What is the reciprocal of $0.\overline{714285}$? Express your answer as a decimal.
To find the reciprocal of a number, I just need to flip it over, so the numerator and denominator switch places. But first, I want to convert this repeating decimal to a fraction, so it's easier to work with. I'll multiply the decimal by $10^6$, which is 1,000,000, to shift the decimal point 6 places to the right. This gives me $714285.\overline{714285} = 0.\overline{714285} \times 10^6$. Now I can subtract the original decimal from both sides, to get rid of the repeating part on the right. This gives me $714285 = 0.\overline{714285} \times 10^6 - 0.\overline{714285}$. I can factor out the common factor of $0.\overline{714285}$ from the right side, and get $714285 = 0.\overline{714285} \times (10^6 - 1)$. Now I can divide both sides by $10^6 - 1$, which is 999,999, to get $0.\overline{714285} = \frac{714285}{999999}$. I can use the Euclidean algorithm to find the greatest common divisor of 714285 and 999999. I start by dividing 999999 by 714285, and get a quotient of 1 and a remainder of 285714. Then I divide 714285 by 285714, and get a quotient of 2 and a remainder of 142857. Then I divide 285714 by 142857, and get a quotient of 2 and a remainder of 0. This means that 142857 is the greatest common divisor of 714285 and 999999. So I can divide both the numerator and denominator by 142857, and get $0.\overline{714285} = \frac{714285}{999999} = \frac{5}{7}$. This is the simplest fraction form of the repeating decimal. Now I can find the reciprocal of this fraction by flipping it over, and get $\frac{7}{5}$. To express this as a decimal, I just need to divide 7 by 5, which gives me 1.4. 