ASSOCIATIVE PROPERTY OF ADDITION OF RATIONAL NUMBERS -
The associative property of addition states that when adding three or more numbers, the grouping of the numbers does not affect the sum. In the context of rational numbers, this means that when adding multiple rational numbers together, you can regroup them in any way without changing the result.
Formally, if a, b, and c are any three rational numbers, then:
(a + b) + c = a + (b + c)
This property holds true for all rational numbers because addition is associative for integers, and rational numbers can be expressed as fractions of integers.
Let's illustrate this with an example using specific rational numbers.
Example.1) Consider the rational numbers:
1 2 3
a = -------, b = -------, c = -------
2 3 4
Ans.) First, we calculate (a + b) + c :-
1 2 3
(------- + -------) + ------ = L.H.S
2 3 4
(1 X 3) + (2 X 2) 3
=> ------------------- + ------- [LCM of denominator 2 & 3 is 6]
(2 X 3) 4
(3 + 4) 3
=> ----------- + -------
6 4
7 3
=> ------- + -------
6 4
(7 X 2) (3 X 3)
=> ----------- + ---------- [LCM of denominator 6 & 4 is 12]
(6 X 2) (4 X 3)
14 9
=> ------- + -------
12 12
(14 + 9) 23
=> ---------- = ------
12 12
Now, a + (b + c) :-
1 2 3
------- + (------ + ------) = R.H.S
2 3 4
1 (2 X 4) + (3 X 3)
=> ------ + {-----------------} [LCM of denominator 3 & 4 is 12]
2 12
1 (8 + 9)
=> ------ + ----------
2 12
1 17
=> ------ + --------
2 12
(1 X 6) (17 X 1)
=> ----------- + ------------ [LCM of denominator 2 & 12 is 12]
(2 X 6) (12 X 1)
6 17
=> -------- + -------
12 12
(6 + 17) 23
=> ---------- = ------
12 12
So, it's proven that, L.H.S = R.H.S
1 2 3 1 2 3 23
(----- + -----) + ----- = ------ + (------ + ------) = -------
2 3 4 2 3 4 12
So, (a + b) + c = a + (b + c) (Proven)
Example.2) Consider the rational numbers:
2 5 7
a = ------, b = ------, c = ------
3 6 12
Ans.) First, we calculate (a + b) + c :-
2 5 7
(------ + ------) + ------ = L.H.S
3 6 12
(2 X 2) (5 X 1) 7
=> {--------- + ---------} + ----- [LCM of denominator 3 & 6 is 6]
(3 X 2) (6 X 1) 12
4 5 7
=> (----- + -----) + ------
6 6 12
(4 + 5) 7
=> ---------- + ------
6 12
9 7
=> ------ + ------
6 12
(9 X 2) (7 X 1)
=> ---------- + ---------- [LCM of denominator 6 & 12 is 12]
(6 X 2) (12 X 1)
18 7
=> ------ + ------
12 12
(18 + 7) 25
=> ----------- = -------
12 12
Now, a + (b + c) :-
2 5 7
------ + (------ + ------) = R.H.S
3 6 12
2 (5 X 2) (7 X 1)
=> ------ + {--------- + ---------}[LCM of denominator 6 & 12 is 12]
3 (6 X 2) (12 X 1)
2 (10 + 7)
=> ------ + -----------
3 12
2 17
=> ------ + ------
3 12
(2 X 4) (17 X 1)
=> ---------- + ----------- [LCM of denominator 3 & 12 is 12]
(3 X 4) (12 X 1)
8 17
=> ------ + ------
12 12
(8 + 17) 25
=> ---------- = ------
12 12
So, it's proven that, L.H.S = R.H.S
2 3 7 2 3 7 25
(----- + -----) + ----- = ----- + (----- + -----) = ------
3 5 12 3 5 12 12
So, (a + b) + c = a + (b + c) (Proven)