CLASS-7
CLOSURE 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.