Conjugate of Complex Number

A complex conjugate is a concept in complex number theory where for any given complex number, a conjugate exists that reverses the sign of the imaginary part while keeping the real part unchanged. If z = a + bi is a complex number, where a and b are real numbers and iii is the imaginary unit (i² = –1), the complex conjugate of z is denoted as $\overline{\mathrm{Z}}$ and is given by: $\overline{\mathrm{z}}=\mathrm{a}-\mathrm{bi}$

1.0What is a Complex Conjugate?

Conjugate of a complex number z=a+i b (where a, b are real numbers) is denoted and defined by

$\bar{z}=a-ib$. In a complex number if we replace i by –i, we get conjugate of the complex number. $\bar{z}$ is the mirror image of z about the real axis on the Argand Plane.

Geometrical Representation of Conjugate of Complex Number

Geometrically, conjugate of z is the mirror image of complex number z w.r.t real axis on Argand plane

Ex:

$z=3+4i\text{ and }\bar{z}=3-4i$

$\mathrm{z}(3,4)\text{ and }\overline{\mathrm{z}}(3,-4)$

In Polar form, the complete conjugate of complex number re^iQ is re^–iQ

2.0Conjugate Modulus of a Complex Number

The modulus (or absolute value) of a complex number z = a + bi is defined as: $|z|=\sqrt{a^2+b^2}$

Interestingly, the modulus of a complex number and its conjugate are the same. For z and $|\mathrm{z}|=|\overline{\mathrm{z}}|=\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$

3.0Complex Conjugate of a Matrix

The complex conjugate of a matrix A with complex entries results in another matrix  in which every entry is the complex conjugate of the corresponding entry in A. Consider the row matrix $A=\left[\begin{array}{lll}1-i& 4+2i& 3+7i\end{array}\right]$. The complex conjugate of matrix A, denoted as $\overline{\mathrm{A}}\text{, is }\mathrm{B}=\left[\begin{array}{lll}1+\mathrm{i}& 4-2\mathrm{i}& 3-7\mathrm{i}\end{array}\right]$, where each entry in B is the conjugate of the corresponding entry in  A . Therefore, we can write $\mathrm{B}=\overline{\mathrm{A}}$. Let's look at another example of a matrix with complex entries to determine its complex conjugate.

For a matrix with complex entries, the complex conjugate is obtained by taking the complex conjugate of each element in the matrix. For example, if A is a complex matrix:

$A=\left(\begin{array}{cc}1+i& 2-3i\\ 4+i& -i\end{array}\right)$

The conjugate of A, denoted as

$\overline{\mathrm{A}},is:\bar{A}=\left(\begin{array}{cc}1-i& 2+3i\\ 4-i& i\end{array}\right)$

4.0Root Theorem on Complex Conjugate

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

5.0Multiplication of Complex Conjugate

Multiplying a complex number by its conjugate yields a real-valued result. Specifically, the product is the square of the modulus of the original complex number. If z = a + bi is a complex number, its conjugate is $\overline{\mathrm{Z}}=\mathrm{a}-\mathrm{bi}$. The multiplication of z by $\overline{\mathrm{Z}}$ is given by:

$\mathrm{z}\cdot \overline{\mathrm{z}}=(\mathrm{a}+\mathrm{bi})(\mathrm{a}-\mathrm{bi})={\mathrm{a}}^2-(\mathrm{bi}{)}^2={\mathrm{a}}^2-{\mathrm{b}}^2(-1)={\mathrm{a}}^2+{\mathrm{b}}^2$

This result is always a non-negative real number.

Example:

For  z = 3 + 4i 

$\overline{\mathrm{z}}=3-4\mathrm{i}$

Multiplying z by $\overline{\mathrm{Z}}$ : 

$z\cdot \bar{z}=(3+4i)(3-4i)=3^2-(4i{)}^2=9-16(-1)=9+16=25$

So, the product of 3 + 4i and its conjugate 3 – 4i is 25. This demonstrates that multiplying a complex number by its conjugate results in a real number equal to the square of the modulus of the original complex number.

6.0Properties of Complex Conjugate $(\bar{z})$

(i) $\text{ If }z=x+iy,\text{ then }x=\frac{z+\bar{z}}{2},y=\frac{z-\bar{z}}{2i}\Rightarrow \mathrm{Re}(z)=\frac{z+\bar{z}}{2},\mathrm{Im}(z)=\frac{z-\bar{z}}{2i}$

(ii) $z=\bar{z}\iff \mathrm{z}\text{ is purely real }$

(iii) $z+\bar{z}=0\iff z\text{ is purely imaginary }$

(iv) $\overline{\bar{z}}=z$

(v) Relation between modulus and conjugate. $|z{|}^2=z\bar{z}$

(vi) $\overline{\left(z_1\pm z_2\right)}={\bar{z}}_1\pm {\bar{z}}_2$

(vii)  $\overline{\left(z_1{z}_2\right)}={\bar{z}}_1{\bar{z}}_2$

(viii) $\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\left({\bar{z}}_1\right)}{\left({\bar{z}}_2\right)}\left(z_2\ne 0\right)$  

(ix) ${\mathrm{z}}_1{\overline{\mathrm{z}}}_2+{\overline{\mathrm{z}}}_1{\mathrm{z}}_2=2\mathrm{Re}\left({\mathrm{z}}_1{\overline{\mathrm{z}}}_2\right)$

(x) ${\mathrm{z}}_1{\overline{\mathrm{z}}}_2-{\overline{\mathrm{z}}}_1{\mathrm{z}}_2=2\mathrm{i}\mathrm{Im}\left({\mathrm{z}}_1{\overline{\mathrm{z}}}_2\right)$

7.0Solved Example of Conjugate of Complex Number

Example 1: The conjugate of 6i – 5 is

(A) (6i + 5) (B) (–6i – 5) (C) (–6i + 5) (D) None of these

Ans. (B)

Solution:

_{6 i-5=(-5)+(6 i)}

⇒ conjugate of 6 i-5=(-5-6 i) 

Example 2: The conjugate of the complex number $\frac{(1+i{)}^2}{1-i}$ is

(A)1 – i (B) 1 + i (C) –1 + i (D) –1 –i

Ans. (D)

Solution:

The given number

$\frac{(1+i{)}^2}{1-i}=\frac{1+i^2+2i}{1-i}\times \frac{1+i}{1+i}=\frac{2i(1+i)}{1-i^2}\left(\because {\mathrm{i}}^2=-1\right)$

$=\frac{2i+2i^2}{1-(-1)}=\frac{2i+2(-1)}{2}=i-1=-1+i$

$\therefore \text{ Conjugate }=-1-i$

Example 3: If the conjugate of $(x+iy)(1-2i)\text{ be }1+i$, then

(A) $x=\frac{1}{5}$ (B) $y=\frac{3}{5}$

(C) $x+iy=\frac{1-i}{1-2i}$ (D) $x-iy=\frac{1-i}{1+2i}$

Ans. (C)

Solution:

$(x+iy)(1-2i)=1-i\Rightarrow x+iy=\frac{1-i}{1-2i}$

8.0Solved Questions for Conjugate of Complex Number

1. What is a Complex Conjugate?

Ans: A complex conjugate of a complex number $\mathrm{z}=\mathrm{a}+\mathrm{bi}\text{ is }\overline{\mathrm{z}}=\mathrm{a}-\mathrm{bi}$, where a and b are real numbers and i(iota) represents the imaginary unit.

2. How do you find the complex conjugate of a complex number?

Ans: To find the complex conjugate of a complex number z = a + bi, change the sign of the imaginary part: $\overline{\mathrm{z}}=\mathrm{a}-\mathrm{bi}$.

3. What is the modulus of a complex conjugate?

Ans: The modulus of a complex conjugate is the same as the modulus of the original complex number z = a + bi. It is given by $|z|=\sqrt{a^2+b^2}$.

4. How does the complex conjugate relate to the complex plane?

Ans: In the complex plane, the complex conjugate $\overline{\mathrm{Z}}$ of z is the reflection of z across the real axis.

5. What are some properties of complex conjugates?

Ans:

Addition: $\overline{{\mathrm{z}}_1+{\mathrm{z}}_2}=\overline{{\mathrm{z}}_1}+\overline{{\mathrm{z}}_2}$

Multiplication: $\overline{{\mathrm{z}}_1\cdot {\mathrm{z}}_2}=\overline{{\mathrm{z}}_1}\cdot \overline{{\mathrm{z}}_2}$

Division: $\overline{\left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right)}=\frac{\overline{{\mathrm{z}}_1}}{\overline{{\mathrm{z}}_2}}$

Modulus: $|\mathrm{z}|=|\overline{\mathrm{z}}|$

6. What is the complex conjugate of a matrix?

Ans: The complex conjugate of a matrix A is another matrix $\overline{\mathrm{A}}$, obtained by taking the complex conjugate of each entry in A.

7. How do you multiply a complex number by its conjugate?

Ans: To multiply a complex number z = a + bi by its conjugate 

$\bar{z}=a-bi:z\cdot \bar{z}=(a+bi)(a-bi)=a^2+b^2$

8. Can you provide an example of a complex conjugate?

Ans: For the complex number z = 2 + 3i:

$\overline{\mathrm{z}}=2-3\mathrm{i}$

9. What happens when you add a complex number and its conjugate?

Ans: When you add a complex number z = a + bi and its conjugate  $\overline{\mathrm{z}}=\mathrm{a}-\mathrm{bi}$

$\mathrm{z}+\overline{\mathrm{z}}=(\mathrm{a}+\mathrm{bi})+(\mathrm{a}-\mathrm{bi})=2\mathrm{a}$

This results in a real number.