# Tagged Questions

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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### Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
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### Find all solutions to the following equation: $x^3=-8i$

Find all solutions to the following equation: $$x^3=-8i$$ I found the modulus, $$r=8$$ $$\operatorname{arg}(x)=\arctan(-8/0)=-π/2+2πk$$ By De Moivre's Theorem: $$2[\cos(-π/6+2/3πk)+i\sin(-π/6+2/3πk)]$$...
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### is 1 greater than i?

I'm not sure this question even makes sense because complex numbers are a plane instead of a line. The magnitudes are obviously the same because i is a unit vector, but is there any inequality you can ...
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### Rotations of complex graphs

Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ counter-...
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### Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
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### Consider the equation $|z + 3i|=3|z|$ for complex z and give a geometric description of the set S of all solutions.

Writing $z$ in the form $a+ib$ and then rearranging gives $-8a^2-8b^2+6b+9=0$. The most promising form I could manage from this is $(b-\frac{3}{8})^2=(\frac{9}{8}-a)(\frac{9}{8}+a)$ but I still do not ...
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### How can I split this into its' real and imaginary parts, and simplify?

Essentially, I want to prove that $| \sum_{k=1}^n e^{ik}|$ is bounded. If I obtain an expression for this sum: $$\sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ I am not sure how to proceed ...