Repeating decimal (recurring decimal)

Definition and characteristics

When a decimal has a finite initial part followed by an infinite repetition of a block of digits, the repeating block is called the repetend or reptend, and the number of digits in that block is the period length. A purely repeating decimal has no non-repeating initial part (e.g., 0.444...), while a mixed repeating decimal has both a non-repeating prefix and a repeating tail (e.g., 0.166 = 0.1666...).

Why repeating decimals occur

Every rational number (a fraction of two integers) has a decimal expansion that either terminates or eventually repeats. The behavior depends on the denominator in lowest terms: if the denominator has only the prime factors 2 and/or 5, the decimal terminates; otherwise the decimal repeats. This connection is a basic fact from elementary number theory; further properties of repetends can be studied using concepts from number theory.

Conversion to and from fractions

There is a straightforward algebraic method to convert a repeating decimal to an exact fraction. Multiply the decimal by a power of 10 that shifts one full repetend to align with the original, subtract to eliminate the repeating part, and solve for the number. The reverse—expressing a rational fraction as a decimal—follows from long division of numerator by denominator. For more on procedures and worked examples, consult resources on conversion techniques.

Examples and notable facts

• Simple examples: 1/3 = 0.333..., 1/6 = 0.1666..., 2/7 = 0.285714285714... where the repetend 285714 has period length 6.
• Terminology: the repeating block is the repetend; its length is the period.
• For prime denominators not dividing 10, the period of 1/p divides p−1 (a consequence of the multiplicative order of 10 modulo p).

For general background on decimal representations and related concepts, see an introductory article on decimal notation. Repeating decimals bridge elementary arithmetic and modular arithmetic and are a standard topic where decimal notation, fractions, and divisibility intersect.