Technique explanation
The Duplex (D) method is a universal way to square any number. The duplex of a single digit is its square (a²), and the duplex of two digits is twice their product (2ab). For more digits, you pair the outer digits, double their product, and add the duplex of the remaining inner digits. To square a number, find the duplex of every possible prefix and suffix, arrange them in columns, and resolve carries.
Method Condition: Universal method for squaring; especially useful when numbers are not near a base.
Standard fallback: Standard long multiplication (N × N) or the identity (a + b)² = a² + 2ab + b².
Worked examples
Example 1: Two-Digit Square
Calculate 43².
- Duplexes: D(4), D(43), D(3).
- D(4) = 16; D(43) = 24; D(3) = 9.
- Arrange: 16 | 24 | 9.
- Resolve carries: 9; 4 (carry 2); 16+2=18.
- Result: 1849.
- Check: 43 × 43 = (40 + 3)² = 1600 + 2(40)(3) + 9 = 1600 + 240 + 9 = 1849.
Example 2: Three-Digit Square
Calculate 123².
- Duplexes: D(1), D(12), D(123), D(23), D(3).
- D(1)=1; D(12)=4; D(123)=2(1×3)+2²=10; D(23)=12; D(3)=9.
- Arrange: 1 | 4 | 10 | 12 | 9.
- Resolve carries: 9; 2 (c1); 10+1=11 (1, c1); 4+1=5; 1.
- Result: 15129.
- Check: 123 × 123 = 15129 (Verified by standard long multiplication).
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