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Sūtra 2 of 16

Nikhilaṁ Navataścaramaṁ Daśataḥ

All from 9 and the last from 10

Multiplication of numbers near a power of 10 (like 98 × 97).

When to use

When both numbers are close to a base such as 10, 100, 1000.

How it works

Pick a base (e.g., 100). Find each number's deficit from the base. Cross-subtract for the left part, multiply the deficits for the right part.

Memory formula

Near base B: left = one number ± the other deviation; right = deviations multiplied.

Worked examples

Problem 1
97 × 96
0/5
    Problem 2
    988 × 994
    0/5

      Try it yourself

      Try to solve, then reveal the steps one at a time.

      easy
      Compute 98 × 97
      medium
      Compute 992 × 996

      Modern applications

      Where this ancient technique still lives in today's world.

      Coding

      Modular arithmetic

      The 'all-from-9-and-last-from-10' rule is exactly the 10's-complement trick used in BCD subtraction and digital signal processing.

      Cryptography

      Modular subtraction

      Forms the basis of the complement operations in two's-complement arithmetic and modular crypto primitives.

      Code in your favourite language

      Multiply two numbers near a power-of-ten base by cross-subtracting.

      Python
      def nikhilam(a: int, b: int, base: int) -> int:
    da, db = base - a, base - b
    left = a - db
    return left * base + da * db

print(nikhilam(98, 97, 100))  # 9506

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