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

NON-ASSOCIATIVE PROPERTY OF DIVISION OF RATIONAL NUMBER -

The non-associative property of division means that when dividing three rational numbers, the grouping (or association) of the numbers affects the result. In other words, the way in which the numbers are grouped in the operation matters, and changing the grouping can change the result. This contrasts with addition and multiplication, which are associative operations.

Non-associative Property of Division:-

For rational numbers a, b, and c:-

             a                   b

      (-----) ÷ c ≠ a ÷ (-----)

          b                   c

Example.1) Let, a = 3/4, b = 5/7, c = 7/9

Ans.)

             a               3/4          7

L.H.S = (-----) ÷ c = (-------) ÷ ------

             b               5/7          9

                             3          5           7

                       = (------ ÷ ------) ÷ ------

                             4          7           9

                             3          7          7

                       = (------ x -----) ÷ ------

                             4          5          9

                                           21         7          21         9

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

                            20         9          20          7

                                                                             27

                                              = ------

                                                   20

                  b           3          5/7

R.H.S = a ÷ (------) = ------ ÷ (------) 

                  c           4          7/9          

                             3            5         7

                       = ------- ÷ (------ ÷ ------)

                             4            7          9

                             3           5         9

                       =  ------ ÷ (----- x ------)

                             4           7         7

                            3          45         3         49         

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

                            4          49         4         45

                                                   49

                                              = ------

                                                   60

 So, L.H.S ≠ R.H.S  (Proven)

Example.2) Example with Whole Numbers:

  Let a = 12, b = 3, and c = 2.

Ans.) 

                  a               12

Calculate:-   (-----) ÷ c = (------) ÷ 2 = 4 ÷ 2 = 2

                  b                3

                                          b                 3                2

Now calculate:-   a ÷ (-----) = 12 ÷ (------) = 12 x ----- = 8

                            c                 2                3

Clearly, 2 ≠ 8 .

 So, L.H.S ≠ R.H.S  (Proven)

Why Division is Non-associative:-

• Order of Operations:- Division does not follow the associative property because the order in which the divisions are performed affects the result. This is due to the fact that division and its inverse operation, multiplication, are not inherently commutative or associative.
• Multiplicative Inverse:- When dividing, you are essentially multiplying by the reciprocal. Changing the grouping changes the placement of the reciprocals, leading to different results.

Importance of the Non-associative Property:-

Understanding that division is non-associative is crucial for:Key Points to Remember

• Simplifying Expressions:- When simplifying expressions involving multiple divisions, it's important to carefully consider the grouping of the operands.
• Solving Equations:- When solving equations that involve multiple division operations, the grouping must be maintained to ensure correct solutions.
• Practical Applications:- In real-world problems, such as calculations involving rates, ratios, and proportions, the order and grouping of division operations can significantly affect the result.

Key Points to Remember:-

1. Grouping Matters:- In division, the grouping of the operands cannot be changed without affecting the result.
2. Order of Operations:- The non-associative nature of division emphasizes the importance of following the correct order of operations.
3. Unique Results:- Each division operation yields a unique result based on the grouping of the operands.

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