1. Number Systems (NCERT)
NCERT
1 : Number System
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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:
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:
3. Show how 
can be represented on the number line .
Solution:
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) 
2. You know that 
. Can you predict what the decimal expansions of 
are , without actually doing the long division ? If so , how ?
3. Express the following in the form 
, where and 
are integers and
.
(i) 
(ii) 
(iii) 
4. Express 
in the form 
. Are you surprised by your answer ? With your teacher and classmates discuss why the answer makes sense .
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 :
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 :
7. Write three numbers whose decimal expansions are non-terminating non-recurring .
Solution :
8. Find three different irrational numbers between the rational numbers 
and 
.
Solution :
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 :
2. Visualise 
on the number line , upto 4 decimal places .
Solution :
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) 
is a rational number .
(iv) 
is an irrational number .
(v) is an irrational number .
2. Simplify each of the following expressions :
(i) 
(ii) 
(iii) 
(iv) 
Solution : (i) 
(ii) 
(iii) 
(iv) 
.
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 :
4. Represent 
on the number line .
Solution :
5. Rationalise the denominators of the following :
(i) 
(ii) 
(iii) 
(iv) 
Solution : (i) 
(ii) 
(iii) 
(iv) 
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, 
Posted 4 years ago