Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

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

Solution: (i) We have 36/100 = 0.36
∴ The decimal expansion of 36/100 is terminating.
(ii) Dividing 1 by 11, we have:

∴  
Thus, the decimal expansion is “non-terminating repeating”.
Note: The bar above the digits indicates the block of digits that repeats. Here, the repeating block is 09.
 (iii) To write in p/q from, we have
Now, dividing 33 by 8, we have :

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 3/13  is “non-terminating repeating”.
(v ) Dividing 2 by 11, we have

Here, the repeating block of digits is 18.

∴

Thus, the decimal expansion of 2/11 is “non-terminating repeating”.
(vi) Dividing 329 by 400, we have

Remainder = 0, means the process of division terminates.

∴  

Thus, the decimal expansion of 329/400  is terminating.

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

∴

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

Ques 3. Express the following in the form (p\q) , where p and q are integers and q ≠ 0.

Solution: (i) Let x =  = 0.6666… Since, there is one repeating digit.
∴  We multiply both sides by 10,
10x = (0.666…) x 10
or 10x = 6.6666…

∴ 10x - x = 6.6666... - 0.6666...

or 9x = 6

or x = 6/9 = 2/3

Thus,  

(ii) Let

∴

or   10x = 4.777       ...(1)
and  100x = 47.777   ...(2)
Subtracting (1) from (2), we have
100x – 10x = (47.777…) – (4.777…)
90x = 43

(iii) Let ..              ...(1)
Here, we have three repeating digits after the decimal point, therefore we multiply by 1000.

∴

or   1000x = 1.001001…    ...(2)
Subtracting (1) from (2), we have
1000x – x = (1.001…) – (0.001…)

or 999x = 1 

∴ x = 1/999

Thus,

Ques 4. Express 0.99999… in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Solution: Let x = 0.99999...       ...(1)
Multiply both sides by 10,
we have [∵ There is only one repeating digit.]
10 x x = 10 x (0.99999…)
or 10x = 9.9999              ...(2)
Subtracting (1) from (2),
we get 10x – x = (9.9999…) – (0.9999…)
or 9x = 9

Thus, 0.9999… = 1 As 0.9999… goes on forever, there is no gap between 1 and 0.9999
Hence both are equal.

Ques 5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of (1/17)? Perform the division to check your answer.
Solution: Since, the number of entries in the repeating block of digits is less than the divisor.
In 1/17, 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 1/17. Hence, our answer is verified.

Ques 6. Look at several examples of rational numbers in the form p/q  (q ≠ 0), 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: Let us look at decimal expansion of the following terminating rational numbers:

We observe that the prime factorization 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.

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

Ques 8. Find three different irrational numbers between the rational numbers (5/7) and (9\11).
 Solution: To express decimal expansion of 5/7 and 9/11, we have:

As there are an infinite number of irrational numbers betweenany three of them can be:
(i) 0.750750075000750…
(ii) 0.767076700767000767…
(iii) 0.78080078008000780… 

Ques 9. Classify the following numbers as rational or irrational:
 (i) 
(ii)  
(iii) 0.3796 
(iv) 7.478478  
(v) 1.101001000100001…
Solution: 
(i) ∵ 23 is not a perfect square.
∴  is an irrational number.
(ii) ∵  = 15 x 15 = 225
∴ 225 is a perfect square. 
Thus, 225 is a rational number.
(iii) ∵ 0.3796 is a terminating decimal,
∴ It is a rational number.

(iv) 7.478478… =  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.

REMEMBER
A rational number is a number we can know exactly, either as a whole number, a fraction or a mixed number, but not always exactly as a decimal.
An irrational number can never be known in any form.