Express: $0.99999\ ....$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Given:

A number $0.99999\ ....$

To do:

We have to express $0.99999\ ....$ in the form $\frac{p}{q}$.

Solution:

Let $x=0.99999.....$_________(i)

Multiplying equation (i) by $10$, we get,

$10x=9.9999.....$_______(ii)

Subtract (i) from (ii), we get,

$10x-x=9.9999.......-0.9999.........$

$\Rightarrow 9x=9$

$\Rightarrow x=\frac{9}{9}=1$

Here, $p=1$ and $q=1$

Therefore, $0.9999....=1$

The difference between 1 and 0.999999 is 0.000001 which is negligible.

Hence, we can conclude that 0.999 is very near to 1, therefore, 1 as the answer can be justified.