A fraction is a division: $\frac{3}{8}$ means $3 \div 8$. With long division you can turn any fraction into a decimal. Sometimes the division stops; sometimes it goes on forever, in a pattern.
Idea clave
Divide the numerator by the denominator. The decimal either terminates (the remainder reaches 0) or repeats (a block of digits repeats forever). A bar marks the repeating block: $\frac{2}{3} = 0.\overline{6}$ and $\frac{1}{6} = 0.1\overline{6}$.
Por qué funciona
When you divide by 6, every remainder is one of $0, 1, 2, 3, 4, 5$. If a remainder is 0, the division ends. If not, a remainder must come back within a few steps, and from then on the division does exactly the same thing again, so the digits repeat. A fraction in lowest terms terminates exactly when its denominator has no prime factors except 2 and 5, the factors of 10.
Worked examples
Ejemplo resuelto: A terminating decimal
Write $\frac{3}{8}$ as a decimal.
- $3 \div 8$: $30 \div 8 = 3$ remainder 6; $60 \div 8 = 7$ remainder 4; $40 \div 8 = 5$ remainder 0.
- The remainder is 0, so it stops: $0.375$.
Respuesta: $0.375$
Ejemplo resuelto: A repeating decimal
Write $\frac{2}{3}$ as a decimal.
- $20 \div 3 = 6$ remainder 2. Then $20 \div 3 = 6$ remainder 2 again.
- The remainder 2 keeps coming back, so the 6 repeats forever: $0.666\ldots = 0.\overline{6}$.
Respuesta: $0.\overline{6}$
Ejemplo resuelto: The bar on part of it
Write $\frac{5}{11}$ as a decimal.
- $50 \div 11 = 4$ remainder 6; $60 \div 11 = 5$ remainder 5.
- Remainder 5 is where we started (5 ÷ 11), so "45" repeats: $0.4545\ldots = 0.\overline{45}$.
Respuesta: $0.\overline{45}$
Error común
Stopping too early and writing $\frac{2}{3} = 0.6$. That is only $\frac{6}{10} = \frac{3}{5}$. Even $0.67$ is a rounded value, not equal to $\frac{2}{3}$. Keep dividing until the remainder is 0 or a remainder repeats.
| Fraction | Decimal |
|---|---|
| $\frac{1}{3}$ | $0.\overline{3}$ |
| $\frac{1}{6}$ | $0.1\overline{6}$ |
| $\frac{1}{9}$ | $0.\overline{1}$ |
| $\frac{1}{11}$ | $0.\overline{09}$ |
| $\frac{1}{7}$ | $0.\overline{142857}$ |