You know that times 0 is 0 and times 1 is the same number. Because division undoes multiplication, we can work out the rules for 0 and 1 in division, and see why one division has no answer.
Why it works
Turn each division into a multiplication. $0 \div 4 = \square$ means $\square \times 4 = 0$, so the box is 0. $9 \div 1 = \square$ means $\square \times 1 = 9$, so the box is 9. $9 \div 9$ means $\square \times 9 = 9$, so the box is 1. Now try $5 \div 0$: we need $\square \times 0 = 5$. But any number times 0 is 0, never 5. No number works, so dividing by 0 has no answer.
| Division | Think | Answer |
|---|---|---|
| $0 \div 4$ | $\square \times 4 = 0$ | 0 |
| $9 \div 1$ | $\square \times 1 = 9$ | 9 |
| $9 \div 9$ | $\square \times 9 = 9$ | 1 |
| $5 \div 0$ | $\square \times 0 = 5$ | no number works |
Key idea
0 divided by any number (not 0) is 0. A number divided by 1 is itself. A number divided by itself is 1. You cannot divide by 0.
Worked example: Zero shared
Find $0 \div 7$.
- Think: $\square \times 7 = 0$.
- Only $0 \times 7 = 0$.
Answer: $0$
Worked example: Dividing by zero
What is $6 \div 0$?
- Think: $\square \times 0 = 6$.
- Every number times 0 is 0, so nothing works.
Answer: It has no answer.
Common mistake
Saying $5 \div 0 = 0$ or $5 \div 0 = 5$. Check by multiplying back: $0 \times 0 = 0$ and $5 \times 0 = 0$, neither is 5. That is why we say you cannot divide by 0. (But $0 \div 5 = 0$ is fine: nothing shared is nothing each.)