Question 9, Exercise 1.2

Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

If $z=3+2i,$ then verify that $-|z|\leq \operatorname{Re}\left( z \right)\leq |z|$

Given $z=3+2i$. Then $|z|=\sqrt{9+4}=\sqrt{13}$ and ${\rm Re}z=3=\sqrt{9}$.
As \begin{align} &-\sqrt{13} \leq \sqrt{9} \leq \sqrt{13}\\ \implies &-|z|\leq \operatorname{Re}\left( z \right)\leq |z|\end{align}

If $z=3+2i,$ then verify that $-|z|\leq \operatorname{Im}\left( z \right)\leq |z|$

Given $z=3+2i$. Then $|z|=\sqrt{9+4}=\sqrt{13}$ and ${\rm Im}z=2=\sqrt{4}$.
As \begin{align} &-\sqrt{13} \leq \sqrt{4} \leq \sqrt{13}\\ \implies &-|z|\leq {\rm Im}(z) \leq |z| \end{align}

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