Chapter 1 : Number Systems - Ncert Solutions for Class 9 Maths CBSE

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Chapter 1 - Number Systems Excercise Ex. 1.1

Solution 1
Yes zero is a rational number as it can be represented in the   form,  where p and q are integers and q  0 as  etc.

Concept Insight: Key idea to answer this question is "every integer is a rational number and zero is a non negative integer".  Also 0 can be expressed in form in various ways as 0 divided by any number is 0. simplest is   .

Solution 2
There are infinite rational numbers in between 3 and 4.  
3 and 4 can be represented as respectively.
 
Now rational numbers between 3 and 4 are    
    

Concept Insight:  Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent  form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted.

Solution 3
There are infinite rational numbers between
        
        
Now rational numbers between are  
         
 
Concept Insight: Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent   form by multiplying and dividing by the number at least 1 more than the rational numbers required.

Alternatively for any two rational numbers a and b,    is also a rational number which lies between a and b.


Solution 4
(i)    True, since collection of whole numbers contains all natural numbers.
    
(ii)    False, integers include negative of natural numbers as well, which are clearly not whole numbers. For example -1 is an integer but not a whole number.

(iii)    False, rational numbers includes fractions and integers as well. For example is a rational number but not whole number.

Concept Insight: Key concept involved in this question is the hierarchy of number systems
 
                      
 
Remember the bigger set consists of the smaller one.
Since Mathematics is an exact science every fact has a proof but in order to negate a statement even one counter example is sufficient.

Chapter 1 - Number Systems Excercise Ex. 1.2

Solution 1
(i)  True, since real numbers consists of rational and irrational numbers.
 
(ii)  False, Since negative integers cannot be expressed as the  square root of any natural number.

(iii)  False, real number includes both rational and irrational numbers. So every real number can not be an irrational number.

Concept Insight: Mentioning the reasons is important in this problem. Real Numbers consists of rational and irrational numbers and not vice versa. Every real number corresponds to a point on number line and vice versa.
Recall real number includes negative numbers also. Square root of negative numbers is not defined.

Solution 2
Square roots of all square numbers are rational.
  For example  
    
 Thus the square roots of all positive integers are not irrational  

Concept Insight: In general only the square root of a prime number is irrational.

Therefore square root of perfect square numbers are rational.

Solution 3
Using Pythagoras Theorem: 5=22+12
Taking positive square root we get
 
 
 
1.  Mark a point 'A' representing 2 units on number line.
2. Now construct AB of unit length perpendicular to OA. Join OB
3. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C.
4. Point C represents   on number line. [length (OB) = length (OC)]
Concept Insight: For a positive integer n,   can be   located on number line ,  if    is located using Pythagoras Theorem . If   is a perfect square then this method is useful. 
 
To represent the irrational number   key idea is to use Pythagoras theorem and create a length of   units by constructing a right triangle of base and perpendicular  of length 2 and 1 units. 

Chapter 1 - Number Systems Excercise Ex. 1.3

Solution 1
(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.

Solution 2
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.
Solution 3
(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

Solution 4
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.

Solution 5
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.

Solution 6
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.

Solution 7
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.

Solution 8
 
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
 
Solution 9
(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.

Chapter 1 - Number Systems Excercise Ex. 1.4

Solution 1
3.765 can be represented
 
 
 
Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part.
Solution 2
= 4.2626
We can visualise 4.2626 as in following steps
 
Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part.

Chapter 1 - Number Systems Excercise Ex. 1.5

Solution 1
  (i)
 
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.
 
 (ii)
 
It can be represented in  form so it is a rational number.
 
(iii) 
 
As it can be represented in  form, so it is a rational number.
 
(iv) 
 
 
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.

(v) 
 
As decimal expansion is non terminating non recurring, so it is an irrational number.
 
Concept Insight: Do the simplifications as indicated and see whether the number is terminating, non terminating recurring or neither terminating nor repeating.  Remember Sum/difference/Product of a rational and irrational number may or may not be  irrational.
Solution 2
 
Concept Insight: Apply the algebraic identities (a+b)2, (a-b)2,(a+b)(a-b) etc to simplify the given expressions.
Equivalent Identities used here are
 
 
Solution 3
There is no contradiction. Since    here circumference or diameter are not given to be integers . When we measure a length with scale or any other instrument, we only get an approximate rational value. We never get an exact value. c or d  may be irrational. So, the fraction   is irrational. Therefore, is irrational.  

Concept Insight: A rational number is the number of the form   where p and q are
 
integers. In    c and d are not integers. Also remember that no measurement is exact.

Solution 4
(i)    Mark a line segment OB = 9.3 on number line.
 
(ii)    Take BC of 1 unit.
 
(iii)    Find mid point D of OC and draw a semicircle on OC while taking D as its centre.
 
(iv)     Draw a perpendicular to line OC passing through point B. Let it intersect semicircle at E. Length of perpendicular BE =  .  
(v)    Taking B as centre and BE as radius draw an arc intersecting number line at F. BF is  i.e point F represents   on number line
 
 
Verification: In EDB
ED2=EB2+DB2      Using Pythagoras theorem
 
 
Concept Insight: This method based on the application of Pythagoras theorem can be used to represent root of any rational number on the number line.
 
The key idea to represent  is to create a length of   units.
In ODB
DB =
 
Solution 5
 
 
 
 
 
Concept Insight: Rationalisation of denominator means converting the irrational  denominator  to rational  i.e .  removing the radical sign from  denominator.A number of the form   can be converted to rational form by multiplying with its conjugate. Remember the algebraic identities  
 
 
 
 

Chapter 1 - Number Systems Excercise Ex. 1.6

Solution 1








Concept Insight: Express the number in exponent notation and use the rule Exponent m must be such that it is divisible by n.
Solution 2
 
 
 
 
Concept Insight: Express the number in exponent notation and use the rule of exponents.
Solution 3
 
 
 
 
Concept Insight: Use the rule of exponents
 
 

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