# CLASS-7CLOSURE PROPERTY OF MULTIPLICATION OF RATIONAL NUMBER

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:-

• a ⋅ c is an integer because the product of two integers is always an integer.
• b ⋅ d is an integer because the product of two non-zero integers is always a non-zero integer.

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