Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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Solving complex equation with absolute: $|z|^3 = z^3 + 1$

I've come across an equation: $|z|^3 = z^3 + 1, \quad z \in \mathbb{C}$ and got stuck. Best I could do was conversion: $$z \overline{z}|z|=(z+1)(z^2 - z + 1)$$ but I don't see an further step for removing final the absolute.
Snake Detection Hypothesis
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$az+b\overline{z}+c=0$ represents a line

When does $az+b\overline{z}+c=0$ represent a line? All of $a,b,c$ are complex numbers. I know that a line in the complex plane is usually represented by $z=a+bt$, where the parameter $t$ runs through all real values. So suppose we're given the…
Paul S.
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What is $0^{i+x}$ where $x$ is negative?

What is $0^{i+x}$, where $x$ is negative and $i$ is the imaginary unit? I saw that Wolfram Alpha says its “complex infinity”, something it often gives when the answer is undefined. Is it undefined and if so, why? (Notably it also says that if $x$ is…
Gabriel Tellez
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Finding modulus of $\sqrt{6} - \sqrt{6}\,i$

I found the real part $=\sqrt{6}$. But I don't know how to find imaginary part. I thought it was whatever part of the function that involved $i$, with the $i$ removed? Therefore the imaginary part would be $-\sqrt{6}$. Meaning the modulus is equal…
user88720
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Purely Algebraic Solution to a complex number problem

Question - Let $z_1 , z_2 ,z_3 $ be 3 complex numbers satisfying $|z|=1$ and $$4z_3=3(z_1+z_2)$$ Then find the value of $|z_1-z_2|$. Looking at the question i could immediately see a geometric solution that we can just rearrange the equation…
Aleph
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Why is multiplication of complex numbers defined the way it is?

I am reading in my linear algebra textbook that for two complex numbers a + bi and c + di, their product is defined to be (ac - bd) + (ad + bc)i. But this definition is stated without motivation and I do not see what that motivation is. It seems…
Joa
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Calculate the following contour integral...

Calculate $$\oint_\gamma \frac 1{z-\sin z} dz$$ where the contour is the unit circle in the complex plane. I do not know how to find the order of the pole at 0, though I believe it is 3. Once I have such an order, however, I do not even know how to…
Johnny Apple
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Using moivre's formula to power a complex number

so i have this complex number $\dfrac1{(1+i)^{100}}$ and i need to use De Moivre's formula formula to power. So how do you calculate it ? For what i know it should go like this : $\dfrac1{(1+i)^{100}} = (1+i)^{-100}$ $a = 1 , b =1 $ $n = -100$…
Vytautas
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if $|z-3-2i| \leq 2 , \text{ find the minimum of } |2z-6+5i|$

if $|z-3-2i| \leq 2 , \text{ find the minimum of } |2z-6+5i|$ My attempt:- Let $z=x+iy$ so we have $(x-3)^2+(y-2)^2 \leq4$ let $|2z-6+5i | \text{ be }\phi $ so ${\phi}^2 =4(x-3)^2+(2y+5)^2$ I squared it as the first term is common, however,I'm not…
math and physics forever
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if $|z|-z=3-i\sqrt{3}$ determine the value of $|z|$

if $|z|-z=3-i\sqrt{3}$ determine the value of $|z|$ Below is the solution that I have, but I don't know how they get the (2) form (1) $$|z|-z=3-i\sqrt{3} \Longleftrightarrow z=|z|-3+i\sqrt{3}\;\; (1)$$ $$ \Rightarrow |z|^2=(|z|-3)^2+\sqrt{3}^2 \;\;…
user579102
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Logarithm of a negative number in a very special case.

It is stated that:- $\log_b(n)$ is undefined, if $n < 0 $. However, $\log_i(-1) = 2$, $i$ represents the imaginary number. Here, $-1 <0$, but the operation is well defined. My question:- Is the logarithm of a negative number well defined only if…
user1174597
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Intuition: why complex multiplication add angles in polar coordinates

(Sorry if the question was already asked. I don't seem to find the right keywords, so after the first few pages of search I gave up) For complex numbers, we know that $$(\cos \phi + i \sin \phi) (\cos \psi + i \sin \psi) = \cos (\phi + \psi) + i…
Dmitry
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How is $\,(e^{\pi})^{{{\left(e^{\pi/2}\right)}^{\left(e^{\pi/2}\right)}}^{\ldots\infty}}=-1\,?$

So, here's how I've arrived to that conclusion. My first question was: can $\,e^{i\pi}\,$ be written as $\,(e^\pi)^i\,?$ Well, by taking $\,\log_e$ to both sides of the equation $\,{e^{i\pi}}=-1\,,\,$ we get $\,\log_e{i}=i\pi/2\,$ which is true. And…
I'm CHILL bro
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Complex numbers $z$, $w$, $u$ with $|z|=1$, $|w|=2$, $|u|=3$, Find the maximum value of $|z-u|$

Let be given three complex numbers $z$, $w$, $u$ satisfying $|z|=1$, $|w|=2$, $|u|=3$, and $|z+w-u|=|u+z-w|$. Find the greatest value of $|z-u|$. I tried $$|z+w-u| \leqslant |z-u| + |w| \leqslant |z + u| + |w| \leqslant |z| + |u| + |w|.$$ This…
minhthien_2016
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Solving two complex equation using two real values

I have a system of two complex equations for which I want to find the intersection ($\Gamma$). $$p_3=Abs[\frac{\Gamma k_{32}+k_{31}}{\Gamma k_{34}+k_{33}}],p_4=Abs[\frac{\Gamma k_{42}+k_{41}}{\Gamma k_{44}+k_{43}}]$$ Where all the $k_{nm}$ are…
Julien
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