Vedic Maths Sutra 02 : Nikhilam Navataścaramam Daśataḥ

Sūtra 2 of 16
निखिलं नवतश्चरमं दशतः
Nikhilam Navataścaramam Daśataḥ
“All from 9 and the last from 10”

Meaning & Essence

This Sūtra means “All from 9 and the last from 10”. It is one of the most powerful and versatile techniques in Vedic Mathematics, especially useful for multiplication of numbers that are close to powers of 10 (such as 10, 100, 1000) and for finding complements (subtraction from powers of 10).

The method is extremely fast and can be performed mentally within seconds when the numbers are near a convenient base.

Mathematical Principle

Let the two numbers be close to a base \( x \) (usually a power of 10). Write them as \( x + a \) and \( x + b \), where \( a \) and \( b \) are the deviations (positive if above the base, negative if below).

\[ (x + a)(x + b) = x(x + a + b) + ab \]

In simple words (for base 100):

  • Find the deviations of both numbers from the base
  • Cross-add (or cross-subtract) one number with the other deviation → this gives the left part
  • Multiply the two deviations → this gives the right part (pad with zeros if needed to match the number of zeros in the base)
  • Combine left and right parts to get the final product

Worked Examples

Example 1 → Find \( 97 \times 94 \) (both below base 100)
  • Base = 100
  • Deviations: \( 97 \rightarrow -3 \), \( 94 \rightarrow -6 \)
  • Cross subtract: \( 97 - 6 = 91 \) (or \( 94 - 3 = 91 \))
  • Multiply deviations: \( (-3) \times (-6) = 18 \)
  • Combine → 91 | 18
Answer: 9118
Example 2 → Find \( 104 \times 107 \) (both above base 100)
  • Base = 100
  • Deviations: \( 104 \rightarrow +4 \), \( 107 \rightarrow +7 \)
  • Cross add: \( 104 + 7 = 111 \) (or \( 107 + 4 = 111 \))
  • Multiply deviations: \( 4 \times 7 = 28 \)
  • Combine → 111 | 28
Answer: 11128
Example 3 → Find \( 96 \times 103 \) (one below, one above base 100)
  • Base = 100
  • Deviations: \( 96 \rightarrow -4 \), \( 103 \rightarrow +3 \)
  • Cross: \( 96 + 3 = 99 \) (or \( 103 - 4 = 99 \))
  • Multiply deviations: \( (-4) \times 3 = -12 \)
  • Since right side is negative, borrow 1 from left: left becomes 98, right becomes \( 100 - 12 = 88 \)
Answer: 9888
Interactive Demo – Multiply two numbers near base 100
Please enter two valid numbers reasonably close to 100 (recommended range: 70–130)

    When to Use This Sūtra

    • Multiplication of numbers close to powers of 10 (10, 100, 1000, …)
    • Finding complements / quick subtraction from powers of 10 (e.g. 1000 − 478)
    • Squaring numbers near a base (via the related corollary)
    • Extremely useful for competitive exams and mental math practice

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