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The image presents the following problem:

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**Consider a complex number \( z = -i \), where \( i = \sqrt{-1} \). If the value of**

\[
\text{arg}(z) + \text{arg}(\overline{z}) + \text{arg}(\text{arg}(z)) + \text{arg}(\Im(z))
\]

(where \( \text{arg}(x) \in (-\pi, \pi] \), \( x \) is any complex number) is equal to \( \frac{p}{q} \pi \), where \( p \) and \( q \) are coprime positive integers, then the value of \( p + q \) is:

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### Solution:

Given \( z = -i \), the complex number can be analyzed:

1. **Argument of \( z \):**
   \[
   z = -i
   \]
   In polar form, \( -i \) has an argument of \( \frac{-\pi}{2} \). Thus,
   \[
   \text{arg}(z) = -\frac{\pi}{2}
   \]

2. **Conjugate of \( z \) (denoted \( \overline{z} \)):**
   \[
   \overline{z} = i
   \]
   The argument of \( i \) is \( \frac{\pi}{2} \), so:
   \[
   \text{arg}(\overline{z}) = \frac{\pi}{2}
   \]

3. **Argument of \( \text{arg}(z) \):**
   Since \( \text{arg}(z) = -\frac{\pi}{2} \), and the argument of a real negative number \( -\frac{\pi}{2} \) is \( \pi \), we have:
   \[
   \text{arg}(\text{arg}(z)) = \pi
   \]

4. **Argument of \( \Im(z) \):**
   The imaginary part of \( z = -i \) is \( -1 \), which is a real negative number. The argument of \( -1 \) is \( \pi \), so:
   \[
   \text{arg}(\Im(z)) = \pi
   \]

### Now, adding all these arguments together:

\[
\text{arg}(z) + \text{arg}(\overline{z}) + \text{arg}(\text{arg}(z)) + \text{arg}(\Im(z)) = -\frac{\pi}{2} + \frac{\pi}{2} + \pi + \pi = 2\pi
\]

We know that \( 2\pi = \frac{2}{1}\pi \), where \( p = 2 \) and \( q = 1 \), and \( p \) and \( q \) are coprime.

### Therefore, \( p + q = 2 + 1 = 3 \).

The value of \( p + q \) is **3**.

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Would you like further details or have any questions?

### Related Questions:
1. How do you find the argument of complex numbers in polar form?
2. What is the conjugate of a complex number and how is its argument calculated?
3. How can you interpret the argument of real and imaginary parts of a complex number?
4. What does it mean for two integers to be coprime?
5. How does the principal argument differ from the general argument of a complex number?

**Tip:** Always remember that the argument of a complex number is typically restricted to the range \( (-\pi, \pi] \).

Consider a complex number z = -i, where i = sqrt(-1). If the value of arg(z) + arg(z̄) + arg(arg(z)) + arg(Im(z)) (where arg(x) ∈ (-π, π], x is any complex number) is p/q π, where p and q are coprime positive integers, then the value of p + q is?

This problem deals with calculating the sum of the arguments of z = -i and its conjugate, argument of argument, and imaginary part. The solution provides detailed steps to solve using polar coordinates and properties of complex numbers.

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