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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Let us consider the following pairs of rational numbers and add them.

(a) 5 3 (b) −7 5 (c) −3 , −5
,
,
9 9 13 8 7 8
5 3 53 8 3 5 35 8
+
+
(a) We have, + = = . Also, + = = .
9 9 9 9 9 9 9 9
5
Thus, + 3 = 3 + 5
9 9 9 9
−7 5 −56 65 9 5  − 7 65 − 56 9
+
(b) We have, + = = . Also, +   = = .
13 8 104 104 8  13  104 104
−7 5 5  7
− 
Thus, + = +  
13 8 8  13 

−3  −  5 −24 + − ( 35 ) −59 −5  −  3 −35 − 24 −59
(c) We have, +   = = . Also, +   = =
7  8  56 56 8  7  56 56
−5  −  3 −3  −  5
Thus, +   = +  
8  7  7  8 

We observe that changing the order of two rational numbers being added does not affect the
result. Hence, addition is commutative for rational numbers. In other words, the commutative
property holds for the addition of rational numbers.

In general,
p r p r r p
if and are two rational numbers, then + = + .
q s q s s q

Associative Property

−5 3 5
Let us take three rational numbers , and , and then add them in a group. We have,
7 14 14

−5  3 5  −5  8   −5 3  5  −10 3  5
+  +  = +   Also,  +  + =  +  +
7  14 14  7  14   7 14  14  14 14  14

= −10 + 8 = −2 or −1 = −7 + 5 = −2 or −1
14 14 14 7 14 14 14 7
−5  3 5   −5 3  5
Thus, +  +  =  +  + .
7  14 14   7 14  14
We observe that rational numbers are grouped when addition does not affect the result. This shows

that addition is associative for rational numbers. In other words, the associative property holds
for the addition of rational numbers.

In general,
p r u p r  u  p r  u
if , and are three rational numbers, then +  +  = + + .
q s v q  s v  q s  v


Mathematics-8 16