Division.-From the commutative law for multiplication, which shows that 3 X 4d.
This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, - which merely means that-1 is commutative with all other quantities.
This is included in the preceding, but it is simpler in that the various operations are commutative.
The method is similar in some respects to the treatment of non commutative algebras.
The effect of these definitions is that the sum and the product of two quaternions are also quaternions; that addition is associative and commutative; and that multiplication is associative and distributive, but not commutative.