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Last updated on July 22nd, 2025

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Complex Conjugate

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A complex conjugate is a complex number formed by changing the sign of the imaginary part of another complex number. They are used in engineering fields, such as control systems, to assess system stability. Now, we will explore the concept of complex conjugates in more detail.

Complex Conjugate for UAE Students
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What is a Complex Conjugate?

Every complex number has a complex conjugate, where the real part remains the same, but the imaginary part has the opposite sign. 

 

 

The formula we use for a complex number is a + bi.

 

Here:
a and b are real numbers
i: an imaginary number equal to the square root of – 1
a: real part
b: imaginary part

 

The complex conjugate of a + bi is a – bi, where a is the real part and the imaginary part is –b
 

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Representation of the Conjugate of Complex Number

The conjugate of a complex number can be mathematically denoted as z. Since z and z  are each other’s conjugates, they are together called a complex-conjugate pair.
Consider z = x + iy as a complex number, so the conjugate of z can be represented as:
z = x – iy
The relationship between z and z  shows us that the conjugate is found by simply changing the sign of the imaginary part.
 

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Geometric Interpretation of Complex Conjugate

In geometry, the conjugate of a complex number z is its reflected or mirrored version across the x-axis in the complex plane, also called the Argand plane, as shown below:

 

 

Multiplication of Complex Number with Its Conjugate


Multiplying a complex number by its complex conjugate always results in a real number. 
For example, We multiply the complex number a + bi by its conjugate a – bi
 (a + bi) (a – bi) =  a2 – i2 b2  = a2 + b2
 

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Properties of Conjugate

Provided that z, z1, and z2 are complex numbers, the properties they possess are stated below:

 

 

When the conjugate of a complex number is equal to the number itself, it indicates the number is real. We represent it as: z = z. 
For example: (5 + 0i) = (5 – 0i) = 5.

 

 

If the sum of a complex number and its conjugate is zero, the number is considered to be imaginary. This can be mathematically expressed as:
                                        (z + z = 0).

 

For example: the conjugate of (0 – 5i) = (0 + 5i) = 5i

(z ) = z

z +  z = 2 Re (z)

z –  z = 2i  Im (z)

z.z = {Re (z)}2 + {lm(z)}2

(z1 + z2)=z1 + z2

(z1-z2)=z1 - z2

(z1 . z2)=z1 z2

z = (z1/ z2)    z = z1/ z2; z2  ≠ 0

 

 

Keep in mind that the conjugate of a complex number can only be found if the complex number is in the form: z = (x + iy). If not, convert it to this form before finding the conjugate.
 

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Complex Conjugate Root Theorem

The complex conjugate root theorem states that if p(x) is a polynomial with real coefficients and a + bi as its root, then the conjugate of the root, a – bi, is also a root of the polynomial. 
 

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Real-Life Applications of Complex Conjugate

Complex conjugates play a vital role in different fields. Students must understand the application of complex conjugates in simplifying complex expressions. Here, we will learn how they can be applied:

 

 

  • Complex conjugates are widely utilized in studying alternating current circuits. For example, engineers apply complex conjugate in calculating impedance, current, and voltage.

 

  • They are used in aerodynamics for structuring airplane wings and analyzing the airflow around objects.

 

  • In the medical field, they enhance image quality and assist in detecting medical conditions in MRI scans.

 

  • We use complex conjugates to analyze sound waves and other mechanical vibrations. 

 

  • They are utilized in signal processing in mobile networks to enhance the transmission of messages.
     
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Common Mistakes and How to Avoid Them in Complex Conjugates

Complex conjugates are widely used in simplifying mathematical expressions. However, students might make mistakes when dealing with complex conjugates. Here are some common mistakes and the ways to avoid them:
 

Mistake 1

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Changing the Sign of the Real Part

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Some students mistakenly believe that the sign of the real part should also be changed along with the imaginary part. Remember that only the sign of the imaginary part changes, while the real part remains the same.
 

Mistake 2

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 Incorrect Division of the Conjugate 
 

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They may misuse the conjugate when rationalizing the denominator, leading to errors. Use the conjugate of the denominator and multiply it by both the numerator and the denominator.
For example: (2 + i)/(3 – i)
Multiplying by the conjugate of the denominator, i.e., (2 + i) × (3 + i)/(3 - i) × (3 + i)
 

Mistake 3

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Not Converting to Standard Form
 

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Trying to find the conjugate without first checking if the complex number is in standard form can lead to mistakes. Make sure the complex number is always in standard form. Otherwise, convert it before finding the conjugate.
 

Mistake 4

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Overlooking the Conjugate Rule in Polynomials

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Failing to recognize that if a polynomial equation has a complex root, its conjugate is also a root. Recall the complex conjugate root theorem, which states that if a + bi is a root, then a – bi must be a root.
 

Mistake 5

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Misapplying Conjugate Multiplication
 

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They might overlook the fact that a real number is always obtained when a complex number is multiplied by its conjugate.
Solution: Apply the formula:
z. z = {Re (z)}2 + {lm(z)}2
For example, given: 
z = 6 + 5i, then z = 6 – 5i
(6+ 5i) (6 – 5i) = 36 – 25i2 
We have: i2 = –1,
36 – 25 (– 1) = 36 + 25 = 61 (a real number).
 

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Solved Examples of Complex Conjugate

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Problem 1

Determine the conjugate of z = –11 + 4i

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z  = –11 – 4i
 

Explanation

We obtain the conjugate of a complex number simply by changing the sign of its imaginary part.
z  = –11 – 4i
 

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Problem 2

Multiply z = 9 + 11i by its conjugate.

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We get 202 as the result, which is a real number.
 

Explanation

Here, the conjugate of z = 9 + 11i is z = 9 – 11i
We will now multiply z by its conjugate:
(9 + 11i) (9 – 11i)
Apply the identity (a + b)(a – b) = a2 – b2:
92 – (11i)2
We substitute i2 = –1 into the expression (11i)2 = 121(–1) = –121:
81 – (– 121) = 81 +121 = 202
Therefore, the value we obtained (202) is a real number.
 

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Problem 3

Simplify 6 + 4i/3 -i applying the complex conjugate

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The expression 6 + 4i/3 -i is simplified to 1.4 + 1.8i 
 

Explanation

Following the conjugate property of multiplication, we multiply the numerator and denominator by the conjugate of the denominator, which is 3 + i:

(6 + 4i) (3 + i)/(3 -i) (3 + i) 

Here, we use the identity (a – b) (a + b) = a2– b2 for the denominator:
(3 – i) (3 + i) = 32 – i2 = 9 – (– 1) = 9 + 1 = 10

Use the distributive property to expand the denominator: 
(6 + 4i) (3 + i) = 6(3) + 6(i) + 4i(3) + 4i(i)
= 18 + 6i + 12i + 4i2
Substituting i2 = –1 into the expression:
= (18 – 4) + (6i + 12 i)
= 14 + 18i
The expression becomes,
(14 + 18 i)/ 10
14/ 10 + 18i/10
= 1.4 + 1.8i
 

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Problem 4

Multiply z = 9 + 5i by its conjugate.

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(9 + 5i) (9 – 5i) = 106.
 

Explanation

Here, z = 9 – 5i is the conjugate of z = 9 + 5i
We now multiply z by its conjugate:
(9 + 5i) (9 – 5i)
Applying the identity, (a+ b) (a – b) = a2– b2:
92 – (5i)2
= 81 – 25i2
Substituting i2 = –1,
= 81 – 25 (– 1)
= 81 + 25
= 106
Since the result is a real number, we conclude that multiplying a complex number by its conjugate will always give a real number.
 

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Problem 5

Check if the product of z = 5 – 2i and its conjugate is a real number.

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 (5 – 2i) (5 +2i) = 29, which is a real number.
 

Explanation

Here,  z = 5 + 2i is the conjugate of z = 5 – 2i
z.z = (5 – 2i) (5 +2i)
Apply the identity (a – b) (a + b) = a2 – b2
52– (2i)2
= 25 – 4i2
Substituting the value of i2 = –1:
= 25 – 4(– 1)
= 25 + 4
= 29
We get 29 as the result, confirming that the product of a complex number and its conjugate will always result in a real number.
 

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FAQs of Complex Conjugate

1.What do you mean by the complex conjugate of a complex number?

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2.What is the mathematical representation of the conjugate of a complex number?

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3.What happens if you multiply a complex number by its conjugate?

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4.Is it possible for a complex number to be equal to its conjugate?

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

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6.How can children in United Arab Emirates use numbers in everyday life to understand Complex Conjugate?

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7.What are some fun ways kids in United Arab Emirates can practice Complex Conjugate with numbers?

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8.What role do numbers and Complex Conjugate play in helping children in United Arab Emirates develop problem-solving skills?

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9.How can families in United Arab Emirates create number-rich environments to improve Complex Conjugate skills?

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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: She loves to read number jokes and games.

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