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Quaternions

 

from Wikipedia

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843[1][2] and applied to mechanics in three-dimensional spaceIn modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and thus also form a domain.

The equations

i2 = j2 = k2 = ijk = −1,

where ij, and k are basis elements of H, determine all the possible products of ij, and k. For example right-multiplying both sides of −1 = ijk by k gives:

All the other possible products can be determined by similar methods, resulting in

which can be expressed as a table whose rows represent the left factor of the product and whose columns represent the right factor, as shown at the top of this article.

 

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