Commutative Law

In Mathematics, commutative law  deals with the arithmetic operations of addition and multiplication. However, it isn't used for the other two arithmetic operations, subtraction and division.. Let’s define commutative: “Commutative” comes from the word “commute” which can be defined as to move around or travel. According to commutative law or commutative property. If a and b are any two integers, the addition and multiplication of a and b produce the same result regardless of the position of a and b. It can be represented symbolically as:

a + b = b + a

a × b = b × a

For example, if 5 and 10 are the two numbers, then;

5 + 10 = 10 + 5 = 15

5 x 10 = 10 x 5 = 50

What is Commutative Law ? 

The commutative law states that when two numbers are added or multiplied, the final value remains the same regardless of the position of the two numbers. Or, to put it another way, the sequence in which we add or multiply any two real numbers has no effect on the outcome.

Let’s explain commutative law

So mathematically, if altering the order of the operands has no effect on the result of the arithmetic operation, that arithmetic operation is commutative.

Consider, A and B are two real numbers, then, as per this law;

A + B = B + A

A . B = B . A

Commutative Law of Addition

According to the commutative law of addition, when two numbers are added, the result is equal to the addition of their interchanged positions.

A + B = B + A

Examples:

1 + 2 = 2 + 1 = 3

4 + 5 = 5 + 4 = 9

-3 + 6 = 6 + (-3) = 6 - 3 = 3

This law does not apply to subtraction because if the first number is negative and the position is altered, the first number's sign is changed to positive, resulting in:

(-A) - B = -A – B ……(1)

When we change the position of the first and second number, we get;

B - (-A) = B + A ….(2)

As a result of equations 1 and 2, we get:

(-A) - B ≠ B - (-A)

For example: (-9) - 2 = -9 - 2 = -11

& 2-(-9) = 2 + 9 = 11

Hence, -11 ≠ 11.

Commutative Law of Multiplication

According to the commutative law of multiplication, the result of the multiplication of two numbers will remain the same, even if the positions of the numbers are interchanged.

Hence, A . B = B . A

Examples:

1 × 2 = 2 × 1 = 2

4 × 5 = 5 × 4 = 20

-3 × 6 = 6 × (-3) = -18

Why is Multiplication Commutative ?

Let's look at an example to understand.

We have a formula in math that says the same thing. It's as follows:

a * b = b * a

The different letters stand for different numbers. Notice how we have a * b on the left side of the equal symbol, while the b comes first on the right side of the equal sign. As a result, this formula also tells us that the sequence in which we multiply our numbers doesn't matter. Still we get the same answer. 

Even though this formula only shows two numbers, the commutative property of multiplication also applies when you multiply more than two numbers.When there are more than two numbers, we can arrange them in whatever order we wish. For instance, suppose we have:

3 * 5 * 2

We can switch it around to 3 * 2 * 5 or 5 * 3 * 2. We will end up with the same answer in either way.

Hence we can say that multiplication is commutative.

Commutative Law in Percentages

If we interchange or switch the order of the values while calculating percentages, the result remains the same.  Mathematically , we can say:

A% of B = B% of A

Example:

10% of 50 = 50% of 10

Since, 

10% of 50 can be written as  (10/100) x 50 = 5

50% of 10 can be written as (50/100) x 10 = 5

Thus, the answer remains the same.

Commutative Law of Sets

The term "set" refers to a grouping of items or objects. We've studied about the numerous types of operations that may be performed on sets, such as intersection, union, and difference.

According to the Commutative Law for Union of Sets and the Commutative Law for Intersection of Sets,

the sequence of the sets in which the operations are performed has no effect on the result.

If A and B are two distinct sets, then, according to commutative rule,

A ∪ B = B ∪ A  [ Represents Union of sets]

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A ∩  B = B ∩ A   [ Represents Intersection of sets]

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Consider an  example, if A = {1, 2, 3} and B = {3, 4, 5, 6}, then;

A Union B is represented as A ∪ B {1, 2, 3, 4, 5, 6}  …….. (i)

B Union A is represented as B ∪ A  {1, 2, 3, 4, 5, 6}   ……… (ii)

From (i) and (ii), we get;

A ∪ B = B ∪ A

Now, 

A intersection B is represented as  A ∩ B = {3}   ……..(iii)

B intersection A is represented as B ∩ A = {3}   ……..(iv)

From (iii) and (iv), we get;

A ∩ B = B ∩ A

As a result, the commutative law for the union and intersection of two sets was established.

Conclusion

We have discussed commutative law in maths, the Commutative property states that "changing the order of the operands does not change the result. We have seen commutative property only applies to multiplication and addition. However, subtraction and division does not follow commutative property.