Conjugates of Complex Numbers - Properties and Solved Examples

Conjugate of complex numbers is an important aspect of the mathematics of complex numbers. The conjugate of a complex number helps in various algebraic operations such as division, finding magnitudes, and solving polynomial equations. The main application of conjugate of complex numbers is solving polynomial equations, signal processing, quantum mechanics, and control systems.

Complex Number

The number which has no real meaning then these numbers are represented in complex forms. The general form of complex numbers are $a+i b$ where i is iota or$\sqrt{-1}$.

A number of the form a + ib is called a complex number (where a and b are real numbers and i is iota). We usually denote a complex number by the letter $z, z_1, z_2$, etc

For example,$z=5+2$i is a complex number.

5 here is called the real part and is denoted by Re(z), and 2 is called the imaginary part and is denoted by Im(z)

Conjugate of a Complex Number

Conjugate of a complex number is another complex number whose real parts $\operatorname{Re}(z)$ are equal and imaginary parts $\operatorname{Im}(z)$ are equal in magnitude but opposite in sign. The Conjugate of a complex number $z$ is represented by $\bar{z}$. while ($z$ & $\bar{z}$. $)$ together are known as a complex-conjugate pair because $z$ and $\bar{z}$ are conjugate to each other.

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

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.

Geometrical Representation of complex conjugate

The geometrical meaning of conjugate of a complex number $\bar{z}$. is the reflection or mirror image of the complex number z about the real axis (x-axis) in the complex plane or argand plane, which is shown in the following figure:

Geometrically complex conjugate of a complex number is its mirror image with respect to the real axis (x-axis).

For example

$z=2+2 i$ and $=2-2 i$

Properties of the conjugate complex numbers

$z_1, z_1, z_2$, and $z_3$ 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. $\overline{\mathrm{z}_1 \pm \mathrm{z}_2}=\overline{\mathrm{z}_1} \pm \overline{\mathrm{z}_2}$

In general, $\overline{\mathrm{z}_1 \pm \mathrm{z}_2 \pm \mathrm{z}_3 \pm \ldots \ldots \ldots \pm \mathrm{z}_{\mathrm{n}}}=\overline{\mathrm{z}_1} \pm \overline{\mathrm{z}_2} \pm \overline{\mathrm{z}_3} \pm \ldots \ldots \ldots \pm \overline{\mathrm{z}_{\mathrm{n}}}$
7. $\overline{z_1 \cdot z_2}=\overline{z_1} \cdot \overline{z_2}$

In general, $\overline{z_1 \cdot z_2 \cdot z_3 \cdot \ldots \ldots \ldots \cdot z_n}=\overline{z_1} \cdot \overline{z_2} \cdot \overline{z_3} \cdot \ldots \ldots \ldots \cdot \overline{z_n}$
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. $\mathrm{z}_1 \cdot \overline{z_2}+\overline{z_1} \cdot z_2=2 \operatorname{Re}\left(z_1 \cdot \overline{z_2}\right)=2 \operatorname{Re}\left(\overline{z_1} \cdot z_2\right)$

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, $z+\bar{z}=\mathrm{a}+\mathrm{ib}+\mathrm{a}-\mathrm{ib}=2 \mathrm{a}$ (which is real)
• When a complex number is multiplied by 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 product, $\mathrm{z} \cdot \bar{z}=(\mathrm{a}+\mathrm{ib}) \cdot(\mathrm{a}-\mathrm{ib})=\mathrm{a}^2-(\mathrm{ib})^2$ $=a^2+b^2$ (which is real)

Summary

The conjugate of complex numbers is used in various areas such as solving complex equations, simplifying the division of complex numbers, and in fields like signal processing, control theory, and quantum mechanics. Understanding the concept of conjugate complex numbers is essential for working effectively with complex numbers and their applications.

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Solved Examples Based On Conjugates of Complex Numbers

Example 1: If $(a+i b)^5=p+i q$, where $i=\sqrt{-1}$ then, $(b+i a)^5=$

Solution:

$(a+i b)^5=p+i d$

Taking conjugate of both sides, we get

$(a-i b)^5=p-i q$

This can be written as

$\left(-i^2 a-i b\right)^5=-i^2 p-i q$

Taking common on both sides

$\begin{aligned} & (-i)^2(i a+b)^3=(-i)(i p+q \\ & (-i)(i a+b)^3=(-i)(i p+q)\end{aligned}$

Now (-i) gets cancelled out from both sides and we are left with

$(b+i a)^5=q+i p$

Hence, the answer is q+ip.

Example 2: A conjugate of $\frac{(3-i)(2+i)}{(1-i)(3+i)}$ will be:

Solution:

As we learned in

Conjugate of a Complex Number -

$z=a+i b \Rightarrow \bar{z}=a-i b$

- wherein

denotes conjugate of z

$\begin{aligned} & \frac{(3-i)(2+i)}{(1-i)(3+i)}=\frac{6-i^2+3 i-2 i}{3-i^2-3 i+i}=\frac{7+i}{4-2 i}=\frac{7+i}{4-2 i}=\frac{26+18 i}{20}= \\ & \frac{13+9 i}{10}=\frac{13}{10}-\frac{9}{10} i\end{aligned}$

its conjugate will be $\frac{13}{10}-\frac{9}{10} i$

Hence, the answer is $\frac{13}{10}-\frac{9}{10} i$.

Example 3: z is a complex number such that $z+\bar{z}=5$ and $z-\bar{z}=7 i$ then z equals

Solution:

As we learned in

Properties of Conjugate of a Complex Number -

$\operatorname{Im}(z)=\frac{z-\bar{z}}{2 i}$

- wherein

Im(z) denotes the Imaginary part of z

denotes conjugate of z

$\begin{aligned}
& z+\bar{z}=5 \\
& z-\bar{z}=7 i
\end{aligned}$

Adding both $2 z=5+7 i \Rightarrow z=\frac{5}{2}+\frac{7}{2} i$

or

$\operatorname{Re}(z)=\frac{z+\bar{z}}{2}=\frac{5}{2}$ and $\operatorname{Im}(z)=\frac{z-\bar{z}}{2 i}=\frac{7}{2}$

Hence, the answer is $\frac{5}{2}+\frac{7}{2} i$.

Example 4: z is a complex number such that $z-\bar{z}=4 i$ , then $\operatorname{Im}(z)$ equals

Solution:

As we learned in

Properties of Conjugate of a Complex Number -

$\operatorname{Im}(z)=\frac{z-\bar{z}}{2 i}$

- wherein

Im(z) denotes the Imaginary part of z

denotes conjugate of z

$\operatorname{Im}(z)=\frac{z-\bar{z}}{2 i}=\frac{4 i}{2 i}=2$

Hence, the answer is 2.

Example 5: Let $\pi_1 z_2$ are two complex numbers such that $z_1-\bar{z}_2=\sqrt{3}+$ then arg $\left(z_1-z_2\right)$ equals

Solution:

As we learned in Properties of Conjugate of Complex Number -

$\operatorname{bar}\{z\} \_1-\operatorname{bar}\{z\} \_2=\operatorname{loverline}\left\{z \_1-z \_2\right\}$

- wherein

$\bar{z}$ denotes conjugate of $z$

$
\Rightarrow \overline{z_1-z_2}=\sqrt{3}+i \Rightarrow z_1-z_2=\sqrt{3}-i
$

$\because z_1-z_2$ lies in the fourth quadrant
so arg

$
\left(z_1-z_2\right)=-\tan ^{-1}\left|\frac{-1}{\sqrt{3}}\right|=\frac{-\pi}{6}
$
Hence, the answer is $\frac{-\pi}{6}$.