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Commutative Property – Definition & Examples

commutative property

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The commutative property is one of the fundamental properties in mathematics. It refers to the order of numbers in arithmetic operations like addition and subtraction.

The commutative property along with associative property and distributive property helps you to solve problems more efficiently and quickly and also helps in reducing the chances of committing mistakes.

What is Commutative Property?

Let us understand the commutative property with the help of an example:

Let us suppose we have to find the sum of $15$ and $25$. To do this, you will write $15 + 25$. What will happen, if you write $25 + 15$? Still, the result remains the same. Here, you notice that $15 + 25 = 25 + 15 = 40$.

The same is the case with multiplication. The product of $3$ and $7$ can be written as $3 \times 7$. It can also be written as $7 \times 3$. In both cases, the product is $21$, i.e., $3 \times 7 = 7 \times 3 = 21$.

The commutative property states that the result of the operation of two numbers remains the same even if the order of numbers is changed or the numbers swap positions.

Mathematically it can be stated as

If $A$ and $B$ are two numbers then,  

  • $A + B = B + A$ (In the case of addition of numbers)
  • $A \times B = B \times A$ (In the case of multiplication of numbers)

Note: The commutative property does not hold for subtraction and division.

Difference Between Infinity & Not Defined

Commutative Property of Addition

The commutative property of addition states that $A + B = B + A$.

Let’s understand this property by this example.

Suppose you have a set of $7$ balls.

commutative property
A set of $7$ balls

And another set of $5$ balls.

commutative property
A set of $5$ balls

Now, you put all these balls of two sets together (set 1 and set 2).

commutative property
Two sets of balls – Set 1 with $7$ balls and Set 2 with $5$ balls

How many balls are there? There are a total $12$ balls

Now, first, keep balls from set 2 and then set 1.

commutative property
Two sets of balls – Set 2 with $5$ balls and Set 1 with $7$ balls

Still, the total number of balls is $12$.

So, what did you notice?

$7 + 5 = 12$ and also $5 + 7 = 12$.

This is the commutative property of addition.

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Maths can be really interesting for kids

Commutative Property of Multiplication

The commutative property of multiplication states that $A \times B = B \times A$.

Let’s again consider an example of balls. You go to a shop to buy some balls with your friend. The shop owner has balls available in two sets.

commutative property
Set 1 has $3$ balls
commutative property
Set 2 has $4$ balls

You buy $4$ sets of set 1 and your friend buys $3$ sets of set 2.

How many balls did you buy? Of course $3 \times 4 = 12$.

And you friend bought $4 \times 3 = 12$.

Here also, you can notice that the product $3 \times 4$ is the same as the product $4 \times 3$.

That is $3 \times 4 = 4 \times 3 = 12$.

This is the commutative property of multiplication.

Is Commutative Property Applicable to Numbers of All Categories?

The commutative property in both the two forms – the commutative property of addition and the commutative property of multiplication works well with any real number

Note: A set of real numbers $R$ is a superset of sets of natural numbers, a set of whole numbers, a set of integers, a set of rational numbers, and a set of irrational numbers.

Natural Numbers and Whole Numbers

Consider any two whole numbers $57$  and $32$.

Commutative Property of Addition

We want to verify that $A + B = B + A$

commutative property

$57 + 32 = 89$.

commutative property

And $32 + 57 = 89$.

So, $57 + 32 = 32 + 57$.

Commutative Property of Multiplication

We want to verify that $A \times B = B \times A$.

commutative property

$57 \times 32 = 1824$.

commutative property

And, $32 \times 57 = 1824$.

Therefore, $57 \times 32 = 32 \times 57$.

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Integers (Signed Numbers)

Consider any two integers $-17$  and $15$.

Commutative Property of Addition

Adding $-17$  and $15$. $-17 + 15 = -2$.

And adding $15$ and $-17$ is $15 + \left( -17 \right) = -2$. For detail see Addition and Subtraction of Integers.

Therefore, $-17 + 15 = 15 + \left(-17 \right)$.

Commutative Property of Multiplication

Multiplying $-17$ by $15$. $-17 \times 15 = -255$.

And multiplying $15$ by $-17$ is $15 \times \left( -17 \right) = -255$. For detail see Multiplication and Division of Integers.

So, $-17 \times 15 = 15 \times \left(-17 \right)$.

Decimal Numbers

Consider any two decimal numbers $A = 2.9$ and $B = 6.3$.

Commutative Property of Addition

Adding $2.9$  and $6.3$. $2.9 + 6.3 = 9.2$.

And adding $6.3$ and $2.9$ is $6.3 + 2.9 = 9.2$. For detail see Addition and Subtraction of Decimals.

Therefore, we observe that $2.9 + 6.3 = 6.3 + 2.9$.

Commutative Property of Multiplication

Multiplying $2.9$  and $6.3$. $2.9 \times 6.3 = 18.27$.

And multiplying $6.3$ and $2.9$ is $6.3 \times 2.9 = 18.27$. For detail see Multiplication and Division of Decimals.

Therefore, we observe that $2.9 \times 6.3 = 6.3 \times 2.9$.

Fractions

Consider any two fractions $A = \frac{3}{5}$ and $B =\frac{7}{9}$.

Commutative Property of Addition

Adding $\frac{3}{5}$  and $\frac{7}{9}$.

$\frac{3}{5} + \frac{7}{9} = \frac{3\times 9 + 7\times 5}{5 \times 9} =  \frac{27 + 35}{45} = \frac{62}{45} = 1\frac{17}{45}$.

And adding $\frac{7}{9}$  and $\frac{3}{5}$.

$\frac{7}{9} + \frac{3}{5} = \frac{7 \times 5 + 3 \times 9}{9 \times 5} = \frac{35 + 27}{45} = \frac{62}{45} = 1\frac{17}{45}$.

For details check Addition and Subtraction of Fractions.

Therefore, $\frac{3}{5} + \frac{7}{9} = \frac{7}{9} + \frac{3}{5}$.

Commutative Property of Multiplication

Multiplying $\frac{3}{5}$  and $\frac{7}{9}$.

$\frac{3 \times 7}{5 \times 9} = \frac{21}{45}$.

Multiplying $\frac{7}{9}$  and $\frac{3}{5}$.

$\frac{7 \times 3}{9 \times 5} = \frac{21}{45}$.

For details check Multiplication and Division of Fractions.

Therefore, $\frac{3}{5} \times \frac{7}{9} = \frac{7}{9} \times \frac{3}{5}$.

$6$ Amazing Facts about Numbers

Irrational Numbers

Consider any two irrational numbers $A = 2\sqrt{3}$, and $C = 5\sqrt{3}$.

Commutative Property of Addition

Adding $2\sqrt{3}$  and $5\sqrt{3}$.

$2\sqrt{3} + 5\sqrt{3} = \left(2 + 5\right) \sqrt{3} = 7\sqrt{3}$.

Adding $5\sqrt{3}$  and $2\sqrt{3}$.

$5\sqrt{3} + 2\sqrt{3} = \left(5 + 2\right) \sqrt{3} = 7\sqrt{3}$.

Therefore, $2\sqrt{3} + 5\sqrt{3} = 5\sqrt{3} + 2\sqrt{3}$.

Commutative Property of Multiplication

Multiplying $2\sqrt{3}$  and $5\sqrt{3}$.

$2\sqrt{3} \times 5\sqrt{3} = 2 \times 5 \times \sqrt{3} \times \sqrt{3} = 10 \times 3 = 30$.

Multiplying $5\sqrt{3}$  and $2\sqrt{3}$.

$2\sqrt{3} \times 5\sqrt{3} = 2 \times 5 \times \sqrt{3} \times \sqrt{3} = 10 \times 3 = 30$.

Therefore, $2\sqrt{3} \times 5\sqrt{3} = 5\sqrt{3} \times 2\sqrt{3}$.

Does the Commutative Property for Subtraction Hold?

Let’s verify whether the cumulative property of subtraction is true, i.e., $A – B = B – A$.

To check this, let’s consider two numbers $A = 17$ and $B = 12$.

$A – B = 17 – 12 = 5$ and $B – A = 12 – 17 = -5$.

Since $5 \ne -5$, therefore, $A – B \ne B – A$.

Does the Commutative Property for Division Hold?

Again consider the two numbers $A = 18$ and $B = 3$.

$A \div B = 18 \div 3 = 6$ and $B \div A = 3 \div 18 = \frac {3}{18} = \frac {1}{6} = 0.1666…$.

Again, we see that the commutative property does not hold for the division also.

Note 

  • The commutative property does not hold for subtraction
  • The commutative property does not hold for division

Conclusion

The commutative property of addition and multiplication states that the result remains the same irrespective of the order in which the operations of addition or subtraction are performed on the numbers. But the commutative property is valid for subtraction and division.

Practice Problems

  1. Verify the commutative property of addition for the following numbers.
    1. $A = 5, B = 17$
    2. $A = 12, B = 19$
    3. $A = 19, B = 28$
    4. $A = 27, B = 42$
  2. Verify the commutative property of multiplication for the following numbers.
    1. $A = 5, B = 19$
    2. $A = 11, B = 14$
    3. $A = 12, B = 10$
    4. $A = 5, B = 38$

Recommended Reading

FAQs

What is commutative property example?

The commutative property applies to addition and multiplication. It means that if you change the position of numbers in addition and multiplication, the result remains the same.

For example, consider two numbers $8$ and $4$, then in the case of addition, $8 + 4 = 4 + 8 = 12$ and in the case of multiplication, $8 \times 4 = 4 \times 8 = 32$.

What is commutative property formula?

The commutative formula for addition is $A + B = B + A$. If you take $A = 3$ and $B = 9$, then $3 + 9 = 9 + 3 = 12$. Similarly, the commutative formula for multiplication is $A \times B = B \times A$. If you take $A = 3$ and $B = 9$, then $3 \times 9 = 9 \times 3 = 27$.

What is commutative property of addition?

The commutative property of addition states that the order in which numbers are taken for addition does not change the result. Mathematically, the commutative property of addition can be stated as $A + B = B + A$.

For example, if you have two numbers $11$ and $17$, then the sum $11 + 17$ will be the same as the sum $17 + 11$. In both cases, the sum is $28$.

What is commutative property of multiplication?

The commutative property of multiplication states that the order in which numbers are taken for multiplication does not change the result. Mathematically, the commutative property of addition can be stated as $A \times B = B \times A$.

For example, if you have two numbers $11$ and $17$, then the sum $11 \times 17$ will be the same as the sum $17 \times 11$. In both cases, the sum is $187$.

What are the major four properties in Maths?

The four major properties in Maths are –
the Commutative property, the Associative property, the Distributive property, and Identity property.

What is the difference between commutative and associative property?

The commutative property deals with the ordering of numbers and states that the ordering of numbers in addition and multiplication does not change the result. The associative property deals with the grouping of numbers and states that the grouping of numbers in addition and multiplication does not change the result.

Can Commutative Property be Used for Subtraction and Division?

No, the commutative property does not hold for subtraction and division. It only holds for the addition and multiplication of numbers.

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