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. Since $a\ne 0,$ it has a multiplicative inverse, which is $\frac{1}{a}.$ Next, multiply both sides of the left-hand equation by $\frac{1}{a}$ and simplify. It was obtained that $b=0.$ Consequently, if the product $ab$ is equal to $0$ and $a\ne 0,$ then $b=0.$ This shows that when the product of two numbers is zero, at least one of them must be zero.