Conjugate of a Complex Number

In any two complex numbers, if only the sign of the imaginary part differs then, they are known as a complex conjugate of each other. Thus conjugate of a complex number a + bi would be a – bi.

What’s the use of a complex conjugate?

Thus we can observe that multiplying a complex number with its conjugate gives us a real number. Thus the division of complex numbers is possible by multiplying both numerator and denominator with the complex conjugate of the denominator.

Examples of Complex Conjugates  

Properties of Complex Conjugates

Property 1:

Property 2:

Property 3:

Property 4:

Property 5:

Division of Two Complex Numbers 

Division of complex numbers is done by multiplying both numerator and denominator with the complex conjugate of the denominator. 

Example 1:

Example 2:

Example 3:

Example 4:

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i² = −1. For example, 5+6i is a complex number, where 5 is a real number and 6i is an imaginary number. Therefore, the combination of both the real number and imaginary number is a complex number. There can be four types of algebraic operations on complex numbers which are mentioned below. The four operations on the complex numbers include:

• Addition
• Subtraction
• Multiplication
• Division

Addition of Complex Numbers 

To add two complex numbers, just add the corresponding real and imaginary parts.

(a + bi) + (c + di) = (a + c) + (b + d)i 

Examples:  

• (7 + 8i) + (6 + 3i)  = (7 + 6) + (8 + 3)i = 13 + 11i
• (2 + 5i) + (13 + 7i) = (2 + 13) + (7 + 5)i = 15 + 12i
• (-3 – 6i) + (-4 + 14i) = (-3 – 4) + (-6 + 14)i = -7 + 8i
• (4 – 3i ) + ( 6 + 3i) = (4+6) + (-3+3)i = 10
• (6 + 11i) + (4 + 3i) = (4 + 6) + (11 + 3)i = 10 + 14i

Subtraction of Complex Numbers 

To subtract two complex numbers, just subtract the corresponding real and imaginary parts.

(a + bi) − (c + di) = (a − c) + (b − d)i 

Examples:

• (6 + 8i)  –  (3 + 4i) = (6 – 3) + (8 – 4)i = 3 + 4i
• (7 + 15i) – (2 + 5i) = (7 – 2) + (15 – 5)i = 5 + 10i
• (-3 + 5i) – (6 + 9i) = (-3 – 6) + (5 – 9)i = -9 – 4i
• (14 – 3i) – (-7 + 2i) = (14 – (-7)) + (-3 – 2)i = 21 – 5i
• (-2 + 6i) – (4 + 13i) = (-2 – 4) + (6 – 13)i = -6 – 7i

Multiplication of Two Complex Numbers 

Multiplication of two complex numbers is the same as the multiplication of two binomials. Let us suppose that we have to multiply a + bi and c + di. We will multiply them term by term.

(a + bi) ∗ (c + di) = (a + bi) ∗ c + (a + bi) ∗ di

                              = (a ∗ c + (b ∗ c)i)+((a ∗ d)i + b ∗ d ∗ −1)

                              = (a ∗ c − b ∗ d + i(b ∗ c + a ∗ d))   

Example 1:  Multiply (1 + 4i) and (3 + 5i).

(1 + 4i) ∗ (3 + 5i) = (3 + 12i) + (5i + 20i²)

                            = 3 + 17i − 20

                            = −17 + 17i

Note: Multiplication of complex numbers with real numbers or purely imaginary can be done in the same manner. 

Example 2: Multiply 5 and (4 + 7i).

5 ∗ (4+7i) can be viewed as (5 + 0i) ∗ (4 + 7i)

                                            = 5 ∗ (4 + 7i)

                                            = 20 + 35i   

Example 3: Multiply 3i and (2 + 6i).

3i ∗ (2 + 6i) can be viewed as (0 + 3i) ∗ (2 + 6i)

                                               = 3i ∗ (2 + 6i)

                                               = 6i + 18i²

                                               = 6i − 18

                                               = −18 + 6i   

Example 4: Multiply (5 + 3i)  and  (3 + 4i).

(5+3i) ∗ (3+4i) = (5 + 3i) ∗ 3 + (5 + 3i) ∗ 4i

                        = (15 + 9i) + (20i + 12i²)

                        = (15 − 12) + (20 + 9)i

                        = 3 + 29i  

Review of Complex Numbers Addition, Subtraction and Multiplication

1. (a + bi) + (c + di) = (a + c) + (b + d)i
2. (a + bi) − (c + di) = (a − c) + (b − d)i
3. (a + bi) ∗ (c + di) = ((a ∗ c − b ∗ d) + (b ∗ c + a ∗ d)i)