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Let's break down and solve the problem step by step.

We are given the complex number \( z = -i \), where \( i = \sqrt{-1} \). The goal is to compute the sum of various arguments:

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

### Step 1: Calculate \( \arg(z) \)

We know that \( z = -i \). In polar form, the argument of \( -i \) corresponds to the angle made with the positive real axis. Since \( -i \) lies on the negative imaginary axis, the argument is:

\[
\arg(z) = -\frac{\pi}{2}
\]

### Step 2: Calculate \( \arg(\overline{z}) \)

The conjugate of \( z \), denoted by \( \overline{z} \), is \( \overline{-i} = i \). The argument of \( i \), which lies on the positive imaginary axis, is:

\[
\arg(\overline{z}) = \frac{\pi}{2}
\]

### Step 3: Calculate \( \arg(|z|) \)

The modulus \( |z| \) of \( z = -i \) is:

\[
|z| = 1
\]

Since \( |z| \) is a real positive number, the argument of any positive real number is 0:

\[
\arg(|z|) = 0
\]

### Step 4: Calculate \( \arg(\arg(z)) \)

We previously found \( \arg(z) = -\frac{\pi}{2} \). Now we need to compute the argument of this real number. Since \( -\frac{\pi}{2} \) is a negative real number, the argument of any negative real number is \( \pi \):

\[
\arg(\arg(z)) = \pi
\]

### Step 5: Calculate \( \arg(\Im(z)) \)

The imaginary part of \( z = -i \) is \( \Im(z) = -1 \), which is a negative real number. Again, the argument of any negative real number is \( \pi \):

\[
\arg(\Im(z)) = \pi
\]

### Step 6: Sum all the arguments

Now, we sum all the computed arguments:

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

### Step 7: Express in the form \( \frac{p}{q} \pi \)

We have \( 2\pi \), which can be written as:

\[
2\pi = \frac{2}{1} \pi
\]

Here, \( p = 2 \) and \( q = 1 \). Since \( p \) and \( q \) are coprime, the sum \( p + q \) is:

\[
p + q = 2 + 1 = 3
\]

Thus, the value of \( p + q \) is **3**.

---

Would you like further details or have any additional questions? Here are some related questions to expand your understanding:

1. What is the geometrical interpretation of the argument of a complex number?
2. How do you compute the modulus of a complex number in general?
3. What is the argument of the conjugate of a complex number, and why?
4. How does the argument behave for purely real numbers, both positive and negative?
5. Can the argument of a complex number ever exceed \( \pi \) or be less than \( -\pi \)?

**Tip:** When dealing with complex numbers, always remember that arguments are taken in the principal range \( (-\pi, \pi] \).

Consider a complex number z = -i, where i = √−1. If the value of arg(z) + arg(¯z) + arg(|z|) + 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

Step-by-step solution to the complex number problem involving arguments and properties of complex numbers like conjugate and modulus. Understand how to calculate arg(z), arg(¯z), and others to find the coprime integers p and q for this problem. Suitable for Grades 11-12.

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