Technique explanation
The recurring decimal digits of a fraction can be generated directly from its remainder cycle. For a divisor ending in 7, multiply each remainder by 7 and take the units digit to find the next decimal digit. The cycle closes when the starting remainder returns. This method requires establishing the remainder cycle first, often via ordinary division, which serves as the standard fallback.
Standard fallback: Conventional long division (1 ÷ d) to generate the decimal digits sequentially.
Worked examples
Example 1: Build 1/7 from its remainder cycle (3, 2, 6, 4, 5, 1).
1. Multiply each remainder by the last digit 7 and take the units digit.
2. 3 × 7 = 21 (1); 2 × 7 = 14 (4); 6 × 7 = 42 (2); 4 × 7 = 28 (8); 5 × 7 = 35 (5); 1 × 7 = 07 (7).
3. Check: The resulting sequence is 142857. 0.142857 × 7 = 0.999999.
Check: Multiply the result by the divisor: 0.142857 × 7 = 0.999999, which represents the recurring fraction 1/7.
Result: 1/7 = 0.142857 recurring.
Example 2: Build the first four digits of 1/17 from remainders (10, 15, 14, 4).
1. The last digit of the divisor 17 is 7.
2. Multiply remainders by 7: 10 × 7 = 70 (0); 15 × 7 = 105 (5); 14 × 7 = 98 (8); 4 × 7 = 28 (8).
3. Check: The digits generated are 0, 5, 8, 8.
Check: Perform standard division for the first four places: 1 ÷ 17 = 0.0588...
Result: The first four digits of 1/17 are 0.0588...
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