What Operations Are Commutative? Explained with 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: Definition, Explanation & Examples
1. What is the commutative law in math and English?
The commutative law in mathematics states that the order of numbers doesn't affect the result in addition and multiplication. This means that a + b = b + a and a × b = b × a. In English grammar, a comparable concept might be the order of adjectives in a sentence sometimes not significantly altering the meaning, though this is not a strict parallel.
2. Can you give examples of the commutative law?
Here are some examples:
Addition: 5 + 2 = 2 + 5 = 7
Multiplication: 4 × 6 = 6 × 4 = 24
These show how switching the numbers around doesn't change the final answer.
3. Does the commutative law apply to subtraction and division?
No, the commutative law does not apply to subtraction or division. The order of the numbers matters significantly: 10 - 4 ≠ 4 - 10 and 12 ÷ 3 ≠ 3 ÷ 12.
4. How is the commutative law used in Boolean algebra?
In Boolean algebra, the commutative law applies to logical operations like AND and OR. A AND B is equivalent to B AND A, and A OR B is equivalent to B OR A. This reflects the commutative property in set theory where A ∪ B = B ∪ A and A ∩ B = B ∩ A.
5. What is the difference between the commutative and associative laws?
The commutative law concerns the order of numbers in an operation (a + b = b + a). The associative law concerns the grouping of numbers in an operation: (a + b) + c = a + (b + c). The commutative law doesn't apply to subtraction and division; the associative law does not always apply to subtraction.
6. Why is the commutative law important in real-life calculations?
The commutative law simplifies calculations. In situations like adding up costs or finding the area of a rectangle, we can change the order to make calculations easier. For example, when adding a series of numbers it is often quicker to combine numbers that add up to easy numbers such as 10.
7. How do programmers use the commutative property in algorithms?
Programmers use the commutative property to optimize algorithms and improve efficiency. For example, when calculating the product of several values, they might rearrange the order for faster processing or to reduce memory requirements.
8. Can the commutative law be extended to matrices or vectors?
Matrix multiplication is generally not commutative (A × B ≠ B × A). Vector addition is commutative, while vector multiplication (dot product and cross product) has different commutative properties depending on the type of product.
9. Is there a linguistic equivalent to the commutative law in English grammar?
While not a direct equivalent, the order of some adjectives in a sentence can be changed without significantly altering the meaning (e.g., "a big red ball" vs. "a red big ball"). However, this is not a strict parallel to the mathematical commutative law which only applies to addition and multiplication.
10. What are common mistakes students make with the commutative law?
Common mistakes include incorrectly applying the commutative law to subtraction and division, confusing it with the associative or distributive laws, or failing to recognize its application in different mathematical contexts (e.g., Boolean algebra).
11. What is the commutative property of sets?
For sets, the commutative property applies to union (A ∪ B = B ∪ A) and intersection (A ∩ B = B ∩ A), meaning the order in which you combine sets doesn't change the resulting set.
