Class IX Math NCERT Solutions For Real Numbers Exercise– 1.3

Exercise– 1.3

1.  Write the following in decidmal form and say what kind of decimal expansion each has:

           

Ans.

       (i)   We have 

              ⇒ The decimal expansion of  is terminating.

       (ii)    Dividing 1 by 11, we have:

                     

Note:

               The bar above the digits indicates the block of digits that repeats. Here, the repeating block is 09.

                

                               

                Remainder = 0, means the process of division terminates.

                

                Thus, the decimal expansion is terminating.

       (iv)   Dividing 3 by 13, we have

                                                  

                Here, the repeating block of digits is 230769.

                

                Thus, the decimal expansion of  is

                “non-terminating repeating”.

       (v)   Dividing 2 by 11, we have

                

                Here, the repeating block of digits is 18.

                

                Thus, the decimal expansion of  is

                “non-terminating repeating”.

       (iv)   Dividing 329 by 400, we have

                                

                Remainder = 0, means the process of division terminates.

                

                Thus, the decimal expansion of  is terminating.

2.  You know that  Can you predict what the decimal expansions of  are, without actually doing the long division? If so, how?

Ans.

           

           Thus, without actually doing the long division we can predict the decimanl expansions of the above given rational numbers.

3.  Express the following in the form 

       where p and q are integers and q ≠ 0.

       

Ans.

       

       Since, there is one repeating digit.

       ⇒ We multiply both sides by 10,

       10x = (0.666...) × 10

       or 10x = 6.6666...

       ⇒ 10x – x = 6.6666... – 0.6666...

       or 9x = 6

       

       (ii)   

       and 100x = 47.777

       Subtracting (1) from (2), we have

       100x – 10x = (47.777. . .) &ndsah; (4.777. . .) 90x = 43

       

       (iii)   

       Here, we have three repeating digits after the decimal point, therefore we multiply by 1000.

       

       → 1000x 0.001001 . . .

       or Subtracting (1) from (2), we have

       1000x – x = (1.001. . .) – (0.001. . .)

       or 999x = 1

       

4.  Let x = 0.99999 . . . in the form  Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Ans.

       Multiply both sides by 10, we have

       [⇒ There is only one repeating digit.]

       10× x = 10× (0.99999 . . .)

       or 9x = 9

5.  What can the maximum number of digits be in the repeating block of digits in the decimal expansion of  Perform the division to check your answer.

Ans.

           Since, the number of entries in the repeating block of digits is less than the divisor. In  the divisor is 17.

           ⇒ The maximum number of digits in the repeating block is 16. To perform the long division, we have

           

           The remainder 1 is the same digit from which we started the division.

           

           Thus, there are 16 digits in the repeating block in the decimal expansion of  Hence, our answer is verified.

6.  Look at several examples of rational numbers in the form  where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Ans.

           Let us look at decimal expansion of the following terminating rational numbers:

           

           We observe that the prime factorisation of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both.

           Note:

           If the denominator of a rational number

           (in its standard form) has prime factors either 2 or 5 or both, then and only then it can be represented as a terminating decimal.

7.  Write three numbers whose decimal expansions are non-terminating

           non-recurring.

Ans.

           

8.  Find three different irrational numbers between the rational numbers 

Ans.

           To express decimal expansion of  we have:

           

           As there are an infinite number of irrational numbers between  any three of them can be:

           (i)      0.750750075000750. . .

           (ii)    0.767076700767000767. . .

           (iii)   0.78080078008000780. . .

9.  Classify the following numbers as rational or irrational:

       

Ans.

       (i)   ⇒23 is not a perfect square.

              ⇒  is an irrational number.

       (ii)   ⇒225 = 15 × 15 = 152

              ⇒225 is a perfect square.

              Thus,  is a rational number.

       (iii)   ⇒0.3796 is a terminating decimal,

              ⇒It is a rational number.

       (iv)   

              Since,  is a non-terminating and recurring (repeating) decimal.

              ⇒It is a rational number.

       (v)   Since, 1.101001000100001... is a non-terminating and non-repeating decimal number.

              ⇒It is an irrational number.