Difference between Associative and Commutative Property
In mathematics, we have certain rules for the numbers we are working on. Properties are the rules that one follows when doing math. There are many properties in mathematics, like commutative, associative, distributive, identity, and inverse properties. The guide is about the difference between associative and commutative property.
These properties have rules for the four basic math operations: addition, subtraction, multiplication, and division. However, this article will only talk about commutative and associative properties. Rules that apply to addition and multiplication operations include commutative and associative characteristics.
These rules are utilized in mathematics to aid with algebra and problem-solving. Both properties seem very similar, so it’s essential to know the difference between them and how they can be used in laws of addition and multiplication.
So let’s look at what commutative and associative properties mean and how they differ from each other.
Commutative Property
The word “commute,” which means to move about, is where the commutative property originates from. According to this property, the numbers we use in addition and multiplication can be moved about or switched without changing the result.
For instance, if we add 6+4, the answer is 10. Similarly, if we add 4+6, the answer is 10. So regardless of how numbers are positioned, the answer always remains the same. This rule or property of addition refers to as commutative property of addition.
Likewise, if we multiply 2×3, the answer is 6, and if we multiply 3×2, the answer is 6 again. It doesn’t matter in which order the numbers are placed; the result would be the same always. This rule or property of multiplication is called the commutative property of multiplication.
Examples
Let’s have a look at some examples of this property to have a better understanding.
Commutative property of addition:
p + q = p + q
6 + 5 =11 similar to 5 + 6 = 11
10 + 20 = 30 similar to 20 + 10 = 30
In the above examples, you see no matter how you place the numbers while adding them. The answer will always remain the same.
Commutative property of multiplication:
p x q + q x p
4 x 5 = 20 similar to 5 x 4 = 20
8 x 10 = 80 similar to 10 x 8 = 80
From the examples, you will understand that how the numbers are positioned or placed will never affect the result of the multiplication process.
Associative Property
The associative property comes into action if we add or multiply three or more numbers. It says that no matter how the numbers are grouped when adding or multiplying three or more integers, the result will always be the same. Whether we add 1 + 2 first, then add 4 into it, or add 2 + 4 first, then put 1 into it, the result of adding 1 + 2 + 4 is always 7.
This property of addition is called the associative property of addition. The same property applies to the multiplication of three or more numbers. The answer of multiplying 5 x 3 x 2 will always be 30 regardless of whether we multiply 5 x 3 first or 3 x 2 first.
This property of multiplication is called the associative property of multiplication. To use this property, we use parenthesis to group numbers. In the examples below, you will be able to understand this better.
Examples:
Let’s have a look at some examples of this associative property to have a better understanding of it.
Associative property of addition:
(p + q) + r = p + (q + r) = p + q + r
(6 + 3) + 2 = 6 + (3 + 2) = 11
(10 + 20) + 30 = 10 + (20 + 30) = 60
As you can see, the sum will always be the same no matter how the numbers are grouped.
Associative property of multiplication:
(p x q) x r = p x (q x r) = p x q x r
(1 x 3) x 2 = 1 x (3 x 2) = 6
(30 x 40) x 10 = 30 x (40 x 10) = 12000
The examples above show that the product will always be the same regardless of how they are grouped or placed.
Difference between Associative and Commutative Properties
As mentioned above, associative and commutative properties seem to be very similar and can be very confusing for learners. That’s why it’s important to understand the key differences between the two. The following table will help you out in this regard.
| Commutative Property | Associative Property |
| The commutative property states that the sum or product of numbers will be the same no matter in which position or order they are written while adding or multiplying. | The associative property states that the sum or product of three or more numbers will be the same regardless of their grouping while adding or multiplying. |
| The commutative property comes from the word commute, meaning moving or going to another place. | The associative property comes from the word associate or grouping of something |
| Rule of commutative property of addition: p + q = p + q | Rule of associative property of addition: (p + q) + r = p + (q + r) |
| Rule of the commutative property of multiplication: p x q = p x q | Rule of associative property of multiplication: (p x q) x r = p x (q x r) |
| Example of commutative property of addition: 8 + 5 =13 similar to 5 + 8 = 13 | Example of associative property of addition: (3 + 2) + 1 = 3 + (2 + 1) = 6 |
| Example of commutative property of multiplication: 5 x 2 = 10 similar to 2 x 5 = 10 | Example of associative property of multiplication: (3 x 2) x 2 = 3 x (2 x 2) = 12 |
| The commutative property of addition involves 2 numbers | Associative property of addition involves grouping of 3 or more numbers |
| By definition, the commutative property of multiplication involves 2 numbers | By definition, the associative property of multiplication involves the grouping of 3 or more numbers |
| Real Life implementation: Survey of counting children in houses in a society. We can start counting from any house; the order of houses won’t affect the result. | Real Life implementation: Going to a shop to buy 3 or more things. For the bill, we can add any 2 things first and then the third. No matter how we group for adding, it won’t affect the total bill. |
Frequently Asked Questions
Can you add and multiply three integers using the commutative property?
The commutative property only applies to two numbers; however, three numbers provide the same outcome. This is because we may utilize this attribute on two out of three integers in different combinations.
What procedures have both commutative and associative properties?
Commutative and associative properties apply to multiplication and addition.
Are the commutative and associative characteristics the same?
The numbers can be rearranged without affecting the result, but changing the grouping of the numbers affects the result.
Which operations are not commutative and associative?
Commutative and associative operations don’t apply to division and subtraction.
What is the key difference between commutative and associative property?
Commutative comes from the word commute, meaning changing or re-positioning. Associative comes from the word associate or grouping. Always remember that by commutative, we mean relocating the numbers without affecting the result. By associative, we mean re-grouping three or more numbers without affecting the result.
The associative and commutative characteristics should be used for what reasons?
The associative and commutative characteristics in mathematics are constant rules that apply to addition and multiplication.
Does the associativity property involve commutativity?
No. Associativity differs from commutativity, which examines whether the placement of two operands influences the outcome.
Conclusion
In summary, Commutative and associative properties can be quite confusing. But if you understand the gist of them, then they are quite simple to understand. Always remember that the commutative word originates from the commute, which means to move.
Associative property originates from the word grouping, so in a commutative property. We rearrange or move the numbers; in an associative property, we regroup the numbers. In both cases, the result remains the same.
Another thing that can be a key difference is that commutative law applies when we are using 2 numbers. In comparison, associative property usually applies when there are 3 or more 3 numbers.
It can be done by answering the question: are we changing the order of the numbers, or are we grouping them?
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