What Is the Commutative Law Formula and Solved Examples
The concept of commutative law plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing how and where the commutative property is used—mainly in addition and multiplication—helps students solve maths problems more quickly and reduces errors, especially in competitive exams and school assignments.
What Is Commutative Law?
The commutative law is a basic arithmetic rule that says the order in which two numbers are added or multiplied does not change their sum or product. In simpler words, swapping the numbers does not affect the answer. You’ll find this law applied in areas such as number operations, algebra, computer science (logic gates), and set theory.
Key Formula for Commutative Law
Here’s the standard formula:
Addition: a + b = b + a
Multiplication: a × b = b × a
Step-by-Step Illustration
- Addition Example:
Let’s add 7 and 12.
1. 7 + 12 = 19
2. Now, swap the numbers: 12 + 7 = 19
The answer remains the same, showing the commutative property. - Multiplication Example:
Multiply 8 and 3.
1. 8 × 3 = 24
2. 3 × 8 = 24
Again, changing the order does not affect the result.
Table: More Examples of Commutative Law
| Operation | Example 1 | Example 2 | Example 3 |
|---|---|---|---|
| Addition | 4 + 5 = 5 + 4 = 9 | 13 + 2 = 2 + 13 = 15 | a + b = b + a |
| Multiplication | 7 × 6 = 6 × 7 = 42 | 2 × 15 = 15 × 2 = 30 | m × n = n × m |
| Set Union | A ∪ B = B ∪ A | {1,2} ∪ {3,4} = {3,4} ∪ {1,2} | - |
Where Does Commutative Law Not Apply?
The commutative property does not work for subtraction or division. This is a common source of mistakes for students. For example:
- Subtraction: 10 − 4 ≠ 4 − 10 (6 ≠ −6)
- Division: 12 ÷ 3 ≠ 3 ÷ 12 (4 ≠ 0.25)
So, always remember: only addition and multiplication (and a few set operations) follow this law!
Commutative Law in Sets and Boolean Algebra
In set theory, commutative property holds for union and intersection:
Union: A ∪ B = B ∪ A
Intersection: A ∩ B = B ∩ A
In Boolean algebra (a branch of algebra for logic gates), it also holds:
A + B = B + A (OR operation) and A · B = B · A (AND operation).
This property helps make simplifications in maths and computer science.
Comparison: Commutative, Associative, and Distributive Laws
| Law | Formula | Which Operations? |
|---|---|---|
| Commutative Law | a + b = b + a a × b = b × a |
Addition, Multiplication |
| Associative Law | (a + b) + c = a + (b + c) (a × b) × c = a × (b × c) |
Addition, Multiplication |
| Distributive Law | a × (b + c) = (a × b) + (a × c) | Multiplication over Addition |
Frequent Errors and Misunderstandings
- Thinking subtraction or division are commutative (they are not!)
- Mixing up commutative with associative or distributive property
- Not checking operation type before applying commutative law
Relation to Other Concepts
The commutative law is closely related to the associative law and distributive law. Mastering commutative property makes it easier to learn about properties of addition, properties of multiplication, and sets in later chapters.
Try These Yourself
- Is 25 × 2 the same as 2 × 25?
- Does 5 − 3 equal 3 − 5? Why or why not?
- Give an example of commutative law using sets.
- Find two real-life cases where changing order does not change the outcome.
Classroom Tip
An easy way to remember the commutative law: "Order Doesn’t Matter" for addition and multiplication. In Vedantu classes, teachers often use a trick: imagine candies on your left and right—no matter which you eat first, the total is the same!
We explored commutative law—from its definition, formula, stepwise examples, and where it does not apply, to how it relates with other rules. Keep practicing with Vedantu to get even better at maths and ace your exams using this important rule!
For more practice and related concepts, check out: Associative Law, Distributive Law, Properties of Addition, and Properties of Multiplication.
FAQs on Commutative Law in Maths with Meaning and Proof
1. What is the commutative law in maths?
The commutative law states that changing the order of numbers does not change the result for certain operations like addition and multiplication.
- For addition: a + b = b + a
- For multiplication: a × b = b × a
- It does not apply to subtraction or division.
2. What is the formula for the commutative property?
The formula for the commutative property is a + b = b + a and a × b = b × a.
- Addition form: Order of addends can switch.
- Multiplication form: Order of factors can switch.
- Valid for real numbers, integers, fractions, and decimals.
3. Can you give an example of the commutative law?
An example of the commutative law is 4 + 7 = 7 + 4 = 11.
- Addition example: 9 + 3 = 3 + 9 = 12
- Multiplication example: 5 × 2 = 2 × 5 = 10
4. Does the commutative law apply to subtraction?
No, the commutative law does not apply to subtraction because changing the order changes the result.
- Example: 8 − 3 = 5
- But: 3 − 8 = −5
5. Does the commutative law apply to division?
No, the commutative law does not apply to division because reversing numbers changes the quotient.
- Example: 12 ÷ 4 = 3
- But: 4 ÷ 12 = 1/3
6. Why is the commutative property important?
The commutative property is important because it simplifies calculations and mental math.
- It allows flexible rearranging of numbers.
- Makes addition and multiplication easier.
- Helps in algebraic simplification and solving equations.
7. What is the difference between commutative and associative laws?
The commutative law changes the order of numbers, while the associative law changes the grouping of numbers.
- Commutative: a + b = b + a
- Associative: (a + b) + c = a + (b + c)
- Both apply to addition and multiplication.
8. Is multiplication always commutative?
Yes, multiplication of real numbers is always commutative, meaning a × b = b × a.
- Example: 6 × 8 = 8 × 6 = 48
- Applies to integers, fractions, decimals, and algebraic terms.
9. How do you prove the commutative property of addition?
The commutative property of addition can be shown using number line reasoning or counting principles.
- Start at 0, move 3 steps, then 5 steps → reach 8.
- Start at 0, move 5 steps, then 3 steps → reach 8.
- Both ways give the same total.
10. Does the commutative law apply to algebraic expressions?
Yes, the commutative law applies to algebraic expressions involving addition and multiplication.
- Addition: x + y = y + x
- Multiplication: 3a × b = b × 3a
- Helps in rearranging terms while simplifying expressions.
