Question

Is sutraction associative in rational numbers? Explain with an example.​

Answer 1

First, let’s clarify what ‘associative’ means:

Associativity means you can perform an operation regardless of the grouping of numbers to achieve the same result, i.e. +(+)=(+)+
a
+
(
b
+
c
)
=
(
a
+
b
)
+
c


e.g.

4+(6+1)=4+(7)=11
4
+
(
6
+
1
)
=
4
+
(
7
)
=
11


(4+6)+1=(10)+1=11
(
4
+
6
)
+
1
=
(
10
)
+
1
=
11


For integers, addition is associative.

Is subtraction associative over rational numbers?

Let’s see. A rational number is one that can be written in the form :,∈ℤ
p
q
:
p
,
q
∈
Z


So, the question is:

Is it always true that −(−)=(−)−
a
b
−
(
c
d
−
f
g
)
=
(
a
b
−
c
d
)
−
f
g
?

We could formally prove that the answer is no, but we don’t need to since we can provide a simple counter-example which shows that the answer is no.

Counter-example:

Consider 123−(93−63)=(123−93)−63
12
3
−
(
9
3
−
6
3
)
=
(
12
3
−
9
3
)
−
6
3


LHS=

123−(93−63)
12
3
−
(
9
3
−
6
3
)


=123−(33)
=
12
3
−
(
3
3
)


=93
=
9
3


=3
=
3


RHS=

(123−93)−63
(
12
3
−
9
3
)
−
6
3


=(33)−63
=
(
3
3
)
−
6
3


=−33
=
−
3
3


=−1
=
−
1


Since 3≠−1
3
≠
−
1
, LHS ≠
≠
RHS

Thus, subtraction over rational numbers is not associative.