What Are the Properties of Division? (Rules and Examples)
What happens if you flip a division problem around? Or change its grouping? Unlike multiplication, division has a few rules that can easily trip you up.
The properties of division describe how numbers behave when they are divided. Important rules include division by 1, division by zero, dividing zero by a nonzero number, and the fact that division is not commutative or associative.
Let's make each one easy to remember.
Division Properties at a Glance
1. Division by 1 Property
Dividing any number by 1 gives the original number.
a ÷ 1 = a
24 ÷ 1 = 24
Think of it this way: if 24 candies are put into one group, that group still contains all 24 candies.
2. Division by Itself
Any nonzero number divided by itself equals 1.
a ÷ a = 1, when a ≠ 0
7 ÷ 7 = 1
25 ÷ 25 = 1
Why? Because 7 × 1 = 7 and 25 × 1 = 25.
3. Zero Divided by a Nonzero Number
Zero divided by any nonzero number is zero.
0 ÷ a = 0, when a ≠ 0
0 ÷ 6 = 0
0 ÷ 100 = 0
0 ÷ 6 = 0, while 6 ÷ 0 is undefined.
4. Division by Zero Is Undefined
5 ÷ 0 = undefined
100 ÷ 0 = undefined
Why? Division asks what number multiplied by the divisor gives the dividend. There is no number that you can multiply by 0 to get 5, 100, or any other nonzero number.
So remember:
5. Division Is Not Commutative
The commutative property means that switching the order does not change the answer. Division does not have this property.
Since the answers are different, division is not commutative.
With division, order matters.
6. Division Is Not Associative
The associative property is about grouping. Division is not associative, meaning changing the grouping can change the answer.
(24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3
But:
24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12
With division, parentheses matter.
7. Is Division Closed?
Closure depends on the number set you're using.
For example:
6 ÷ 2 = 3
but
5 ÷ 2 = 2.5
Since 2.5 is not a whole number, whole numbers are not closed under division.
5 ÷ 2 = 2.5
Does Division Distribute Over Addition?
There is an important detail here.
Division can be distributed when the same divisor applies to each term in the numerator:
(12 + 8) ÷ 4 = 20 ÷ 4 = 5
and
12 ÷ 4 + 8 ÷ 4 = 3 + 2 = 5
12 ÷ (2 + 4) ≠ 12 ÷ 2 + 12 ÷ 4
This is a common division mistake.
What Are the Division Properties of Exponents?
When you divide powers with the same nonzero base, subtract the exponents.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
x⁷ ÷ x³ = x⁷⁻³ = x⁴
Another example:
2⁵ ÷ 2² = 2³ = 8
This is called the quotient of powers property. It is an exponent rule rather than a basic arithmetic property such as division by 1 or division by zero.
Common Mistakes to Avoid
Keep these five traps in mind:
- ✓ Don't switch the dividend and divisor.
- ✓ Don't assume division is commutative.
- ✓ Don't change the grouping without checking the answer.
- ✓ Never divide by zero.
- ✓ When dividing powers with the same base, subtract the exponents rather than adding them.
