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Let $z\in\mathbb{C}$.

What is the correct way of square rooting both sides of the inequality $\text{Im}(z)^2 < 3\text{Re}(z)^2\;\text{?}$

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    Both sides of the inequality are real numbers, so you handle them "as usual".2012-11-24

2 Answers 2

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$ |\text{Im}\,(z)| < \sqrt3\,|\text{Re}\,(z)|. $

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    The common practice is that when you write the square root of a positive real number, you mean the positive square root. Because of that, $\sqrt{x^2}=|x|$.2012-11-24
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As in any other such case:

$\forall\,x,a\in\Bbb R\,\,,\,a>0\;\;\;,\;x^2

So here

$Im(z)^2<3\,Re(z)^2\Longrightarrow |Im(z)|<\sqrt 3\,|Re(z)|$