Number System – Exercise 1.3 – Class IX

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

(i) ³⁶/₁₀₀

(ii)¹/₁₁

(iii) 4¹/₈

(iv) ³/₁₃

(v) ²/₁₁

(vi) ³²⁹/₁₀₀

Solution:

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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?

[Hint : Study the remainders while finding the value of ¹/₇ carefully.]

Solution:

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3. Express the following in the form ^p/_q , where p and q are integers and q ≠ 0.

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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:

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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:

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.

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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:

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.

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7. 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.

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8. 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

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9. 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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