How to find modulus of complex number?

djeetThis article has been authored by Djeet,the Maths editor at Xamplified,who has over four years of experience in teaching Mathematics.His write-ups at Xamplified can truly be termed ‘Examplified’ as he breaks down complex maths concepts into simple how-to steps.

Table Of Content

Brush Up Basics

Step 1: Recap basic facts

  • Argand Plane is used to represent the complex number. The horizontal axis of Argand plane represents the real part while vertical axis represents the imaginary part of complex number.
  • The modulus of complex number is distance of a point P (which represents complex number in Argand Plane) from the origin.

Step 2: Plot the complex number in Argand plane

In geometrical representation, complex number z = (x + iy) is represented by a complex point P(x, y) on the complex plane or the Argand Plane. Join this point ‘P’ with the origin ‘O’. Let this ray be called OP.


Geometrical representation of complex number in Argand Plane

Depending upon the sign of real part ‘x’ and imaginary part ‘y’, complex point ‘P’ can exist in  any quadrant.

Step 3: Calculate the length of ray ‘OP’

The length OP is calculated using Pythagoras Theorem.
In  Triangle OAP,

OP² = OA² + AP²
OP² = x² + y²



The length of OP is the modulus of complex number and is denoted by |z|.


Observations to give you insight

  • It does not matter in which quadrant complex number lies, its modulus will always be positive.
  • Two different complex numbers can have same modulus. For example 4 + i3, 4 – i3, -4 + i3, -4 – i3, 3 + i4, 3 – i4, -3 + i4 and -3 – i4 all have five as their modulus.