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Express $ 0.99999 \ldots $ 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.
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