When you write a repeating decimal into a fraction why does the fraction always have only 9s or 9s and 0s as digits in the denominator?

Answer 1

When converting a repeating decimal into a fraction, the reason the denominator consists of 9s or a combination of 9s and 0s is rooted in the nature of the decimal system. Each digit in the repeating part corresponds to a division by powers of 10, while the repeating cycle creates a geometric series. The formula for converting a repeating decimal to a fraction effectively captures this series, resulting in a denominator that is a series of 9s for each repeating digit, and 0s for any non-repeating digits that precede the repeating section. For example, in the decimal 0.666..., the repeating '6' creates a fraction with a denominator of 9, while a decimal like 0.1(23) would result in a denominator of 990, reflecting both the repeating and non-repeating parts.