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Continued fraction calculator

Expand a number into its continued fraction [a₀; a₁, a₂, …] and list the convergents — the best rational approximations at each step. Feed it a decimal, an exact fraction a/b, or a square root √k: rationals terminate, √k is periodic, and irrationals stream on to the term cap. Continued fractions give the closest fractions for a given size of denominator and reveal the hidden structure of numbers like the golden ratio and π.

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Need Continued fraction, Convergents, Best approximation (decimal)? The Continued fraction calculator derives it from Input type, Value / numerator a / k, Denominator b (fraction only) in one step. For instance, with Input type = Decimal number, Value / numerator a / k = 415 and Denominator b (fraction only) = 93 it returns Continued fraction = [415], Convergents = 415/1 and Best approximation (decimal) = 415.000000000.

How to use it

  1. Enter your values: Input type, Value / numerator a / k, Denominator b (fraction only).
  2. Read the result instantly: Continued fraction, Convergents, Best approximation (decimal).

Frequently asked questions

What does the Continued fraction calculator actually compute?

It takes Input type, Value / numerator a / k and Denominator b (fraction only) and derives Continued fraction, Convergents and Best approximation (decimal) from them. The calculation is live as you type, so the result updates on every change.

What information do I need to provide?

3 values: Input type, Value / numerator a / k and Denominator b (fraction only). Nothing else is required — no account, no file upload.

Can you show a worked example?

With Input type = Decimal number, Value / numerator a / k = 415 and Denominator b (fraction only) = 93, the calculator returns Continued fraction = [415], Convergents = 415/1 and Best approximation (decimal) = 415.000000000. Those figures come from running this exact tool, so you can reproduce them by entering the same values.

Which “Input type” option should I choose?

You can pick between « Decimal number », « Fraction a/b » and « Square root √k ». Each one changes what the calculator works out, so switch and compare — the default is « Decimal number ».

When would I actually use this?

Reading a number written another way: a Roman numeral on a monument, a hexadecimal colour, a binary byte, or a fraction that needs reducing.

What is the most common mistake?

Assuming a decimal fraction survives a change of base. A tenth is exact in decimal and infinitely repeating in binary, which is why 0.1 + 0.2 is not 0.3 in most programming languages.

How accurate is it, and what are the limits?

For √k, the recurring block repeats; for irrational decimals, terms are capped and depend on input precision.

What is the difference between the Continued fraction calculator and the Decimal to fraction calculator?

This one returns Continued fraction and Convergents; the Decimal to fraction calculator returns Fraction. That is the whole difference — open the one whose figure you need.

Is there a tool for the next step?

Fraction Calculator is the closest one after this: Add, subtract, multiply or divide two fractions and get the reduced result plus its decimal value.

What else is worth having open alongside it?

Egyptian fraction calculator and Multiple fraction calculator — they come up in the same task often enough to be worth a second tab.

Further reading

All guides →
How-toHow to Convert a Fraction to a DecimalDivide the top number by the bottom. Here's the rule, why some fractions repeat, how to round them, and how to turn the result into a percentage.How-toHow to Multiply and Divide Fractions: Multiply Across, Flip to DivideMultiplying fractions needs no common denominator — just multiply across. Dividing means multiplying by the reciprocal. Learn both with worked examples and how to simplify.How-toHow to Add and Subtract Fractions: Common Denominator, Combine, SimplifyYou can only add or subtract fractions that share a denominator. Learn to find a common denominator, combine the numerators, and simplify the result, with a worked example.How-toHow to Convert a Percentage to a Fraction: Over 100, Then ReduceA percentage is just a fraction out of 100. Put the number over 100, reduce to lowest terms, and handle decimal percentages like 12.5% by clearing the decimal first. With examples.How-toHow to Round Numbers: Round Half Up, Round Half to Even, and MoreRound to any decimal place or number of significant figures, and learn when to use round half up versus round half to even to avoid bias.How-toHow to Find Prime Factors: Factor Trees and the Division MethodBreak any whole number into its prime building blocks with a factor tree or repeated division — and write the result as a clean prime factorization with exponents.