Fraction to Decimal Converter

A decimal terminates when the long division eventually produces a remainder of exactly 0. That only happens when the fraction, once fully simplified, has a denominator whose only prime factor is 2, only 5, or some combination of the two — because 2 and 5 are the prime factors of 10, and a decimal is really just a way of writing tenths, hundredths, thousandths, and so on.

Because 3 is not built only from 2s and 5s. Dividing 1 by 3 with long division gives a remainder of 1 at every single step, forever, so the digit 3 keeps getting produced with no way to ever land on a remainder of 0. The division simply never terminates — it repeats the same digit endlessly instead.

Yes — always check the denominator after simplifying. 6/8 looks like it has a denominator of 8 (which would terminate), but once reduced it's 3/4, still denominator 4, so it does terminate here anyway. A trickier case: 4/6 reduces to 2/3, and 3 is not a power of 2 or 5, so it repeats even though the original denominator, 6, is even.

Once the whole-number part of the division is finished, there's nothing left to bring down from the original numerator, so a 0 is attached after the decimal point instead. That 0 gets divided along with whatever remainder is left over, effectively asking "how many tenths, then hundredths, then thousandths" fit into the remainder — one more decimal digit at a time.

For a final answer that needs a fixed number of digits, rounding is fine and standard practice — just be clear it's an approximation. The mistake is treating the rounded value as exact partway through further math: 0.67 × 3 = 2.01, not the 2 that 2/3 × 3 actually equals, because the rounding introduced a small error that a repeating decimal, by definition, never has.

Check the denominator's prime factors first — it settles the question instantly, with no division needed. If you're dividing by hand and want to confirm by observation instead, a fraction with denominator d will either hit a remainder of 0 within d steps (terminating) or start repeating a remainder it has already seen within d steps (repeating), so d steps is always enough to be certain either way.