This converter turns a decimal number into the simplest fraction that approximates it within a denominator you choose. It is useful for recovering a clean ratio from a measurement, finding a gear or pulley ratio, or expressing a constant like pi as a friendly fraction such as 355/113.
How it works
The tool walks the Stern-Brocot tree, a structure that lists every reduced
positive fraction exactly once. It separates the whole part from the fractional
part, then searches the fractional part between the bounds 0/1 and 1/1. At
each step it computes the mediant of the current bounds:
mediant(a/b, c/d) = (a + c) / (b + d)If the mediant is below the target it becomes the new lower bound, otherwise the new upper bound, narrowing in like a binary search. The search stops once the denominator would exceed your limit, keeping the best (lowest-error) fraction seen. Because every Stern-Brocot fraction is already in lowest terms, no extra reduction is needed.
Classic examples worth trying
Some of the most famous rational approximations are fun to verify with this tool:
| Decimal | Denominator limit | Best fraction | Error |
|---|---|---|---|
| 3.14159265 | 10 | 22/7 | +0.000402… |
| 3.14159265 | 10,000 | 355/113 | -0.00000027… |
| 1.41421356 (√2) | 100 | 99/70 | -0.0000051… |
| 1.61803399 (φ) | 100 | 89/55 | +0.0000083… |
| 0.333333… (1/3) | 10 | 1/3 | 0 (exact) |
| 0.142857… (1/7) | 10 | 1/7 | 0 (exact) |
The approximation 355/113 for pi is remarkably accurate — known since the 5th century by the Chinese mathematician Zu Chongzhi — and cannot be improved until you allow a denominator of over 33,000. This makes it the best “simple-denominator” approximation of pi in that range.
When this tool is genuinely useful
Gear and pulley ratios. If you measure a gear ratio as a decimal (for example, 0.4286 from a tachometer reading), converting to 3/7 immediately tells you the tooth counts involved and confirms the mechanism is working correctly.
Recipe scaling. A scaling factor of 0.6667 is exactly 2/3. Recognising this avoids imprecise measurements — pour to the 2/3 mark rather than trying to measure 0.6667 of a cup.
Music theory. Frequency ratios in Western tuning are rational numbers. A perfect fifth is 3/2 (1.5), a major third is 5/4 (1.25), and equal temperament’s whole tone is 2^(2/12), which is irrational. Finding the closest simple ratio reveals how well a tuning approximates just intonation.
Display and screen ratios. Screen aspect ratios like 1.7778 resolve immediately to 16/9.
Engineering measurements. A measured dimension of 0.625 inches simplifies to 5/8, which is a standard drill or fastener size that you can then actually buy.
Denominator limit and accuracy trade-off
The denominator limit is the single control over accuracy versus simplicity. Some practical values:
- 10: very simple fractions only — halves, thirds, quarters, fifths, eighths. High error on irrational numbers.
- 100: most simple engineering fractions (1/16, 3/7, etc.). Good for measurements in imperial units.
- 1,000: close approximations of most common constants.
- 10,000: excellent approximations — 355/113 for pi, 99/70 for √2.
- 1,000,000: near-perfect; mainly useful for checking whether a decimal is “almost exactly” a simple fraction.
Raising the limit always gives a result at least as accurate as a lower limit, never worse. If your decimal is exactly representable as a fraction within the limit (for example 0.375 = 3/8), the result is exact with zero reported error. All computation runs locally in your browser.