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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Making sense of $(-1)^\frac{1}{2}$

I just opened a thread that sparked a new question that I now want to discuss (making a new thread however to not clog the other one). Obviously looking at $(-1)^{\frac{1}{2}}$ one might have the number $i$ in mind. But as I've now learned defining…
RedLantern
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Maximum value of $|Z_1-Z_2|^2 +|Z_2-Z_3|^2+|Z_3-Z_1|^2$

Given three complex numbers $|Z_1|= 2 , |Z_2|= 3, |Z_3| = 4$ find the maximum value of $$|Z_1-Z_2|^2 +|Z_2-Z_3|^2+|Z_3-Z_1|^2$$ If we treat them as three vectors $a, b, c$ centred at zero the above expression becomes…
user659291
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Powers of $i$ in Complex Numbers.

Since $i = \sqrt{-1}, i^2 = -1, i^3=-i, i^4 =1$ I understand to calculate, say, $i^{999}$ I just have to $i^{999} = i^{4 \cdot 249 + 3} = (i^4)^{249} \cdot i^{3} = -i$ But I have a question here, why can't I do something like $i^{999} =…
William
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Circuit Analysis - series RC Circuit with AC supply

If anyone is familiar with Horowitz and Hill... its exercise 1.19 Show that all the average power delivered to the preceding circuit winds up in the resistor. Do this by computing the value of $V^2/R$. What is the power, in watts for a series…
JetRex
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If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then prove that $\sin(2x-y-z)+\sin(2y-z-x)+\sin(2z-x-y)=0$

If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then prove that $$\sin(2x-y-z)+\sin(2y-z-x)+\sin(2z-x-y)=0\\ \cos(2x-y-z)+\cos(2y-z-x)+\cos(2z-x-y)=0$$ My…
Sooraj S
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Comparing complex numbers

If $a+ib$, $c+id$, $e+if$ are three complex numbers, than can we tell which one is greater or smaller between them? If yes, then how and if no then why not? Can somebody give explanation on this.... I will be grateful to him.
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If $z_1,z_2$ satisfy $z+\bar{z}=2|z-1|$, $\arg(z_1-z_2)=\dfrac{\pi}{4}$, then find $\Im(z_1+z_2)$

If $z_1$ and $z_2$ both satisfy $z+\bar{z}=2|z-1|$, $\arg(z_1-z_2)=\dfrac{\pi}{4}$, then find $\Im(z_1+z_2)$ My Attempt $$ z_1+\bar{z}_1=2|z_1-1|\quad\&\quad z_2+\bar{z}_2=2|z_2-1|\quad\&\quad…
Sooraj S
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Some weird results in complex number computing

The question I met is to show that if $z=\cos (\theta)+i\sin(\theta)$ with $i=\sqrt{-1}$,then $ Re(\frac{z-1}{z+1})=0$ In the normal way, we found that:…
yuanming luo
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Find every $z$s that fit $\cos(z) = -2$

I couldn't find any, I tried to write $\cos(z)$ as $\cos(x)\cos(iy)-\sin(x)\sin(iy)$ which then gave me $\cos(x)\cosh(y) - i\sin(x)\sinh(y) = -2$ $\sin(x)=0$ so that imaginary part become $0$ now we have to find $\cosh(y) = -2$ which is not true…
no0ob
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If $z=4+i \sqrt{7}$ then find the value of $z^3 -4z^2 -9z + 91$.

So i was learning complex numbers and i came across this problem. In the solution they have made $z-4=i\sqrt{7}$ and then they squared the above equation resulting in $z^2 -8z+16=-7$ then they proceeded by sending $-7$ to the LHS resulting in $z^2…
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Is this a pure imaginary number or real number?

Is $\dfrac{0}{2yi}$ a pure imaginary number or a real number? I'm debating, $0$ is a real number but if you divide by $i$, it's imaginary.
Tom S
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Complex number problem $i \tan (\theta)$ proof.

Given $z = \cos (\theta) + i \sin (\theta)$, prove $\dfrac{z^{2}-1}{z^{2}+1} = i \tan(\theta)$ I know $|z|=1$ so its locus is a circle of radius $1$; and so $z^{2}$ is also on the circle but with argument $2\theta$; and $z^{2}+1$ has argument…
Rik
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Solve $|z^2|+\text{Re}(az)+b = 0.$

Fix $a\in\mathbb{C}$ and $b\in\mathbb{R}.$ Show that the equation $|z^2|+\text{Re}(az)+b = 0$ has a solution iff $|a^2|\geq 4b.$ When solutions exist, show the solution set is a circle. A seemingly easy problem. Letting $z=x+yi$ and $a = c+di$ the…
Fabled
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Complex numbers - shading on the Argand diagram

Find the modulus of the complex number $z$ that lies in the region $|z-1| \leq |z-i|$ and $|z-(2+2i)| \leq 1$ for which $\arg(z)$ is least. I am having trouble obtaining the required answer ($\sqrt(7)$) I first drew a rough sketch of the region…
PhysicsMathsLove
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complex numbers equation deg 4

Find the sum of squares of elements of set $A$ if: $$A=\Big{\{}\big|z^n+\frac{1}{z^n}\big|;\:n\in\mathbb{N},\:z\in\mathbb{C},\: z^4+z^3+z^2+z+1=0\Big{\}}.$$
Numbers
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