Multiplicative of zero.
Commutativity means that we can change the order of (i.e., we can exchange or "commute") the numbers when we do operations. For the real numbers, both addition and multiplication is commutative:
Associativity means that it doesn't matter how we "group" (or "associate") things when doing the same operation:
Multiplication is distributive over addition (for the real numbers), which means we can distribute one factor to each term in the other:
(left distributivity)
and
(right distributivity)
Zero (0) is called the additive identity because it is a number which satisfies
Similarly, one (1) is the multiplicative identity because
We may also speak of additive and multiplicative inverses. The additive inverse of is because and the multiplicative inverse of is because
From the above properties, we can deduce others. Notably, . This would be the multiplicative of zero.
Note that not all sets and operations satisfy these properties (for example, the integers generally do not have multiplicative inverses, and the set of even integers lacks a multiplicative identity). Those sets with binary operations that do satisfy these properties (such as the real numbers) are called fields.