Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers
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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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $9$