"All from 9 and the last from 10."
97 × 96 → deficiencies 3,4 → (97-4)=93, 3×4=12 → 9312
Subtracting from powers of 10, multiplying numbers close to a base (10, 100, 1000).
To subtract from a power of 10, subtract all digits from 9 except the last digit which is subtracted from 10. For multiplication near a base, use cross-subtraction with complements.
This is one of the most powerful Vedic sutras. When subtracting any number from a power of 10 (100, 1000, 10000 etc.), apply "all from 9 and the last from 10" — each digit is subtracted from 9, except the last non-zero digit which is subtracted from 10. For example: 10000 − 7648 → 9−7=2, 9−6=3, 9−4=5, 10−8=2 → answer 2352. For multiplication of numbers near a base: find each number's deficiency from the base, cross-subtract to get the left part, and multiply deficiencies for the right part. This works because (base−a)(base−b) = base(base−a−b) + ab.
The "All from 9 and the last from 10" principle reflects the ancient Indian fascination with the number 9 and the decimal system. India invented the decimal place-value system (circa 500 CE), and this sutra exploits its structure brilliantly. The technique was used extensively by medieval Indian accountants and astronomers for rapid subtraction from round numbers. Bharati Krishna Tirtha considered this one of the most fundamental sutras, noting its deep connection to the concept of complements — a principle independently discovered in European mathematics only in the 17th century for use in mechanical calculators (Pascal's Pascaline used nines-complement arithmetic).
The mathematical basis is the complement principle: for any n-digit number N, its complement from 10ⁿ is found by subtracting each digit from 9 except the last from 10. This is equivalent to (10ⁿ − N). For multiplication: (B−a)(B−b) = B(B−a−b) + ab, where B is the base. This factors multiplication into simpler operations. In computer science, this is directly analogous to two's complement arithmetic used in all modern processors. The nines-complement is also the foundation of check-digit algorithms (ISBN, credit card validation via Luhn algorithm).