Tech

Polynomial Operations and The Decimal System

1. Mathematical Modeling

In a decimal system, any two-digit number can be represented as a first-order polynomial where x=10:

N = 10a + b, P(x) = ax + b

Where a is the tens-place coefficient and b is the units-place coefficient.

2. Polynomial Expansion

When calculating the square of this number, it is equivalent to the binomial expansion of the polynomial:

P(x)^2 = (ax + b)^2 = a^2x^2 + 2abx + b^2

3. Carrying Logic and Coefficient Mapping

In pure algebra, the coefficients a^2, 2ab, and b^2 are independent. However, in decimal arithmetic, if a coefficient exceeds the base 10, a carrying operation is triggered.

Case A: Without Carrying

Take 13^2 as an example, where a=1, b=3:

1
2
3
4
5
6
7
8
- a^2 = 1^2 = 1
    
- 2ab = 2(1)(3) = 6
    
- b^2 = 3^2 = 9
    
    The final result is 625.

Case B: With Carrying

Take 25^2 as an example, where a=2, b=5:

1
2
3
4
5
6
7
8
9
- a^2 = 4, \quad 2ab = 20, \quad b^2 = 25
    
- b^2=25, keep 5,  2.
    
- 2ab=20, 20+2=22，keep 2，carry 2.
    
- a^2=4, 4+2=6，got 6.
    
    The final result is 625.

4. Conclusion

Arithmetic operations in positional notation are essentially evaluations of polynomials at x=10. This discovery reveals the isomorphism between the Place-Value Principle and Algebraic Expansion.