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 
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 = 1/23/2 = (1  3)/2 = (2)/2 = −1
R.H.S b – a = 3/2 1/2 = (3  1)/2 = 2/2 = 1

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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