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Express 0.99999 .... in the form p/q. Are you surprised by your answer?

4.  Express 0.99999 .... in the form \frac{p}{q} . Are you surprised by your answer?

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Let x = 0.\overline{9}= 0.9999....             -(i)

Now, multiply by 10 on both sides

10x= 9.999....

\Rightarrow 10x = 9 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))

\Rightarrow 9x = 9

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

Therefore,  \frac{p}{q}  form of  0.999....  is  1

The difference between 1 and 0.999999 is o.000001 which is almost negligible.

Therefore, 0.999 is too much closer to 1. Hence, we can write 0.999999.... as 1

 

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