Repeating Decimal
A repeating decimal is a decimal in which one or more digits after the decimal point repeat infinitely in a regular pattern.
Definition
A repeating decimal is a decimal where a digit or group of digits keeps repeating forever after the decimal point; we write a bar (vinculum) over the repeating part to show it goes on forever, so $0.333\ldots$ is written $0.\overline{3}$. Formally, a repeating decimal has a block of digits (the repetend) that cycles infinitely, and every repeating decimal represents a rational number while every rational number has a repeating (or terminating) decimal expansion. A rational $p/q$ in lowest terms has a purely repeating expansion iff $\gcd(q, 10) = 1$; otherwise it has a non-repeating initial part of length $\max(v_2(q), v_5(q))$ followed by a repeating block of length equal to the multiplicative order of $10$ modulo $q' = q/(2^{v_2(q)} \cdot 5^{v_5(q)})$.
Example
$1/3 = 0.333\ldots$ (the $3$ repeats) and $2/11 = 0.181818\ldots$ ($18$ repeats), written $0.\overline{3}$ and $0.\overline{18}$. To convert a repeating decimal back to a fraction: let $x = 0.2727\ldots$, then $100x = 27.2727\ldots$, and subtracting gives $99x = 27$, so $x = 27/99 = 3/11$. For $1/12$: since $12 = 2^2 \times 3$, the non-repeating part has length $\max(2,0) = 2$ and $q' = 3$ has $\text{ord}_3(10) = 1$, so $1/12 = 0.08\overline{3}$, two non-repeating digits followed by a period-$1$ repeat.
Key Insight
Every fraction eventually turns into either a terminating decimal or a repeating decimal when you divide, because the division process must eventually repeat, there are only so many possible remainders. The length of the repeating block of $1/p$ (for prime $p \neq 2$ or $5$) equals the multiplicative order of $10$ modulo $p$, a result from number theory; for $1/7$, $\text{ord}_7(10) = 6$, so the repeat length is $6$ digits ($142857$). The theory of repeating decimals is in fact equivalent to the theory of linear recurrences over $\mathbb{Z}/10\mathbb{Z}$, and the Berlekamp-Massey algorithm can find the minimal repeating pattern of any sequence, connecting elementary fraction arithmetic to modern sequence analysis and coding theory.