Question

If z is a non-zero complex number, then \[\arg \left( z \right) + \arg \left( {\overline z } \right)\] equals
A. 0
B. \[\pi \]
C. \[2\pi \]
D. None of these

Answer 1

Hint: In this question, we have been given that our number z is a complex number with some value, i.e., it is not equal to 0. We have to evaluate the value of the sum of the argument of the complex number and the argument of the conjugate of the complex number. The conjugate of a complex number is obtained by changing the sign (plus to minus and minus to plus) of only the complex number part of the given number. Then, the argument of a complex number is obtained by dividing the coefficient of the complex part by the coefficient of the real part. So, in this question we just write the complex number, then we write its conjugate, then we calculate the argument of the complex number and the argument of the conjugate of the complex number and then we just add the two, and we are going to have our answer.

Complete step-by-step answer:
Let the given complex number z be
\[z = ax + {\rm{i}}y\]
Then the argument of z (obtained by dividing the coefficient of the complex part by the coefficient of the real part) is
\[\arg \left( z \right) = \dfrac{y}{x}\]
Now, the conjugate (obtained by changing the sign of only the complex number part of the given number) of z is
 \[\overline z = ax - {\rm{i}}y\]
And the argument of \[\overline z \] is
\[\arg \left( {\overline z } \right) = - \dfrac{y}{x}\]
So, \[\arg \left( z \right) + \arg \left( {\overline z } \right) = \dfrac{y}{x} + \left( { - \dfrac{y}{x}} \right) = \dfrac{y}{x} - \dfrac{y}{x} = 0\]

Hence, the correct option is A).

Note: So, we saw that in solving questions like these, we first write what has been given to us, then we write what we have to evaluate. Then we apply the required formulae so as to calculate what needs to be evaluated. We assume the unknown quantities if they are needed to be assumed in order to calculate the result, like we did in this question. The only thing that is required from our side is that we know what the exact formula is.