# CLASS-7CLOSURE PROPERTY OF SUBTRACTION OF RATIONAL NUMBER

CLOSURE PROPERTY OF SUBTRACTION OF RATIONAL NUMBER -

The closure property for an operation within a set means that performing that operation on members of the set results in a member of the same set. When we say that the rational numbers are closed under subtraction, we mean that subtracting any two rational numbers results in another rational number.

Closure Property of Subtraction for Rational Numbers -

Explanation -

Statement: If a and b are any two rational numbers, then their difference a−b is also a rational number.

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. More formally, a rational number a can be written as p/q where p and q are integers and q ≠ 0.

Let a and b be two rational numbers:-

p

Let a = ----------

q

r

Let b = ---------

s

where p, q, r, and s​ are integers, and q ≠ 0 and  s ≠ 0.

Subtracting Two Rational Numbers:-

p               r

To subtract b from a:   (a − b) =  ---------- - ----------

q               s

First, find a common denominator, which is (q X s)​:

ps              rq                 (ps - rq)

a − b = ---------- - ----------- =  --------------------

qs              qs                     qs

Now perform the subtraction in the numerator:-

(ps - rq)

a − b----------------

qs

Conclusion :-

(ps - rq)

The result of the subtraction, ----------------​​, is a quotient of two integers

qs

(since p, q, r, and ​are all integers, and integer multiplication and subtraction are closed operations). Moreover, the denominator qs​ is not zero because q​ and s​ are both non-zero.

Thus, the difference (a − b) is a rational number. This demonstrates that the set of rational numbers is closed under subtraction.