From Real To Hyper Complex Numbers

The group of transformation matrices from which a "spinor" is realised is related to a set of "Hyper-complex" numbers discovered by Hamiltonian called quaternians. Quaternians are used in programming to efficiently effect rotations in CGI laden films and games.

One way to view quaternians is as higher (four-) dimensional versions of the number line of Real numbers. Just as Complex numbers are 2-dimensional extensions of 1-dimensional Reals, quaternians are to be regarded as three-dimensional Hyper complex numbers.

View multiplication of one real number by another as the expansion (re-scaling) of the 1-dimensional (1-d) Real number line. Multiplication by a complex number then represents rotation in the 2-d Argand x-i plane.

We then think of quaternians as representing the set of 4 possible manipulations (pitching, rolling and yawing) of an object (think of an aeroplane) in 3-d space plus the rescaling of space as for the Reals.

In combination these three objects make up an Algebra. Very important in Supersymmetry and in dealing with fermions such as electrons in a quantum Field theory. The algebra is called a Grassman Algebra:

Grassman Algebra: {Real, Complex, Quaternians} numbers