Number Systems
Chapter 1
Exercise 1.3
EX 1.3 QUESTION 1.
Write the following in decimal form and say what kind of decimal expansion each has :
(i) 36/100
(ii)1/11
(iv) 3/13
(v) 2/11
(vi) 329/400
Solution:
EX 1.3 QUESTION 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:
EX 1.3 QUESTION 3.
3. Express the following in the form p/q, where p and q are integers and q 0.
(i) 0.
(ii) 0.4
(iii) 0.
Solution:
(i) 0.
Let x = 0.
⇒ x = 0.6666……….. (i)
multiplying equation (i) by 10 on both sides,
10x = 6.666…
10x = 6 + x [From equation (I)]
9x = 6
x = 6/9
x = 2/3
(ii) 0.4
Let x = 0.4
⇒ x = 0.47777………….. (i)
multiplying equation (i) by 10 on both sides,
⇒ 10x = 4.7777 ………….. (ii)
multipmultiplying equation (ii) by 100 on both sides,
100x = 47.7777…
100x = 43 + 47.7777
100x = 43 + 10x [From equation (Ii)]
100x – 10 x = 43
90x = 43
x = 43/90
(iii) 0.
Let x = 0.
⇒ x = 0.001001001…………(i)
multiplying equation (i) by 1000 on both sides,
1000x =1.001001001……
⇒ 1000x = 1 + 0.001001001…..
⇒ 1000x =1 + x [from equation (i)]
⇒ 1000x – x = 1
⇒ 999x = 1
⇒ x = 1/ 999
EX 1.3 QUESTION 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= 0.99999………..(i)
multiplying equation (i) by 10 on both sides,
10x = 9.99999
⇒ 10x = 9+ 0.99999
⇒ 10x = 9+ x [from equation (i)]
⇒ 10x – x = 9
⇒ 9x = 9
⇒ x = 9/9 = 1
⇒ x = 1
The difference between 1 and 0.999999 is 0.000001 which is very close.
Hence, we can say that 0.99999 = 1.
EX 1.3 QUESTION 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:
1/17
1/17 =
There are maximum16 digits in the repeating block of the decimal expansion of 1/17.
EX 1.3 QUESTION 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:
- 2/5 = 0.4
- 1/10 = 0.1
- 3/2 = 1.5
- 7/8 = 0.875
The denominator of all the rational numbers is in front of 2m x 5n , where m and n are integers.
EX 1.3 QUESTION 7.
Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution:
- √3 = 1.732050807568
- √5 = 2.23606797
- √26 =5.099019513592
EX 1.3 QUESTION 8.
8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Solution:
Three different irrational numbers between 0.
- 0.73073007300073000073…
- 0.75075007300075000075…
- 0.76076007600076000076…
EX 1.3 QUESTION 9.
Classify the following numbers as rational or irrational according to their type:
(i) √23
(ii)√225
(iii) 0.3796
(iv) 7.478478…..
(v) 1.101001000100001
Solution:
(i) √23
√23 = 4.79583152331…
Irrational number.
(ii)√225
√225 = 15 = 15/1
Rational number.
(iii) 0.3796
Rational number.
(iv) 7.478478…..
Rational number.
(v) 1.101001000100001
Irrational number.
