Question

If z represents a complex number then find the value of $\arg \left( z \right) + \arg \left( {\overline z } \right).$
A.$\dfrac{\pi }{4}$
B.$\dfrac{\pi }{2}$
C.0
D.$- \dfrac{\pi }{4}$

Answer 1

Hint: We are going to use the basic properties of argument of complex numbers to solve the given problem.

Given z is a complex number.
We need to find the value of $\arg \left( z \right) + \arg \left( {\overline z } \right).$
[ $\because$arg (x) + arg (y) = arg (xy)]
$= \arg \left( {z\overline z } \right)$
$= \arg \left( {{{\left| z \right|}^2}} \right)$
= arg (real number) = 0

Note: The argument of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the complex plane. Let’s take a complex number a = x+iy, then $\left| a \right| = \sqrt {{x^2} + {y^2}}$ .