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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.

Consider a complex number z = -i, where i = √-1. If the value of arg(z) + arg(\overline{z}) - arg(|z|) + arg(arg(z)) + arg(Im(z)) = \frac{p}{q}\pi - \pi, where p and q are coprime positive integers, then the value of p + q is?

Explore how to solve a complex number argument expression involving z = -i and its components. This detailed solution breaks down arguments of complex numbers, conjugates, and moduli to find the value of p + q, where p and q are coprime integers.

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