Vedic Squares, Square Roots & Digit Sum Verification

1. Fast Squaring Algorithms

A. Numbers Ending in 5: Ekadhikena Purvena (“By one more than the previous”)

Core Formula: For any number ending in 5 with preceding digits N:

Square = [ N × (N + 1) ] followed by 25

B. Numbers Near a Base: Yavadunam Tavadunikritya Varganka

“Whatever the deficiency, lessen by that amount and set up the square of the deficiency.”

• Rule for Base 100: (Number - Deficiency) followed by (Deficiency²)
• Example 1: 97² (Base = 100, Deficiency = 3)
  • LHS: 97 - 3 = 94
  • RHS: 3² = 09
  • Result = 9,409
• Example 2: 106² (Base = 100, Excess = 6)
  • LHS: 106 + 6 = 112
  • RHS: 6² = 36
  • Result = 11,236

2. Square Roots by Inspection: Vilokanam

For perfect squares up to 4 or 5 digits, square roots can be extracted mentally by observing unit digits and grouping pairs:

Note: A perfect square can never end in 2, 3, 7, or 8.

Example: Find √6724

1. Pair from right: (67) (24).
2. Find tens digit: The highest perfect square below 67 is 64 (8²). Thus, the tens digit is 8.
3. Find units digit: The number ends in 4, so the units digit is either 2 or 8 (Root is either 82 or 88).
4. Test middle baseline (85²): 8 × 9 followed by 25 = 7,225.
5. Compare: Since 6,724 is less than 7,225, the root is the smaller number: 82.
   • Verification: 82 × 82 = 6,724.

3. Method of Quick Verification from Digit Sum (Beejank)

The Digit Sum (Beejank) of a number is the single-digit sum obtained by adding all its digits together repeatedly until one digit remains. Casting out 9s (treating 9 as 0) speeds up this calculation.