1. Key Concepts and Formulas

• Complex Conjugate: For a complex number $z = x+iy$, its conjugate is denoted by $\bar{z} = x-iy$.
• Argument of a Complex Number: For a non-zero complex number $z = x+iy$, its argument, $\arg(z)$, is the angle $\theta$ (in radians) that the line segment from the origin to $(x,y)$ makes with the positive real axis in the Argand plane.
• Division of Complex Numbers: To divide complex numbers in the form $\frac{a+bi}{c+di}$, we multiply both the numerator and the denominator by the conjugate of the denominator: $\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}$.

2. Step-by-Step Solution

Step 1: Identify Given Information and the Goal

We are given $z = 1+i$. Our goal is to find the value of $\frac{12}{\pi} \arg \left(z_{1}\right)$, where $z_{1}=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$.

Step 2: Calculate the Conjugate of $z$

Since $z = 1+i$, the complex conjugate is obtained by changing the sign of the imaginary part: $\bar{z} = 1-i$

Step 3: Simplify the Numerator of $z_1$

The numerator of $z_1$ is $1 + i\bar{z}$. Substituting $\bar{z} = 1-i$, we get: $1 + i(1-i) = 1 + i - i^2 = 1 + i - (-1) = 1 + i + 1 = 2 + i$

Step 4: Simplify the Denominator of $z_1$

The denominator is $\bar{z}(1-z)+\frac{1}{z}$. First, we find $1-z$: $1 - z = 1 - (1+i) = -i$ Then, we compute $\bar{z}(1-z)$: $\bar{z}(1-z) = (1-i)(-i) = -i + i^2 = -i - 1 = -1-i$ Next, we find $\frac{1}{z}$: $\frac{1}{z} = \frac{1}{1+i} = \frac{1}{1+i} \cdot \frac{1-i}{1-i} = \frac{1-i}{1 - i^2} = \frac{1-i}{1 - (-1)} = \frac{1-i}{2}$ Finally, we add the two terms: $\bar{z}(1-z) + \frac{1}{z} = (-1-i) + \frac{1-i}{2} = \frac{-2-2i + 1-i}{2} = \frac{-1-3i}{2}$

Step 5: Calculate the Value of $z_1$

Now we have $z_1 = \frac{2+i}{\frac{-1-3i}{2}} = \frac{2(2+i)}{-1-3i} = \frac{4+2i}{-1-3i}$. To simplify, we multiply the numerator and denominator by the conjugate of the denominator: $z_1 = \frac{4+2i}{-1-3i} \cdot \frac{-1+3i}{-1+3i} = \frac{(4+2i)(-1+3i)}{(-1-3i)(-1+3i)} = \frac{-4 + 12i - 2i + 6i^2}{1 - 9i^2} = \frac{-4 + 10i - 6}{1 + 9} = \frac{-10 + 10i}{10} = -1+i$

Step 6: Calculate the Argument of $z_1$

We have $z_1 = -1+i$. The real part is -1 and the imaginary part is 1. Since the real part is negative and the imaginary part is positive, $z_1$ lies in the second quadrant. The reference angle is $\alpha = \tan^{-1}\left|\frac{1}{-1}\right| = \tan^{-1}(1) = \frac{\pi}{4}$. Therefore, the argument of $z_1$ is $\arg(z_1) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.

Step 7: Calculate the Final Expression

We need to find $\frac{12}{\pi} \arg(z_1)$. Substituting $\arg(z_1) = \frac{3\pi}{4}$, we get: $\frac{12}{\pi} \cdot \frac{3\pi}{4} = \frac{12 \cdot 3}{4} = 3 \cdot 3 = 9$

3. Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs, especially when dealing with complex conjugates and multiplying complex numbers.
• Quadrant Awareness: Always check the quadrant of the complex number to determine the correct argument. Using only $\arctan(y/x)$ can lead to errors.
• Conjugate Multiplication: Remember to multiply both the numerator and denominator by the conjugate of the denominator when dividing complex numbers.

4. Summary

We simplified the expression for $z_1$ by finding the conjugate of $z$, performing complex number arithmetic, and using the fact that $i^2 = -1$. We then found the argument of $z_1$ by considering its location in the complex plane. Finally, we calculated the value of the given expression $\frac{12}{\pi} \arg \left(z_{1}\right)$, which resulted in 9.

5. Final Answer

The final answer is \boxed{9}.