The image presents the following problem:

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:

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.

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 $(-\pi, \pi]$.