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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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Complex numbers - solving for smallest positive value of $n$

Given that $z_1=2\sqrt{3}\operatorname{cis}\left(\frac{3\pi}{2}\right)$ and $z_2=2\operatorname{cis}\left(\frac{2\pi}{3}\right)$ find the smallest positive value of $n$ such that $\left(\frac{z_1}{z_2}\right)^n \in \Bbb{R}^+$. My…
IM_LOST
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the minimum possible value of absolute $|z|^2 + |z-3|^2+|z-6i|^2$,where z is a complex number

The minimum possible value of $|z|^2 +|z-3|^2+|z-6i|^2$,where z is a complex number I do not know how to do it. here $|z|$ refers to absolute value of $z$. Here $i=\sqrt {-1}$.
pagal
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Show that $\cos(z)=z$ has a solution for some complex number $z$ (non-real)

I've tried using the exponential form of cosine but the equation was harder than it seemed to solve. I've seen a solution on this site using a Picard Theorem but I don't have any knowledge of analysis yet (I am in high school). Is there a way to…
kelvin
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How do I simplify $\prod_{r=1}^{45} \left(-1+e^{\frac{\pi i}{90}\cdot(2r-1)} \right)$

The original question was to find $$\prod_{r=1}^{45} \sin\left(\left(2r-1\right)^\circ\right)$$ Using $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}=\frac{e^{-ix}}{2i}\cdot(e^{2ix}-1)$ and simplifying a little I ultimately got $$\frac{1}{2^{45}i}\cdot…
Popular Power
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Necessary and sufficient condition for $ \left|\sum_{j=1}^n \lambda_j e^{i\theta_j}\right|^2 = 1 $

Is it true that $$ \left|\sum_{j=1}^n \lambda_j e^{i\theta_j}\right|^2 = 1, $$ where $\sum_{i=j}^n |\lambda_j|^2 = 1$ if, and only if $|\lambda_{j^{\star}}| = 1$ for some $j^{\star}$?
SRichoux
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Show that the equation, $\tan(i \log\frac{ x−iy }{x+iy} ) = 2$ represents the rectangular hyperbola $x^2 − y^2 = xy$

Show that the equation, $\;\tan\left(i \log\dfrac{x−iy}{x+iy}\right)=2\;$ represents the rectangular hyperbola $\;x^2 − y^2 = xy\;$. What I could do is to simplify the expression $\tan\left(i\log\dfrac{x−iy}{x+iy}\right)=2\quad$ to the following one…
Arthur
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value of $a_1|z_1|^2+a_2|z_2|^2+a_3|z_3|^2+a_4|z_4|^2$

If $z_{1},z_{2},z_{3},z_{4}$ are concyclic and $a_{1}+a_{2}+a_3+a_4=0$ and $a_1z_1+a_2z_2+a_3z_3+a_4z_4=0$. Then value of $a_1|z_1|^2+a_2|z_2|^2+a_3|z_3|^2+a_4|z_4|^2=$, where $a_1,a_2,a_3,a_4\in\mathbb{R}$ Here from figure, We…
jacky
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Prove $1 \leq |z^2-5| \leq 9$ if $|z| \leq 2$, $z = a + bi$ (complex number)

Prove $1 \leq |z^2-5| \leq 9$ if $|z| \leq 2$, $z = a + bi$ (complex number) I guess this is really easy question, but I still having troubles with it. Here is my try: $$z^2 = (a+bi)^2 = (a^2 - b^2) + 2abi$$ $$|z| = \sqrt{a^2 + b^2} \leq 2 \implies…
KratosTheFarmer
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Can we extend the complex numbers to include infinite points in these ways?

This is a follow up to question Is there an effective way to generalize this approach of affinely extending the number line?. This question is about possible ways of extending this system as to go beyond the reals, in particular extending the…
Zuhair
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$z^4=\bar z$ complex number equation

Find all $z\in\mathbb{C}$ that satisfy $z^4=\bar z$ and display them on the complex plane. I started with the exponential forms of the two and got to $r^3e^{i4\varphi}=e^{-i\varphi}$. Is it wrong to assume, that $r=1$ and $4\varphi =-\varphi…
fluffy
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$a^2=3b$ is a necessary and sufficient condition for the roots of $x^3+ax^2+bx+c=0$ to constitute an equilateral triangle.

The three roots of the equation $x^{3}+ax^{2}+bx+c=0$ (where $a, b, c$ are given complex numbers) are represented on the Argand diagram by the points $A, B, C$. Prove that $ABC$ is an equilateral triangle if and only if $a^{2}=3b$. Could someone…
spring2000
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Equation with a fraction

Given the following equation: $ \left(\tfrac{z-1}{z+1} \right)^3=i-1 $ It's possible to write: $ \tfrac{z-1}{z+1}=\sqrt[3]{i-1} $ But the following step is corrent or I have to calculate the roots of $ i-1 $ first? $ z-1=(z+1)\sqrt[3]{i-1} $
tizzzato
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Is it possible for two non real complex numbers a and b that are squares of each other? ($a^2=b$ and $b^2=a$)?

Is it possible for two non real complex numbers a and b that are squares of each other? ($a^2=b$ and $b^2=a$)? My answer is not possible because for $a^2$ to be equal to $b$ means that the argument of $b$ is twice of arg(a) and for $b^2$ to be equal…
Magenta
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If $z$ is a non-zero complex number and $p,q,m,n$ are positive jntegers where $\frac{p}{q}=\frac{m}{n}$ with $gcd(m,n)=1$, then

If $z$ is a non-zero complex number and $p,q,m,n$ are positive jntegers where $\frac{p}{q}=\frac{m}{n}$ with $gcd(m,n)=1$, then $z^{\frac{p}{q}}=z^{\frac{m}{n}}$. The proof given in the book is as follows: Here, $np=qm$. Let $w=z^{p/q}$. Then…
Arthur
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Write $ z = \frac{(1-i)^3(√3+i)}{4i}$ to polar form

Write the complex number in polar form: $$ z = \frac{(1-i)^3(\sqrt 3+i)}{4i}$$ So my try goes as follows: \begin{align} \frac{(1−i)^3(\sqrt 3+i)}{4i} &= \frac{(1−i)^3(\sqrt 3+i) \times -4i}{16}\\& = \frac{(1-3i-3+i)(\sqrt 3+i)\times-4i}{16}\\&…
The anime kid
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