Fraction to Decimal Converter

A fraction in lowest terms terminates exactly when its denominator factors only into 2s and 5s. That is the same as saying you can multiply the denominator by some power of 10 and land on a whole number. So 3/8 terminates (8 = 2^3, and 3/8 = 375/1000 = 0.375), 7/25 terminates (25 = 5^2), and 1/40 terminates (40 = 2^3 * 5). The moment any other prime sneaks in, 3, 7, 11, 13, and so on, the decimal expansion repeats forever. 1/3 repeats because 3 is not 2 or 5. 1/12 repeats because 12 = 2^2 * 3 and that lone 3 is enough to break termination. Always reduce the fraction first: 6/15 looks like it should repeat, but it reduces to 2/5, which terminates as 0.4.

Long division of 1 by 3 produces remainder 1, then 1 again, then 1 again, indefinitely. Each step writes a 3 in the quotient and leaves the same remainder you started with, so the loop never breaks. That is exactly the definition of a repeating decimal: a remainder reappears, so the digits between the two appearances form the repeating block. For 1/3 the block is a single digit, '3'. For 1/7 the block is six digits, '142857'. The calculator surfaces the block separately as the recurringDigits field, and renders the value as 0.333... with three trailing dots so you can tell at a glance you are looking at an exact repeating decimal rather than a truncated approximation.

Irrational numbers like pi, e, and sqrt(2) have no exact fraction representation, period. Any fraction you write down for pi is wrong in some decimal place, so the converter uses continued-fraction expansion to find the best rational approximation whose denominator stays under your maxDenominator cap. With cap 10, pi rounds to 22/7. With cap 1000, you get 355/113, which matches pi to six decimal places. Repeating decimals you typed as truncated values (like 0.333333 without indicating it should repeat) are also approximated rather than recovered exactly: at cap 1000 you will get 1/3 back, but the converter still has to guess that intent from the input.

They are two notations for the same value. The improper fraction 11/4 has a numerator larger than its denominator. The mixed number 2 3/4 splits that out into a whole part (2) and a fractional remainder (3/4). They evaluate to the same thing: 11 / 4 = 2.75 and 2 + 3/4 = 2.75. Pick whichever reads better for your audience: mixed numbers are friendlier in cooking and construction (2 1/2 cups, 3 7/8 inches), improper fractions are friendlier for algebra and computation (multiplying 11/4 by 5/3 is one step; multiplying 2 3/4 by 1 2/3 is the same operation after you convert both to improper). The converter shows both, so you can copy whichever you need.

To reduce a fraction to lowest terms, divide both the numerator and denominator by their greatest common divisor. The Euclidean algorithm finds the GCD by repeatedly replacing the larger number with its remainder mod the smaller one: gcd(10, 15) = gcd(15, 10) = gcd(10, 5) = gcd(5, 0) = 5. So 10/15 reduces to (10/5) / (15/5) = 2/3. The reduced form is the unique simplest representation, and it is what the calculator uses to decide whether the decimal terminates or repeats, since only the reduced denominator's prime factors matter. Reduction also normalizes the sign: a negative denominator is flipped onto the numerator, so -3/-6 always comes out as 1/2.