Complex Numbers

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

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

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

$(1+i{)}^8=\text{\hspace{0.17em}⁣}{\left[\sqrt{2}\right]}^8\cdot \text{\hspace{0.17em}⁣}(\cos 2\pi +i\sin 2\pi )=16\cdot (1+0)=16$.

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

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

$\mathrm{\mid }z\mathrm{\mid }=\sqrt{1+3}=2$. Argument: $\tan \theta ={\displaystyle \frac{\sqrt{3}}{-1}}=-\sqrt{3}$, with $z$ in Q2, so $\theta ={\displaystyle \frac{2\pi }{3}}$. Hence $z=2\text{\hspace{0.17em}⁣}(\cos {\displaystyle \frac{2\pi }{3}}+i\sin {\displaystyle \frac{2\pi }{3}})$.

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

Convert each base to modulus–argument form: $$1+i\sqrt{3}=2\mathrm{cis}{\displaystyle \frac{\pi }{3}},1-i=\sqrt{2}\mathrm{cis}\text{\hspace{0.17em}⁣}(-{\displaystyle \frac{\pi }{4}})$$ Apply De Moivre to each power: $$(1+i\sqrt{3}{)}^5=2^5\mathrm{cis}{\displaystyle \frac{5\pi }{3}}=32\mathrm{cis}{\displaystyle \frac{5\pi }{3}}$$ $$(1-i{)}^3=(\sqrt{2}{)}^3\mathrm{cis}\text{\hspace{0.17em}⁣}(-{\displaystyle \frac{3\pi }{4}})=2\sqrt{2}\mathrm{cis}\text{\hspace{0.17em}⁣}(-{\displaystyle \frac{3\pi }{4}})$$ Divide moduli and subtract arguments: $$\mathrm{\mid }z\mathrm{\mid }={\displaystyle \frac{32}{2\sqrt{2}}}=8\sqrt{2}$$ $$\arg z={\displaystyle \frac{5\pi }{3}}-(-{\displaystyle \frac{3\pi }{4}})={\displaystyle \frac{29\pi }{12}}\equiv {\displaystyle \frac{5\pi }{12}}(\text{mod }2\pi )$$ $$z=8\sqrt{2}\mathrm{cis}{\displaystyle \frac{5\pi }{12}}$$

$\mathrm{\mid }z-1\mathrm{\mid }=\mathrm{\mid }z-(-1)\mathrm{\mid }$ 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 $\mathrm{Re}(z)=0$.