Last updated on July 22nd, 2025
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.
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
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.
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
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.
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.
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 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:
Determine the conjugate of z = –11 + 4i
z = –11 – 4i
We obtain the conjugate of a complex number simply by changing the sign of its imaginary part.
z = –11 – 4i
Multiply z = 9 + 11i by its conjugate.
We get 202 as the result, which is a real number.
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.
Simplify 6 + 4i/3 -i applying the complex conjugate
The expression 6 + 4i/3 -i is simplified to 1.4 + 1.8i
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
Multiply z = 9 + 5i by its conjugate.
(9 + 5i) (9 – 5i) = 106.
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.
Check if the product of z = 5 – 2i and its conjugate is a real number.
(5 – 2i) (5 +2i) = 29, which is a real number.
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.
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.
: She loves to read number jokes and games.