Conjugates of Complex Numbers - Practice Questions & MCQ

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

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. $\mathrm{z}=\mathrm{a}+$ $\mathrm{ib}, \bar{z}=\mathrm{a}-\mathrm{ib}$

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

Then the product, $z \cdot \bar{z}=(a+i b) \cdot(a-i b)=a^2-(i b)^2$

$
=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 } \bar{z}=2-2 i
$

Properties of the conjugate complex numbers:

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

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

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

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