NCERT Class 9 Maths Solutions Chapter - 1 Number Systems, Ex 1.3

Ex - 1.3

Question 1.  Write the following in decimal form and say what kind of decimal expansion each has:
(i)             (ii)             (iii) 

(iv)              (v)            (vi)    

Solution

(i) 

     terminating

 (ii) 

      non terminating repeating
  
 (iii) 

      Terminating

 (iv) 

       non terminating repeating

 (v) 

non terminating repeating decimal

 (vi) 

      Terminating decimal

Concept Insight: The decimal expansion of a rational number is either terminating or non terminating recurring.
Decimal expansion  terminates in case the prime factors of denominator includes 2 or 5 only.

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

Solution

Yes it can be done as follows:

Concept Insight: Multiples of the given decimal expansion can be obtained by simple multiplication with the given constant. Cross check the answer by performing long division.

Question 3.  Express the following in the form  , where p and q are integers and .

Solution

(i)      

       Let x = 0.666 ...    (i)
      Multiplying by 10 we get
         10x = 6.666 ...    (ii)
      (ii)  - (i) gives

          9x = 6
      Or  x = 


(ii)     


        Let x = 0.4777 ...   (i)
          10x = 4.777 ...
        100x = 47.777 ... (ii)
        (ii) - (i) gives
        99 x = 43

             x =  

(iii)     

           Let x = 0.001001 ...(i)
          1000x = 1.001001 ...(ii)
          (ii) - (i) gives

            999x = 1
                  x =  
Concept Insight: The key idea to express a recurring decimal in the p/q form is to multiply the number by the 10n where n = number of digits repeating.
This is done to  make the repeating block a whole number part of the decimal. By subtracting the two expressions x can be expressed in the P/q form

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

Solution

Let x = 0.9999 .. .. ..(i)
  10x = 9.9999 ... ...(ii)
(ii) - (i) gives
  9x = 9
    x = 1

Concept Insight: .9999999 ..... is nothing but 1 when expressed in p/q form.

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

Solution

Expressing   in the  decimal form we

            
There are 16 digits in repeating block of decimal expansion of .

Concept Insight: Maximum number of digits that can repeat will be 1 less than the prime number in denominator.

Question 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?

Solution

Terminating decimal expansion will come when denominator q of rational number  , is either of 2, 4, 5, 8, 10, and so on ... ...

Terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions are having power of 2 only or 5 only or both.

Concept Insight: A rational number in its simplest form will terminate only when prime factors of its denominator consists of 2 or 5 only.

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

Solution

3 numbers whose decimal expansion is non terminating non recurring are ... ... ,
0.505005000051509 ... ... ...
0.72012009200011500007200000 ... ... ...
7.03124509761202 ... ... ... ... ... ...

Concept Insight: Recall that a non terminating non recurring decimal is an irrational number.  Answer to such questions is not unique.

Question 8.  Find three different irrational numbers between the rational numbers 

Solution

3 irrational numbers are -
0.73073007300073000073 ... ... ...

0.75075007500075000075 ... ... ...
0.79079007900079000079 ... ... ...

Concept Insight: There is infinite number of rational and irrational numbers between any two rational numbers. Convert the number into its decimal form to find irrationals between them.

Alternatively following result can be used to answer

Irrational number between two numbers x and y

Question 9.  Classify the following numbers as rational or irrational:

Solution

(i) 

As decimal expansion of this number is non-terminating non recurring. So it is an irrational number.

(ii)  

Rational number as it can be represented in   form.

(iii)   0.3796
As decimal expansion of this number is terminating, so it is a rational number.

(iv)   

As decimal expansion of this number is non terminating recurring so it is a rational number.

(v)  

As decimal expansion of this number is non terminating non repeating so it is an irrational number.

Concept Insight:  A number is rational if its decimal expansion is either terminating or non terminating but recurring. A number which cannot be expressed in p/q  form is irrational. Square root of prime numbers is always irrational.

Ex-1.4