CLOSURE PROPERTY OF MULTIPLICATION OF RATIONAL NUMBER -
The closure property states that the product of any two rational numbers is also a rational number. This property ensures that when you multiply two rational numbers, the result will always be a rational number, meaning it can be expressed in the form a/b, where a and b are integers and b≠0.
Proof of the Closure Property -
Let's take two rational numbers:-
a c
------- & -------
b d
where a, b, c, and d are integers, b ≠ 0 and d ≠ 0
The product of these two rational numbers is:-
a c a x c
------- X ------- = ----------
b d b x d
In this product:-
Thus, the result (a ⋅ c)/(b ⋅ d) is a rational number because both the numerator and the denominator are integers, and the denominator is not zero.
Sample Illustration -
Example.1) Simple multiplication -
2 3
Consider ------- & -------
5 4
Ans.)
2 3
We have, ------- & -------
5 4
Their product is -
2 3 6 3
------- X ------- = ------- = ------
5 4 20 10
3/10 is a rational number. (Ans.)
Example.2) Negative Rational Number -
- 5 4
Consider ------- & -------
8 7
Ans.)
- 5 4
As per given condition, we have ------- & -------
8 7
Their product is -
- 5 4 - 5 - 5
------- X ------- = --------- = --------
8 7 (2 X 7) 14
- 5/14 is a rational number. (Ans.)
Example.3) Mixed signs -
6 - 2
Consider ------ & -------
- 7 9
Ans.)
6 - 2
As per the condition, we have ------- & -------
- 7 9
6 - 2 2 X (- 2) - 4
------- X ------- = ------------- = -------
- 7 9 (- 7) X 3 - 21
4
= ------- (Ans.)
21