### A Quaternion Algebra Tool Set

Here is a compilation of basic algebra for quaternions.  It should look very similar to complex algebra, since it contains three sets of complex numbers, t + x i, t + y j, and t + z k.  To strengthen the link, and keep things looking simpler, all quaternions have been written as a pair of a scalar t and a 3-vector V, as in (t, V).  All these relations have been tested in a C library and a Java quaternion calculator.

Technical note: it is vital that every tool in this set can be expressed as working with a whole quaternion q.  This will make doing quaternion analysis with automorphic functions fruitful.

#### Multiplication

The Grassman product as defined here uses the same rule Hamilton developed.  The Euclidean product takes the conjugate of the first of the two elements (following a tradition from quantum mechanics).

#### Trigonometry

Note: since the unit vectors of sine and cosine are the same, these two commute so the order is irrelevant.

#### Quaternion exponential multiplication

Andrew Millard suggested the result for the Grassman product.