Belongs to the unit Real Numbers

- 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:**

- 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)2.00~~12~~

**Solution:**

Let r = 2.00~~12~~…….

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) **0.11 2233 **

**Solution:**

Let r = 0.11~~2233~~

100r = 11.2233……

1000000r = 112233.2233……

(10^{6}r – 10^{2}r) = 112233.2233 – 11.2233 = 112222.00

999900r = 112222

r = (112222)/(999900)

**r **= (5101)/(45450)

- Construct rational numbers with period of lengths

(i) 10

**Solution:**

Let r = 4.1234543213

10^{10} r = 41234543213. 1234543213

(10^{10 } – 1) r = 41234543209.0

r = (41234543209)/(10^{10}−1)

(ii) 12

**Solution:**

Let r = 8.431354789265

10^{12} r = 8431354789265. 431354789265

(10^{12} – 1) r = 8431354789257.0

r = (8431354789257)/(10^{12 }−1)

(iii) 15

**Solution:**

Let r = 2.345464748494142

10^{15} r = 2345464748494142. 345464748494142

(10^{15 } – 1) r = 2345464748494140.0

r = (2345464748494140.0)/(10^{15}−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

10^{6} r = 142857. 142857

(10^{6 } – 1) r = 142857.0

r = 142857/999999

We observe that r = 1/7

The two rational numbers are not the same.

- 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~~