A Veritable Almanac!
Sir William Rowan Hamilton(1805 – 1865) came up with the following revelation in dealing with extended complex number theory that you need to have 4 dimensions in order to rotate points in three dimensions:-
For complex numbers in two dimensional form:
In four dimensions called Quaternions by Hamilton :-
In a flash of inspiration Hamilton came up with the following which correctly observes all the properties of quaternions, he was so excited about it he carved this on a bridge in Ireland:-
Properties of quaternions can be remembered with the aid of the diagram above:-
Starting with i and going in a clockwise direction:
Starting with i and going in an anticlockwise direction:-
Note multiplication is not commutative!