The provided text in Arabic and mathematical symbols translates to:

"If Arg(z̅) = Arg(z), then Arg(z) equals π."

Here:

• $z$ is a complex number.
• $\overline{z}$ is the conjugate of $z$.
• $\text{Arg}(z)$ represents the principal argument (angle) of $z$.

Explanation

For a complex number $z$ represented as $z = x + yi$ (where $x$ is the real part and $y$ is the imaginary part), the conjugate $\overline{z}$ is $x - yi$.

1. The argument $\text{Arg}(z)$ gives the angle of $z$ in the complex plane relative to the positive real axis.

2. The argument of $\overline{z}$, $\text{Arg}(\overline{z})$, reflects the angle $\text{Arg}(z)$ over the real axis, so: $\text{Arg}(\overline{z}) = -\text{Arg}(z).$

3. The condition $\text{Arg}(\overline{z}) = \text{Arg}(z)$ implies: $-\text{Arg}(z) = \text{Arg}(z).$

4. Solving for $\text{Arg}(z)$, we find: $\text{Arg}(z) = 0 \quad \text{or} \quad \text{Arg}(z) = \pi.$

• Since the problem states that $\text{Arg}(z)$ equals $\pi$, the argument of $z$ corresponds to an angle of $\pi$ radians, meaning $z$ lies on the negative real axis.

Summary

• $\text{Arg}(z) = \pi$ indicates that $z$ is a real number with a negative value.

Would you like a graphical explanation or further clarification?

Related Questions:

1. What is the argument of a complex number, and how is it calculated?
2. How is the conjugate of a complex number geometrically represented?
3. Why does $\text{Arg}(\overline{z}) = -\text{Arg}(z)$ hold true?
4. What are the principal and general arguments of a complex number?
5. Can a complex number have an argument equal to $\pi$ and still have a non-zero imaginary part?

Tip: For any complex number $z$ on the negative real axis, its argument is always $\pi$.