Math

Decimal to Fraction Calculator

What this does

Convert any decimal into its simplest fraction using continued fractions, with mixed-number output and honest notes when only an approximation exists.

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Calculator inputs

How to use it

Using the decimal to fraction calculator

  1. 01

    Enter the decimal

    Positive, negative or zero; terminating or long-repeating alike.

  2. 02

    Read the simplified fraction

    Already reduced to lowest terms via the GCD.

  3. 03

    Check the extras

    Values over one also show a mixed-number form; approximate matches say so explicitly.

Good to know

How the search stays fast and correct

Brute-forcing denominators from 1 to a million works but wastes effort; continued-fraction convergents jump straight to the best candidate at each size, because number theory guarantees no smaller denominator fits between consecutive convergents. That is why π collapses to 355/113 so quickly.

Exact versus approximate, stated plainly

Terminating decimals like 0.375 map to exact fractions (3/8). Irrational-ish inputs cannot match exactly within any denominator cap, so the note tells you the result is the best approximation found; no silent fudging either way.

Where mixed numbers help

Cooking and carpentry live in mixed units: 11/4 cups reads awkwardly while “2 and 3/4 cups” pours cleanly. Both forms are offered whenever the improper fraction exceeds one.

How it's calculated

The math behind this calculator

Continued-fraction convergents p/q minimise |x − p/q| with q ≤ 1,000,000

The decimal is expanded into a continued fraction whose convergents give the best rational approximations at every denominator size. We take the first convergent reproducing your input within floating-point precision, capped at a denominator of one million; values with no exact small fraction (like π) get the closest achievable approximation, flagged honestly.

Assumptions & limitations

  • Denominators above 1,000,000 are not searched.
  • Floating-point input itself limits precision beyond about fifteen significant digits.
  • Mixed numbers appear only for values whose absolute value exceeds one.

Worked example

2.75 converts exactly to 11/4, shown as the mixed number 2 3/4.

FAQ

Frequently asked questions

Why did I get an approximation notice?
Your decimal likely came from an irrational quantity or was rounded somewhere upstream; the tool returns the closest simple fraction below the denominator cap and tells you.
Are negative decimals supported?
Yes; the sign attaches to the numerator, e.g. −0.75 becomes −3/4.
What about repeating decimals like 0.333…?
Enter enough digits (e.g. 0.3333333333) and the convergent search recovers 1/3 exactly within tolerance.
Is the fraction always in lowest terms?
Yes; convergents arrive reduced, and a final GCD pass guarantees it regardless.

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