Continued Fraction Calculator

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Walk-through

How to Use This Calculator

4 steps

1

Enter a value

Type a decimal number (like 3.14159265) or a ratio in a/b form (like 415/93) into the Value field. The calculator updates instantly as you type.

2

Read the continued fraction

The result card shows the notation [a0; a1, a2, ...] — the partial quotients of the expansion — plus the best rational approximation found so far, right below it.

3

Open the Convergents tab

Switch to Convergents to see the full table of pn/qn approximations, each one closer to your input than the last, along with the decimal value and error for every row.

4

Open the Steps tab to see the work

Steps shows the Euclidean-style extraction that produces each partial quotient, plus the convergent recurrence applied numerically term by term.

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Reference

Formula & Methodology

2 formulas

Integer and fractional part extraction

a_n = floor(x_n), x_(n+1) = 1 / (x_n - a_n)

Take the integer part of the current value as the next partial quotient a_n. Subtract it off and invert the remaining fractional part to get the next value x_(n+1), then repeat. This is the same integer-division-with-remainder process the Euclidean algorithm uses to find a GCD, which is why continued fractions and the Euclidean algorithm are two views of the same computation.

Convergent recurrence

p_n = a_n * p_(n-1) + p_(n-2), q_n = a_n * q_(n-1) + q_(n-2)

Each convergent p_n/q_n builds on the previous two, starting from p_(-2)=0, p_(-1)=1, q_(-2)=1, q_(-1)=0. Every convergent is in lowest terms and is the best rational approximation to the input among all fractions with a denominator no larger than q_n.

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Glossary

Key Terms Explained

7 terms

Continued fraction A way of writing a number as a0 + 1/(a1 + 1/(a2 + 1/(a3 + ...))), shortened to the notation [a0; a1, a2, a3, ...]. Every real number has a continued fraction expansion, and every rational number's expansion is finite.

Convergent A fraction pn/qn formed by truncating the continued fraction after n terms. Convergents alternate above and below the true value and are the best possible rational approximations for their denominator size.

Euclidean algorithm A method of finding the GCD of two integers by repeated division with remainder. Applying the same division-with-remainder process to a real number instead of two integers is exactly how a continued fraction expansion is generated.

Rational approximation A fraction used to stand in for a number that may not itself be rational, chosen to be as accurate as possible for how simple (small-denominator) it is. Convergents are, by construction, the best rational approximations available.

Partial quotient Each integer a0, a1, a2, ... appearing in the continued fraction notation [a0; a1, a2, ...]. The first term a0 can be negative or zero; every later term is a positive whole number.

Golden ratio The number (1 + sqrt(5)) / 2, approximately 1.6180339887. It has the simplest possible infinite continued fraction, [1; 1, 1, 1, ...], because it is the number furthest from being well-approximated by simple fractions.

Terminating expansion A continued fraction that ends after finitely many terms rather than continuing forever. Every rational number a/b has a terminating expansion; every irrational number has an infinite one.

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Scenarios

Real-World Examples

3 worked examples

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Approximating pi

Expand pi (3.14159265358979) to see how quickly it can be approximated

Value 3.14159265358979 Max terms 6

Pi expands as [3; 7, 15, 1, 292, 1, ...], and its early convergents are famous: 22/7 is accurate to about 0.04%, while 355/113 (from just 4 terms) is accurate to roughly 3 parts in 10 million. The huge partial quotient 292 signals that 355/113 is an unusually good approximation for how small its denominator is — a pattern that shows up whenever a partial quotient is much larger than its neighbors.

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The golden ratio

Expand the golden ratio (1.6180339887498949) to see the all-1s pattern

Value 1.6180339887498949 Max terms 10

The golden ratio expands as [1; 1, 1, 1, 1, ...] — every partial quotient is 1, the smallest value allowed. Its convergents are exactly the ratios of consecutive Fibonacci numbers (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...), and because every term is as small as possible, the golden ratio converges more slowly than any other number, which is why it is called the 'most irrational' number.

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A terminating rational

Expand the ratio 415/93 to see an exact, finite expansion

Value 415/93 Max terms 10

415/93 expands to exactly [4; 2, 6, 7] and stops there, because the Euclidean-style extraction eventually hits a remainder of 0. The final convergent, 415/93, exactly reconstructs the original ratio — this always happens for rational inputs, and it is the cleanest way to tell a terminating rational number apart from an irrational one that has simply been rounded to a finite number of decimal digits.

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Deep Dive

Understanding Continued Fractions

3 chapters · 4 min read

A continued fraction rewrites a number as a nested stack of integer divisions, [a0; a1, a2, ...], and in the process reveals the best possible rational approximations to that number at every level of precision. This guide explains how the expansion is built, what convergents are good for, and how continued fractions relate to two calculators you may already be using: the Euclidean algorithm behind the GCF & LCM Calculator, and simple fractions.

How the expansion is built

The expansion works by repeatedly separating a number into its whole-number part and its fractional remainder, then flipping the remainder upside down and repeating. Formally, a_n = floor(x_n) and the next value is x_n+1 = 1 / (x_n - a_n). Feeding an exact fraction through this process is identical to running the Euclidean algorithm on its numerator and denominator — the sequence of quotients produced is exactly the sequence of partial quotients in the continued fraction. That is also why every rational number's expansion terminates: the Euclidean algorithm always reaches a remainder of 0 in finitely many steps.

What convergents are good for

Truncating the expansion after n terms gives a convergent p_n/q_n, computed from the recurrence p_n = a_n·p_n-1 + p_n-2 and q_n = a_n·q_n-1 + q_n-2. Convergents are provably the best rational approximations available for their denominator size — no other fraction with a denominator that small gets closer. This is exactly why 22/7 and 355/113 are the classic approximations for pi, why calendar reform proposals use continued-fraction convergents of the solar year to decide leap-year rules, and why mechanical gear trains are designed around convergents when an exact ratio is impractical to cut.

Terminating vs. infinite expansions

Every rational number a/b has a finite continued fraction, and the last convergent always reconstructs a/b exactly — this calculator's Steps tab shows that termination happening live. Irrational numbers, by contrast, have infinite expansions that never repeat into an exact match; this tool caps the display at your chosen max-terms setting (up to 30) and labels the result as capped rather than exact whenever the true expansion keeps going. If you're working with plain fractions rather than expansions, the Fraction Calculator and Prime Factorization Calculator cover related ground, and the Modulo Calculator exposes the remainder operation this expansion is built from.

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Questions

Frequently Asked Questions

6 questions

What is the formula behind a continued fraction expansion?+

Each term is the integer part of the current value, found by repeatedly extracting a whole number and inverting the leftover fraction: a_n = floor(x_n), then x_(n+1) = 1 / (x_n - a_n). This is the same integer/fractional-part extraction used by the Euclidean algorithm, just applied to a real number instead of two integers.

What are convergents, exactly?+

Convergents are the fractions you get by cutting the continued fraction short after n terms. Each one, written p_n/q_n, is the closest possible rational approximation to the original value among all fractions with a denominator no bigger than q_n — which is why they are called the best rational approximations.

What happens when I enter a rational number?+

The expansion terminates after finitely many terms. The Euclidean-style extraction eventually hits a remainder of exactly 0, and the last convergent computed reconstructs your original fraction exactly — a direct check that the number really was rational.

What happens with an irrational number like pi or e?+

The expansion never terminates — there is always a nonzero remainder to keep extracting. Since a decimal input is itself a finite approximation, this calculator computes as many terms as fit within your max-terms setting (up to 30) and marks the result as capped rather than exact.

Why does the golden ratio have every term equal to 1?+

The golden ratio is defined by phi = 1 + 1/phi, so extracting its integer part always leaves exactly 1/phi as the remainder, forever. A continued fraction made entirely of the smallest possible term (1) converges more slowly than any other expansion, which is why the golden ratio is famously called the "most irrational" number.

What are continued fractions actually used for?+

Finding the simplest fraction that is accurate enough for a purpose: calendar and leap-year design (approximating the length of a year), gear-ratio design when a mechanism can only use whole-number teeth counts, and general best rational approximation whenever a precise decimal needs to be replaced with a simple fraction.

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Value to expand

Convergents — successive best rational approximations

Euclidean-style extraction trace

Convergent recurrence, step by step

How to Use This Calculator

Enter a value

Read the continued fraction

Open the Convergents tab

Open the Steps tab to see the work

Formula & Methodology

Key Terms Explained

Real-World Examples

Approximating pi

The golden ratio

A terminating rational

Understanding Continued Fractions

How the expansion is built

What convergents are good for

Terminating vs. infinite expansions

Frequently Asked Questions

Get a branded PDF of your results

Continued Fraction Calculator

Value to expand

Convergents — successive best rational approximations

Euclidean-style extraction trace

Convergent recurrence, step by step

Related Calculators

How to Use This Calculator

Enter a value

Read the continued fraction

Open the Convergents tab

Open the Steps tab to see the work

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding Continued Fractions

How the expansion is built

What convergents are good for

Terminating vs. infinite expansions

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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