How does the Stern-Brocot search work?

The Stern-Brocot tree organizes every positive fraction in lowest terms. Starting from the bounds 0/1 and 1/1, the tool repeatedly takes the mediant (sum of numerators over sum of denominators) and moves toward the target, which finds the simplest fraction at each level of accuracy.

What does the maximum denominator control?

It caps how large the bottom of the fraction may grow. A small limit like 100 gives a rough but tidy fraction, while a large limit like 1000000 finds a very close approximation at the cost of larger numbers.

Will the fraction be in lowest terms?

Yes. The Stern-Brocot construction only ever produces fractions that are already fully reduced, so the numerator and denominator share no common factor.

Can it represent a number exactly?

If the decimal is itself a fraction whose denominator fits within your limit, the tool returns it exactly with zero error. Irrational numbers like pi can only be approximated, so a small residual error remains.

Does it handle negative numbers and values above one?

Yes. The whole-number part is separated out and the fractional part is approximated, then they are recombined with the correct sign, so values like -2.75 or 12.333 work fine.

Float to Rational Fraction Converter

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.

Tips and example

Try 3.14159265 with a maximum denominator of 10000: the tool returns 355/113, a famously accurate rational approximation of pi with an error under a millionth. Raising the denominator limit always gives an approximation at least as good, while lowering it produces simpler but coarser fractions. If your decimal is exactly representable — for instance 0.375 with a limit of 8 — the result is exact with zero reported error.