Nov. 11, 2019

Argument of complex number And its Properties:


Argument/Amplitude of Complex Number in Different Quadrants

Let z = x+y be a non-zero complex number and can be represented in the form z = r(cosθ+ sinθ) where, r is the modulus and θ is the argument of z. From the figure, let z be represented by a line OP inclined at an angle θ with the positive direction of x-axis or the angle measured in the counter-clockwise direction and the distance of the point from O in the direction is r i.e. |z| Then, in right angled triangle OPM, right angled at M, we have

To measure the argument of any complex number (non-zero) in any quadrant, we must first calculate the argument of the respective complex number in first Quadrant. 

let z = x +y be a complex number lying in any of the quadrants

and z' = |x| +|y| be a complex number lying in first quadrant 

such that

= arg(z') = {argument calculated in first quadrant}

then the arg(z) calculated in respective quadrants can be calculated as


a) Argument is measured along +iv direction of x-axis

b) Argument can also be measured in counter-clockwise direction.

c) Principle value of argument lies between

d) Angle(s) measured in anticlockwise direction is taken as +iv and those measured in clockwise direction are taken as -iv

Properties of Argument

P-1 arg

P-2 arg

P-3 arg

P-4 arg(z) = arg(|z|2) = arg (POSITIVE REAL NUMBER) = 0

P-5 arg {using P-2 and then P-3}


P-7 If then where n