Operation Commutative Closed or not
Addition a  + b = b + a True
Subtraction a  − b = b − a False
Multiplication a × b = b × a  True
Division a/b = b/a  False

 

Commutative Property for numbers.jpg

 

So commutativity is always possible for addition &

multiplication, but not for subtraction & division.

 

For Rational Numbers

Let us take two rational numbers 1/2 & 3/2

Operation

Number

Remark

  Addition

a + b = b + a

Take a = 1/2 & b = 3/2

 

L.H.S

a + b

= 1/2+3/2

= (1 +  3)/2

= 4/2

= 2

 

R.H.S

b + a

= 3/2 + 1/2 

= (3 +  1)/2

= 4/2

= 2

 

∴ a + b = b + a

 

Since a + b = b + a,

∴ Addition is commutative.

Subtraction

 

a − b = b − a

Take a = 1/2 & b = 3/2

 

L.H.S

a − b

= 1/2-3/2

= (1 -  3)/2

= (-2)/2

= −1

 

R.H.S

b – a

= 3/2- 1/2 

= (3 -  1)/2

= 2/2

= 1


∴ a − b ≠ b – a

Since a − b ≠ b − a,

∴ Subtraction is not commutative .

Multiplication

 

a ×  b =  b × a

Take a = 1/2, b = 3/2

 

L.H.S

a × b

= 1/2×  3/2

= (1 × 3)/(2 × 2)

= 3/4

 

R.H.S

b × a

= 3/2×1/2 

= (3  ×  1)/(2  ×  2)

= 3/4

 

∴ a × b = b × a

Since, a × b = b × a

∴ Multiplication is commutative.

Division

 

a/b=b/a

Take a = 1/2 , b = 3/2

 

L.H.S

 a/b

= (1/2  )/(3/2)

= 1/2×2/3

= 1/3

 

R.H.S

 b/a

= (3/2  )/(1/2)

= 3/2×2/1

= 3

 

∴ a/b≠b/a

Since a/b≠b/a

 ∴ Division is not commutative .

 

To summarize

Numbers

Commutative for

Addition

Subtraction

Multiplication

Division

Natural numbers

Yes

No

Yes

No

Whole numbers

Yes

No

Yes

No

Integers

Yes

No

Yes

No

Rational Numbers

Yes

No

Yes

No

 

So commutativity is always possible for addition & multiplication,

but not for subtraction & division.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.