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

From ProofWiki
Jump to navigation Jump to search

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 $\beta$

$\blacksquare$


Sources