+400 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.)
+394 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.
