#### ByManali

May 28, 2022

The beauty of mathematics lies in its wonderful way of coming up with universal truths about the worlds of numbers and figures. One such property is commutative property. The property belongs to the two arithmetic operations of addition and multiplication. Simply put, it states that the order in which two numbers are placed while adding or multiplying them won’t change their sum or product respectively.

To put it in other words,

• The commutative property of addition states that the sum of any two numbers remains the same, whatever order they may be added in.

In other words, for any two given numbers of whatever value‘a’ and ‘b’

a+b=b+a

For example, for two numbers ‘3’ and ‘4’,

3+4=7=4+3

• The commutative property of multiplication states that the product of two numbers remains the same, in what order they may be multiplied.

In other words, for any two given numbers of whatever value ‘a’ and ‘b’

a*b=b*a

For example, for two numbers ‘3’ and ‘4’,

3*4=12=4*3

Where does this property apply?

This property applies to the addition and multiplication of all the

• Natural Numbers
• Whole Numbers
• Fractions
• Decimals
• Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers

Where does the theory not apply?

The reader must note that this property doesn’t apply to them the other arithmetic operation – that is, subtraction and division. That is,

• For any two given numbers pf whatever Value, ‘a’ and ‘b’

a-b is not equal to b+a unless a=b

For example, for two numbers ‘3’ and ‘4’,

3-4=-1, which is not equal to 4-3 =1

• For any two given numbers of whatever value, ‘a’ and ‘b’

a/b=b/a unless a=b

For example, for two numbers ‘3’ and ‘4’,

¾ is not equal to 4/3

How to provide commutative property for two given numbers?

The reader can prove the commutative property for any two given numbers by adding them or multiplying them (as is required by the question) in the two different possible orders.

So, for example, to prove the commutative property of the addition of two given numbers 5 and 6, one must check if 5+6=6+5.

To solve this, one takes both the sides of equation one by one.

Left hand side (L.H.S) = 5+6 = 11

Right hand side (R.H.S.) = 6+5 = 11

Left-hand side (L.H.S) = Right-hand side (R.H.S.)

Hence the commutative property is proved.

And, to prove, to prove the commutative property of multiplication of two given numbers 5 and 6, one must check if 5*6=6*5.

To solve this, one takes both sides of the equation one by one.

Left hand side (L.H.S) = 5*6 = 55

Right hand side (R.H.S.) = 6*5 = 55

Left-hand side (L.H.S) = Right-hand side (R.H.S.)

Hence the commutative property is proved.

Associative property of addition and multiplication

A property very similar to the commutative property of addition and multiplication is the associative property of the same two arithmetic operations. The distributive property is another related property.

The associative property of addition and multiplication states that the sums and multiplication of any three or more numbers remain unchanged, no matter whatever the order they may be added.

Associative property of addition states that any three or more numbers can’t be added in any groups whatsoever, and their sum will remain the same.

That is, for any three given numbers a, b and c

(a+b)+c=a+(b+c)

The associative property of multiplication states that any three or more numbers can’t be multiplied in any groups whatsoever, and their product will remain the same.

That is, for any three given numbers a, b and c

(a*b)*c=a*(b*c)