Executing long division steps manually clarifies exactly where the repeating sequence originates. Pencil-and-paper arithmetic follows a strict sequence of division, multiplication, subtraction, and bringing down trailing zeroes:
First, inspect the initial digits. 14 cannot divide into 1, nor can it divide into 10. You evaluate the full integer 100. 14 enters 100 exactly 7 times. Multiplying 7 by 14 gives 98. Subtracting 98 from 100 leaves 2.
Place a decimal point directly above the quotient line next to 7, and append a zero to the remainder of 2, creating 20. 14 enters 20 once (1 × 14 = 14). Subtracting 14 from 20 leaves 6.
Bring down another zero to make 60. 14 fits into 60 four times (4 × 14 = 56). Subtracting 56 leaves 4.
Bring down another zero to make 40. 14 fits into 40 two times (2 × 14 = 28). Subtracting 28 leaves 12.
Bring down a zero to make 120. 14 enters 120 eight times (8 × 14 = 112). Subtracting 112 leaves 8.
Bring down a zero to make 80. 14 enters 80 five times (5 × 14 = 70). Subtracting 70 leaves 10.
Bring down a zero to make 100. 14 enters 100 seven times (7 × 14 = 98). Subtracting 98 leaves 2. At this exact stage, the remainder resets back to 2, causing the cycle 1-4-2-8-5-7 to repeat continuously.
