Hamilton's Quaternions

 Let's take a break from Hamilton's Theory of Systems of Rays and take a look at his quaternions while they are still fresh on my mind from some recent work.

We start with the unit vectors of a Cartesian coordinate system, i, j, and k, and try to construct a table for a binary operation, ◦, on them. We first choose i◦j=k, j◦k=i, and k◦i=j. In analogy with complex numbers where i=√-̅1̅ and i²=-1 we let i◦i=-1, j◦j=-1, and k◦k=-1. Using the multiplication rules of ordinary multiplication we deduce j◦i=-k, k◦j=-i, and i◦k=-j and so we have a table for the binary operations on the vectors.

This table contains the element 1 which is an ordinary number or scalar and so we can include scalar multiplication for closure and obtain a more complete table involving all the elements.

A quaternion is a linear combination of these four elements multiplied by four coefficients so q=1q₀+iq₁+jq₂+kq₃ like ordinary vectors. If you have problems with the logical consistency of adding scalars and vectors we can think of the identity element 1 as a fourth independent vector. Using the rules of ordinary algebra we can define the "product" of two quaternions p=q2◦q1 and use the table to find the products of the elementary units.

Here's a sample calculation involving the product of two quaternions.

One can deduce the following formulas involving only the coefficients to help simplify the calculations in Excel.

Supplemental (Aug 7): The starting point in constructing the operator table was identifying a set of normal vectors for the three pairs of vectors. One can think of the operator ◦ as a composite operator defined on a composite field. Hamilton may have been influenced by Galois' earlier work. One can associate 90° rotations with the multiplication of one vector element by another. The quaternions form a group.

Supplemental (Aug 7): Here are some early works on complex numbers

  Wallis, On Negative Squares (Latin) 1673

  Wessel, On the Analytical Representation of Direction (Danish) 1797

  Smith, A Source Book in Mathematics, English translations