In a commutative law we can swap numbers around and still get the same answer when added. This also holds for multiplication.

A fundamental property of many binary operations is commutative property, there are many mathematical proofs which depend on it. Also known to be associated with functions.

Among basic arithmetic operations division and subtraction are not commutative where as addition and multiplication are said to be commutative. Here changing the order will not change the result and the term 'Commutative' can also be used in several related senses.

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Two elements a and b of a set are said to be commutative under a binary operation '+' if they satisfy

**a + b = b + a**Real numbers are said to be commutative under addition.Any finite sum is unaltered by reordering its terms or factors. Commutative property of addition holds for many systems such as the real or complex numbers.

Two elements a and b of a set are said to be commutative under a binary operation * if they satisfy:

**a * b = b * a**Real numbers are said to be commutative under multiplication. A product is not changed at rearrangement of its factors. For a p * p matrices commutativity of multiplication is invalid. Scalar multiplication of two vectors is said to be commutative.

A mathematical concept generalizes the concept of a set of all vectors of ordinary three-dimensional space.

**Addition of vectors obeys the commutative law:**

From the above figure we see that

a + b = b + a

In terms of the following components

$\vec{a}$ = (a$_{1}, a_{2}, a_{3}$) $\vec{b}$ = (b$_{1}, b_{2}, b_{3}$)

For addition, $\vec{a + b}$ = (a$_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}$)

**Some problems based on commutative law are given below:**

**Example 1: ** Is ( 3 * 5) * 4 = 4 * (3 * 5)?

**Solution:**

Consider LHS = (3 * 5) * 4

(3 * 5) = 15

and 15 * 4 = 60

Now consider 3 * (5 * 4)

Consider 5 * 4 = 20

now 3 * 20 = 60

We see that LHS = RHS = 60

Therefore (3 * 5) * 4 = 3 * (5 * 4), commutative law of multiplication is satisfied.

**Example 2:** Is (4 + 5) + (5 + 2) = (5 + 2) + (4 + 5)?

**Solution:**

Consider LHS (4 + 5) + (5 + 2),

On adding all the terms , (4 + 5) + (5 + 2) = 9 + 7

we get 16

Similarly consider the RHS (5 + 2) + (4 + 5)

On adding all the terms above, we get 7 + 9 = 16

Clearly it is evident that LHS =RHS

There (4 + 5) + (5 + 2) = (5 + 2) + (4 + 5) commutative law of addition is satisfied.

Two elements a and b of a set are said to be commutative under a binary operation * if they satisfy:

A mathematical concept generalizes the concept of a set of all vectors of ordinary three-dimensional space.

From the above figure we see that

a + b = b + a

In terms of the following components

$\vec{a}$ = (a$_{1}, a_{2}, a_{3}$) $\vec{b}$ = (b$_{1}, b_{2}, b_{3}$)

For addition, $\vec{a + b}$ = (a$_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}$)

Consider LHS = (3 * 5) * 4

(3 * 5) = 15

and 15 * 4 = 60

Now consider 3 * (5 * 4)

Consider 5 * 4 = 20

now 3 * 20 = 60

We see that LHS = RHS = 60

Therefore (3 * 5) * 4 = 3 * (5 * 4), commutative law of multiplication is satisfied.

Consider LHS (4 + 5) + (5 + 2),

On adding all the terms , (4 + 5) + (5 + 2) = 9 + 7

we get 16

Similarly consider the RHS (5 + 2) + (4 + 5)

On adding all the terms above, we get 7 + 9 = 16

Clearly it is evident that LHS =RHS

There (4 + 5) + (5 + 2) = (5 + 2) + (4 + 5) commutative law of addition is satisfied.

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