## Task A-4.1.

Determine the argument of the complex number $z$ if

$$
\operatorname{Re} \frac{1}{z}=\operatorname{Im} \frac{1}{z}=2020
$$
## First solution.

From the condition of the problem, we have $\frac{1}{z}=2020+2020 i$.

It follows that

$$
z=\frac{1}{2020+2020 i}=\frac{1}{2020} \cdot \frac{1}{1+i} \cdot \frac{1-i}{1-i}=\frac{1}{4040}(1-i). \quad 3 \text { points }
$$

![](https://cdn.mathpix.com/cropped/2024_05_30_a374c05c2f1967f14be6g-19.jpg?height=708&width=928&top_left_y=1502&top_left_x=495)

The argument of the number $z$ is the same as the argument of the number $1-i$, which we can determine by drawing in the complex plane. The desired angle is $\varphi=315^{\circ}=\frac{7 \pi}{4}$.