What is the Difference Between Associative and Commutative Properties?
The concept of Associative Property plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Associative Property?
The Associative Property in maths means that the way numbers are grouped when adding or multiplying does not change the final result. For example, grouping numbers like (a + b) + c or a + (b + c) gives the same total. You’ll find this concept applied in properties of addition, properties of multiplication, and rational number operations.
Key Formula for Associative Property
Here’s the standard formula for associative property:
Addition: (a + b) + c = a + (b + c)
Multiplication: (a × b) × c = a × (b × c)
Cross-Disciplinary Usage
Associative property is not only useful in Maths but also plays an important role in Physics for groupings in equations, in Computer Science for programming logic, and in daily logical reasoning. Students preparing for JEE or other board exams often see associative property in several calculation-based questions.
Step-by-Step Illustration
- Let’s check associative property in addition:
Numbers: 2, 3, 4
(2 + 3) + 4 = 5 + 4 = 9
2 + (3 + 4) = 2 + 7 = 9
So, (2 + 3) + 4 = 2 + (3 + 4) - Now check associative property in multiplication:
Numbers: 2, 3, 4
(2 × 3) × 4 = 6 × 4 = 24
2 × (3 × 4) = 2 × 12 = 24
So, (2 × 3) × 4 = 2 × (3 × 4)
Associative Property: Non-Examples
Associative property does not apply for subtraction and division. See the table below:
| Operation | Is Associative? | Example | Result |
|---|---|---|---|
| Addition | Yes | (2 + 3) + 4 = 2 + (3 + 4) | 9 = 9 |
| Multiplication | Yes | (2 × 3) × 4 = 2 × (3 × 4) | 24 = 24 |
| Subtraction | No | (7 − 3) − 2 ≠ 7 − (3 − 2) | 2 ≠ 6 |
| Division | No | (16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷ 2) | 2 ≠ 8 |
Associative Property of Addition
When you add three or more numbers, changing how they are grouped does not change the sum. Example:
Let’s add 5, 7, and 2:
(5 + 7) + 2 = 12 + 2 = 14
5 + (7 + 2) = 5 + 9 = 14
So, (5 + 7) + 2 = 5 + (7 + 2). The sum is the same no matter how you group them.
Associative Property of Multiplication
The product of numbers also does not change even if you change the grouping. Example:
(4 × 3) × 2 = 12 × 2 = 24
4 × (3 × 2) = 4 × 6 = 24
Again, the product remains the same even if the brackets are placed differently.
Difference: Associative vs Commutative Property
| Property | Definition | Example |
|---|---|---|
| Associative | Grouping can change; result stays the same. | (a + b) + c = a + (b + c) |
| Commutative | Order can change; result stays the same. | a + b = b + a |
Tip: "Associative" means "association" or grouping. "Commutative" means "commute" or change of position/order.
Learn more at Commutative Property.
Classroom Tip
A quick way to remember the associative property: Whenever you see a sum or product with more than two numbers and brackets are shifting places, it’s likely the associative property. Vedantu’s teachers use songs and hand movements to show regrouping in fun ways during live online classes.
Try These Yourself
- Show that (8 + 9) + 5 = 8 + (9 + 5)
- Is (6 × 2) × 10 = 6 × (2 × 10)?
- Does the associative property work for 18 − (7 − 2)? Try both groupings.
- Identify the property applied in: (a × b) × c = a × (b × c)
Frequent Errors and Misunderstandings
- Confusing associative property with commutative property—remember, associative involves grouping, not order.
- Using it with subtraction or division, which is incorrect.
- Forgetting that at least three numbers are needed for the associative property.
Relation to Other Concepts
The idea of associative property connects closely with topics such as distributive property and maths properties. Mastering this helps with understanding algebraic expressions and solving equations in later chapters.
We explored Associative Property—from its definition, formula, and examples, to mistakes and connections to other math and science concepts. Continue practicing with Vedantu to become confident in solving problems using this important property.
See also: Commutative Property, Properties of Addition, Properties of Multiplication, Properties of Rational Numbers
FAQs on Associative Property in Maths: Addition, Multiplication, Examples & Non-Examples
1. What is the associative property in maths?
The associative property in math states that you can regroup numbers when adding or multiplying without changing the result. This means the order in which you perform the operations doesn't matter, as long as the operations are all addition or all multiplication. The grouping of numbers is done using parentheses.
2. How do you identify the associative property in a sum or product?
Look for expressions where three or more numbers are either added or multiplied. If the numbers are grouped differently using parentheses but the final answer remains the same, then the associative property is being applied. For example: (2 + 3) + 4 = 2 + (3 + 4) and (2 × 3) × 4 = 2 × (3 × 4).
3. What are some examples of the associative property?
Addition: (1 + 2) + 3 = 1 + (2 + 3) = 6
Multiplication: (2 × 4) × 5 = 2 × (4 × 5) = 40
4. For which operations does the associative property work?
The associative property only applies to addition and multiplication. It does not work for subtraction or division.
5. What is the difference between associative and commutative properties?
The associative property deals with the grouping of numbers in addition or multiplication, while the commutative property deals with the order of numbers. The commutative property states that the order of numbers does not change the result (a + b = b + a; a × b = b × a). The associative property doesn't change the order, only the grouping.
6. Does the associative property apply to rational numbers?
Yes, the associative property applies to rational numbers for both addition and multiplication. For example: (1/2 + 1/4) + 1/8 = 1/2 + (1/4 + 1/8) and (1/2 × 1/4) × 1/8 = 1/2 × (1/4 × 1/8).
7. How is the associative property useful in simplifying calculations?
The associative property allows you to rearrange numbers to make calculations easier. For example, 98 + 2 + 100 is easier to solve as (98 + 2) + 100 = 200 rather than adding left to right. This is particularly helpful for mental math.
8. Why doesn't the associative property work for subtraction or division?
Because subtraction and division are not commutative operations. The order of the numbers significantly affects the outcome. Changing the grouping alters the order of operations, leading to different results. For example, (10 - 5) - 2 = 3, but 10 - (5 - 2) = 7.
9. Can you give an example of a real-world application of the associative property?
Imagine you're buying groceries: If you buy items costing ₹20, ₹30, and ₹50, the total cost remains the same regardless of how you group the prices: (₹20 + ₹30) + ₹50 = ₹20 + (₹30 + ₹50) = ₹100.
10. What are some common mistakes students make when applying the associative property?
A common mistake is applying the associative property to subtraction or division. Another is confusing the associative property with the commutative property, failing to understand the difference between changing the order and changing the grouping of numbers.
11. How can the associative property be used to simplify algebraic expressions?
The associative property helps simplify algebraic expressions by allowing you to regroup terms to combine like terms more easily. For example, (x + 2y) + 3x can be simplified as x + 3x + 2y = 4x + 2y.
12. Does the associative property apply to matrices?
Matrix multiplication is associative; that is, for matrices A, B, and C for which the products are defined, (AB)C = A(BC). However, matrix addition is also associative: (A+B)+C = A+(B+C).
