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| 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 = 1/2-3/2 = (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.
