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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
Solution:-


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



Example 12:


Solution:-

Taking the modulus on both sides

Using

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

 

​Example13:
Find the argument of

Solution:



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

 

​Example 13:


Solution:


 

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

Note:-
If you do by the 1st method, you will get





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