Technique explanation
The duplex square root method (Dwandwa Yoga) reverses the duplex squaring process. For a number N, group digits in pairs from the right. The first group gives the first root digit. The remaining steps involve dividing the 'gross dividend' by twice the first digit and subtracting the duplex of the root digits found so far (excluding the first) from the dividend before the next division.
This method is optimized for perfect squares. If the number is not a perfect square, the method will produce a non-zero remainder. Standard fallback: The conventional long division square root method or Newton's iteration for non-perfect squares.
Worked examples
Example 1: Square root of 1849
- Group: 18 | 49. The root has two digits.
- First Digit: Largest square = 18 is 16 (4^2). First root digit is 4. Remainder is 18 - 16 = 2.
- Divisor: 2 4 = 8.
- Gross Dividend: The remainder 2 prefixes 4 to make 24.
- Next Digit: 24 / 8 = 3. Remainder is 0. Second root digit is 3.
- Final Check: The remainder 0 prefixes 9 to make 09. Subtract duplex D(3) = 9. 9 - 9 = 0.
- Result: 43.
- Check: 43 × 43 = 1849.
Example 2: Square root of 54756
- Group: 5 | 47 | 56. Root has three digits.
- First Digit: Largest square = 5 is 4 (2^2). First root digit is 2. Remainder 5 - 4 = 1.
- Divisor: 2 2 = 4.
- Next Digit: 14 / 4 = 3. Remainder 2. Second root digit is 3.
- Next Step: Gross dividend 27. Subtract D(3) = 9. Net dividend 18.
- Next Digit: 18 / 4 = 4. Remainder 2. Third root digit is 4.
- Final Step: Gross dividend 25. Subtract D(34) = 24. Net dividend 1. 1 prefixes 6 to make 16. Subtract D(4) = 16. 16 - 16 = 0.
- Result: 234.
- Check: 234 × 234 = 54756.
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