Question

What is imaginary part of the complex number that is equal to the additive inverse of the complex number 23+73i+-53+23i-83+93i ?

Answer 1

Dear student

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{additive}\mathrm{inverse}\mathrm{of}\mathrm{a}\mathrm{complex}\mathrm{number}\mathrm{is}\mathrm{given}\mathrm{by} \\ \mathrm{a}+\mathrm{ib}=-\left(\mathrm{a}+\mathrm{ib}\right) \\ \mathrm{Let}\mathrm{the}\mathrm{required}\mathrm{number}\mathrm{be}\mathrm{x}+\mathrm{iy} \\ \mathrm{A}.\mathrm{T}.\mathrm{Q} \\ \mathrm{x}+\mathrm{iy}=-\left[\left(\frac{2}{\sqrt{3}}+\frac{7\mathrm{i}}{\sqrt{3}}\right)+\left(-5\sqrt{3}+\frac{2}{\sqrt{3}}\mathrm{i}\right)-\left(\frac{8}{\sqrt{3}}+9\sqrt{3}\mathrm{i}\right)\right] \\ =-\left[\frac{2}{\sqrt{3}}+\frac{7\mathrm{i}}{\sqrt{3}}-5\sqrt{3}+\frac{2}{\sqrt{3}}\mathrm{i}-\frac{8}{\sqrt{3}}-9\sqrt{3}\mathrm{i}\right] \\ =-\left[\frac{2}{\sqrt{3}}-5\sqrt{3}-\frac{8}{\sqrt{3}}+\mathrm{i}\left(\frac{7}{\sqrt{3}}+\frac{2}{\sqrt{3}}-9\sqrt{3}\right)\right] \\ =-\left(\frac{2}{\sqrt{3}}-5\sqrt{3}-\frac{8}{\sqrt{3}}\right)-\mathrm{i}\left(\frac{7}{\sqrt{3}}+\frac{2}{\sqrt{3}}-9\sqrt{3}\right) \\ \mathrm{So},\mathrm{y}=\frac{21}{\sqrt{3}}+\mathrm{i}\left(\frac{18}{\sqrt{3}}\right) \\ \mathrm{y}=\frac{1}{\sqrt{3}}\left(21+18\mathrm{i}\right) \end{gathered}$$
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