Mathematics

The key insight is that a repeating decimal is really a geometric series in disguise. When you multiply $0.\overline{8}$ by 10, you shift the decimal point one place—and because the pattern repeats forever, you get back almost the same number. This lets you set up an equation to trap the infinite repetition and solve for the fraction directly.

The key insight is that a repeating decimal is really a geometric series in disguise. When you multiply $0.\overline{8}$ by 10, you shift the decimal point one place—and because the pattern repeats forever, you get back almost the same number. This lets you set up an equation to trap the infinite repetition and solve for the fraction directly.