How this instrument works
The fraction a⁄b is not a separate kind of object waiting to be translated into a decimal — the bar between the two numbers is a division sign, and always has been. Long division simply carries out the instruction the notation already gives: bring down a zero, divide by b, keep the remainder, repeat. Each pass produces exactly one digit of the answer, so 3⁄4 and 0.75 are not two different numbers that happen to agree; they are one number written two ways.
Whether that process ever finishes comes down to the remainders it produces along the way. At every step the remainder is some whole number from 0 up to b minus 1, so there are only b possible remainders to draw from. Division stops the moment a remainder of exactly 0 turns up. If 0 never appears, some earlier remainder must eventually reappear instead — there are only finitely many to choose from — and once a remainder repeats, the digits that follow it repeat too, forever. Reduce b to lowest terms and check what it is built from: only when every prime inside it is a 2 or a 5, the pair that also builds the number ten, does the division ever land on that closing zero.
The repeating case can be stranger than it sounds. One-seventh divides out to 0.142857142857…, a six-digit block that cycles endlessly — and that block does something unexpected: multiply 142857 by 2, 3, 4, 5, or 6 and the digits rearrange into the very same six digits, just starting from a different point in the loop (285714, 428571, 571428, 714285, 857142). Some denominators push the repeating block out even further: 1⁄97 does not settle into a pattern until 96 digits have gone by, the longest cycle a denominator that size can produce.
- Type the fraction's numerator — the count of parts you have — into Numerator.
- Type the fraction's denominator — how many equal parts the whole was split into — into Denominator; zero is not allowed there.
- Decimal updates at once, carrying the division out far enough to show a terminating result or reveal a repeating block.
- Change either field to check another fraction — there is nothing to reset, the division simply reruns on the new numbers.
Worked example — three makes out of four free throws
A basketball player attempts 4 free throws in a game and makes 3 of them. Put 3 into Numerator and 4 into Denominator, and long division works through 3 ⁄ 4 one digit at a time: 30 divided by 4 gives 7 with remainder 2, then that remainder becomes 20, and 20 divided by 4 gives 5 with remainder 0 — division stops there because the remainder hit zero. Decimal reads 0.75 exactly, the same figure a scoreboard would post as a .750 shooting mark, since box scores report this ratio as a decimal with the leading zero dropped by convention.
Change the numbers and the same mechanism shows the other outcome. A player who instead makes 2 of 3 attempts produces 2 ⁄ 3: 20 divided by 3 gives 6 with remainder 2, and then 20 divided by 3 gives 6 with remainder 2 again — the identical remainder has come back, so the digit 6 repeats without end, 0.666…, which a scoreboard rounds to .667. Nothing about basketball decided that difference; it came down entirely to whether 4 or 3 divides evenly into a power of ten.
Questions
How do you convert a fraction to a decimal?
Divide the numerator by the denominator — a ⁄ b literally means a ÷ b, so there is no separate rule beyond ordinary division. For 3 ⁄ 4, dividing 3 by 4 gives 0.75 directly; nothing needs memorizing beyond how to divide, because the fraction bar already names the operation.
Why do some fractions produce a decimal that never ends?
Because long division can only produce as many distinct remainders as the denominator allows, and once a remainder repeats, every digit after it repeats too. A fraction's decimal terminates only if its denominator, in lowest terms, breaks down into 2s and 5s and nothing else — the primes hidden inside ten. Thirds, sevenths, and elevenths all fail that test and repeat forever.
What is a repeating decimal's period, and how long can it get?
The period is the length of the block that repeats, and it is always shorter than the denominator itself. One-seventh repeats with a six-digit block, 142857 — the longest period a denominator of 7 can produce. Larger denominators can push it further: 1 ⁄ 97 does not cycle back to its start until 96 digits have passed.
Does a negative numerator or an improper fraction change how the division works?
No, the same division runs unchanged either way. A negative numerator carries its sign straight into the decimal, so −3 ⁄ 4 gives −0.75. An improper fraction, where the numerator exceeds the denominator, simply divides out past 1.0: 5 ⁄ 4 gives 1.25, a whole number plus a fractional remainder, exactly as long division would produce for any other pair.
How many decimal digits does this calculator show before it stops?
Enough to display an exact terminating result, or enough to make a repeating pattern visible when the division never lands on a zero remainder. Because termination depends entirely on the denominator's prime factors, no single digit count fits every fraction — 1 ⁄ 2 needs one digit, 1 ⁄ 16 needs four, and a repeating fraction needs however many digits it takes to show the cycle.
Is a fraction with a denominator of 1 already a decimal?
Effectively yes — dividing by 1 leaves the numerator unchanged, so 7 ⁄ 1 converts to the whole number 7.0. It is the simplest case the division handles, and a useful sanity check: any method that gets this case wrong has a flaw somewhere in how it treats the denominator.