FractionReference

Rational approximation

Type a decimal, get the fractions worth knowing about it: the exact conversion of your digits, then the sequence of best simple approximations, each with its error stated.

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Common questions

What is a rational approximation?
A fraction with a small denominator that lands close to a number. For 3.14159, the classics are 22/7 and 355/113 — each is the best possible fraction at or below its denominator, and each is shown here with its exact error.
How are the candidates found?
By continued fractions: the number is repeatedly split into a whole part and a remainder, and the convergents of that expansion are provably the best rational approximations. Deterministic, and documented on the methodology page.
Is the exact fraction the same as an approximation?
No, and this page keeps them apart. 3.14159 typed in IS exactly 314159/100000 — that is a conversion. The approximations are different, simpler fractions that land nearby, each labeled with how far off it is.
Where this comes from. Exact rational arithmetic on the values shown — fractions are pairs of integers until display, and inch–metric conversion uses 25.4 mm to the inch, exact by definition since 1959. Convergents are computed by the standard continued-fraction recurrence. See the full methodology.