"By the completion or non-completion."
98×97 → Base 100: (98-3)=95, 2×3=06 → 9506
Simplifying multiplication by completing to the nearest round number.
Complete a number to the nearest convenient value, perform the operation, then adjust. Works beautifully for numbers near bases like 100.
This sutra is about "completing the whole" — a natural mental ability to see how much a number differs from a round number. When adding 38 + 5, you complete 38 to 40 (adding 2), then add the remaining 3 to get 43. For subtraction: 55 − 19 = 55 − 20 + 1 = 36. For multiplication near 100: 98 × 97 → both below 100 by 2 and 3. Cross-subtract: 98 − 3 = 95. Multiply deficiencies: 2 × 3 = 06. Answer: 9506. The Vedic text says "The Sutra By the Completion or Non-Completion describes the ability we all have to see and use wholeness." This is used extensively in mental addition, subtraction, and multiplication throughout Vedic Mathematics.
The principle of "completing to a whole" is deeply embedded in Indian architectural and spiritual traditions. The Shulba Sutras describe completing geometric shapes to calculate areas. In Ayurveda, dosage calculations often round to convenient base amounts and adjust. The mathematical formalization in Bharati Krishna Tirtha's system draws from the Vedic concept of "purna" (fullness/completeness) — the Isha Upanishad opens with "Purnamadah Purnamidam" (That is complete, this is complete). This philosophical concept directly translates to the mathematical technique of completing to the nearest whole unit.
The technique exploits the algebraic identity (B−a)(B−b) = B(B−a−b) + ab. This is factoring via difference from a base. In computer science, this is the principle behind "rounding and adjusting" — a common optimization in floating-point arithmetic. The completion principle also appears in completing the square (used to derive the quadratic formula) and in Gaussian integers. Psychologically, this technique leverages the brain's preference for round numbers — cognitive load research shows that operations with round numbers are processed 40% faster than with arbitrary numbers.