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.

This Story also Contains

1. Complex Number
2. Conjugate of a Complex Number
3. Multiplication of Complex Conjugate
4. Complex Conjugate of a Matrix
5. Complex Conjugate Root Theorem
6. Geometrical Representation of complex conjugate
7. Properties of the conjugate complex numbers

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.

Multiplication of Complex Conjugate

When a complex number is multiplied by its complex conjugate, the product is a real number whose value is equal to the square of the magnitude of the complex number. If the complex number a + ib is multiplied by its complex conjugate a - ib, we have

$
(a+i b)(a-i b)=a^2-(i b)^2=a^2-i^2 b^2=a^2+b^2
$

Let us consider an example and multiply a complex number $5+\mathrm{i}$ with its conjugate $5-\mathrm{i}$ $(5+i)(5-i)=5^2-(i)^2=5^2-i^2=25+1=26=$ Square of Magnitude of $5+i$

Complex Conjugate of a Matrix

The complex conjugate of a matrix A with complex entries is another matrix whose entries are the complex conjugates of the entries of matrix A. Consider a row matrix A = [4-i 8+2i 9+7i], the complex conjugate of matrix A is B = [4+i 8-2i 9-7i] where each entry in matrix B is the conjugate of each entry in matrix A. The complex conjugate of matrix A is denoted by ¯AA¯. So, B = ¯AA¯.

Complex Conjugate Root Theorem

The complex conjugate root theorem states that if f(x) is a polynomial with real coefficients and a + ib is one of its roots, where a and b are real numbers, then the complex conjugate a - ib is also a root of the polynomial f(x).

Let us take an example of a polynomial with complex roots. Consider $f(x)=x^3-7 x^2+41 x-87$. Now, the roots of the polynomial $f(x)$ are $3,2+5 i, 2-5 i$. Here $2+5 i$ and $2-5 i$ are the roots of $f(x)$ and conjugates of each other. This implies that non-real roots, that is, the complex roots of a polynomial come in pairs. Hence, if we know one complex root of a polynomial, then we can say that its complex conjugate is also a root of the polynomial without calculating it.

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.

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 $-i$ 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

$\bar{z}$ denotes conjugate of z

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

$\therefore$ 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

$\bar{z}$ 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

$\bar{z}$ 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 -

$\bar{z}_1 - \bar{z}_2 = \overline{z _1-z _2}$

- 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 $\frac{-\pi}{6}$.