Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 192126 by universe last updated on 08/May/23

$$\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}$$ $$\mid \mathit{z}\mid >\frac{\mid \mathit{R}\mathit{e}\left(\mathit{z}\right)\mid +\mid \mathit{I}\mathit{m}\left(\mathit{z}\right)\mid }{2},\mathrm{\forall }\mathit{z}\in \mathbb{C}$$

Commented byYork12 last updated on 09/May/23

$$sirhowcanIreachyouout,Ineedtoaskseveralquestions$$

Answered by AST last updated on 08/May/23

$$Letz=x+yi;Re\left(z\right)=\frac{z+\stackrel{-}{z}}{2};Im\left(z\right)=\frac{z-\stackrel{-}{z}}{2i}=\frac{(z-\stackrel{-}{z})i}{-2}$$ $$\Rightarrow \mid Re\left(z\right)\mid =\mid x\mid ;\mid Im\left(z\right)\mid =\frac{\mid z-\stackrel{-}{z}\mid }{2}=\mid y\mid ;\mid z\mid =\sqrt{x^2+y^2}$$ $$Requiredtoprove\sqrt{x^2+y^2}>\frac{\mid x\mid +\mid y\mid }{2}$$ $$\equiv x^2+y^2>\frac{\mid x{\mid }^2+\mid y{\mid }^2+2\mid x\mid \mid y\mid }{4}\equiv 3(x^2+y^2)>2\mid x\mid \mid y\mid$$ $$Itisknownthat{x}^2+y^2⩾2\sqrt{x^2{y}^2}=2\mid x\mid \mid y\mid$$ $$Hence,3(x^2+y^2)>2\mid x\mid \mid y\mid \iff \mid z\mid >\frac{\mid Re\left(z\right)\mid +\mid Im\left(z\right)\mid }{2}$$

Answered by mehdee42 last updated on 08/May/23

$$let:z=a+ib$$ $$\parallel z\parallel =\sqrt{a^2+b^2}⩾\sqrt{a^2}=\mid a\mid \left(i\right)$$ $$\parallel z\parallel =\sqrt{a^2+b^2}⩾\sqrt{b^2}=\mid b\mid \left(ii\right)$$ $$\left(i\right)+\left(ii\right)\Rightarrow 2\parallel z\parallel >\mid a\mid +\mid b\mid \Rightarrow \parallel z\parallel >\frac{\mid a\mid +\mid b\mid }{2}✓$$ $$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com