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**Q.1. Write the following in decimal form and say what kind of decimal expansion each has:****(i) 36/100****(ii) 1/11****(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:

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.

∴ 329/400 = 0.8225

Thus, the decimal expansion of 329/400 is terminating.**Q.2. You know that ****Can 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

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.**

**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)**

∴ 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,

**(iii)**

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,**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.

Solution.

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.**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:

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.**

As there are an infinite number of irrational numbers betweenany 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)****(ii) ** **(iii) **0.3796** ****(iv)** 7.478478** ****(v) **1.101001000100001…**Solution. ****(i)** ∵ 23 is not a perfect square.

∴ is an irrational number.

**(ii)**

∴ 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.

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