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==Short bio==
[[es:Cuaterniones]][[ja:四元数]]
Mexican mathematics grad school student.
'''Quaternions''' are an extension of the [[real number]]s, similar to the [[complex number]]s. While the real numbers are extended to the complex numbers by adding a number ''i'' such that ''i''<sup>2</sup> = -1, quaternions are extended by adding elements ''i'', ''j'' and ''k'' to the real numbers such that ''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = -1. A quaternion then is a number of the form ''a'' + ''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]]:
I'm also a major contributor of [http://planetmath.org Planetmath], the free mathematics encyclopedia.
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==Images uploaded to wikipedia==
These products form the [[quaternion group]] of order 8, ''Q''<sub>8</sub>.
* [[Image:Drini-nonuniformconvergence.png|thumb|300px|left|Used on [[Uniform convergence]]]]
* [[Image:Drini-conjugatehyperbolas.png|thumb|300px|left|Used on [[Hyperbola]]]]
Unlike real or complex numbers, multiplication of quaternions is not [[commutative]]: ''ij'' = ''k'', ''ji'' = -''k'', ''jk'' = ''i'', ''kj'' = -''i'', ''ki'' = ''j'', ''ik'' = -''j''. The quaternions are an example of a [[skew field]], an algebraic structure similar
to a [[field]] except for commutativity of multiplication. In particular, multiplication is still [[associative]] and every non-zero element has a unique inverse. They form a 4-dimensional [[associative algebra]] over the reals (in fact a [[division algebra]]) 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 skew fields over the field of real numbers.
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 ''z''<sup>2</sup> + 1 = 0 for instance has the infinitely many quaternions ''z'' = ''bi'' + ''cj'' + ''dk'' with ''a''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> = 1 as solutions.
The ''conjugate'' of the quaternion ''z'' = ''a'' + ''bi'' + ''cj'' + ''dk'' is defined as ''z''<sup>*</sup> = ''a'' - ''bi'' - ''cj'' - ''dk'', and the ''absolute value'' of ''z'' is the non-negative real number defined by |''z''| = √(''zz''<sup>*</sup>) = √(''a''<sup>2</sup> + ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup>). Note that (''wz'')<sup>*</sup>= ''z''<sup>*</sup>''w''<sup>*</sup>, which is not in general equal to ''w''<sup>*</sup>''z''<sup>*</sup>. The multiplicative inverse of the non-zero quaternion ''z'' can be conveniently computed as ''z''<sup>-1</sup> = ''z''<sup>*</sup> / |''z''|<sup>2</sup>.
By using the distance function ''d''(''z'',''w'') = |''z'' - ''w''|, the quaternions form a [[metric space]] 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 algebra]].
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 '''R'''<sup>3</sup> consisting of quaternions with real part equal to zero:
it is not hard to see that the conjugation by a unit quaternion (a quaternion of absolute value 1) with real part ''cos t'' is a rotation by an angle ''2t'',
the axis of the rotation being the direction of the imaginary part.
Quaternions are sometimes used in [[computer graphics]] (and associated geometric analysis) to represent rotations or orientations of objects in [[3d]] space. The advantages are: non singular representation (compared with [[Euler angles]] for example), more compact (and faster) than [[matrix (mathematics)|matrices]].
The set of all unit quaternions forms a [[3-sphere|3-dimensional sphere]] ''S''<sup>3</sup> and a [[mathematical_group|group]] (even a [[Lie group]]) under multiplication.
''S''<sup>3</sup> is the double cover of the group ''SO''(3,'''R''') of real orthogonal 3x3 [[matrix|matrices]] of [[determinant]] 1 since ''two'' unit quaternions correspond to every rotation under the above correspondence. The group ''S''<sup>3</sup> is isomorphic to ''SU''(2), the group of complex [[unitary matrix|unitary]] 2x2 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 [[integer|integers]] or all [[rational number|rational numbers]] with odd numerator and denominator 2. The set ''A'' is a [[ring (mathematics)|ring]] and a lattice. There are 24 unit quaternions in this ring and they are the vertices of a [[regular polytope]] called {3,4,3} in Schlafli's notation.
== Representing quaternions by matrices ==
There are at least two ways of representing quaternions as [[matrix (mathematics)|matrices]], in such a way that quaternion addition and multiplication correspond to matrix addition and [[matrix multiplication]].
One is to use 2x2 [[complex number|complex]] matrices, and the other is to use 4x4 [[real number|real]] matrices.
In the first way, the quaternion ''a + bi + cj + dk'' is represented as:
: <math>\begin{pmatrix} a-di & -b+ci \\ b+ci & a+di \end{pmatrix}</math>
This representation has several nice properties:
* All [[complex number]]s (''c'' = ''d'' = 0) correspond to matrices
with only real entries.
* 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''<sup>3</sup> and SU(2). The latter group is important in [[quantum mechanics]] when dealing with [[spin (physics)|spin]]; see all [[Pauli matrices]].
In the second way, the quaternion ''a + bi + cj + dk'' is represented as:
: <math>\begin{pmatrix} a & -b & d & -c \\
b & a & -c & -d \\
-d & c & a & -b \\
c & d & b & a \end{pmatrix}</math>
In this representation, the conjugate of a quaternion corresponds to the [[transpose]] of the matrix.
== History ==
Quaternions were discovered by [[William Rowan Hamilton]] 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 ''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = -1 suddenly occurred to him; he then promptly carved this equation into the side of nearby Brougham bridge.
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, ''Elements of Quaternions'', had 800 pages and was published shortly after his death.
Even by this time there was controversy about the use of quaternions. Some of Hamilton's supporters viciously opposed the growing fields of vector algebra and vector calculus (from developers like [[Oliver Heaviside]] and [[Willard Gibbs]]), maintaining that quaternions provided a superior notation. While this may be true in three dimensions plus time (i.e., [[spacetime]]), quaternions cannot be used in other dimensions (though other deriverative exist like [[Octonions]] and [[Clifford algebra]]s for this). Their scientific recognition compared to vectors has therefore decreased over time. They are today still used in [[computer graphics]] and [[Plasma physics]].
== Generalizations ==
If ''F'' is any [[field]] and ''a'' and ''b'' are elements of ''F'', one may define a four-dimensional unitary [[associative algebra]] over ''F'' by using two generators ''i'' and ''j'' and the relations ''i''<sup>2</sup> = ''a'', ''j''<sup>2</sup> = ''b'' and ''ij'' = -''ji''. These algebras are either isomorphic to the algebra of 2-by-2 matrices over ''F'', or they are [[division algebra]]s over ''F''. They are called [[quaternion algebra]]s.
See also: [[Octonion]], [[Hypercomplex number]], [[Division algebra]]
== Calculator ==
* [http://theworld.com/~sweetser/java/qcalc/qcalc.html Quaternion Calculator] [Java]
== Related resources ==
==Personal reminder==
* [http://world.std.com/~sweetser/quaternions/qindex/qindex.html Doing Physics with Quaternions]
{{Opentask}}
* [http://arxiv.org/pdf/math-ph/0201058 The Physical Heritage of Sir W. R. Hamilton] (PDF)
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