What happens when you switch two numbers around in an addition problem? Or change the way three numbers are grouped? Sometimes the answer stays exactly the same—and sometimes it doesn't.
That is where the properties of rational numbers come in. They describe how rational numbers behave when we add, subtract, multiply, or divide them.
The six commonly taught properties are closure, commutative, associative, distributive, identity, and inverse.
The Six Properties of Rational Numbers at a Glance
| Property | What it means | Example |
|---|---|---|
| Closure | The result stays rational | 1/2 + 1/3 = 5/6 |
| Commutative | Changing the order does not change the answer | 1/2 + 1/3 = 1/3 + 1/2 |
| Associative | Changing the grouping does not change the answer | (1/2 + 1/3) + 1/6 = 1/2 + (1/3 + 1/6) |
| Distributive | Multiplication can be distributed over addition or subtraction | 2(3 + 4) = 2×3 + 2×4 |
| Identity | A special number leaves the original number unchanged | 5/7 + 0 = 5/7 |
| Inverse | A number can be paired with another to produce an identity | 3/5 + (-3/5) = 0 |
1. Closure Property
The closure property means that performing an operation on rational numbers gives another rational number.
Rational numbers are closed under:
For example:
1/2 + 2/3 = 7/6
Since 7/6 is rational, closure holds.
For division:
2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6
The answer is also rational.
The important catch? You cannot divide by zero.
2. Commutative Property
For rational numbers, addition and multiplication are commutative:
a + b = b + a
a × b = b × a
For example:
2/5 + 1/3 = 1/3 + 2/5
But subtraction and division are not commutative.
2/3 − 1/3 ≠ 1/3 − 2/3
So, changing the order does not always work.
3. Associative Property
Addition and multiplication of rational numbers are associative:
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
For example:
(1/2 + 1/3) + 1/6 = 1/2 + (1/3 + 1/6)
Both sides equal 1.
Subtraction and division are not associative.
Quick memory trick
Commutative → change the order
Associative → change the grouping
4. Distributive Property
The distributive property tells us that multiplication can be distributed over addition or subtraction.
a(b − c) = ab − ac
For example:
2(3 + 4) = 2×3 + 2×4
14 = 6 + 8
14 = 14
This property becomes especially useful when simplifying algebraic expressions.
5. Identity Property
An identity is a number that leaves another number unchanged.
There are two important identities:
For any rational number a:
a + 0 = a
a × 1 = a
For example:
7/8 + 0 = 7/8
and
7/8 × 1 = 7/8
The easiest way to remember this is:
0 does nothing when you add it; 1 does nothing when you multiply by it.
A Quick Way to Remember the Properties
Before a test, use this cheat sheet:
- Closure: Does the answer stay in the set?
- Commutative: Can I change the order?
- Associative: Can I change the grouping?
- Distributive: Can I distribute multiplication?
- Identity: What number leaves it unchanged?
- Inverse: What number undoes it?
Which property is shown here?
Answer: The commutative property of addition, because the order of the numbers changed but the answer stayed the same.
