Convert fractional numbers to different bases and detect repeating digit sequences with detailed derivation
Input Fraction (e.g., 1/3 or 0.333...)
Target Base (2-36)
Result
Result with Repeating Notation
Step-by-Step Derivation
Repeating Fraction Detection Principle
① Multiply fractional part by base, extract integer digit
② Track remainders—repeat detected when remainder repeats
③ Notation: non-repeating.(repeating), e.g., 0.123(456)
A fraction terminates if denominator (in lowest terms) shares all prime factors with base. Otherwise, it repeats. Remainder repetition indicates cycle start!
⚠Some fractions have very long repeating cycles. We limit to 20 digits for practical display. Parentheses () indicate the repeating part.
What Are Repeating Radix Fractions?
Repeating radix fractions have digit sequences that repeat indefinitely. Detected by tracking remainders during conversion—when a remainder repeats, the cycle repeats!
💡 Teaching Example: 1/3 in binary: 0.(01). Multiply by 2: 0.666...→0, 1.333...→1, 0.666...→0 (remainder repeats!). Cycle: 01.
Applications
Number TheoryMathematicsEducationComputer ScienceRecreational Math
Frequently Asked Questions
What is a repeating radix fraction?▼
A repeating radix fraction is a fractional number in some base where a sequence of digits repeats indefinitely. In decimal, 1/3 = 0.(3). In binary, 1/3 = 0.(01). The repeating part is called the repetend.
How do you detect repeating digits?▼
While converting the fractional part, track remainders. When a remainder repeats, the digits between the two occurrences form the repeating cycle. This is because remainders determine subsequent digits.
What is vinculum notation?▼
Vinculum notation uses a horizontal line (or parentheses) over repeating digits. For example, 0.123456456456... is written as 0.123(456) where (456) is the repeating cycle.
Why do some fractions repeat in some bases but not others?▼
A fraction terminates in base b if, when in lowest terms, the denominator has no prime factors other than those of b. Otherwise it repeats. Example: 1/3 terminates in base 3 but repeats in decimal.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.