Conjugates of Complex Numbers - Practice Questions & MCQ

The conjugate of a complex number $z=a+ib(a,b$ are real numbers) is $a-ib$. It is denoted as $\overline{z}$. i.e. if $z=a+ib$, then its conjugate is $\overline{z}=a-ib$.

The conjugate of complex numbers is obtained by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.
Note:
When a complex number is added to its complex conjugate, the result is a real number. i.e. $\mathrm{z}=\mathrm{a}+$ $\mathrm{ib},\overline{z}=\mathrm{a}-\mathrm{ib}$

Then the sum, $z+\overline{z}=a+ib+a-ib=2a$ (which is real)
When a complex number is multiplied by its complex conjugate, the result is a real number i.e. $z=$ $\mathrm{a}+\mathrm{ib},\overline{z}=\mathrm{a}-\mathrm{ib}$

Then the product, $z\cdot \overline{z}=(a+ib)\cdot (a-ib)=a^2-(ib{)}^2$

$=a^2+b^2(\text{ which is real })$
Geometrically complex conjugate of a complex number is its mirror image with respect to the real axis (x-axis).

For example

$\mathrm{z}=2+2\mathrm{i}\text{ and }\overline{z}=2-2i$

Properties of the conjugate complex numbers:

z, z₁, z_2, and z₃ be the complex numbers

1. $\overline{(\overline{z})}=z$
2. $\mathrm{z}+\bar{\mathrm{z}}=2\cdot \mathrm{Re}(\mathrm{z})$
3. $\mathrm{z}-\bar{\mathrm{z}}=2\mathrm{i}\cdot \mathrm{Im}(\mathrm{z})$
4. $\mathrm{z}+\bar{\mathrm{z}}=0\Rightarrow \mathrm{z}=-\bar{\mathrm{z}}\Rightarrow \mathrm{z}$ is purely imaginary
$5.\mathrm{z}-\bar{\mathrm{z}}=0\Rightarrow \mathrm{z}=\bar{\mathrm{z}}\Rightarrow \mathrm{z}$ is purely real
6. $\overline{z_1\pm z_2}=\overline{z_1}\pm \overline{z_2}$

In general, $\overline{z_1\pm z_2\pm z_3\pm \dots \dots \dots \pm {\mathrm{z}}_n}=\overline{z_1}\pm \overline{z_2}\pm \overline{z_3}\pm \dots \dots \dots \pm \overline{z_n}$
7. $\overline{{\mathrm{z}}_1\cdot {\mathrm{Z}}_2}=\overline{{\mathrm{z}}_1}\cdot \overline{{\mathrm{z}}_2}$

In general, $\overline{z_1\cdot z_2\cdot z_3\cdot \dots \dots \dots \cdot \cdot }\overline{z_n}=\overline{z_1}\cdot \overline{z_2}\cdot \overline{z_3}\cdot \dots \dots \dots \cdot \overline{z_n}$
8. $\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z_1}}{\overline{z_2}},z_2\ne 0$
9. $\overline{{\mathrm{z}}^{\mathrm{n}}}=(\bar{\mathrm{z}}{)}^{\mathrm{n}}$
10. ${\mathrm{z}}_1\cdot \overline{{\mathrm{z}}_2}+\overline{{\mathrm{z}}_1}\cdot {\mathrm{z}}_2=2\mathrm{Re}({\mathrm{z}}_1\cdot \overline{{\mathrm{z}}_2})=2\mathrm{Re}(\overline{{\mathrm{z}}_1}\cdot {\mathrm{z}}_2)$