Questions tagged [complex-numbers]

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

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
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Solve for $z$ in equation $|z-a|+|z+a|=2|c|$

I'm trying to solve for $z$ in the equation $$|z-a|+|z+a|=2|c|.$$ My idea is to square both sides $$|z-a|^2+|z+a|^2+2|(z-a)(z+a)|=4|c|^2$$ Using $|z|^2=z\bar{z}$, this becomes $$|z|^2+|a|^2+|(z-a)(z+a)|=2|c|^2$$ I'm not sure how to continue from…
Mika H.
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How to solve z^6 = -15625?

$z^6 = -15625$ has six solutions. $z^6 + 15625 = (z^2+25)(z^4-25z^2+625)$ $z^2+25 = 0$ $\Rightarrow x_{1} = -5i$ $\Rightarrow x_{2} = 5i$ That's easy, but I just don't find a way to get the other 4 solutions. Thanks in advance
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having trouble with a complex numbers problem

I'm pretty good at this stuff, but sometimes I come across questions like this: $$e^z = e^w, \quad z, w \in \mathbb{C}.$$ choose the correct answers: A: $z=w$ B: $|z| = |w|$ C: $e^{iz} = e^{iw}$ D: $e^{2z} = e^{2w}$ can anyone explain what are the…
user2970357
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Isomorphic mapping of $\Bbb C\to \Bbb C$

Prove that the mapping $z\mapsto\bar z$ of $\Bbb C\to \Bbb C$ is an isomorfism of $\Bbb C$ to itself that punctually fixes $\Bbb R$. That's not so hard to prove, if $\bar z=a-ib=x-iy=\bar w \Rightarrow a=x$ and $-b=-y\Rightarrow b=y\Rightarrow…
Ana Galois
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Given that $z= \cos \theta + i\sin \theta$, prove that $\Re\left\{\dfrac{z-1}{z+1}\right\}=0$

Given that $z= \cos \theta + i\sin \theta$, prove that $\Re\left\{\dfrac{z-1}{z+1}\right\}=0$ How would I do this?
Joe Slater
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Sixth Root of Unity

We have to solve the following task of determing the sum and products of all sixth unit roots. I have started as follows: $z^6 = 1 \Longleftrightarrow (z^3-1)(z^3+1) = 0 \Longleftrightarrow (z-1)(z^2+z+1)*(z+1)(z^2-z+1) = 0$ And hence the roots…
user66280
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Graphing Complex Number on Argand Diagram

Can someone please answer me, HOW does $Im(z^2) = 4$ get graphed like this? and not like a normal parabola? Like $Re(z^2) = 4$ But of course on the Imaginary axis. It has been eating my mind up - and I just can't explain why the graph does that.…
MATHSUSER
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finding the product of $z_1z_2z_3$ if $(z+m)^3=-27i$

in complex numbers, if: $(z+m)^3=-27i$ Need to find the term $z_1z_2z_3$ and the term $z_1+z_2+z_3$ by $m$. While $z_1,z_2,z_3$ are the roots of the equation. Tried to simplify but got messy. i thought about vieta formulas but we didn't learn that…
bero
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What is the period

How to find the period of the function \begin{array}{cc} e^{z}\end{array} where "z" is a complex number?? Does this function really have any period??
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Complex Numbers - Sketching on Argand Diagram

Sketch the subsets of the Argand diagram - Draw near labelled sketched to indicate each of the subsets of the Argand diagram described below. $\{z: |z|\ge 1\text{ and }0\le\operatorname{Arg} z\le\frac\pi3\}$ $\{z:z+\bar z\gt 0\} $ I can solve…
MATHSUSER
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finding all the roots (including complex)of the equation

Find all the roots of $z^4=16(z+2i)^4$. Can someone help me teach/ guide to solve this equation?
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What line are inverse functions on the complex plane reflected over?

On the real plane (xy plane) inverse functions are reflections of their original functions over y=x. Is there such line for complex functions and their inverses?
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Any complex number can be written as a sum of complex numbers of modulus 1?

I found this problem in a text-book, no solution offered. I'm curious because it seems like a very interesting result. Full statement is: Let $M \subseteq \mathbb{C}$, a set with the following properties: 1. if $x\in{\mathbb{C}}$ with $|x|=1$, then …
Lisa
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Using de Moivre's Theorem to derive the relation...

I must use de Moivre's Theorem to derive the following relation although I'm not exactly sure where to start: $$\sin(3 \theta) = -4 \sin^3(\theta) + 3\sin(\theta)$$ Thanks in advance.
Simon M
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solve: $z^2+i=|z|$

Solve this: $$z^2+i=|z|$$ I started playing the square then I high again at the square! $$(z^2+i)^2=|z|^2$$ but after that the calculations become many! there is a better way to solve it?
malloc
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