26 Feb 2009

# Recursive Equation for Calculating Squares

Here's a recursive equation for calculating squares that came to mind the other day when I was thinking about the relationship between the addition and multiplication operators in rings.

### Equation

n^2 = (n-1)^2 + (n-1) + (n-1) + 1

### Proof

(n-1)^2 + (n-1) + (n-1) + 1 = (n-1)*(n-1) + (n-1) + (n) = n^2 - 2*n + 1 + (n-1) + (n) = n^2 - 2*n + 2*n + 1 - 1 = n^2 QED

### In English

Suppose you already know that 12 squared is 144, but you cannot remember what 13 squared is. To use this equation to find 13 squared, start with 144 (12 squared), add 12, then add 13. The result is 169 -- exactly 13 squared.

### Examples

14^2 = 13^2 + 13 + 14 = 196 3^2 = 2^2 + 2 + 3 = 9