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I know how to convert most rational decimal but I am having trouble to convert 1.8768686868... or 0.95287928792879...

I did end up solving one of these tricky decimals problems but I was wondering if there is a quick simple technique for this?

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  • $\begingroup$ One lengthy but fun method is to make a geometric progression. E.g in$1.87686868\ldots$ you can make a geometric progression out of the repeating part with general term $T_r=\frac{68}{100^{r+1}}$ $\endgroup$ – AvZ Feb 17 '15 at 5:11
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$$1.87\overline{68} = \dfrac{187}{100} + \dfrac{1}{100} \dfrac{68}{100-1} = \dfrac{18581}{9900}$$

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  • $\begingroup$ This is really nice. Separating them into two pieces is just brilliant idea. Thank you. $\endgroup$ – Sadij Feb 18 '15 at 14:15
  • $\begingroup$ Is there some literature that explores this concept further and explains why ABCD.../9999... = 0.ABCD...ABCD...ABCD... ? How can this concept be expressed in proper mathematical notation? $\endgroup$ – Dmitri Jun 8 '17 at 13:58
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    $\begingroup$ @Dmitri Geometric series: $\sum_{n=1}^\infty x/10^{nk} = x/(10^k-1)$. $\endgroup$ – Robert Israel Jun 8 '17 at 15:31
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Let $x = 1.8768686868...….$ eqn (1)

The repetition starts from $2$ decimal places. This means we need to multiply (1) by $10^2$ and get

$100x = 187.686868……$ eqn (2)

To cancel the $.686868…$ we need another supporting equation:-

$10000x = 18768.686868….$ eqn(3)

Do (3) – (2) and solve for x.

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$ \text{ Repetition part starts in the thousandths spot and ends in the 10 thousandths spot. Let } x=1.876868686868.... \text{ then } 10,000x=18768.686868686868.... \text{ Now find the difference } 10,000x-x=18768.68686868...-1.8768686868....=18768.68-1.87 \\ 9,999x=18766.81 \\ \text{ Now multiply both sides by 100: } 999900x=1876681 \text{ now you can solve for } x $

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Count the length of the repeating part. In your first example it is two digits. Multiply by $10$ to that power and subtract. So for your first example, let $x=1.87686868\dots $ Then $10^2x=100x=187.686868\dots $ That gives $99x=187.686868\dots - 1.87686868\dots =185.79$ The point of counting the repeat is to get the infinite tail to subtract to nothing.

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