The document Ex 1.3 NCERT Solutions - Number System Class 9 Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.

All you need of Class 9 at this link: Class 9

**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

∴ 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:**

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) **

(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

Arationalnumber 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.

Anirrationalnumber can never be known in any form.

192 videos|230 docs|82 tests

### Ex 1.4 NCERT Solutions - Number System

- Doc | 1 pages
### Video: Representing Real Numbers on the Number Line (Hindi)

- Video | 02:31 min
### Ex 1.5 NCERT Solutions - Number System

- Doc | 1 pages
### Video: Locating Irrational Numbers on a Number Line (Hindi)

- Video | 27:27 min
### Ex 1.6 NCERT Solutions - Number System

- Doc | 1 pages
### Video: Laws of Exponents for Real Numbers

- Video | 10:48 min

- Video: Representing Real Numbers on the Number Line
- Video | 06:20 min
- Test: Representation On Number Line
- Test | 10 ques | 10 min