Dividing by Zero Is Undefined

While dividing a number by zero is not allowed, dividing zero by a (non-zero) number is meaningful! In fact, there is a straightforward, simple rule for this situation:

Zero divided by any (non-zero) number is zero.

$\frac{0}{a}=0$ for all $a\neq 0$

For example, $\frac{0}{5}=0$. This is an abstract mathematical fact we can take as a given rule, but you can also understand it intuitively when you relate it to an everyday experience. Let's return to our pizza party, where we have zero pizzas and 80 students show up. Each student will receive $\frac{0}{80}=0$ slices. In fact, this party is disappointing no matter how many students show up. Everyone will receive zero amount of pizza, indicating that 0/(any number) = 0.

There is another way to understand this division fact. Let's invoke the language of multiplicative inverses. Dividing by a number $a$ is the same operation as multiplying by its reciprocal $\frac{1}{a}$. It now follows that $0$ divided by any number $a$ equals $0$ multiplied by any number $\frac{1}{a}$. Based on your work in the previous section, we know this equals zero.