Special Rules: Dividing by 1 and 0 - Third Grade Mathematics

While most division problems involve sharing items among groups, the numbers 1 and 0 follow unique mathematical rules that require special attention. In this chapter, you will discover why dividing any number by 1 leaves it unchanged, why zero divided by any number is always zero, and the profound mathematical reason why dividing by zero is completely impossible.

The Rules for Dividing by 1

There are two distinct situations when working with the number 1 in division:

Case 1: Dividing a Number by 1

When you divide any number by 1, the quotient is always that original number.

Rule: a ÷ 1 = a

Why does this make sense? Imagine you have 8 slices of pizza and only 1 person eating. That single person gets all 8 slices!
- 5 ÷ 1 = 5

- 19 ÷ 1 = 19

- 250 ÷ 1 = 250


Dividing by 1 means you have only 1 group, so the single group contains the entire total.

Case 2: Dividing a Number by Itself

When any non-zero number is divided by itself, the quotient is always 1.

Rule: a ÷ a = 1  (for any number a not equal to 0)

Why does this make sense? Imagine you have 6 stickers and you share them equally among 6 friends. Each friend gets exactly 1 sticker!
- 4 ÷ 4 = 1

- 9 ÷ 9 = 1

- 100 ÷ 100 = 1


When the number of items equals the number of groups, every group gets exactly one item.

The Rules for Zero in Division

Zero behaves in two very different ways depending on whether it is the dividend (the number being divided) or the divisor (the number you are dividing by).

Case 3: Zero Divided by Any Number

When zero is divided by any non-zero number, the quotient is always 0.

Rule: 0 ÷ a = 0  (where a is not 0)

Think of a real-world story: Suppose you have an empty cookie jar containing 0 cookies. If you invite 4 friends over and share the cookies equally, how many cookies does each friend receive? Each friend receives 0 cookies!
- 0 ÷ 3 = 0

- 0 ÷ 8 = 0

- 0 ÷ 75 = 0


If you start with nothing, every share is nothing.

Case 4: Dividing by Zero Is IMPOSSIBLE (Undefined!)

In mathematics, you can NEVER divide by zero. Mathematicians say that dividing by zero is "undefined."

Rule: a ÷ 0 is UNDEFINED (Impossible!)

Why is dividing by zero impossible? Let us look at it using two logical tests:

Test 1: The Cookie Sharing Test

Imagine you have 6 cookies. Can you share them among 0 people? Who would you give the cookies to? If there are no people, the action of sharing cannot even happen! It makes no sense in the real world.

Test 2: The Multiplication Inversion Test

Remember that division can always be checked using multiplication: If 6 ÷ 0 = ?, then ? x 0 must equal 6! Can you think of any number that, when multiplied by 0, equals 6? No! Remember the Zero Property of Multiplication: ANY number times 0 equals 0. Nothing times 0 can ever equal 6. Therefore, no answer can possibly exist!

+-----------------------+---------------------+-------------------------------+
| Division Expression   | Result              | Reason                        |
+-----------------------+---------------------+-------------------------------+
| 7 ÷ 1                 | 7                   | 1 group holds everything      |
| 7 ÷ 7                 | 1                   | Equal items and groups = 1    |
| 0 ÷ 7                 | 0                   | Nothing shared is nothing     |
| 7 ÷ 0                 | UNDEFINED           | You cannot divide by zero!    |
+-----------------------+---------------------+-------------------------------+

Chapter Practice Exercises

  1. Solve each problem (or write "undefined" if impossible): a. 9 ÷ 1 b. 8 ÷ 8 c. 0 ÷ 6 d. 12 ÷ 0 e. 45 ÷ 1 f. 0 ÷ 10
  2. Fill in the missing numbers: a. ___ ÷ 1 = 23 b. 15 ÷ ___ = 1 c. 0 ÷ 14 = ___ d. 38 ÷ ___ = 38
  3. Explain in your own words why 0 ÷ 5 = 0, but 5 ÷ 0 is impossible.
  4. An art studio has 0 pencils in a box. 6 students need a pencil. How many pencils does each student receive? Write the division equation.
  5. True or False: 10 ÷ 10 = 0. If false, write the correct equation and explain why.

Solutions and Step-by-Step Answers

  1. Solutions: a. 9 ÷ 1 = 9 b. 8 ÷ 8 = 1 c. 0 ÷ 6 = 0 d. 12 ÷ 0 is undefined (cannot divide by zero). e. 45 ÷ 1 = 45 f. 0 ÷ 10 = 0
  2. Missing values: a. 23 ÷ 1 = 23 b. 15 ÷ 15 = 1 c. 0 ÷ 14 = 0 d. 38 ÷ 1 = 38
  3. 0 ÷ 5 means splitting 0 items among 5 groups; each group gets 0 items, which is a sensible answer. But 5 ÷ 0 means taking 5 items and placing them into 0 groups, which cannot be done. Furthermore, checking with multiplication requires ? x 0 = 5, which has no solution because any number times 0 is 0.
  4. 0 pencils shared among 6 students: 0 ÷ 6 = 0 pencils per student.
  5. False. 10 ÷ 10 = 1. When a number is divided by itself, the result is always 1 because 10 items shared equally among 10 people gives each person exactly 1 item.