RATIONAL NUMBERS ARE NOT COMMUTATIVE UNDER SUBTRACTION -
Indeed, subtraction is not commutative for rational numbers. This means that the order in which you subtract two rational numbers affects the result. Formally, for any two rational numbers a and b:-
(a − b) ≠ (b − a)
Examples and Explanation:-
Example.1)
3 5
Let, a = ------, and b = -------
4 7
3 5
Now, L.H.S = (a − b) = -------- − -------
4 7
(3 X 7) − (5 X 4)
= -------------------- [LCM of 4 & 7 is 28]
(4 X 7)
(21 − 20) 1
= ------------- = --------
28 28
5 3
Now, R.H.S = (b − a) = -------- − --------
7 4
(5 X 4) − (3 X 7)
= -------------------- [LCM of 7 & 4 is 28]
(7 X 4)
(20 − 21) − 1
= ------------- = --------
28 28
We can observe that, L.H.S ≠ R.H.S
So, (a − b) ≠ (b − a)
Example.2)
5 2
Let, a = − -------, and b = -------
6 9
5 2
Now, L.H.S = (a − b) = − ------- − -------
6 9
− (5 X 3) − (2 X 2)
= ------------------- [LCM of 6 & 9 is 18]
18
− 15 − 4 − 19
= ------------- = ---------
18 18
2 5
Now, R.H.S = (b − a) = − ------- − (− ------)
9 6
2 5
= − ------ + ------
9 6
− (2 X 2) + (5 X 3)
= ------------------- [LCM of 9 & 6 is 18]
18
− 4 + 15 11
= ------------ = --------
18 18
We can observe that, L.H.S ≠ R.H.S
So, (a − b) ≠ (b − a)
General Explanation:-
When subtracting two rational numbers a and b, the order in which you subtract them makes a difference because:
a − b = a + (−b)
This changes the sign of b and adds it to a. Conversely,
b − a = b + (−a)
changes the sign of a and adds it to b. Unless a and b are the same number, the results of these two operations will not be equal. The difference in the order changes which number is being negated and affects the final result.
Conclusion:-
Subtraction of rational numbers is not commutative because the order in which the numbers are subtracted affects the outcome. This non-commutativity is a fundamental property of subtraction and distinguishes it from commutative operations like addition and multiplication.