Real Numbers Exercise 1.2.2 – Class IX

1. Write down the decimal expansion of the following number:

(i) 13/31;

(ii) 123/35;

(iii) 103/625;

(iv) 68/35;

(v) 2013/1024;

(VI) 2/17;

(vii) 123/750;

Which of them have terminating expansion? What are the periods?

(i) 13/31

Solution:

This has a non terminal expansion.

(ii) 123/35

Solution:

(iii) 103/625;

Solution:

(iv) 68/35;

Solution:

(v) 2013/1024;

Solution:

(VI) 2/17;

Solution:

(vii) 123/750;

Solution:

1. Find the rational number which is represented by the following decimal numbers:

(i) 0.999999

Solution:

Let, r = 0.9999 …….

Then, 10r  = 9.9999 ……

10r – r = 9.0000

9r = 9

r = 9/9 = 1

(ii)2.0012

Solution:
Let r = 2.0012…….
100r = 200.1212 ……
10000 r = 20012.1212
10000 r – 100r = 19812.00
9900r = 19812
r = 19812/9900
r = 𝟏𝟔𝟓𝟏/𝟖𝟐𝟓

(iii) 2013.13

Solution:

Let r = 2013.1313…….

100r = 201313.13……

100r – r = 199300.00

99r = 199300

r = 𝟏𝟗𝟗𝟑𝟎𝟎/𝟗𝟗

(iv) 0.112233

Solution:

Let r = 0.112233

100r = 11.2233……

1000000r = 112233.2233……

(106r – 102r) = 112233.2233 – 11.2233 = 112222.00

999900r = 112222

r = (112222)/(999900)

r = (5101)/(45450)

1. Construct rational numbers with period of lengths

(i) 10

Solution:

Let r = 4.1234543213

1010 r  = 41234543213. 1234543213

(1010  – 1) r = 41234543209.0

r = (41234543209)/(1010−1)

(ii) 12

Solution:

Let r = 8.431354789265

1012  r = 8431354789265. 431354789265

(1012  – 1) r = 8431354789257.0

r = (8431354789257)/(1012 −1)

(iii) 15

Solution:

Let r = 2.345464748494142

1015  r = 2345464748494142. 345464748494142

(1015  – 1) r = 2345464748494140.0

r = (2345464748494140.0)/(1015−1)

1. Find the rational numbers whose decimal expansions are: (i) 0.142857 (ii) 0.142857. Are these two same?

Solution:

(i) 0.142857

We have r = 0.142857 = (142857)/(1000000)

r = (142857)/(1000000)

(ii) 0.142857

Let r = 0.142857

106  r = 142857. 142857

(106  – 1) r = 142857.0

r  = 142857/999999

We observe that r = 1/7

The two rational numbers are not the same.

1. Write 1 as an infinite decimal.

Solution:

We can write 1as infinite decimal as 0.9999…… both 1.00 and 0.9999….. are rational numbers which are one and the same. The first one terminates and the second one is periodic hence infinite. i.e., 1 = 0.9