Vedic Maths Sutra 10 : Yāvadūnam

Sūtra 10 of 16
यावदूनम्
Yāvadūnam
“Whatever the extent of its deficiency”

Meaning & Essence

This Sūtra means “Whatever the extent of its deficiency”.

Together with its corollary (“whatever the deficiency, lessen it still further by that amount and set up the square of the deficiency”), it gives a very fast method for squaring numbers that are close to a power of 10 (or any convenient base).

Mathematical Principle

Let the number be close to a base \( x \) (usually 10, 100, 1000…). Let the deficiency (or surplus) be \( d \).

\[ (x \pm d)^2 = (x \pm 2d) \cdot x + d^2 \] (or more simply in the Vedic form)

Practical steps:

  • Find how much the number is deficient from (or in excess of) the base → this is \( d \)
  • Lessen (or increase) the original number by the same \( d \)
  • Write that result as the left part of the answer
  • Square the deficiency \( d \) and write it as the right part (with the correct number of digits)

Worked Examples

Example 1 → \( 9^2 \) (base 10)
  • 9 is 1 less than 10 → deficiency \( d = 1 \)
  • Lessen further by 1: \( 9 - 1 = 8 \)
  • Square of deficiency: \( 1^2 = 1 \)
  • Answer: 81
Answer: 81
Example 2 → \( 96^2 \) (base 100)
  • 96 is 4 less than 100 → deficiency \( d = 4 \)
  • Lessen further by 4: \( 96 - 4 = 92 \)
  • Square of deficiency: \( 4^2 = 16 \)
  • Answer: 9216
Answer: 9216
Example 3 → \( 103^2 \) (base 100, surplus)
  • 103 is 3 more than 100 → surplus \( d = 3 \)
  • Increase further by 3: \( 103 + 3 = 106 \)
  • Square of surplus: \( 3^2 = 09 \)
  • Answer: 10609
Answer: 10609
Example 4 → \( 994^2 \) (base 1000)
  • 994 is 6 less than 1000 → deficiency \( d = 6 \)
  • Lessen further by 6: \( 994 - 6 = 988 \)
  • Square of deficiency: \( 6^2 = 036 \)
  • Answer: 988036
Answer: 988036
Interactive Demo – Square a number near a base

Enter a number close to 10, 100 or 1000. The demo applies the Yāvadūnam method.

Please enter a valid number

    When to Use This Sūtra

    • Squaring numbers close to 10, 100, 1000, etc.
    • Quickly finding squares of numbers just below or just above a convenient base
    • Mental calculation of squares in competitive exams
    • Extends naturally to the related Nikhilam methods for multiplication near a base

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