Page 6 - 9 Math 2

2. Real and Complex Numbers eLearn.Punjab

e.g., = 1.414213562..., p= 3.141592654..., e= 2.718281829..., etc.

Obviously these decimal representations are neither terminating
nor recurring.
We consider the following example.

Example

Express the following decimals in the form p where p, q d Z
q

and q m0

(a) 0.3 = 0.333... (b) 0.23 = 0.232323...

Solution
(a) Let x = 0.3 which can be rewritten as
x = 0.3333... …… (i)
Note that we have only one digit 3 repeating indefinitely.
So, we multiply both sides of (i) by 10, and obtain
10x = (0.3333...) x 10
or 10x = 3.3333... …… (ii)
Subtracting (i) from (ii), we have
10x – x = (3.3333...) – (0.3333...)

or 9x = 3 ⇒ x = 1
3

Hence 0.3 = 1
3

(b) Let= x 0=.23 0.23 23 23...

Since two digit block 23 is repeating itself indefinitely, so we multiply
both sides by 100 .
Then 100x = 23.23

Version: 1.1 100x =23 + 0.23 =23 + x 6
⇒ 100x - x =23

⇒ 99x =23

⇒ x =23
99