CLASS-7
NON-COMMUTATIVE PROPERTY OF DIVISION OF RATIONAL NUMBER

NON-COMMUTATIVE PROPERTY OF DIVISION OF RATIONAL NUMBER -

The non-commutative property of an operation indicates that changing the order of the operands affects the result. In the context of rational numbers, this property is particularly evident in the division operation. Unlike addition and multiplication, which are commutative (i.e., a + b = b + a and a × b = b × a), division does not generally yield the same result when the order of the operands is changed.

Non-commutative Property of Division-

For rational numbers a and b, the non-commutative property of division can be stated as:-

       a           b

  ------- ≠ -------

      b           a

​ unless a = b or both a and b are reciprocals of each other (i.e., ab = 1).

Examples.1) Let, a = 4/5, b = 3/7

Ans.)

            a         4/5         4          3          4          7            

L.H.S = ----- = ------- = ------ ÷ ------ = ------ x ------ 

            b         3/7         5          7          5          3   

                                                                                       28

                                                    = -------

                                                         15

             b         3/7          3          4          3           5

R.H.S = ------ = ------- = ------ ÷ ------ = ------- x ------

             a         4/5          7          5          7           4

                                                                                          15

                                                      = ------- 

                                                           28

        So, L.H.S ≠ R.H.S     (Proven)

Examples.2) Let, a = 3/4, b = 2/5

Ans.)

            a         3/4         3          2          3          5            

L.H.S = ----- = ------- = ------ ÷ ------ = ------ x ------ 

            b         2/5         4          5          4          2   

                                                                                                                                           15

                                                    = -------

                                                          8

             b         2/5          2          3          2           4

R.H.S = ------ = ------- = ------ ÷ ------ = ------- x ------

             a         3/4          5          4          5           3

                                                                                                                                              8

                                                      = ------- 

                                                           15

           So, L.H.S ≠ R.H.S     (Proven)

Importance of the Non-commutative Property:-

Understanding that division is non-commutative is crucial in various mathematical contexts, such as:Key Points to Remember

• Simplifying Expressions:- When simplifying expressions involving division, it is important to maintain the correct order of operands.
• Solving Equations:- When solving equations involving division, the order of division must be carefully preserved to obtain correct solutions.
• Practical Applications:- In real-world problems, such as rate calculations, the order in which quantities are divided can significantly affect the result.

1. Order Matters:- In division, the order of the operands cannot be swapped without changing the result.
2. Reciprocals:- Division can be interpreted as multiplication by the reciprocal, but the non-commutative nature still holds.
3. Unique Results:- Each division operation yields a unique result based on the order of the operands.

In summary, the non-commutative property of division in rational numbers highlights the importance of operand order in division operations, ensuring accurate calculations and interpretations in both mathematical and practical applications.