# 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 Î¸ . 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

*Î¸*is called the argument or amplitude of the complex number.

â€‹

# Geometrical meaning of Modulus and argument of a complex number

**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.

*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

**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*â‰ 0, there corresponds only one value of Î¸ in the interval which is called the principal argument of

*z*such that â€‹ (polar form), Î¸ is the argument such that real part of

*z*= rcos Î¸ and imaginary part of

*z*= rsin Î¸ . The value of Î¸ is taken according to the quadrant in which the complex number exists. See diagram below:

**Properties of argument**

**â€‹Example13:**

Find the argument of

**Solution:**

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

**â€‹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 1

^{st}method, you will get