# 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}$

Last updated date: 23rd Mar 2023

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Answer

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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}} $ .

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}} $ .

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