CLASS-7
PROPERTY OF 1 OF DIVISION OF RATIONAL NUMBER

PROPERTY OF 1 OF DIVISION OF RATIONAL NUMBER -

The "Property of 1" in the context of division of rational numbers states that any rational number divided by 1 remains unchanged. This property can be formally stated as follows:

Property of 1:-

For any rational number a,

          a

      ------- = a

          1

Explanation:-

• Rational Numbers:- 

       A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0.

• When you divide a rational number by 1, you are essentially saying that the number is being split into one part, which is itself.

Examples:-

  1. Positive Rational Number:-

       Let, a = 3/4

                     3               3

         a ÷ 1 = ------ ÷ 1 = ------

                     4               4

  2. Negative Rational Number:-

      Let, a = - 5/7         

                   - 5            - 5

         a ÷ 1 = ------ ÷ 1 = ------

                     7               7

   3. Whole Number:-      

      Let, a = 99         

                     99               99

         a ÷ 1 = -------  ÷ 1 = ------ = 99 

                      1                1

    4. Zero:-

        Let, a = 0                    

         a ÷ 1 = 0 ÷ 1 = 0

Proof:-

To understand why this property holds, consider the definition of division and rational numbers. Any rational number a can be expressed as p/q. Dividing p/q by 1 means multiplying by the reciprocal of 1, which is still 1:-

    a          p/q          p              p          1          p

 ------ = ------- = ------ ÷ 1 = ------ x ------ = ------ = a

    1           1           q              q          1          q

Thus, for any rational number a, dividing it by 1 results in the number itself. This demonstrates the "Property of 1" for division of rational numbers.

Importance:-

The identity property of division is fundamental because it helps in simplifying expressions and solving equations. Knowing that dividing by 1 does not change the value of a number allows us to manipulate and simplify rational expressions more easily.

In summary, the property of 1 in the division of rational numbers states that any rational number divided by 1 remains the same, highlighting the identity property of division.