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

Class 9 Mathematics by VP Classes

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

The document Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev is a part of the Class 9 Course Class 9 Mathematics by VP Classes.
All you need of Class 9 at this link: Class 9

Q.1. Write the following in decimal form and say what kind of decimal expansion each has:
(i) 36/100
(ii) 1/11
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
(iv) 3/13
(v) 2/11
(vi) 329/400
Solution. 
(i) We have 36/100 = 0.36
∴ The decimal expansion of 36/100 is terminating.
(ii) Dividing 1 by 11, we have:

Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
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 Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRevin p/q from, we have
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Now, dividing 33 by 8, we have :

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

Remainder = 0, means the process of division terminates.
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Thus, the decimal expansion is terminating.

(iv) Dividing 3 by 13, we have

Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Here, the repeating block of digits is 230769.
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

Thus, the decimal expansion of 3/13  is “non-terminating repeating”.

(v) Dividing 2 by 11, we have

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

Here, the repeating block of digits is 18.
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Thus, the decimal expansion of 2/11 is “non-terminating repeating”.

(vi) Dividing 329 by 400, we have

Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Remainder = 0, means the process of division terminates.
∴ 329/400 = 0.8225
Thus, the decimal expansion of 329/400  is terminating.

Q.2. You know that Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRevCan you predict what the decimal expansions of  2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?
Solution. 
We are given that
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

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

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

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

Solution.
(i) Let x = Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev = 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,  
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

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

∴ 10x = 10 x (0.4777...)
or   10x = 4.777       ...(1)
and  100x = 47.777   ...(2)
Subtracting (1) from (2), we have
100x – 10x = (47.777…) – (4.777…)
90x = 43
or x = 43/90
Thus,
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

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

Here, we have three repeating digits after the decimal point, therefore we multiply by 1000.
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

or   1000x = 1.001001   …...(2)
Subtracting (1) from (2), we have
1000x – x = (1.001…) – (0.001…)
or 999x = 1
∴ x = 1/999
Thus,
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

Q.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
or x = 9/9 = 1
Thus, 0.9999… = 1 As 0.9999… goes on forever, there is no gap between 1 and 0.9999
Hence both are equal.

Q.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

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

The remainder 1 is the same digit from which we started the division.
Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev

Thus, there are 16 digits in the repeating block in the decimal expansion of 1/17. Hence, our answer is verified.

Q.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:

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

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

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.

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

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

Q.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:

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

As there are an infinite number of irrational numbers betweenEx 1.3 NCERT Solutions - Number System Class 9 Notes | EduRevany three of them can be:
(i) 0.750750075000750…
(ii) 0.767076700767000767…
(iii) 0.78080078008000780…

Q.9. Classify the following numbers as rational or irrational:
(i)Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
(ii) Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev 
(iii) 0.3796 
(iv) 7.478478  
(v) 1.101001000100001…
Solution. 
(i) ∵ 23 is not a perfect square.
∴  is an irrational number.

(ii) Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev
∴ 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… = Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev 
Since, Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev 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.

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