Absolute Value of Components of Complex Number no greater than Root 2 of Modulus

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Theorem

Let $z = x + i y \in \C$ be a complex number.

Then:

$\size x + \size y \le \sqrt 2 \cmod z$

where:

$\size x$ and $\size y$ denote the absolute value of $x$ and $y$
$\cmod z$ denotes the complex modulus of $z$.


Proof

Let $z = x + i y \in \C$ be an arbitrary complex number.

Aiming for a contradiction, suppose the contrary:

\(\displaystyle \size x + \size y\) \(>\) \(\displaystyle \sqrt 2 \cmod z\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {\size x + \size y}^2\) \(>\) \(\displaystyle 2 \cmod z^2\) squaring both sides
\(\displaystyle \leadsto \ \ \) \(\displaystyle \size x^2 + 2 \size x \, \size y + \size y^2\) \(>\) \(\displaystyle 2 \cmod z^2\) multiplying out
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2 + 2 \size x \, \size y + y^2\) \(>\) \(\displaystyle 2 \cmod z^2\) Definition of Absolute Value
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2 + 2 \size x \, \size y + y^2\) \(>\) \(\displaystyle 2 \paren {x^2 + y^2}\) Definition of Complex Modulus
\(\displaystyle \leadsto \ \ \) \(\displaystyle 2 \size x \, \size y\) \(>\) \(\displaystyle x^2 + y^2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 2 \size x \, \size y\) \(>\) \(\displaystyle \size x^2 + \size y^2\) Definition of Absolute Value
\(\displaystyle \leadsto \ \ \) \(\displaystyle \size x^2 - 2 \size x \, \size y + \size y^2\) \(<\) \(\displaystyle 0\) rearranging
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {\size x - \size y}^2\) \(<\) \(\displaystyle 0\) factoring

But as $\size x$ and $\size y$ are both real this cannot happen.

Thus our initial assumption $\size x + \size y > \sqrt 2 \cmod z$ is false.

Hence the result.

$\blacksquare$


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