Question

Find the rotional number whose decimal expansion is $0.\overline{4}23$.

Answer 1

Let the rational number S be $0.\overline{4}23$

$\because S=0.\overline{4}23=0.4+\mathrm{0.0232323........}$

$=0.4+0.023+0.00023+.........\infty$

Clearly, S is a geometric series with the first term a being 1 and the common ratio, r being (10^{-2})

$\therefore S=0.4+\frac{23}{{10}^3}+\frac{23}{{10}^5}+...$

$\Rightarrow S=0.4+\frac{23}{{10}^3}(1+\frac{1}{{10}^2}+\frac{1}{{10}^4}+....)$

$\Rightarrow S=0.4+\frac{23}{1000}\left(\frac{1}{1-\frac{1}{100}}\right)$

$\Rightarrow S=0.4+\frac{23}{1000}\left(\frac{100}{99}\right)$

=$\frac{4}{10}+\frac{23}{990}=\frac{396+23}{990}=\frac{419}{990}$