# Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers

## Theorem

Let $\alpha, \beta \in \C$ be complex numbers.

Then:

$\cmod {\alpha + \beta}^2 + \cmod {\alpha - \beta}^2 = 2 \cmod \alpha^2 + 2 \cmod \beta^2$

## Proof

Let:

$\alpha = x_1 + i y_1$
$\beta = x_2 + i y_2$

Then:

 $\ds$  $\ds \cmod {\alpha + \beta}^2 + \cmod {\alpha - \beta}^2$ $\ds$ $=$ $\ds \cmod {\paren {x_1 + i y_1} + \paren {x_2 + i y_2} }^2 + \cmod {\paren {x_1 + i y_1} - \paren {x_2 + i y_2} }^2$ Definition of $\alpha$ and $\beta$ $\ds$ $=$ $\ds \cmod {\paren {x_1 + x_2} + i \paren {y_1 + y_2} }^2 + \cmod {\paren {x_1 - x_2} + i \paren {y_1 - y_2} }^2$ Definition of Complex Addition, Definition of Complex Subtraction $\ds$ $=$ $\ds \paren {x_1 + x_2}^2 + \paren {y_1 + y_2}^2 + \paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2$ Definition of Complex Modulus $\ds$ $=$ $\ds {x_1}^2 + 2 x_1 x_2 + {x_2}^2 + {y_1}^2 + 2 y_1 y_2 + {y_2}^2 + {x_1}^2 - 2 x_1 x_2 + {x_2}^2 + {y_1}^2 - 2 y_1 y_2 + {y_2}^2$ Square of Sum, Square of Difference $\ds$ $=$ $\ds 2 {x_1}^2 + 2 {x_2}^2 + 2 {y_1}^2 + 2 {y_2}^2$ simplifying $\ds$ $=$ $\ds 2 \paren { {x_1}^2 + {y_1}^2} + 2 \paren { {x_2}^2 + {y_2}^2}$ simplifying $\ds$ $=$ $\ds 2 \cmod {x_1 + i y_1}^2 + 2 \cmod {x_2 + i y_2}^2$ Definition of Complex Modulus $\ds$ $=$ $\ds 2 \cmod \alpha^2 + 2 \cmod \beta^2$ Definition of $\alpha$ and $\beta$

$\blacksquare$