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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Solving $\overline{z}\cdot|z|\cdot z^5=8\sqrt{2}\left(-\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^8$

I have this equation to solve: $$\overline{z}\cdot|z|\cdot z^5=8\sqrt{2}\left(-\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^8$$ Since $\overline{z}\cdot z = |z|^2$ and utilizing the de Moivre's formula this can be simplified…
tomnookey
  • 93
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Finding the minimum value of $|z-3+i|$ given $z$ satisfies $\arg{(z-2i)}=\frac{\pi}{6}$

I need to find the the minimum value of $|z-3+i|$ given $z$ satisfies $\arg{(z-2i)}=\frac{\pi}{6}$. My issue is that just looking at the graph of $\arg{(z-2i)}=\frac{\pi}{6}$ (which is a ray from $(0,2)$ on the Argand diagram) and $|z-3+i|$ (a…
Pen and Paper
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Find all complex numbers which make the following equations true: $ |z+1| =1 $ and $ |z^2+1| =1 $

Find all complex numbers which make the following equations true: $$ |z+1| =1 $$ $$ |z^2+1| =1 $$ Solution: If $ |z+1| =1 $ holds true, then $$z+1 = 1.e^{i2n\pi}$$ $$z = 1.e^{i2n\pi}-1$$ $$z = 1.e^{i2n\pi}-1e^{i2m\pi}$$ If $ |z^2+1| =1 $ holds true,…
Anwesa Roy
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complex solutions

Find all $|z|=1$ such that $|z^4+4| = \sqrt{5}.$ I've tried doing $$|z^4+4|^2 = 5 \implies (z^4+4)(\overline{z^4}+4) = 5 \implies |z|^8 + 4(z^4+\overline{z^4}) +11=0,$$ but i'm not sure how to solve that.
fds-mqdwqmqkdwl
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Calculating $\sum^{10}_{k=1}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$

Find the value of $$\sum^{10}_{k=1}\left (\sin\left (\frac{2k\pi}{11} \right )+i\cos\left (\frac{2k\pi}{11}\right ) \right)$$ My approach: Since $\cos\theta + i\sin\theta = e^{i\theta}$, we can write the given equation as: $$\begin{align*} &i…
Sachin
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complex exponential equation

I am trying to solve the following exponential equation: $z^{1+i} = 4$ where the argument of $z$ is between $-\pi$ and $\pi$. Here is what I have gotten so far: If $z = a + bi$ then the magnitude of $z$ is $2$ and $arctan(\frac{b}{a}) = -2$,…
imranfat
  • 10,029
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Calculating the roots of a complex polynomial

the problem is the following: Given a complex polynomial P = z4 - $\alpha$, with $\alpha \in \mathbb {C} $. We know that $ e^{i\frac{2\pi}{3}} $ is a complex root of 2P. Select all the correct options: (A) 1 is a complex root of P (B) $…
Gonçalo Silva
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How I can write this in polar form?

$\cos x +i(1+\sin x)$ I already know that: $\bbox[yellow] {\sin x=2\sin(\frac{x}{2})\cos(\frac{x}{2})}$ and $\bbox[yellow]{\cos x=2\cos^2(\frac{x}{2})-1}$ But this doesn't help me in this case.
Zakaria
  • 87
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Sum of the powers of roots of unity

I was reading this proof of the roots of unity filter and I point can't understand is why: $$ S = \sum_{j=0}^{n-1} w^{kj} =0$$ If $ \omega $ is the nth root of unity and $k$ is some positive integer not equal to zero and n doesn't divide k I…
tryst with freedom
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Formula for the sum of the value of a rational function over roots of unity

Let $n,k$ be integers, and let $U$ be the set of all $n$-ths roots of unity (so there are exactly $n$ elements in $U$). Let $U'=U \setminus \lbrace 1 \rbrace$. Are there simple formulas (in terms of $n$ and $k$) for the sums $$ S_k=\sum_{\varepsilon…
Ewan Delanoy
  • 61,600
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Finding $n$ such that $(3-3w+2w^2)^{4n+3}+(2+3w-3w^2)^{4n+3}+(-3+2w+3w^2)^{4n+3}=0$ for $w\neq1$ a cube root of unity

Let $${\left( {3 - 3\omega + 2{\omega ^2}} \right)^{4n + 3}} + {\left( {2 + 3\omega - 3{\omega ^2}} \right)^{4n + 3}} + {\left( { - 3 + 2\omega + 3{\omega ^2}} \right)^{4n + 3}}=0$$ If $\omega\ne1$ be the complex cube root of unity, then the…
Samar Imam Zaidi
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Geometrically, why are complex numbers both linear maps that perform rotation/dilation, and points on the plane?

The standard geometric interpretation of complex numbers is to draw the complex plane by identifying the horizontal axis as the real axis and the vertical axis as the imaginary axis, so that the complex number $z = x + iy$ is identified with the…
IssaRice
  • 1,155
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Complex number such that affixes of itself, its squared and its cube form a right triangle. $z,z^2,z^3$ form a right triangle in argand plane

$\DeclareMathOperator{\aff}{Aff}$ Here is a funny problem I stumbled upon recently I call the affix of a complex number the corresponding point in the argand plane, that is $$\aff(\mathrm{i})=(0;1)$$ Find all $z \in \mathbf{C}$ such that the…
T.D
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5 answers

Need help with complex equation: $|z−i|+|z+i|=2$

I am trying to solve this equation: |z−i|+|z+i|=2 and don't know how to do it. This what I have: $$\sqrt{(x+1)^2+y^2}+ \sqrt{(x-1)^2+y^2} = 2 /^2$$ $$(x+1)^2+y^2+(x-1)^2+y^2 + 2\sqrt{[(x+1)^2+y^2][(x-1)^2+y^2]} = 4$$ $$x^2 +2x + 1+y^2+x^2-2x + 1+y^2…
cocacola
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Let $s,t,r$ be non zero complex numbers and $L$ is is set of solutions of $z=x+iy$ of the equation $sz + t\bar z+r=0$

Prove that $L$ is a singleton set if $|s|\not =|t|$ And, prove that $z$ is a straight line if $L$ is not singleton Solving the equation, I got $z=\frac{\bar s r -\bar r t}{|t|^2-|s|^2}$ I personally cannot see any reason why $z$ will have a…
Aditya
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