Zero Product Property

Let a and b be two real numbers such that ab=0. The goal is to show that at least one of them is zero. If a=0 or b=0, there is nothing to prove and the property is true. The interesting case is when one of the numbers is different from zero. Therefore, assume that a is a non-zero number. ab=0 and a≠ 0 Since a≠ 0, it has a multiplicative inverse, which is 1a. Next, multiply both sides of the left-hand equation by 1a and simplify.

It was obtained that b=0. Consequently, if the product ab is equal to 0 and a≠ 0, then b=0. This shows that when the product of two numbers is zero, at least one of them must be zero.