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In mathematics, a quaternion is a four-dimensional object important in group theory and geometry. As with the complex numbers, the quaternions can be viewed as an extension of the real number line. Unlike the complex numbers, however, the quaternions are not a field, since multiplication is not commutative. Instead, the quaternions are a skew field.
The quaternions obey all the usual arithmetic operations. Quaternions may be added, subtracted, multiplied, and divided. Addition is associative and commutative, while multiplication is only associative. Moreover, addition distributes over multiplication and nonzero quaternions have multiplicative inverses, and so the quaternions are termed a skew field.
Quaternions are basically an extension of the complex plane to four dimensions. In addition to an imaginary unit quaternions also have j and k, which all follow the relation:
Hamilton made this discovery while on a walk and wrote it down on the Brougham Bridge.
A quaternion, can be written as
If and are two quaternions then their sum is . Multiplicaton is more complicated. To find a find a product one makes repeated use of the relations given of i, j and k. It turns out that quanternion multiplication is messy to write. As mentioned above, quanternion multiplication is noncommutative.
Weisstein, Eric W. "Quaternion." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Quaternion.html