​ Express 0. bar(123) as a rational number. ​

Question

Express 0.\(\overline{123}\) as a rational number.

Answer 1

Let, x = 0.123123123…. 

⇒ x = 0.123 + 0.000123 + 0.000000123 + …∞ 

⇒ x = 123(0.001 + 0.000001 + 0.000000001 + …∞)

⇒ x = 123 \(\left(\frac{1}{10^3} + \frac{1}{10^6} + \frac{1}{10^9} + \frac{1}{10^12} + ..... \infty\right)\)

This is an infinite geometric series.

Here, a = \(\frac{1}{10^3}\) and r = \(\frac{1}{10^3}\)

0.\(\overline{123}\) = \(\frac{123}{999}\)