Tukuk-OS🔁 Repeating Decimal
ℹ️ What this tool does & how it works
A decimal either terminates or repeats — there is no third option, and the proof fits in one page of long division. Remainders are integers between 0 and the divisor, so after at most d−1 steps one must appear twice, and the digits between the two sightings are the cycle. This page runs the division in BigInt (so 1/97 comes out exact, not float-rounded), marks the cycle, reports its length, and shows the remainder trail so you can see the pigeonhole working.
The pigeonhole principle, sold as arithmetic
Long division on a remainder that has appeared before must produce the same digit and the same next remainder as the first time — the loop is forced, not observed. That single fact gives every property above: cycles of length at most d−1, the 2-and-5 terminating rule (those factors are exactly the ones that can be "absorbed" by multiplying the remainder by 10 until it is coprime), and the reason 1/7 and 2/7 are the same six digits rotated: the cycle is a property of the denominator, and the numerator only chooses the starting point inside it.