Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Graphical representation of Complex numbers and its conjugate

Example:

Let z=3+7i, \quad \overline{z}=3-7i

 \begin{aligned} \text{then } (i) Re z &=\dfrac{z+\overline{z}}{2} =\dfrac{(3+7i)+(3-7i)}{2}\\&=\dfrac{6}{2}=3 \end{aligned}

\begin{aligned} (ii) Im z&=\dfrac{z-\overline{z}}{2i}=\dfrac{(3+7i)-(3-7i)}{2i}\\&=\dfrac{14i}{2i}=7 \end{aligned}

 |z|=\sqrt{3^2+7^2}=\sqrt{9+49}=\sqrt{58}

 |\overline{z}|=\sqrt{3^2+(-7)^2}=\sqrt{9+49}=\sqrt{58}

 \therefore |z|=|\overline{z}|

NOTE:

(i)  Re(\overline{z})=Re(z)

(ii)  Im(\overline{z})=-Im(z)

(iii)  \overline{z}=z

The multiplicative inverse of the non-zero complex number  z is given by  z^{-1}=\dfrac{1}{a+ib}

\begin{aligned} \implies z^{-1}&=\dfrac{1}{a+ib}=\dfrac{a}{a^2+b^2}+i\dfrac{-b}{a^2+b^2}\\&=\dfrac{a+i(-b)}{a^2+b^2}\\&=\dfrac{a-ib}{a^2+b^2}\\&=\dfrac{\overline{z}}{|z|^2} \end{aligned}

 \implies z \cdot z^{-1}=\dfrac{z \cdot \overline{z}}{|z|^2} \implies 1=\dfrac{z \cdot \overline{z}}{|z|^2}

 \implies z \cdot \overline{z}=|z|^2

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