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) 41/8
(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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