Fraction to decimal without guesswork
Core idea
Divide numerator by denominator. If, after simplification, the denominator has only 2s and 5s as prime factors, the decimal terminates; otherwise it repeats.
How it works
- Simplify a/b first to reduce work.
- Perform long division for the decimal expansion.
- Track remainders to detect repetition and the repeating block.
Sanity checks
- Sign: only on the final decimal; denominator stays positive.
- Back‑multiply: decimal × denominator ≈ numerator (within displayed rounding).
- Repeat detection: remainder seen before → cycle found.
Shortcuts
- Denominators 2,4,5,8,10,16,… terminate quickly.
- 1/3=0.(3), 2/3=0.(6), 1/6=0.1(6), 1/7=0.(142857).
- Use precision control to format, not compute.
Pitfalls
- Rounding the value instead of the display.
- Stopping long division before the repeat is clear.
- Skipping simplification; it may hide a terminating result.
Micro‑examples
- 3/8 = 0.375 (terminates)
- 5/12 = 0.41(6) (repeats)
- −7/4 = −1.75
Mini‑FAQ
- Why do some terminate? Because denom in lowest terms is 2^a·5^b.
- How to show repeating? Use an overline on the repeating digits.
- Exact vs rounded? Exact structure is from remainders; rounding only affects display.
Action tip
Enter the fraction, toggle repeating notation if helpful, and use the step output to see long‑division reasoning and verify the result.