Repeating Decimals

Summary: Decimal fractions in which one or more digits repeat indefinitely; every vulgar fraction produces either a terminating or a repeating decimal, and any repeating decimal equals a vulgar fraction with denominator consisting of 9s.

Sources: chapter-1.3.12

Last updated: 2026-05-02

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Why decimals repeat

When dividing numerator $a$ by denominator $b$, the remainder at each step belongs to $\{0,1,\dots ,b-1\}$. If the remainder is ever 0 the decimal terminates; otherwise, after at most $b-1$ steps a remainder must recur, and from that point the digit pattern repeats cyclically (§533).

Period length

The period (length of the repeating block) divides $b-1$ for prime denominators $b$ coprime to 10. For denominator 7 the six possible remainders $\{1,2,3,4,5,6\}$ give period exactly 6: $1/7=0.\overline{142857}$.

Converting repeating decimals to fractions

Multiply $s$ by $10^k$ where $k$ is the period length, then subtract $s$ (§537–538):

$k=1:0.\overline{a}=\frac{a}{9}$ $k=2:0.\overline{ab}=\frac{ab}{99}$ $k=3:0.\overline{abc}=\frac{abc}{999}$

In general, a repeating block of $k$ digits gives denominator $\underset{k}{\underbrace{99\cdots 9}}$, which equals $10^k-1$.

Examples

Decimal	Fraction
$0.\overline{3}$	$3/9=1/3$
$0.\overline{6}$	$6/9=2/3$
$0.\overline{142857}$	$142857/999999=1/7$
$0.\overline{09}$	$9/99=1/11$
$0.\overline{296296}$	$296/999=8/27$
$9.\overline{9}$	$9+9/9=10$

The last example shows that $9.999\dots =10$ exactly (§524).

Connection to geometric series

A repeating decimal is an infinite geometrical progression. For example, $0.\overline{142857}$ equals the series with first term $142857/10^6$ and ratio $1/10^6$; the sum formula $a/(1-r)$ gives $142857/(10^6-1)=1/7$.

Related pages

• decimal-fractions
• geometrical-progressions
• fractions
• ch1.3.12-infinite-decimal-fractions
• ch1.3.11-geometrical-progressions