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.

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.