Modulus of Complex Number

A complex number in mathematics is an expression of the form a+bi, where i is the imaginary unit (√-1) and a and b are real numbers.
 

The square root of the sum of the squares of the complex number’s real and imaginary parts is its modulus. |z| or r Represents the modulus of a complex number z=x+iy, which is defined as follows:
|z|=x2+y2.
 

In mathematics, a complex number’s conjugate is created by inverting the sign of its imaginary component while leaving the real component unaltered. When a complex number z=a+bi is given, with ‘a’ standing for the real component and ‘b’ for the imaginary component, and ‘i’ for the imaginary component, and ‘i’ for the imaginary unit (√-1), its conjugate, z is defined as a-bi.
For example,
If z=5+3i,
Then its conjugate is z = 5-3i
 

Simply changing the sign of the imaginary part of a complex number while maintaining the same sign for the real part gives the conjugate.
Formula:
Complex number is written as,
z=a+bi
Conjugate is,
z = a-bi
For example,
z=6+2i
Now, change the sign of the imaginary part of a complex number.
Conjugate,
z = 6-2i
 

The distance between the complex number’s coordinates and the center of the complex plane is said to be the modulus of the complex number when it is shown on a graph. A complex number’s modulus is the distance between two complex numbers. It is presented as a point on the Argand plane (a, b). This distance is measured as r=(a2+b2) and is a linear distance from the center (0, 0) to the point (a, b). Moreover, the Pythagoras theorem can be used to determine the modulus of a complex number. Here, the hypotenuse represents the modulus, the real component is the base, and the imaginary part seems to be the perpendicular of the right-angled triangles.  
     r=a2+b2
 

Some of the properties of a complex number are listed below. Given two complex numbers, z and w, we have:

• Complex numbers z and -z have the same modulus, so |z|=|-z|.
• A complex number’s modulus is zero only if it is zero; that is, |z|=0 if z=0.
• The modulus of a product equals the product of moduli: zw = zw
   

The modulus of a complex number is used in real-world applications. Let us see how it is used.

Ac circuits in electrical engineering
The magnitude of impedance, or how much a circuit resists the flow of alternating current, is found using the modulus of a complex number in AC circuit analysis.

Processing of signals
When signals are expressed as complex numbers in the frequency domain (such as in Fourier transforms), the modulus indicates the strength of each frequency, or, to put it another way, the strength of the signal at that frequency.

Systems of control
Engineers use the modulus to examine the distances between specific points (referred to as poles and zeros) and the complex plane’s center. They can determine whether the system will behave steadily or become unstable.

Wave machines in physics
The probability amplitude or wave intensity is represented by the modulus of a complex number in quantum mechanics and wave physics.

Navigation and GPS
The modulus helps in determining the separation between two points on a plane when locations are expressed as complex numbers.