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# 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.

Unlike real or complex numbers, multiplication of quaternions is not commutative: ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j.

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 occured to him; he then proceeded to carve this equation into the side of a bridge!)

Quaternions are sometimes used in [computer graphics]? (and associated geometric analysis) to represent the orientation of an object in 3d space. A unit quaternion, a quaternion of length 1, can represent a 3d rotation (in fact the unit quarternions form a [double cover]? of the space of 3d rotations). The advantages are: non singular representation (compared with [Euler angles]? for example), more compact (and faster) than matrices.

See also: Hypercomplex numbers

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Last edited August 13, 2001 3:28 am (diff)
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