Commutative Property – Definition & Examples

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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.

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.

And another set of $5$ balls.

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

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

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

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.

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.

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 $A + B = B + A$

$57 + 32 = 89$.

And $32 + 57 = 89$.

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

Commutative Property of Multiplication

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

$57 \times 32 = 1824$.

And, $32 \times 57 = 1824$.

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

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

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

• Distributive Property – Meaning & Examples
• Associative Property – Meaning & Examples
• Decimal Number System – With Types & Properties

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