We are given the equation involving a non-zero complex number $z$ which satisfies:

$\frac{\text{Re}\left(z + \bar{z} + |z|\right)}{\text{Im}\left(z - \bar{z} - i |z|\right)} = 1,$ where $i = \sqrt{-1}$ is the imaginary unit and $\bar{z}$ denotes the complex conjugate of $z$.

Step-by-step Analysis:

Let $z = x + iy$ where $x$ and $y$ are the real and imaginary parts of $z$, respectively. Therefore, the following holds:

• $\bar{z} = x - iy$ (complex conjugate of $z$),
• $|z| = \sqrt{x^2 + y^2}$ (modulus of $z$).

Substitute these into the equation:

1. Real part:

$\text{Re}(z + \bar{z} + |z|) = \text{Re}(x + iy + x - iy + |z|) = \text{Re}(2x + |z|).$ Thus, the real part is: $2x + |z| = 2x + \sqrt{x^2 + y^2}.$

2. Imaginary part:

$\text{Im}(z - \bar{z} - i|z|) = \text{Im}(x + iy - (x - iy) - i|z|) = \text{Im}(2iy - i|z|) = 2y - |z|.$

3. Equation:

Substitute these expressions into the original equation:

$\frac{2x + \sqrt{x^2 + y^2}}{2y - \sqrt{x^2 + y^2}} = 1.$

4. Solving the equation:

Cross-multiply to get: $2x + \sqrt{x^2 + y^2} = 2y - \sqrt{x^2 + y^2}.$

Move terms involving $\sqrt{x^2 + y^2}$ to one side: $2x + \sqrt{x^2 + y^2} + \sqrt{x^2 + y^2} = 2y,$ $2x + 2\sqrt{x^2 + y^2} = 2y,$ $x + \sqrt{x^2 + y^2} = y.$

Square both sides to eliminate the square root: $\left(x + \sqrt{x^2 + y^2}\right)^2 = y^2,$ $x^2 + 2x\sqrt{x^2 + y^2} + x^2 + y^2 = y^2.$ Simplify: $2x^2 + 2x\sqrt{x^2 + y^2} = 0.$ Thus, $x(x + \sqrt{x^2 + y^2}) = 0.$

This gives us two cases:

1. $x = 0$,
2. $x + \sqrt{x^2 + y^2} = 0$.

Case 1: $x = 0$

If $x = 0$, the equation reduces to: $0 + \sqrt{0^2 + y^2} = y \quad \Rightarrow \quad y = y,$ so no restriction on $y$. Hence, $z = iy$ is purely imaginary.

Case 2: $x + \sqrt{x^2 + y^2} = 0$

This implies: $\sqrt{x^2 + y^2} = -x.$ However, since $\sqrt{x^2 + y^2} \geq 0$, this case does not give valid solutions.

Conclusion:

The set of exhaustive values of $\arg(z)$ (argument of $z$) is $\frac{\pi}{2}$ or $\frac{3\pi}{2}$, meaning that $z$ lies on the imaginary axis. Therefore, $z$ is purely imaginary.

Do you have any questions or need further details?

Related questions:

1. What is the geometrical interpretation of the argument of a complex number?
2. How does the modulus of a complex number influence its properties?
3. How can complex conjugates help in solving equations involving complex numbers?
4. What role does the imaginary unit $i$ play in complex number calculations?
5. How does the polar form of a complex number simplify the calculation of arguments?

Tip: When solving complex number equations, expressing the complex number in terms of real and imaginary parts can significantly simplify the solution process.