What is a continued fraction?

It's a way of writing a number as an integer part plus 1 divided by (another integer plus 1 divided by ...), written compactly as [a₀; a₁, a₂, a₃, ...].

What is a convergent?

A convergent is the fraction you get by truncating a continued fraction expansion after a certain number of terms — each one is the best possible rational approximation of the original number with a denominator that size.

Does every number have a continued fraction?

Every real number does, but only rational numbers (fractions and finite or repeating decimals) have a finite continued fraction — irrational numbers like π or √2 have infinite, non-repeating expansions, which is why this calculator, which requires an exact finite input, works with rationals.

Why does 355/113 approximate π so well?

It's a convergent in π's continued fraction expansion [3; 7, 15, 1, 292, ...] — truncating right before the unusually large term 292 produces an exceptionally accurate approximation for how small its denominator is.

Can I enter a fraction directly?

Yes — enter it in the form p/q, like 355/113, and the calculator will reduce it and expand it directly without needing a decimal approximation first.

Why does this calculator only support non-negative numbers?

Continued fraction expansion relies on the floor function, which behaves differently for negative numbers in a way that would complicate the term-by-term explanation — this calculator keeps the scope to non-negative values for clarity.

How large can my input be?

Decimal inputs support up to 30 digits after the decimal point, and the expansion runs up to 200 terms, more than enough for any reasonably sized rational number.

Where are continued fractions used in real life?

They're used in finding best rational approximations (calendar systems, gear ratios, musical tuning), solving Pell's equation in number theory, and in some algorithms for factoring integers and rational approximation.

Continued Fraction Calculator — Expand a Number and See Its Convergents

Our Continued Fraction Calculator takes any non-negative decimal number or fraction (p/q) and expands it into its continued fraction terms [a₀; a₁, a₂, ...] using the Euclidean algorithm, then shows every convergent — the best rational approximations you get by truncating the expansion early — as both an exact fraction and a decimal, computed with BigInt precision throughout.

Quick Answer

Enter a decimal number or fraction below to instantly see its continued fraction expansion [a₀; a₁, a₂, ...] and a table of every convergent — its best rational approximations.

Number or fraction

Enter a non-negative decimal (e.g. 3.14159) or a fraction in the form p/q (e.g. 355/113).

How to Use the Continued Fraction Calculator — Expand & Evaluate Online

1
Enter a non-negative decimal number (like 3.14159) or a fraction in the form p/q (like 355/113).
2
Click 'Expand' to compute the continued fraction terms.
3
Review the convergents table showing each successively better rational approximation.

Why Use Continued Fraction Calculator — Expand & Evaluate Online?

Every rational number has a finite continued fraction expansion, found the same way the Euclidean algorithm finds a gcd — by repeatedly taking the integer part and inverting the remainder. Truncating that expansion at any point gives a 'convergent,' the best possible rational approximation with a denominator that small — the same principle behind why 22/7 and 355/113 are famous approximations of π. This calculator performs the exact expansion with BigInt precision and lists every convergent along the way, letting you see exactly how the approximation improves at each step.

Frequently Asked Questions

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