**Number System – Exercise 1.3 – Class IX**

**Write the following in decimal form and say what kind of decimal expansion each has:**

**(i) ^{36}/_{100} **

**(ii) ^{1}/_{11}**

**(iii) 4 ^{1}/_{8}**

**(iv) ^{3}/_{13}**

**(v) ^{2}/_{11}**

**(vi) ^{329}/_{100}**

Solution:

**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?

**[Hint : Study the remainders while finding the value of ^{1}/_{7} carefully.]**

Solution:

**Express the following in the form**^{p}/_{q}, where p and q are integers and q ≠ 0.

**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:

**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:

17) 1 (0.0588235294117647

0

————————

10

00

————————

100

85

————————

150

136

———————–

140

136

———————–

40

34

———————–

60

51

————————-

90

85

—————————-

50

34

—————————

160

153

—————————–

70

68

—————————-

20

17

——————————

30

17

—————————–

130

119

——————————

110

102

——————————-

80

68

——————————

120

119

——————————-

1

Therefore, the maximum number of digits in the repeating block is 16. (<17)

Division gives

The repeating block has 16 digits.

**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:

If the denominator is product of 2 and 5 only then it will always give terminating number

For example 3/2 = 1.5, 12/10 = 1.2, 22/5 = 4.5, 1/25 = 0.04, 1/ 10 = 0.1 and many more.

**Write three numbers whose decimal expansions are non-terminating non-recurring.**

Solution:

All rational numbers has non terminating and non recurring decimal expression. So √2, √3, √5 all will give such decimal expansions.

**Find three different irrational numbers between the rational numbers**^{5}/_{7}and^{9}/_{11}.

Solution:

We know,

^{5}/_{7} = 0.7142857143

^{9}/_{11} = 0.8181818182

So we can write number of irrational numbers between ^{5}/_{7} and ^{9}/_{11} .

0.72020020002200…

0.73131113112311

0.750100100010001

**Classify the following numbers as rational or irrational :**

**(i) √23**

**(ii) √225**

**(iii) 0.3796**

**(iv) 7.478478…**

**(v) 1.101001000100001…**

Solution:

(i) √23 – irrational number

(ii) √225 – rational number

(iii) 0.3796 – rational number

(iv) 7.478478… – irrational number

(v) 1.101001000100001… – irrational number

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