Hamilton in his letter of 17 October 1843 to John Graves is very confused about the relationship between i, j, +1, and -1. He asks what are we to do with ij, when i and j are unequal roots of a common square. In fact there is no law of arithmetic which makes ij equal to anything but +1. It is these doubts of Hamilton which are the source of his fallacious theory of the non-commutative properties of the multiplication of imaginary numbers. All multiplication whether of real or imaginary numbers is commutative.