Because division does not enjoy the commutative property, dividing into zero and dividing by zero are two different questions. One of these computations is easy to carry out, while the other is impossible. Can you guess which is which before you read more?

Let's begin by considering division by zero. Many mathematics educators model the operation of division using shared pizzas or pies. If two students show up to a party where there are 10 pizzas, then they can each receive an equal number of pizzas, namely \frac{10}{2}=5 pizzas each; if 80 students show up to that same party, each student will receive \frac{10}{80}=\frac{1}{8} of a single pizza, or one (small) slice each.

Thinking about our pizza party example suggests why the "answer" to this "division by zero" question is so strange: imagine we have 10 pizzas to share among the students at a party, only now zero students show up! How much pizza does each student receive? A reasonable reply is that the question does not make any sense since no students are present. There is no answer or number we can offer in response to this question!

The mathematical rule for division by zero reads like this:

Whenever we divide a number by zero, the answer is undefined.

For example, \frac{5}{0} = undefined. The technical response is, "we cannot assign a meaningful value or answer to the expression \frac{5}{0}".
Back to '1.8 Dividing by Zero Is Undefined'