Complex numbers

Compute $(4 - 3i)(2 + 5i)$ and enter its real part.

What is $i^{2027}$?

Compute $\left|(3 + 4i)(5 - 12i)\right|$ — ideally without multiplying out first.

Which is the polar form of $z = -1 + i\sqrt 3$?

Solve $z^2 - 4z + 13 = 0$.

Which of the following are sixth roots of unity, i.e. solutions of $z^6 = 1$? (Choose all that apply.)

What is wrong with this "proof" that $1 = -1$? $1 = \sqrt 1 = \sqrt{(-1)(-1)} = \sqrt{-1}\,\sqrt{-1} = i \cdot i = -1.$

In the Lean module, cis θ means $\cos\theta + i\sin\theta$ and n : ℕ. Which Lean statement is De Moivre's theorem?

Write nth_roots(w, n) that returns a list of the $n$ complex $n$-th roots of a non-zero complex number w, using polar form: if $w = R\operatorname{cis}\varphi$, the roots are $R^{1/n}\operatorname{cis}\frac{\varphi + 2\pi k}{n}$ for $k = 0, \dots, n-1$. Python's cmath.polar(w) returns $(R, \varphi)$ and cmath.rect(r, theta) returns $r\operatorname{cis}\theta$.