Let's break down the problem step by step. The given complex number is $z = -i$, and the goal is to find the value of $p + q$ where $\frac{p}{q} \pi$ represents the value of a specific expression involving the arguments of the complex number and its components.

Information provided:

1. $z = -i$, where $i = \sqrt{-1}$.
2. The expression for which we need to find the argument: $\text{arg}(z) + \text{arg}(\overline{z}) - \text{arg}(|z|) + \text{arg}(\text{arg}(z)) + \text{arg}(\text{Im}(z)) = \frac{p}{q}\pi - \pi$ where $p$ and $q$ are coprime positive integers.

Step-by-step Solution:

1. Finding $\text{arg}(z)$: $z = -i$ The argument of $-i$ is $\text{arg}(z) = -\frac{\pi}{2}$ because $-i$ lies on the negative imaginary axis.

2. Finding $\text{arg}(\overline{z})$: The complex conjugate of $z$ is $\overline{z} = i$, so: $\text{arg}(\overline{z}) = \frac{\pi}{2}$

3. Finding $\text{arg}(|z|)$: The modulus $|z|$ of any complex number is a positive real number. The argument of any positive real number is zero, so: $\text{arg}(|z|) = 0$

4. Finding $\text{arg}(\text{arg}(z))$: The argument of $\text{arg}(z)$ is a bit more subtle. We know that $\text{arg}(z) = -\frac{\pi}{2}$, and since this is a real number, its argument is zero: $\text{arg}(\text{arg}(z)) = 0$

5. Finding $\text{arg}(\text{Im}(z))$: The imaginary part of $z = -i$ is $\text{Im}(z) = -1$. The argument of a negative real number is $\pi$, so: $\text{arg}(\text{Im}(z)) = \pi$

Summing all the values:

$\text{arg}(z) + \text{arg}(\overline{z}) - \text{arg}(|z|) + \text{arg}(\text{arg}(z)) + \text{arg}(\text{Im}(z)) = -\frac{\pi}{2} + \frac{\pi}{2} - 0 + 0 + \pi = \pi$

This simplifies to $\pi$, and we are given that this is equivalent to: $\frac{p}{q}\pi - \pi$ Thus, we have: $\pi = \frac{p}{q}\pi - \pi$ Adding $\pi$ to both sides: $2\pi = \frac{p}{q}\pi$ Dividing both sides by $\pi$, we get: $2 = \frac{p}{q}$ Thus, $p = 2$ and $q = 1$. Since $p$ and $q$ are coprime, this is valid.

Final Answer:

The value of $p + q = 2 + 1 = 3$.

Would you like more details or have any further questions on this? Here are some related questions:

1. How do you find the modulus of a complex number?
2. What is the geometric interpretation of the argument of a complex number?
3. How does the argument of a negative real number behave?
4. Can you explain how the complex conjugate affects the argument?
5. What are some properties of the argument function in different quadrants?

Tip: When working with arguments of complex numbers, always consider their position in the complex plane, as it helps determine the correct angle.