Quaternions

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Quaternions are an extension to the real numbers, similar to the complex numbers. While the real numbers are extended to the complex numbers by adding a number i such that i2 = -1, quaternions are extended by defining i, j and k such that i2 = j2 = k2 = 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.

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 stucture similar to a field except for commutativity of multiplication. In particular, multiplication is still associative and every non-zero element has a unique inverse. The quaternions, along with the complex and real numbers, are the only finite dimensional skew fields over the field of real numbers. They form a 4-dimensional associative algebra over the reals and a 2-dimensional associative algebra over the complex numbers.

Quaternions were discovered by William Rowan Hamilton. 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 when the solution in the form of equation i2 = j2 = k2 = ijk = -1 suddenly occurred to him; he then proceeded to carve this equation into the side of a bridge!)

The absolute value of the quaternion z = a + bi + cj + dk is defined to be |z| = a2 + b2 + c2 + d2. 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.

The set of quaternions of absolute value 1 forms a 3-dimensional sphere S3 and a group under multiplication. This group acts by conjugation on the copy of R3 consisting of quaternions with real part equal to zero: it is not hard to see that the conjugation by a unit quaternion of real part cos t is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part. Thus, S3 is the double cover of the group SO(3) of real orthogonal 3x3 matrices of determinant 1; it is isomorphic to SU(2), the group of complex 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 integers or all rational numbers with odd numerator and denominator 2. The set A is a 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.

Quaternions are sometimes used in computer graphics (and associated geometric analysis) to represent the orientation of an object in 3d space. The advantages are: non singular representation (compared with Euler angles for example), more compact (and faster) than matrices.

See also: Hypercomplex numbers