Continued Fraction Calculator

Frequently Asked Questions

What is a continued fraction?

A number written as a0 + 1/(a1 + 1/(a2 + …)). Every real number has such an expansion; rationals terminate and quadratic irrationals repeat.

What are convergents?

The fractions you get by truncating the expansion. Each convergent is the best rational approximation of its size - that is how 22/7 and 355/113 for π arise.

Why is it better than rounding a decimal?

A convergent p/q is closer to the true value than any other fraction with a denominator ≤ q, so you get maximum accuracy for the smallest denominator.

What does this calculator show?

The coefficient list [a0; a1, a2, …], each convergent as a fraction and decimal, and the approximation error at every step.

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How This Calculator Works

A continued fraction writes a number as a₀ + 1÷(a₁ + 1÷(a₂ + …)), compactly noted [a₀; a₁, a₂, …]. The integer part a₀ is the floor of the value; subtract it, take the reciprocal of the remaining fraction, and repeat. For a rational p÷q this is exactly the Euclidean algorithm: each quotient floor(p÷q) is the next term a_i, and the remainder feeds the next step - so the expansion always terminates for rationals. The convergents h_n÷k_n are built by the recurrence h_n = a_nh_n−1 + h_n−2 and k_n = a_nk_n−1 + k_n−2. Each convergent is the best rational approximation for its denominator size: no fraction with a smaller denominator gets closer.

Worked Example: 415 ÷ 93

415 = 4 × 93 + 43, so a₀ = 4. Now flip: 93 = 2 × 43 + 7, giving a₁ = 2. Then 43 = 6 × 7 + 1, so a₂ = 6, and 7 = 7 × 1 + 0, so a₃ = 7. The expansion is [4; 2, 6, 7]. The convergents are 4÷1, 9÷2, 58÷13, and finally 415÷93 ≈ 4.4624. The famous case is π ≈ [3; 7, 15, 1, 292, …]: the convergent after [3; 7] is 22÷7, and after [3; 7, 15, 1] is 355÷113 - accurate to six decimals, the approximation Chinese mathematician Zu Chongzhi found in the 5th century.

Where Continued Fractions Are Used

Best rational approximation: gearing ratios, calendar design (leap-year rules), and music tuning all pick a convergent that balances accuracy against a small denominator.

Number theory: solving Pell's equation x² − Dy² = 1 and detecting whether a number is rational (finite expansion) or irrational (infinite).

Cryptography: Wiener's attack on RSA recovers a small private exponent from the convergents of the public key.

The golden ratio φ = [1; 1, 1, 1, …] is the "most irrational" number - its all-1 expansion makes it the hardest to approximate by fractions.

Common Mistakes & Edge Cases

A finite continued fraction has two forms: […, a_n] = […, a_n−1, 1]. Convention prefers the shorter, with the last term ≥ 2.

Decimals are rounded to a rational first, so a long tail of large terms usually signals floating-point noise, not structure - this tool trims it.

Convergents alternate: even-indexed ones lie below the true value, odd-indexed ones above. They do not approach monotonically.

Negative numbers use floor, not truncation: −0.7 has a₀ = −1, not 0, so every later term stays positive.

Why Convergents Are Optimal

Each convergent h_n÷k_n satisfies the striking bound |x − h_n÷k_n| < 1÷k_n² - the approximation error shrinks faster than the square of the denominator. A theorem of Lagrange guarantees these are the best possible rational approximations: no fraction with denominator ≤ k_n lies closer to x. That is why 22÷7 and 355÷113 are the famous π approximations rather than arbitrary fractions - they are convergents, and the unusually large term a₄ = 292 right after 355÷113 explains why that one is so astonishingly accurate (a big term means the previous convergent was already very good). The numerators and denominators each satisfy the same recurrence as the convergents themselves, a fact used in Stern−Brocot tree navigation and the Euclidean GCD.

Related Tools & Limits

This complements a plain fraction simplifier and a decimal−to−fraction converter (which returns one fraction, not the whole approximation ladder). Because doubles carry about 15−16 significant digits, an irrational entered as a decimal yields a truncated expansion - early terms trustworthy, the deep tail floating-point noise. For exact symbolic work (e.g. √2 = [1; 2, 2, 2, …]) a closed-form generator is needed rather than a numeric input.

Continued Fraction Calculator

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