Decimal to Continued Fraction Calculator

How it is calculated

The tool runs the standard continued-fraction loop. Take the integer part of the current value x as the next term a, then set x = 1 / (x − a) and repeat. The result is written as [a0; a1, a2, …]. A typed decimal is treated as the exact fraction it spells (3.14159 is 314159/100000), so the terms are exact integer arithmetic with no floating-point noise, and the expansion simply ends when the remainder reaches zero. The loop also stops at the maximum number of terms. The default is 10; a blank box or a value below 1 uses 10, a fraction is cut down to a whole number, and the limit never goes above 50. Negative numbers use the floor, so −2.75 gives [−3; 4]. Input that is not a number, or whose size is above 10¹⁵, shows no result.

Common mistakes

• Typing a rounded decimal, such as 3.14159 for π, and expecting π's expansion. The tool expands exactly the number you typed, so after the first few terms the digits differ from π's and the expansion ends early.
• Expecting a never-ending list for an irrational number. A typed decimal is always a finite fraction, so the expansion terminates; only the first terms say something about the irrational number behind it.
• Reading the semicolon as a comma. The first number is the whole part, a0; only the later ones are the inverted steps.
• Expecting a negative number to give a negative first term followed by the positive digits of its fraction part as you would write by hand. The tool uses the floor, so −2.75 is [−3; 4].
• Setting the maximum to 0 or a fraction to get a shorter result. A value below 1 falls back to 10 terms, and a fraction is cut down to a whole number.