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Find the first ten digits after the decimal point in the decimal expansion of \frac{7}{11}=0.abcdefghij\ldots without a calculator.
(Express your answer as a ten digit number.)

 Feb 23, 2024
 #1
avatar+394 
+1

Since 11 is a factor of 99, we suspect this is a repeating decimal.

We multiply top and bottom by 9,  getting \(\frac{63}{99}\).

Our suspicions are confirmed. In general, \(0.\overline{abc\dots n}\) is represented by \(\frac{\overline{abc \dots n}}{{10}^{n}-1}\), where n is the number of terms.

So, \(\frac{63}{99}=0.\overline{63}\), so our answer is 6363636363.

 Feb 23, 2024

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