1.
Write the following in decimal form and say what kind of decimal expansion each
has:
36
4
8
100
11
329
3
(iv)
(vi)
(v)
400
13
11
2 3
You know that
= 0.142857. Can you predict what the decimal expansions of
7.
77
4 5 6
7
are, without actually doing the long division? If so, how?
-al
2.
ד יךיך
[Hint : Study the remainders while finding the value of carefully.]
3. Express the following in the form , where p and q are integers and q +0.
9
(i) 0.6
(ii) 0.47
(iii) 0.001
4.
Express 0.99999 ... in the form . Are you surprised by your answer? With you
teacher and classmates discuss why the answer makes sense.
9
Answers
Answer:
Question 1
Write the following in decimal form and say what kind of decimal expansion each has:
i) 36100
ii) 111
iii) 418
iv) 313
v) 211
vi) 329400
Answer
i) 36100 = 0.36. It is a terminating decimal.
ii) 111= 0.09¯¯¯¯¯. It is a non-terminating repeating decimal expansion.
iii) 418 = 4.125. It is a terminating decimal.
iv) 313 = 0.230769¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯. It is a non-terminating repeating decimal expansion.
v) 211 = 0.18¯¯¯¯¯. It is a non-terminating repeating decimal expansion.
vi) 329400 = 0.8225. It is a terminating decimal.
Question 2
You know that the value of 17 = 0.142857¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯. Can you predict what the decimal expansions of 27, 37, 47, 57, and 67 are, without doing the long division? If so, how?
Solution
Question 3
Express the following in the form pq, where p and q are integers and q ≠ 0:
i) 0.6¯¯¯
ii) 0.47¯¯¯
iii) 0.001¯¯¯¯¯¯¯¯
Solution
Question 4
Express 0.9¯¯¯ in the form pq:
Solution
Question 5
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 117? Perform the division to check.
Solution
Question 6
Look at several examples of rational numbers in the form pq (q ≠ 0), where p and q are integers with no common factors other than 1 and have terminating decimal expansions. What property must q satisfy?
Solution
Question 7
Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer
1. 0.1401400140001400001400000...
2. 0.080080008000080000080000008...
3. 0.40400400040000400000...
Question 8
Find three different irrational numbers between the rational numbers 57 and 911.
Answer
First, let us find the decimal expansions of 57 and 911.
57=0.714285¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
911=0.81¯¯¯¯¯
We have to find irrational numbers between 0.714285¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ and 0.81¯¯¯¯¯
If we write three numbers that are non-terminating and non-recurring, we will have written three irrational numbers between the two given fractions.
0.73073007300073..., 0.7890789007890007890000..., and 0.80800800080000... are irrational numbers between the two given fractions.
Question 9
Classify the following numbers as rational or irrational:
i) √23
ii) √225
iii) 0.3769
iv) 7.478478....
v) 1.101001000100001...
Answer
i) √23 = 4.753... It is a non-terminating and non-recurring decimal. It is an irrational number. It cannot be expressed in the form pq where p and q are integers and q ≠ 0.
ii) √225 = 15. It is a terminating decimal. It is a rational number. It can be expressed in the form pq as 151.
iii) 0.3769. It is a terminating decimal. It is a rational number. It can be expressed in the form pq as 376910000.
iv) 7.478478.... It is a non-terminating but recurring decimal. It is a rational number. It can be expressed in the form pq as 7478999 or 7471999.
v) 1.101001000100001... It is a non-terminating and non-recurring decimal. It is an irrational number. It cannot be expressed in the form pq where p and q are integers and q ≠ 0.