Convert and simplify fractions in any base, with repeating pattern detection
Fraction (e.g., 1/3, 5/8)
Source Base
Target Base
Result
Fraction in Target Base
Step-by-Step Derivation
Fraction Conversion Principle
① Convert fraction to decimal first
② Multiply fractional part by target base
③ Extract integer as digit, repeat with remainder
④ Detect repeating patterns or termination
Fractions behave differently in different bases! Some terminate, others repeat with interesting patterns based on the base's prime factors.
⚠This tool detects repeating patterns! If a cycle is detected, the repeating digits will be indicated with parentheses or overbars conceptually.
What Are Fractional Numbers in Different Bases?
Just like decimal fractions use tenths, hundredths, etc., other bases use 1/b, 1/b², 1/b³, etc., for their fractional places.
Terminating Fractions
A fraction terminates if denominator's prime factors are all in the base. 1/8 terminates in decimal (8 = 2³).
Repeating Fractions
Otherwise, it repeats! 1/3 repeats in decimal (3 not factor of 10), but terminates in base3!
Cycle Detection
By tracking remainders, we find when a state repeats, indicating the start of a cycle!
Prime Factors
Base 10 = 2×5, base12 = 2²×3, base60 has many factors! More factors = more terminating fractions.
💡 Teaching Example: 1/3 in decimal = 0.(3) repeating. In base3, 1/3 = 0.1 which terminates perfectly! The base changes everything!
Applications
Number TheoryMathematicsEducationRecreational MathComputer Science
Frequently Asked Questions
What is a fractional number in a given base?▼
A fractional number in base b has an integer part and a fractional part after the radix point. Each digit to the right represents 1/b, 1/b², 1/b³, etc.
What are repeating fractions in different bases?▼
Just like decimal 1/3 = 0.333..., fractions in other bases repeat too if the denominator has prime factors not in the base. The pattern repeats forever!
How do you convert a fraction to another base?▼
Multiply the fractional part by the target base, extract the integer as a digit, and repeat with the new fractional part until it terminates or repeats.
Why do some fractions terminate and others repeat?▼
A fraction terminates in base b if the denominator (in simplest form) has only prime factors that are also factors of b. Otherwise, it repeats indefinitely!
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