If `z ne0`, find Arge `z+Arg bar z`

Answer

To solve the problem of finding \( \text{Arg}(z) + \text{Arg}(\bar{z}) \) where \( z \neq 0 \), let's follow these steps:

### Step 1: Define the complex number
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit.

### Step 2: Find the conjugate of \( z \)
The conjugate of \( z \) is given by:
\[ \bar{z} = x - iy \]

### Step 3: Calculate the arguments
The argument of a complex number \( z \) is defined as:
\[ \text{Arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]
For the conjugate \( \bar{z} \), the argument is:
\[ \text{Arg}(\bar{z}) = \tan^{-1}\left(\frac{-y}{x}\right) \]

### Step 4: Add the arguments
Now, we need to find \( \text{Arg}(z) + \text{Arg}(\bar{z}) \):
\[ \text{Arg}(z) + \text{Arg}(\bar{z}) = \tan^{-1}\left(\frac{y}{x}\right) + \tan^{-1}\left(\frac{-y}{x}\right) \]

### Step 5: Use the property of the tangent function
Using the property of the tangent function, we know that:
\[ \tan^{-1}(a) + \tan^{-1}(-a) = 0 \]
for any real number \( a \). Therefore:
\[ \tan^{-1}\left(\frac{y}{x}\right) + \tan^{-1}\left(\frac{-y}{x}\right) = 0 \]

### Conclusion
Thus, we conclude that:
\[ \text{Arg}(z) + \text{Arg}(\bar{z}) = 0 \]
This holds true as long as \( z \neq 0 \).