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
 

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 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.