Complex Numbers

$\overline{a + bi} = a - bi$, so $\overline{2 - 3i} = 2 + 3i$.

$|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

$(1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$.

$(1 + i)^8 = \!\left[\sqrt{2}\right]^{8} \cdot \!\left(\cos 2\pi + i\sin 2\pi\right) = 16 \cdot (1 + 0) = 16$.

$\dfrac{1 + i}{1 - i} \cdot \dfrac{1 + i}{1 + i} = \dfrac{(1 + i)^2}{1 - i^2} = \dfrac{2i}{2} = i$.

By the Fundamental Theorem of Algebra (or Moivre), $z^n = 1$ has exactly $n$ distinct complex roots, the $n$th roots of unity $\varepsilon_k = e^{2\pi i k/n}$, $k = 0, \ldots, n-1$.

$|z| = \sqrt{1 + 3} = 2$. Argument: $\tan\theta = \dfrac{\sqrt{3}}{-1} = -\sqrt{3}$, with $z$ in Q2, so $\theta = \dfrac{2\pi}{3}$. Hence $z = 2\!\left(\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}\right)$.

Since $13 = 4\cdot 3 + 1$, we have $$i^{13} = i^{4\cdot 3 + 1} = (i^4)^3 \cdot i = i.$$ So $z = 4 + 3i$, and $$|z| = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.$$

Convert each base to modulus–argument form: $$1+i\sqrt{3}=2\,\operatorname{cis}\dfrac{\pi}{3},\qquad 1-i=\sqrt{2}\,\operatorname{cis}\!\left(-\dfrac{\pi}{4}\right)$$ Apply De Moivre to each power: $$(1+i\sqrt{3})^{5}=2^{5}\,\operatorname{cis}\dfrac{5\pi}{3}=32\,\operatorname{cis}\dfrac{5\pi}{3}$$ $$(1-i)^{3}=(\sqrt{2})^{3}\,\operatorname{cis}\!\left(-\dfrac{3\pi}{4}\right)=2\sqrt{2}\,\operatorname{cis}\!\left(-\dfrac{3\pi}{4}\right)$$ Divide moduli and subtract arguments: $$|z|=\dfrac{32}{2\sqrt{2}}=8\sqrt{2}$$ $$\arg z=\dfrac{5\pi}{3}-\left(-\dfrac{3\pi}{4}\right)=\dfrac{29\pi}{12}\equiv\dfrac{5\pi}{12}\ (\text{mod }2\pi)$$ $$z=8\sqrt{2}\,\operatorname{cis}\dfrac{5\pi}{12}$$

$|z - 1| = |z - (-1)|$ describes the set of points equidistant from $1$ and $-1$. That's the perpendicular bisector of the segment between them, i.e. the imaginary axis $\operatorname{Re}(z) = 0$.