# CLASS-7CLOSURE PROPERTY OF DIVISION OF RATIONAL NUMBER

CLOSURE PROPERTY OF DIVISION OF RATIONAL NUMBER -

The closure property for division of rational numbers states that the division of two rational numbers (where the divisor is not zero) results in another rational number.

Explanation:-

A rational number is any number that can be expressed as the quotient or fraction a/b of two integers, where a and b are integers and b ≠ 0.

For two rational numbers a/b and c/d​, where b ≠ 0 and d ≠ 0, their division is given by:-

a/b          a          c           a          d         (a x d)

-------- = ------ ÷ ------ = ------ x ------ = ----------

c/d           b          d          b          c          (b x c)

Since the product of two integers is an integer, and the product of two

a x d

non-zero integers is non-zero, -------- is a rational number.

b x c

Example:-

3             2

Consider two rational numbers ------ and ------ :

4             5

3/4         3          2          3          5          15

------- = ------ ÷ ------ = ------ x ------ = -------

2/5         4          5          4          2           8

15

Here, ​------- is a rational number, demonstrating that the division of two

8

rational numbers results in another rational number.

Formal Proof:-

a           c

Let,----- and ----- be any two rational numbers where b ≠ 0 and d ≠ 0.

b           d

a           c

The division of ------ by ------ is defined as:

b           d

a/b           a           c          a          d          ad

-------- = ------- ÷ ------ = ------ x ------ = -------

c/d           b           d          b          c          bc

Since a, b, c, and d are all integers, adadad and bcbcbc are also integers. Furthermore, since b ≠ 0 and c ≠ 0, bc ≠ 0. Therefore, ad/bc​ is a rational number.

Examples:-

1. Positive Rational Number:-

Let, a = 2/3, and b = 4/5

a          2/3          2          4

So, ------ = -------- = ------ ÷ ------

b          4/5          3          5

2         5         10

= ----- x ------ = ------ is a rational number

3         4         12

2. Negative Rational Number:-

Let, a = - 5/6, and b = 3/4

a        - 5/6       - 5          3

So, ------ = -------- = ------ ÷ ------

b          3/4          6          4

- 5         4        - 10

= ----- x ------ = ------ is a rational number

6         3          9

3. Whole Numbers:-

Let, a = 8, and b = 4

a          8          8          4

So, ------ = ------ = ------ ÷ ------

b          4          1          1

8           1

= ------- x ------- = 2 is a rational number

1           4

This demonstrates that the division of two rational numbers, where the divisor is not zero, always results in another rational number, confirming the closure property of division for rational numbers.