To solve for the argument $\text{Arg}(z)$ where $z = 1 - i$, let's proceed step by step.

Step 1: Recall the definition of $z$ and argument

The complex number $z = 1 - i$ can be written as: $z = x + yi$ where $x = 1$ (real part) and $y = -1$ (imaginary part).

The argument of $z$ is the angle $\theta$ that $z$ makes with the positive real axis, measured counterclockwise. The formula for the argument is: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Step 2: Substitute values into the formula

Here:

• $x = 1$
• $y = -1$

Thus: $\theta = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1)$

Step 3: Evaluate $\tan^{-1}(-1)$

The angle whose tangent is $-1$ lies in the fourth quadrant because:

• In the fourth quadrant, the tangent is negative.

The principal angle for $\tan^{-1}(-1)$ in the fourth quadrant is: $\theta = -\frac{\pi}{4}$

Step 4: Conclusion

The argument of $z = 1 - i$ is: $\text{Arg}(z) = -\frac{\pi}{4}$

Final Answer:

a) $-\frac{\pi}{4}$

Do you need more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. How do you find the magnitude $|z|$ of a complex number $z$?
2. What is the general formula for the argument of a complex number in different quadrants?
3. How do you express $z = 1 - i$ in polar form?
4. What is the relationship between $\text{Arg}(z)$ and the complex conjugate of $z$?
5. If $z = -1 + i$, what would be its argument?

Tip: Always identify the quadrant of the complex number to determine the correct angle for the argument.