Quaternion

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	rotational symmetry

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In Mathematics , the quaternions are a Non-commutative extension of the Complex Number s. They were first described by the Irish Mathematician Sir William Rowan Hamilton in 1843 and applied to Mechanics in three-dimensional space. At first, the quaternions were regarded as Pathological , because they disobeyed the commutative law ''ab'' = ''ba''. Although they have been superseded in most applications by Vector s, they still find uses in both theoretical and applied mathematics, in particular for calculations involving Three-dimensional Rotations . In modern language, the quaternions form a 4-dimensional Normed Division Algebra over the Real Number s. The algebra of quaternions is often denoted by H (for Hamilton), or in Blackboard Bold by $\backslash mathbb\; H$. DEFINITION While the complex numbers are obtained by adding the element ''i'' to the real numbers which satisfies ''i''2 = −1, the quaternions are obtained by adding the elements ''i'', ''j'' and ''k'' to the real numbers which satisfy the following relations. :$i^2\; =\; j^2\; =\; k^2\; =\; ijk\; =\; -1\backslash ,$ If the multiplication is assumed to be Associative (as indeed it is), the following relations follow directly: :$\backslash begin\{matrix\}$ ij & = & k, & & & & ji & = & -k, \ jk & = & i, & & & & kj & = & -i, \ ki & = & j, & & & & ik & = & -j. \end{matrix} (these are derived in detail below). Every quaternion is a real Linear Combination of the basis quaternions 1, ''i'', ''j'', and ''k'', i.e. every quaternion is uniquely expressible in the form ''a'' + ''bi'' + ''cj'' + ''dk'' where ''a'', ''b'', ''c'', and ''d'' are real numbers. In other words, as a Vector Space over the Real Number s, the set '''H''' of all quaternions has Dimension 4, whereas the complex number plane has 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 above for the basis quaternions. Under this multiplication, the basis quaternions, with their negatives, form the Quaternion Group of order 8, ''Q''8.The Scalar part of the quaternion is ''a'' while the remainder is the vector part. Thus a '''vector''' in the context of quaternions has zero for scalar part. Example Let :$\backslash begin\{matrix\}$ x & = & 3 + i \ y & = & 5i + j - 2k \end{matrix} Then :$\backslash begin\{matrix\}$ x + y & = & 3 + 6i + j - 2k \ \ xy & = & (3 + i)(5i + j - 2k) \ & = & 15i + 3j - 6k + 5i^2 + ij - 2ik \ & = & 15i + 3j - 6k - 5 + k + 2j \ & = & -5 + 15i + 5j - 5k \ \ yx & = & (5i + j - 2k)(3 + i) \ & = & 15i + 5i^2 + 3j + ji - 6k - 2ki \ & = & 15i - 5 + 3j - k - 6k - 2j \ & = & -5 + 15i + j - 7k \end{matrix} Arithmetic Unlike real or complex numbers, multiplication of quaternions is not Commutative : e.g. :$\backslash begin\{matrix\}$ ij & = & k \ ji & = & -k \ jk & = & i \ kj & = & -i \ ki & = & j \ ik & = & -j \ \end{matrix} The quaternions are an example of a Division Ring , 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. Quaternions 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 associative division algebras 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 distinct solutions than the degree of the polynomial. The equation $z^2\; +\; 1\; =\; 0$, for instance, has the infinitely-many quaternion solutions $z\; =\; bi\; +\; cj\; +\; dk$ with $b^2\; +\; c^2\; +\; d^2\; =\; 1$. of the quaternion $z\; =\; a\; +\; bi\; +\; cj\; +\; dk$ is defined as = a - bi - cj - dk \, and the ''absolute value'' of ''z'' is the non-negative real number defined by
	Note That <math>(wz)^	z^ w^</math>, which is not in general equal to <math>w^ z^</math> The multiplicative inverse of the non-zero quaternion ''z'' can be conveniently computed as ''z''<sup>&minus1</sup> = ''z''<sup></sup> / ''z''<sup>2</sup>
	By Using The	"http://wwwinformationdelightinfo/encyclopedia/entry/distance_function" class="copylinks">Distance Function ''d''(''z'', ''w'') = ''z'' &minus ''w'', the quaternions form a Metric Space (isometric to the usual Euclidean metric on '''R'''<sup>4</sup>) 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
	:<math>p	\sqrt{p \cdot p} = \sqrt{p^p} = \sqrt{a^2 + b^2 + c^2 + d^2}</math>

  Natural Exponential: <math>\exp(p) \exp(a)(\cos(ec{u}) + \sgn(ec{u})\sin(ec{u}))</math>
  Natural Logarithm: <math>\ln(p) \ln(p) + \sgn(ec{u})\arg(p)</math>
  Sine: <math>\sin(p) \sin(a)\cosh(ec{u}) + \cos(a)\sgn(ec{u})\sinh(ec{u})</math>
  Cosine: <math>\cos(p) \cos(a)\cosh(ec{u}) - \sin(a)\sgn(ec{u})\sinh(ec{u})</math>
  Hyperbolic Sine: <math>\sinh(p) \sinh(a)\cos(ec{u}) + \cosh(a)\sgn(ec{u})\sin(ec{u})</math>
  Hyperbolic Cosine: <math>\cosh(p) \cosh(a)\cos(ec{u}) + \sinh(a)\sgn(ec{u})\sin(ec{u})</math>