Technique explanation
Teaching Explanation
Complementary division treats the divisor as a "near-miss" of a round base like 10, 100, or 1000. When dividing by 98, for instance, we recognize that for every 100 in the dividend, we have a "bonus" of 2 that wasn't accounted for by the round base. The method involves splitting the dividend into a quotient zone and a remainder zone based on the number of zeros in the base. We then propagate the "bonus" (the complement) by multiplying it by each quotient digit and adding the result to the subsequent digits. This iterative process allows for rapid calculation while maintaining a clear structure for verification.
Method Condition
Best suited for divisors slightly below a power of 10 (e.g., 9, 89, 997).
Standard Fallback
Standard long division.
Misconception: Forgetting the leading zero in a multi-digit complement (e.g., using 2 instead of 02 for 98).
Correction: The complement must have as many digits as there are zeros in the chosen base to maintain correct place value.
Worked examples
| Step | Example 1 | Example 2 |
|---|---|---|
| 1. Base/Comp | Base 10, Complement 1. | Base 100, Complement 02. |
| 2. Split | $1 | 2 | 1$ (1 digit for remainder) | $13 | 21$ (2 digits for remainder) |
| 3. Process | Bring down 1. $1 \times 1 = 1$. Add to 2: $2+1=3$. | Bring down 13. $13 \times 02 = 26$. |
| 4. Finalize | $3 \times 1 = 3$. Add to 1: $1+3=4$. | Add 26 to 21: $21+26=47$. |
| 5. Result | 13 R 4 | 13 R 47 |
| 6. Check | $13 \times 9 + 4 = 117 + 4 = 121$. | $13 \times 98 + 47 = 1,274 + 47 = 1,321$. |
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