About the Continued Fraction Calculator
A continued fraction writes a number as a whole part plus one over another whole part plus one over another, and so on. This calculator takes a decimal such as 3.14159265 or a fraction such as 355/113 and gives the list of terms, written [a0; a1, a2, …]. It also lists the convergents, which are the best fractions you can get with small denominators. Type a decimal or a fraction and the continued fraction appears at once, with a few related figures underneath, and you can copy the result with one tap. The sums run in your browser, so nothing you enter is uploaded.
Updated October 2026
How to use the Continued Fraction Calculator
- 1Enter a decimal or a fraction.
- 2Pick the units or options from the menus if the continued fraction tool offers them.
- 3Read the continued fraction in the large result box.
- 4Look at the extra rows for related figures.
- 5Press Copy result to keep the answer.
When you need the continued fraction
Continued fractions explain why 22/7 and 355/113 are good approximations for pi, and why a gear ratio or musical interval can be matched with small whole numbers.
A big term in the list means that the previous convergent is an especially good approximation. For pi, the term 292 is why 355/113 is so precise. Rational numbers end after a finite number of terms, while irrational ones go on forever.
How the continued fraction is worked out
Take the whole part of the number as the next term, subtract it, invert what is left, and repeat. The convergents are built from the terms with hₙ = aₙ × hₙ₋₁ + hₙ₋₂ and the same rule for the denominators. 3.14159265 gives [3; 7, 15, 1, 288, …] (the true value of pi continues 3; 7, 15, 1, 292). The first convergents are 3/1, 22/7 and 333/106. 355/113 comes next and is accurate to six decimal places.
Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.
- Decimals and fractions
- Shows convergents with their decimal values
- Exact whole-number arithmetic
Mistakes to avoid with continued fraction
A decimal that has been rounded gives terms that are only correct up to a point: terms far along reflect the rounding, not the true number. Type more digits for more correct terms. Negative numbers have a negative first term.
On phone and computer
On a phone, type into the boxes one after another and read the continued fraction underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the continued fraction move. The continued fraction calculator works the same on both.
Checking your continued fraction by hand
A good habit is to test the continued fraction calculator with a case you already know. 3.14159265 gives [3; 7, 15, 1, 288, …] (the true value of pi continues 3; 7, 15, 1, 292). The first convergents are 3/1, 22/7 and 333/106. 355/113 comes next and is accurate to six decimal places.
If your own figures give a surprising answer, repeat the working yourself: Take the whole part of the number as the next term, subtract it, invert what is left, and repeat. The convergents are built from the terms with hₙ = aₙ × hₙ₋₁ + hₙ₋₂ and the same rule for the denominators. Small differences usually come from rounding, from mixing units, or from a typing slip in one of the inputs. Continued fractions connect to several other number ideas. Use the Fraction Calculator for everyday fraction sums, see unit-fraction splits with the Egyptian Fraction Calculator, and solve the equation behind a golden-ratio style pattern with the Quadratic Equation Solver.
Features
- Rounded decimals give false late terms
- Large terms mean great approximations
- Rational numbers terminate
- Clear messages when the continued fraction inputs make no sense
- Copy the result with one tap
- Private: nothing is uploaded
Frequently asked questions
How do I calculate continued fraction?
Take the whole part of the number as the next term, subtract it, invert what is left, and repeat. The convergents are built from the terms with hₙ = aₙ × hₙ₋₁ + hₙ₋₂ and the same rule for the denominators. 3.14159265 gives [3; 7, 15, 1, 288, …] (the true value of pi continues 3; 7, 15, 1, 292). The first convergents are 3/1, 22/7 and 333/106. 355/113 comes next and is accurate to six decimal places.
What is the formula for continued fraction?
Take the whole part of the number as the next term, subtract it, invert what is left, and repeat. The convergents are built from the terms with hₙ = aₙ × hₙ₋₁ + hₙ₋₂ and the same rule for the denominators. The continued fraction calculator applies it exactly and rounds only the displayed answer to six significant figures.
What is a convergent?
The fraction you get by cutting a continued fraction after a certain term.
Why is 355/113 so close to pi?
Because the next term after it in pi's continued fraction is the large number 292.
Do all numbers have a continued fraction?
Yes. Fractions end after finitely many terms and irrational numbers go on forever.
Is this continued fraction calculator free and private?
Yes. It is free, needs no account and runs in your browser, so the numbers you type are not uploaded or stored.