Chapter 1 : Number Systems

Important Note of Number Systems

1. For positive real numbers  and  ,then the identities :

(i)

(ii) 

(iii)   

(iv)

(v)  

(vi)  

2. Let  be a real number and and  be rational numbers . Then

(i)    

(ii)  

(iii)   

(iv)  

(v)  

(vi)    

(vii)    

(viii)  

 Notes : (i) A number  is called a rational number , if it can be written in the form  , Where  and  are integers .

(ii) A number  is called a irrational number , if it can not be written in the form  , Where  and  are integers .

(iii) There are infinitely many rational numbers between any two given rational numbers .

(iv) A number whose decimal expansion is terminating or non-terminating recurring is rational .

(v)  A number whose decimal expansion is non-terminating non recurring is irrational .

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 EXERCISE 1.1

1. Is zero a rational number ? Can you write it in the form   , where  and  are integers .

Solution:  Yes , zero is a rational number .  , where 0 and 1 are the integers .

2. Find six rational number between 3 and 4 .

Solution:  We have ,   and

Therefore, the six rational number between 3 and 4 are  and

3. Find five rational number between and  .

Solution:  We have,    and 

Therefore, the five rational number between and  are   and  .

4. State whether the following statements are true or false . Given reasons for your answers .

(i) Every natural number is a whole number .   (ii) Every integer is a whole number .     (iii) Every rational number is a whole number.

Solution:  (i) True , because natural number and zero is include in whole number .

(ii)  False , because negative number is not a whole number .

(iii) False , because negative numbers and fraction is not a whole number . 

EXERCISE  1.2

1.  State whether the following statements are true or false . Justify your answers .

(i) Every irrational number is a real number .

(ii) Every  point on the number line is  of the form  , where  is a natural number .

(iii) Every real number is an irrational number.

Solution: (i) True, because the collection of real numbers is made up of rational and irrational numbers.

(ii) False, Because no negative number can be the square root of any natural number .

(iii) False , because 2,3,4,….., etc are real number but not irrational .

2. Are the square roots of all positive integers irrational ? If not, give an example of the square root of a number that is a rational number .

Solution: The square roots of all positive integers are not irrational number .  For example: is a rational number (i.e., The perfect square numbers is a rational number) .

3. Show how  can be represented on the number line.

Solution:  We draw a number line such that OAB is a right angled triangle at A on it .

So, OA=2 units ,  AB=1 unit

In  , we have

 

 

 

 

Using a compass with centre O and radius  , draw an arc intersecting the number line at the point E . Then E corresponding to  .

EXERCISE 1.3

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

(i)          (ii)      (iii)      (iv)     (v)      (vi) 

Solution: (i) We have,    is a terminating decimal expansion .

(ii) We have,    is a non- terminating repeating decimal expansion .

(iii) We have, is a terminating decimal expansion .

(iv)  We have, is a non- terminating repeating decimal expansion .

(v) We have, is a non- terminating repeating decimal expansion .

(vi) We have, is a terminating decimal expansion .

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

Solution: We have, 

 

and

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

(i)             (ii)              (iii) 

Solution:  (i)        

Let

     

(ii) We have,        

Let

  

 (iii) We have,

Let

 

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

Solution: let

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 .

Solution: We have,

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

Solution: If  be a rational number, then prime factorization of q has only powers of 2 or powers of 5 or both .

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

Solution:  The three numbers whose decimal expansions are non-terminating non-recurring  are

 0.23023002300023…….. , 1.71071007100071……… and 2.92092009200092……

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

Solution:  We have,  and 

Therefore, the three different irrational numbers between the rational numbers and are

0.73073007300073……. , 0.781078100781…….. and 0.79107910791……… .

9. Classify the following numbers as rational or irrational :

    (i)       (ii)      (iii)      (iv)   (v)

Solution:  (i)   is an irrational number .       

 (ii)   is a rational number .    

 (iii)   is a rational number .

 (iv) is a rational number .

 (v)   is an irrational number .

EXERCISE 1.4

1. Visualise  on the number line , using successive magnification .

Solution : We have,  3.765

Using successive magnification :

2. Visualise  on the number line , upto 4 decimal places .

Solution : We have, = 4.2626 [4 decimal places]

Using successive magnification :

EXERCISE 1.5

1. Classify the following numbers as rational or irrational :

 (i)     (ii)    (iii)     (iv)     (v)

Solution: (i)  is a rational numbers .

(ii)   is an irrational numbers  

(iii) We have,

    is a rational number .  

(iv) We have,

  is an irrational number .  

(v)   is an irrational number .

2. Simplify each of the following expressions :

  (i)      (ii)     (iii)     (iv)

Solution:  (i) We have, 

   

 (ii) We have,

 

 (iii) We have,

 

 (iv) We have,

 .

3. Recall,  is defined as the ratio of the circumference ( say ) of a circle to its diameter ( say ) . That is, . This seems to contradicts the fact that  is irrational . How will you resolve this contradiction ?

Solution: We have,

      

There is no contradiction. Remember that when we measure a length with a scale or any other device , we only get an approximate rational  value . So, we may realize that either  and  is irrational .

4. Represent  on the number line .

Solution: We construct a number line :

Mark the distance 9.3 units from a fixed point A on a given line to obtain a point B such that AB = 9.3 units . From B, mark a distance of 1 unit and mark the new point as C . Find the mid-point of AC and mark that point as O .

Draw a semicircle with centre O and radius OC . Draw a line perpendicular to AC passing through B and intersecting the semicircle at D . Then,   . We draw an arc with centre B and radius BD , which intersects the number line in E . Then, E represents  .

5. Rationalise the denominators of the following :

   (i)      (ii)      (iii)        (iv) 

Solution:  (i) We have,

  

 (ii) We have, 

 (iii) We have,

 (iv) We have,

EXERCISE 1.6

1. Find :   (i)       (ii)       (iii)   

Solution:

(i) We have,             

(ii) We have,           

(iii) We have,          

2. Find : (i)        (ii)      (iii)      (iv)   

Solution: (i) We have ,          

 (ii)  We have,            

(iii) We have,     

(iv)  We have,

3. Simplify :  (i)        (ii)     (iii)         (iv)       

Solution : 

(i)  We have,    

 (ii) We have,      

(iii)  We have,       

 (iv) We have,