Question

Prove that 0.8333..... is a rational number. How can I prove?

Answer 1

Let,

$\displaystyle\longrightarrow\sf{x=0.8333\dots\quad\quad\dots(1)}$

$\displaystyle\longrightarrow\sf{10x=8.333\dots\quad\quad\dots(2)}$

Subtracting (1) from (2),

$\displaystyle\longrightarrow\sf{10x-x=8.333\dots\ -\ 0.8333\dots}$

$\displaystyle\longrightarrow\sf{9x=(8.3+0.0333\dots)\ -\ (0.8+0.0333\dots)}$

$\displaystyle\longrightarrow\sf{9x=8.3+0.0333\dots\ -\ 0.8-0.0333\dots}$

$\displaystyle\longrightarrow\sf{9x=8.3-0.8}$

$\displaystyle\longrightarrow\sf{9x=7.5}$

$\displaystyle\longrightarrow\sf{x=\dfrac{7.5}{9}}$

$\displaystyle\longrightarrow\sf{x=\dfrac{5}{6}\quad\quad\dots(3)}$

From (1) and (3),

$\displaystyle\longrightarrow\sf{\dfrac{5}{6}=0.8333\dots}$

Here $\displaystyle\sf{0.8333\dots}$ is represented as a fraction, i.e., in $\displaystyle\sf{\dfrac{p}{q}}$ form. Thus we can say that it is a rational number.

Hence Proved!