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 $\overline{Z}$ and is given by: $\overline{z} = a - bi$

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

$\overset{z}{ˉ} = a - ib$ . In a complex number if we replace i by –i, we get conjugate of the complex number. $\overset{z}{ˉ}$ 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:

$z = 3 + 4 i and \overset{z}{ˉ} = 3 - 4 i$

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

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: $∣ z ∣ = a^{2} + b^{2}$

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

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 $A = [1 - i 4 + 2 i 3 + 7 i]$ . The complex conjugate of matrix A, denoted as $\overline{A}, is B = [1 + i 4 - 2 i 3 - 7 i]$ , where each entry in B is the conjugate of the corresponding entry in A . Therefore, we can write $B = \overline{A}$ . 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:

$A = (1 + i 4 + i 2 - 3 i - i)$

The conjugate of A, denoted as

$\overline{A}, i s : \overset{ˉ}{A} = (1 - i 4 - i 2 + 3 i i)$

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 $\overline{Z} = a - bi$ . The multiplication of z by $\overline{Z}$ is given by:

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

This result is always a non-negative real number.

Example:

For z = 3 + 4i

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

Multiplying z by $\overline{Z}$ :

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

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 $(\overset{z}{ˉ})$

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

(ii) $z = \overset{z}{ˉ} \Leftrightarrow z is purely real$

(iii) $z + \overset{z}{ˉ} = 0 \Leftrightarrow z is purely imaginary$

(iv) $\overline{\overset{z}{ˉ}} = z$

(v) Relation between modulus and conjugate. $∣ z ∣^{2} = z \overset{z}{ˉ}$

(vi) $\overline{(z_{1} \pm z_{2})} = \overset{z}{ˉ}_{1} \pm \overset{z}{ˉ}_{2}$

(vii) $\overline{(z_{1} z_{2})} = \overset{z}{ˉ}_{1} \overset{z}{ˉ}_{2}$

(viii) $\overline{(\frac{z _{1}}{z _{2}})} = \frac{( z ˉ _{1} )}{( z ˉ _{2} )} (z_{2} \neq = 0)$

(ix) $z_{1} \overline{z}_{2} + \overline{z}_{1} z_{2} = 2 Re (z_{1} \overline{z}_{2})$

(x) $z_{1} \overline{z}_{2} - \overline{z}_{1} z_{2} = 2 i Im (z_{1} \overline{z}_{2})$

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

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

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

$∴ Conjugate = - 1 - i$

Example 3: If the conjugate of $(x + i y) (1 - 2 i) be 1 + i$ , then

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

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

Ans. (C)

Solution:

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

8.0Solved Questions for Conjugate of Complex Number

What is a Complex Conjugate?

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

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: $\overline{z} = a - bi$ .

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 $∣ z ∣ = a^{2} + b^{2}$ .

How does the complex conjugate relate to the complex plane?

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

What are some properties of complex conjugates?

Ans:

Addition: $\overline{z_{1} + z_{2}} = \overline{z_{1}} + \overline{z_{2}}$

Multiplication: $\overline{z_{1} \cdot z_{2}} = \overline{z_{1}} \cdot \overline{z_{2}}$

Division: $\overline{(\frac{z _{1}}{z _{2}})} = \frac{z _{1}}{z _{2}}$

Modulus: $∣ z ∣ = ∣ \overline{z} ∣$

What is the complex conjugate of a matrix?

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

How do you multiply a complex number by its conjugate?

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

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

Can you provide an example of a complex conjugate?

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

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

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

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

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

This results in a real number.