Noncommutativity

In standard mathematics, when there are two numbers a and b that are considered to be equal (represented in standard mathematics by a = b), what is left after a has been taken from b (represented in standard mathematics by a – b = 0) is the same as what is left after b has been taken from a (represented in standard mathematics by b – a = 0). The two operations of taking one number from the other are commutative. It is as if each number a and b has turned the other into one and the same number, absolute nothing.

In the new, non-commutative mathematics, what is left after b has been taken from a is not the same as what is left after a has been taken from b: the two operations afford different nothings. It is as if each of the numbers a and b has turned the other into a different nothing.