
If ## See also## Related resources
- Doing Physics with Quaternions
- Quaternion Calculator [Java]
- The Physical Heritage of Sir W. R. Hamilton (PDF)
- Kuipers, Jack (2002).
*Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality*(Reprint edition). Princeton University Press. ISBN 0691102988
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