1.Write the following in decimal form and say what kind of decimal expansion each has :
(i) 36/100
Solution:
Hence, the decimal expansion of 36/100 is terminating.
(ii)1/11
Solution:
Hence, the decimal expansion of 1/11 is non-terminating repeating.
(iii) 4 1/8
Solution:
Hence, the decimal expansion of 4 1/8 is terminating.
(iv) 3/13
Solution:
Here, the repeating block of digits is 230769
Hence, the decimal expansion of 3/13 is non-terminating repeating.
(v) 2/11
Solution:
Here, the repeating block of digits is 18.
Hence, the decimal expansion of 2/11 is non-terminating repeating.
(vi) 329/400
Solution:
Hence, the decimal expansion of 329/400 is terminating.
2.You know that 1/7 = 0.142857. 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:
3.Express the following in the form p/q where p and q are integers and q ≠ 0.
Solution:
(i)
We can write given question as 0.6666….
Let, x = 0.6666…. equation(i)
Now, multiply both side by 10.
10 x = 6.6666….equation(ii)
Substracting equation (i) from equation(ii), we get
9x = 6
x = 6/9 = 2/3
Hence, required answer is 2/3
(ii)
We can write given question as 0.47777….
Let, x = 0.477777…. equation(i)
Now, multiplying both side by 10.
10 x = 4.7777….equation(ii)
Substracting equation (i) from equation(ii), we get
9x = 4.3
x = 43/90
Hence, required answer is 43/90
(iii)
We can write given question as 0.001001….
Let, x = 0.001001…. equation(i)
Now, multiply both side by 1000.
1000 x = 1.001001….equation(ii)
Substracting equation (i) from equation(ii), we get
999x = 1
x = 1/999
Hence, required answer is 1/999
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 = 0.9999….. say equation (a)
Multiplying both sides by 10,
10x = 9.9999…. say equation (b)
Now, equation(b) – equation(a)
x = 1
The difference between 1 and 0.999999 is 0.000001 which is negligible.
Hence, we can conclude that, 0.999 is too much near to 1.
Therefore, 1 as the answer can be justified.
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:
According to question: In 1/17, the divisor is 17
Therefore, there are 16 digits in the repeating block of the decimal expansion of 1/17.
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 decimal expansion of the following terminating rational numbers:
1/2 = 0. 5, denominator q = 21
8/125 = 0.064, denominator q = 53
7/8 = 0. 875, denominator q =23
4/5 = 0. 8, denominator q = 51
Hence, we can observe that the terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has only powers of 2 or powers of 5 or powers of both.
7. Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution:
According to question:
√2 = 1.414213562
√3 = 1.732050807568
√5 = 2.23606797
√26 =5.099019513592
√101 = 10.04987562112
8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Solution:
5/7
9/11
Hence, the required three different irrational numbers between 5/7 and 9/11 are:
(i)0.73073007300073000073…
(ii)0.75075007300075000075…
(iii)0.78080078008000…
9. Classify the following numbers as rational or irrational according to their type:
(i)√23
Solution:
√23 = 4.79583152331…
Since the number is non-terminating non-recurring, not perfect square.Hence, it is an irrational number.
(ii)√225
Solution:
√225 = 15 = 15/1
Since the number can be represented in p/q form, and it perfect square. Hence, it is a rational number.
(iii) 0.3796
Solution:
Since the number,0.3796, is terminating, and perfect square. Hence it is a rational number.
(iv) 7.478478
Solution:
The number,7.478478, is non-terminating but recurring. Hence, it is a rational number.
(v) 1.101001000100001…
Solution:
Since the number,1.101001000100001…, is non-terminating non-repeating (non-recurring). Hence, it is an irrational number.
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