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 and applied to mechanics in three-dimensional space. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and thus also form a domain.
- i2 = j2 = k2 = ijk = −1,
where i, j, and k are basis elements of H, determine all the possible products of i, j, 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.