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.

19229 questions
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A simple clarification of Polar form and Euler form

I have always been wondering why the complex number of the form $\cos\theta + i\sin\theta$ is represented as $e^{i\theta}$. I fully understand how the polar form and euler form works. My only doubt is how the polar is worked out in euler form. For…
Ja jack
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Find the three cube roots of $z = -2+2i$

$n$th roots of a complex number: For a positive integer $n$, the complex number $z=r(\cos(\theta) + i\sin(\theta))$ has exactly $n$ distinct $n$th roots given by $$\sqrt[n]r\left(\cos\left(\frac{\theta + 2\pi k}n\right) + i\sin\left(\frac{\theta +…
user1091990
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Evaluating a Complex expression

Still very new to complex variables and complex analysis so apologies for the probably very simple question. What is the value(s) of this: $$(1+i)^i$$ How to go about it? Also I wanted to confirm that for $$\log(-i)$$ I got $$\ln1 +…
Mathstatsstudent
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Minimum value of a complex function

What is the minimum value of $|z+2+i|+|z-2-i|+|3-z|+|2-z|?\;\;$ $2, 3, 5, 6$ I was trying to solve this problem by assuming $z= r(\cos x+i\sin x)$ but then the calculations become lengthy and complicated. Can someone help me please?
user118413
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Proof that for every $z \in \mathbb C$: $z=\frac{2}{n}\sum_{k=0}^{n-1}(Re(u_{k}\overline z)u_{k})$

Let $n \in \mathbb N$, $n>2$ and $u_{1},u_{2},...,u_{n-1}$ are all root of unity for n. Proof that for every $z \in \mathbb C$: $$z=\frac{2}{n}\sum_{k=0}^{n-1}(Re(u_{k}\overline z)u_{k})$$ I know that $Re(u_{0})=Re(u_{1})$, $Re(u_{2})=Re(u_{3})$…
MP3129
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Number of integral values of 'a' for which the following is true..

Question If it is given that the equation $$|z|^2-2iz+2c(a+i)=0$$ possesses solution for all $c∈R$, then the number of integral values of 'a' for which it is true is ______. Attempt Wrote $z=x+iy$ and substituted in the given equation to get $x=c$…
jayant98
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a problem on complex numbers

Let $w\neq 1$ and $w^{13} = 1$. If $a = w+ w^3 + w^4 + w^{-4} + w^{-3} + w^{-1}$ and $b = w^2+ w^5 + w^6 + w^{-6} + w^{-5} + w^{-2}$, then the quadratic equation whose roots are $a$ and $b$ is ... ? I got…
Rituraj Tripathy
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simplification of complex numbers

How does $$2πi(1/e^{5πi/6} +1/e^{3πi/6}+1/e^{πi/6})$$ reduce to π/3. i tried using exponential form but for some reason this does not give me the answer. what approaches could you take to simplifying this?
Vicem0n
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Complex number roots, complex plane

1.What figure complex roots of this equation create $$z^4+16z^2+100=0$$ 2.What are the equations of straight lines that link vertexes of this figure I don't know how to proceed $z^2=x$ , $x^2+16x+100=0$ , $\delta=-144=144i^2$ ,$\sqrt{\delta}=12i$…
mashhc
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Find a complex number with a real part of $24$

Let $a_1=5-3i$ and let $a_2=2-8i$ be a geometric series. a. Show $\arg(a_{n+8})=\arg(a_n)$ b. let there be an element with a real part of $24$ find its imaginary part and index c. how much elements do we need to get to a sum of $-515-133i$ a.…
newhere
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Finding real part of complex number in exponential form in fraction

Given this complex number, $$e^{9ix/2} \frac{\sin 4x }{ (\sin (x/2) }$$ The real part of this complex number can be worked out easily, by replacing the $e^{9ix/2}$ with $\cos(9x/2)$ However if I'm given the complex number, $$\frac{3} {3 - e^{ix}…
Trunkersn
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Solve $z^5 - iz^3+iz^2+1 =0$

Solve for $z$: $$z^5 - iz^3+iz^2+1 =0$$ I've ruled out the approach with euler's representation, de moivre's way led me to $\cos4\phi + \sin3\phi-\sin2\phi +1 = 0$ (with the assumption that the length of $z$ is $1$, which I derived from the fact…
uros kozole
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Why is $e^{2πi} = e^0$ true while $2πi = 0$ is false?

I came across a perplexing thing while testing some assumptions in Wolfram|Alpha, and as I don't have a math education beyond college algebra, I thought this would be a good place to ask. I would just like to emphasize that this is not a homework…
Braden Best
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Does this equation have a complex number solution?

Does this equation have any solutions: $$\sqrt{z^2+z-7}=\sqrt{z-3}?$$ I know it does not have any real number solutions, but how about complex number solutions? I understand that when you solve this problem algebraically, you get $z=\pm 2$ as…
Sam Tilly
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How are the complex numbers $z_1$ and $z_2$ related if $\arg(z_1) = \arg(z_2)$?

How are the complex numbers $z_1$ and $z_2$ related if $\arg(z_1) = \arg(z_2)$? My attempt : Let $z_1$,$z_2\in \mathbb{C}$ such that $z_1=r_1(\cos\theta + i\sin\theta)=a+ib$ and $z_2=r_2(\cos\gamma+i\sin\gamma)=c+id$ Then, we have…
rcoder
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