# Modulus Amplitude Of A Complex Number

Let z = x + iy where x and y are real numbers and i = √(-1). Then the non-negative square root of (x2 + y2) is known as the modulus or absolute value of z. Modulus or absolute value of z is denoted by |z| and read as mod z.

Hence if z = x + iy, then |z| = |x+iy| = +√x2 + y2.

For example, if z = -3 + 4i then, |z| = |-3 + 4i |= √(-3)2 + 42 = 5.

AMPLITUDE (OR ARGUMENT) OF A COMPLEX NUMBER:

Let z = x + iy where x and y are real numbers and i = √(-1) and x2 + y2 ≠ 0, then the value of θ for which the equations x = |z| cosθ ........(1) and y = |z| sin θ .......(2) are concurrently satisfied is named as the amplitude or argument of z and is denoted by Amp z or Arg z.

Equations (1) and (2) are satisfied for infinitely many values of θ, any of these infinite values of θ is the value of amp z. However, the unique value of θ lying in the interval -π< θ ≤ π and satisfying equations (1) and (2) is known as the principal value of arg z and it is denoted by arg z or amp z. Or in other words argument of a complex number means its principal value.

Since, cos(2nπ + θ)= cos θ and sin(2nπ + θ)= sin θ (where n is an integer), hence

Amp z = 2nπ + amp z                 where -π < amp z ≤ π.

GEOMETRICAL REPRESENTATION OF MODULUS AND AMPLITUDE:

Let point P(x, y) in the z-plane represent the complex number z = x + iy. Drawing perpendicular on (and joining we get,

If = r and ∠XOP = θ, then from the right-angled triangle PON we get,

x = rcosθ and y = rsinθ

Hence or θ =

and r2 = OP2 = ON2 + PN2 = x2 + y2

or r = = √ x2+ y2

Hence z = x + iy = rcosθ + irsinθ = r(cosθ + isinθ)

where r = √ x2 + y2= |z| and θ = = Arg z

The form of representation z = r(cosθ + isinθ), where r = |z| and θ = Arg z is known as the

polar or modulus-amplitude form of z.

Some important points to be noted are given below:

1) 0 < principal value of θ < when P lies in the first quadrant;

2) < principal value of θ < π when P lies in the second quadrant;

3) - π< principal value of θ < - when P lies in the third quadrant;

4) - < principal value of θ < 0 when P lies in the fourth quadrant;

Particularly principal values of θ are 0, π, and - when P lies on respectively.

Example 1:

Find the amplitude and modulus of

Solution:

Since

Hence the required modulus of (Answer)

Now it is clear, that in the z-plane the point lies in the second quadrant. Hence if amp z = θ then,