Question 7, Exercise 1.2

Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Verify that $\sqrt{2}|z| \geq|\operatorname{Re}(z)|+|\operatorname{Im}(z)| \quad$ Hint: (Start with $\left(|x|-|y|)^{2} \geq 0\right)$

Solution.

As $$\begin{align} &\left(|x|-|y|)^{2} \geq 0\right) \\ \implies & |x|^2+|y|^2-2|x||y| \geq 0 \\ \implies & |x|^2+|y|^2 \geq 2|x||y| \\ \implies & 2|x|^2+2|y|^2 \geq |x|^2+|y|^2+2|x||y| \\ \implies & 2(x^2+y^2) \geq\left(|x|+|y|\right)^2 \quad \because |x|^2=x^2\\ \implies & 2|z|^2 \geq \left(|Re(z)|+|Im(z)|\right)^2 \end{align}$$ This ultimately gives us $$\sqrt{2} |z| \geq |Re(z)|+|Im(z)|.$$

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