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.

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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

Therefore,

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

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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.