Calculating Argument of Complex Number Division

We are asked to find the argument of the division of two complex numbers, $z_1 = -1 + i$ and $z_2 = 2i$. We can use the property that the argument of a quotient is the difference of the arguments: Arg($z_1/z_2$) = Arg($z_1$) - Arg($z_2$). Let's find the argument for each complex number.

Argument of $z_1$

The complex number $z_1 = -1 + i$ has a real part of $-1$ and an imaginary part of $1$. This point lies in the second quadrant of the complex plane.

• The reference angle $\alpha$ is calculated using $\tan(\alpha) = \frac{|\text{Imaginary Part}|}{|\text{Real Part}|}$.
• $\tan(\alpha) = \frac{|1|}{|-1|} = 1$.
• Thus, $\alpha = \frac{\pi}{4}$.
• Since $z_1$ is in the second quadrant, its principal argument is Arg($z_1$) = $\pi - \alpha$.
• Arg($z_1$) = $\pi - \frac{\pi}{4} = \frac{3\pi}{4}$.

Argument of $z_2$

The complex number $z_2 = 2i$ can be written as $0 + 2i$. This point lies on the positive imaginary axis.

• The argument for a complex number on the positive imaginary axis is $\frac{\pi}{2}$.
• Therefore, Arg($z_2$) = $\frac{\pi}{2}$.

Argument of $z_1/z_2$

Now, we can find the argument of the quotient using the property Arg($z_1/z_2$) = Arg($z_1$) - Arg($z_2$).

• Arg($z_1/z_2$) = $\frac{3\pi}{4} - \frac{\pi}{2}$.
• To subtract, we find a common denominator: $\frac{\pi}{2} = \frac{2\pi}{4}$.
• Arg($z_1/z_2$) = $\frac{3\pi}{4} - \frac{2\pi}{4} = \frac{\pi}{4}$.

The argument of $z_1/z_2$ is $\frac{\pi}{4}$.