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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Is complex conjugate distributive over exponentiation

The complex conjugate is distributive over addition, subtraction, multiplication and division: $$ \overline{z+w} = \bar z + \bar w, $$ $$ \overline{z*w} = \bar z * \bar w, $$ etc. Is it also distributive over exponentiation, i.e., is…
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What was the motivation for the complex plane?

I've read a bit about the history of the complex numbers, and many seem to credit Caspar Wessel with the idea of associating the complex numbers as points on a 2-dimensional plane (or at least the first to explicitly publish the idea). But what was…
nilcit
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Is $\sqrt[\pi]{-1}$ a real or complex number?

Is $\large\sqrt[\pi]{-1}$ a real, imaginary, or complex number? Or maybe, is it something else entirely? Is there even a way to evaluate this?
user311559
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Complex quadratic equation having one purely imaginary root

Question: If $a$ is a complex number such that $\vert a\vert=1$, then find the values of $a$ such that the equation $az^2+z+1=0$ has one purely imaginary root. The equation can be written as…
Aditya Dev
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Solve for z is Complex number : $z^8 + z^4 - 12 = 0$

As far as I came is that I have two solutions to do now: $z^4 = 3$ and $z^4 = -4$ We've done something like this with polar form in school but I can't remember how. Anyone can show me how it's done?
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If $z_1=iz_2$ and $(z_1-z_3)=i(z_3-z_2)$, prove that $\vert z_3\vert=\sqrt 2 \vert z_1\vert$

If $ z_1=iz_2$ and $(z_1-z_3)=i(z_3-z_2)$, prove that $ |z_3|=\sqrt{2} |z_1|$ Rearranging both given equations and taking Euler form of complex numbers, $$\arg\frac{z_1}{z_2}=\frac{\pi}{2}$$and…
Aditya Dev
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An exercise concerning complex numbers

Assume that $| z + 1 | > 2$. Show that $|z^3 + 1| > 1$. My try was: $$|z^3 + 1| = |z + 1| |z^2 - z + 1| > 2 |z^2 - z + 1| $$ but I'm stuck proving that $|z^2 - z + 1| > \frac 1 2$
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Complex equation: $z^8 = (1+z^2)^4$

What's up with this complex equation? $ z^8 = (1+z^2)^4 $ To start with, there seems to be a problem when we try to apply root of four to both sides of the equation: $ z^8 = (1+z^2)^4 $ $ z^2 = 1 + z^2 $ which very clearly doesn't have any…
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Equations with complex numbers

Could anyone help me to solve two equations with complex numbers? I would like to know if there is a way to solve them avoiding the usual substitution $z=a+ib$ because calculations are not very easy $1)$$z^3\bar{z}+3z^2-4=0$ I tried in this…
Gianolepo
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Let z be a complex number such that $z^2 +z + 1/z^2 + 1/z + 1=0 $

If n is a natural number then find the value of $ z^{2012n} + z^{1006n} + 1/z^{2012n} +1/z^{1006n} $ is equal to. I tried rewriting it as $ t^2+t-1=0 $ where $ t=z+1/z $ and then find roots but I don't know how to use it to get required value.
Mathematics
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Function of complex conjugate equal complex conjugate of function?

Very simply, for what type of functions $f: \mathbb{C} \rightarrow \mathbb{C}$ is the following true? $f(\bar{z})=\overline{f(z)}$ Does Schwarz reflection principle imply this is true for all analytic functions?
staple
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Solving $\sqrt{i}$

The results I get not the same as in the book, basically I need to get only this and the $x$'s of course: $$y = \frac{\sqrt{2}}{2}$$ So this is what I did: $$\sqrt{i} = x+yi$$ $$i = x^2 +2xyi -y^2$$ $$\begin{cases} x^2-y^2 = 0 \\ 2xy = 1…
LiziPizi
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Are irrational complex numbers possible?

I am asking because I was reading this and the mathematics is a little over my head. The title of the paper is Rational Approximations to Irrational Complex Number, and I didn't think that complex irrational numbers could exist.
yiyi
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Complex number loci and min/max argument

Question: The point $P$ represents a complex number $z$ in an Argand diagram. Given that $|z+2-2 \sqrt{3} i|=2$ a sketch the locus of $P$ on an Argand diagram. b Write down the minimum value of $\arg(z)$. c Find the maximum value of $\arg(z)$. I…