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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Write $\frac{1}{i}i$ in the form $xi +y$

This was a test problem that I did not understand at all. I know it is converting complex numbers, but I need help. How do I write $\frac{1}{i}i$ in the form $xi +y$?
Tsangares
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Familys of curves in z-plane depending on 1 parameter

Describe the family of curves depending on $C>0$ $$\left|\frac{z-z_{1}}{z-z_{2}}\right| = C $$ and $$arg\frac{z-z_{1}}{z-z_{2}} = C $$ What I got: let $z=x+iy, z_{1}=a+ib, z_{2}=c+id$ $$\left|\frac{z-z_{1}}{z-z_{2}}\right| =…
Mykolas
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Minimizing $|a+bw+cw^2|$ where $a,b,c\in\mathbb Z$ and $w = \zeta_3$.

If $a,b,c\in\mathbb Z$ are not all equal and $w$ is a cube root of unity $(w\neq 1$), then the minimum value of: $$|a+bw+cw^2|$$ is what? I'm pretty stuck with the above problem. Could someone help me out?
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Squared magnitude of two complex numbers

I believe one easy questions for maths experts. If I have two complex numbers x=a+ib and y=c+di is the squared magnitude of their sum equal to: \begin{equation} |x+y|^2=|x|^2+2|xy|+|y|^2 \end{equation}
Cali
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Angle between two segments described using complex numbers

Assume we have two segments, $AC$ and $BC$. We can represent points $A$, $B$ and $C$ on the complex plane with three complex numbers, respectively $a,b$ and $c\in\mathbb{C}$. My question is: is there a nice formula for the angle between these two…
user263286
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How to find results for $-z + \frac {3} {\overline {z}}=2$

Can you just please help me solve this problem, because i don't know how to solve it: $$-z + \frac {3} {\overline z}=2$$
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What is the sum of imaginary roots of this equation?

What is the sum of imaginary roots of equation: $$x^3+3x^2+3x+3$$ Here is my attempt: $$x^3+3x^2+3x+3=(x^3+3x^2+3x+1)+2=(x+1)^3+2$$ Since $f(x')=0$ if x' is a root of…
user237454
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least value of |w − z|

On an Argand diagram, sketch the loci of points representing complex numbers $w$ and $z$ such that $|w − 1 − 2i|= 1$ and $\arg(z-1)=\dfrac{3}{4}\pi$. Find the least value of $|w-z|$ for points on these loci. My attempt, I've already drawn the loci…
Mathxx
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Simplifying trig expressions with imaginary coefficients

Can anyone give me some clues on how to simplify this expression while eliminating all imaginary terms? he general solution is given as $$x=A_1e^{(-3+i)t}+A_2e^{(-3-i)t}$$ where $$A_1=\frac{1-3i}{2} \quad A_2=\frac{1+3i}{2}$$ The question then…
Buddy
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Roots of $z^5 + {\sqrt2}z^3 + 1 = 0$

What would be a good strategy to tackle this problem? $$z^5 + {\sqrt2}z^3 + 1 = 0$$ where $z \in \Bbb C$. This question comes out of my complex numbers text on finding roots of polynomials. I have tried to brute force this problem by factorising,…
JJH
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Complex numbers

Here's the question: z is a complex, and if $z^5 + z^4 + z^3 + z^2 + z + 1 = 0$ then $z^6=1$. use this fact to calculate how many answers is there for: $$z^5 + z^4 + z^3 + z^2 + z + 1 = 0$$ Thanks.
Mohamed
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$1-2^z+3^z-4^z=0$ with $z\in\mathbb{C}$

I'm looking for characteristics of the solutions for $$1-2^z+3^z-4^z=0$$ with $z\in\mathbb{C}$. I want to compare it with the simple case $1-2^z=0$. The problem is, that I haven‘t the technical possibility to check this, therefore this post. Can…
user90369
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Visualizing complex plane inequality

Can anyone help me visualize and understand how this sort of inequality works? $$\Big\{z:\left|z-1\right|<2\Big\}$$ It's a set of complex numbers, so I know this corresponds to a certain area in the complex plane, but I need some help to draw the…
user351447
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Let $z$ be a complex number $\ne 0$. What is the absolute value of $z\sqrt{z}$?

$\color{red}{\mathbf{EDIT}}$ The question was misinterpreted - it was actually: 'what is the absolute value of $z/\bar{z}$?'; I'am grateful for the answers given on the original problem though and will keep this up as is in case someone else has a…
user335936
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Geometric interpretation of the solution of the complex equation $z^4+z+2=0$

Question Statement:- If $z$ is a complex number such that $z^4+z+2=0$, show that $z$ cannot lie in the interior of the circle $|z|=1$ Attempt at a solution :- We are given with the equation $z^4+z+2=0$, so we can write $$z^4+z+2=0\implies…
user350331
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