The Duplex Method for Squaring Numbers

The Duplex method is a fast and efficient way to calculate the squares of numbers, a technique derived from Vedic Mathematics. At its core is the concept of a "duplex," which is a specific calculation performed on a digit or group of digits.

The Duplex Rules

Duplex of a single digit (a): D(a) = a²
Duplex of two digits (ab): D(ab) = 2 × a × b
Duplex of three digits (abc): D(abc) = 2 × a × c + b²

The General Process

The process for squaring a number using the Duplex method is consistent, regardless of the number of digits. Follow these steps:

Determine the "Duplex Columns": The number of columns and the duplexes to be calculated follow a mirrored pattern, starting from a single digit on the right and expanding outwards.
- For a 2-digit number (ab): The columns are D(a) / D(ab) / D(b)
- For a 3-digit number (abc): The columns are D(a) / D(ab) / D(abc) / D(bc) / D(c)
Calculate the Duplex for Each Column: Go through each column and calculate its value using the rules above.
Combine the Results with Carries (from right to left):
- Start with the rightmost column. Write down the last digit of the result.
- Carry over any remaining digits to the next column on the left.
- Add the carry-over to the value of that next duplex column.
- Repeat this process until you reach the leftmost column. The final sum of this column, including any carry-over, forms the first digits of your answer.

Example: Squaring a 2-Digit Number (47)

Number: ab = 47

Duplex Columns: D(4) / D(47) / D(7)

1. Calculate Duplexes:

            D(4) = 4² = 16

            D(47) = 2 × 4 × 7 = 56

            D(7) = 7² = 49

2. Combine with Carries (Right to Left):

Column 3 (D(7)): The value is 49. Write down 9, carry over 4.
Column 2 (D(47)): The value is 56. Add the carry: 56 + 4 = 60. Write down 0, carry over 6.
Column 1 (D(4)): The value is 16. Add the carry: 16 + 6 = 22. Write down 22.

Final Answer: 2209

Example: Squaring a 3-Digit Number (147)

Number: abc = 147

Duplex Columns: D(1) / D(14) / D(147) / D(47) / D(7)

1. Calculate Duplexes:

            D(1) = 1² = 1

            D(14) = 2 × 1 × 4 = 8

            D(147) = 2 × 1 × 7 + 4² = 14 + 16 = 30

            D(47) = 2 × 4 × 7 = 56

            D(7) = 7² = 49

2. Combine with Carries (Right to Left):

Column 5 (D(7)): The value is 49. Write down 9, carry over 4.
Column 4 (D(47)): The value is 56. Add the carry: 56 + 4 = 60. Write down 0, carry over 6.
Column 3 (D(147)): The value is 30. Add the carry: 30 + 6 = 36. Write down 6, carry over 3.
Column 2 (D(14)): The value is 8. Add the carry: 8 + 3 = 11. Write down 1, carry over 1.
Column 1 (D(1)): The value is 1. Add the carry: 1 + 1 = 2. Write down 2.

Final Answer: 21609

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