Continued Fraction Calculator

How it is calculated

The tool repeats one step: take the integer part a = floor(x), subtract it, and invert the leftover, x = 1 / (x - a). The collected integers form [a0; a1, a2, ...]. It stops when the remainder drops below 1e-9 (the number has ended) or when it reaches the number of terms you ask for. That limit must be between 1 and 30; a value below 1 falls back to 10 and a value above 30 is cut to 30.

The fraction is built with the recurrence h(n) = a(n)*h(n-1) + h(n-2) and k(n) = a(n)*k(n-1) + k(n-2), giving h / k. The decimal shown is that fraction divided out to six places, so you can see how close it is to your input. Floating-point input means very long expansions of a decimal like 0.333333 end in huge terms: the tool expands the decimal you typed, not the fraction you had in mind.

Common mistakes

• Asking for more terms to get a 'cleaner' fraction. More terms give a closer fraction but usually with a much larger denominator; for a gear ratio or a ruler fraction a few terms are often the useful answer.
• Typing a rounded decimal such as 0.333 or 0.33333 and expecting 1/3. The tool expands exactly what you typed, so you get a different, ugly fraction instead.
• Reading the bracket as a list of equal parts. The first number is the whole part and the rest, after the semicolon, are the nested denominators.
• Treating the six-place decimal as the input. It is the value of the fraction found, so compare it with your number rather than copying it back.