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Modulus and Argument

If,then the modulus of z is , It also denoted by the symbol 'r'. i.e.,
If, then the argument of z is . It also denoted by the symbol θ . i.e
When a complex number is expressed in terms of its modulus and argument, it is said to be in the polar form.

In general if a complex number is in the form it is said to be in the polar form. Here 'r' called the modulus and θ is called the argument or amplitude of the complex number.

Geometrical meaning of Modulus and argument of a complex number

  1. Modulus of a complex number

Let . Modulus or magnitude of z, represented by is the length of the point (x, y) measured from the origin.

Let P represent (x, y) in the Argand plane. Then

Properties of modulus of a complex number


Example: Find the modulus of

Example 11:
If z represent a variable point P in the complex plane, find the locus of P if

Example 12:


Taking the modulus on both sides


Squaring both sides, we get
  1. Argument or amplitude of a complex number
It is the angle between the line joining the given complex number with origin and the x - axis, in the positive direction (i.e antiticlockwise direction)

For any complex number z 0, there corresponds only one value of θ in the interval which is called the principal argument of z such that

(polar form), θ is the argument such that real part of z = rcos θ and imaginary part of z = rsin θ . The value of θ is taken according to the quadrant in which the complex number exists. See diagram below:

Properties of argument


Find the argument of


[You can take any one of these values unless specified in the question]


​Example 13:



Example 14. Find the argument of
Note:- This can be done by multiplying and dividing by the conjugate of the denominator and proceeding as in example given above.

Alternate solution using property of argument

If you do by the 1st method, you will get

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