Continued Fraction calculator

Any real number x can be represented uniquely by a continued fraction:

x is equal to a sub 0 plus 1 over a sub 1 plus 1 over a sub 2 plus 1 over a sub 3 plus etcetera

where a1, a2, a3, ... are integer numbers greater than zero. A more compact representation is:

x is equal to a sub 0 plus double slash a sub 1, a sub 2, a sub 3, etcetera double slash

If the number to be represented is rational, there is a finite number of terms in the continued fraction. If the number is a quadratic irrationality of the form fraction whether the numerator is a plus the square root of b and the denominator is c , then the continued fraction is periodic. This calculator can find the continued fraction expansions of rational numbers and quadratic irrationalities.

You can type numbers or numerical expressions on the input boxes at the left.

The calculator accepts numbers of up to 10000 digits.

If you need the square root to subtract the number at the left, just negate both a and c.

If b is negative, the result is not a real number, so it cannot be represented as a continued fraction.

For rational numbers the calculator finds all convergents, but for quadratic irrationalities the calculator stops after the 100000th convergent if the period is larger.


You can use the following operators and parentheses for the expressions:

  • + for addition
  • - for subtraction
  • * for multiplication
  • / for integer division
  • % for modulus (remainder of the integer division)
  • ^ or ** for exponentiation (the exponent must be greater than or equal to zero).
  • <, ==, >; <=, >=, != for comparisons. The operators return zero for false and -1 for true.
  • AND, OR, NOT for binary logic.
  • n!: factorial (n must be greater than or equal to zero).
  • p#: primorial (product of all primes less or equal than p).
  • B(n): Previous probable prime before n
  • F(n): Fibonacci number Fn
  • L(n): Lucas number Ln = Fn-1 + Fn+1
  • N(n): Next probable prime after n
  • P(n): Unrestricted Partition Number (number of decompositions of n into sums of integers without regard to order).
  • Gcd(m,n): Greatest common divisor of these two integers.
  • Modinv(m,n): inverse of m modulo n, only valid when gcd(m,n)=1.
  • Modpow(m,n,r): finds mn modulo r.
  • IsPrime(n): returns zero if n is not probable prime, -1 if it is.
  • NumDigits(n,r): Number of digits of n in base r.
  • SumDigits(n,r): Sum of digits of n in base r.
  • RevDigits(n,r): finds the value obtained by writing backwards the digits of n in base r.

You can use the prefix 0x for hexadecimal numbers, for example 0x38 is equal to 56.

Source code

You can download the source of the current program and the old continued fraction applet from GitHub. Notice that the source code is in C language and you need the Emscripten environment in order to generate Javascript.

Written by Dario Alpern. Last updated 15 January 2018.

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