Question

The value of 1.9999… in the form



, where are integers and ≠ 0​

Answer 1

Let,

$\longrightarrow x=1.999\dots\quad\quad\dots(1)$

Multiplying by 10,

$\longrightarrow 10x=19.999\dots\quad\quad\dots(2)$

Subtracting (1) from (2),

$\longrightarrow 10x-x=19.999\dots\,-1.999\dots$

Neglecting the decimal part,

$\longrightarrow 9x=19-1$

$\longrightarrow 9x=18$

$\longrightarrow x=\dfrac{18}{9}$

$\longrightarrow\underline{\underline{x=2}}$

Now $x=1.999\dots$ is in the form $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q\neq0.$

In $x=2=\dfrac{2}{1},$ we have $p=2k$ and $q=k$ for any non - zero integer $k.$