Express 0.99999.... in the form $$\frac{p}{q}$$. Are you surprised by your answer? Discuss why the answer makes sense with your teacher and classmate.

Asked by Vishal kumar | 2 years ago |  215

##### Solution :-

Let x = 0.99999.....  ......(a)

We need to multiply by 10 on both sides, we get

10x = 9.9999....  ........(b)

Subtract the equation (a) from (b), to get

$$\cfrac{\begin{matrix}10\mathrm x=9.99999....\\-\mathrm x = 0.99999....\end{matrix}}{9\mathrm x=9}$$

9x = 9 as x = $$\frac{9}{9}$$ or x = 1.

Therefore, on converting 0.99999..... $$=\frac{1}{1}$$ which is in the $$\frac{p}{q}$$ form,

Yes, at a glance we are surprised at our answer.

But the answer makes sense when we observe that 0.9999……… goes on forever.

So, there is no gap between 1 and 0.9999……. and hence they are equal.

Answered by Shivani Kumari | 2 years ago

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