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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.

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What does $\arg(z-3)= \frac{-3\pi}{4}$ mean? How to visualize it?

What does $\arg(z-3)= \frac{-3\pi}{4}$ mean? How to visualize it? I can't seem to visualize what it looks like. Could someone please teach me this. I'm currently trying to solve this question and want to break it down into pieces to understand the…
Munchies
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Solving the complex equation $(1+z)^5=z^5$

I must to find $z\in\mathbb{C}$ such that: $\boxed{(1+z)^5=z^5}$ Is the following equivalence correct? $(1+z)^5=z^5\Leftrightarrow 1+z=z$ If this is not correct, how can solve this problem?
yemino
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Why is $\operatorname{Im}{\frac{1-e^{iny}}{-2i\sin{\frac{y}{2}}}}=\frac{1}{2\sin{\frac{y}{2}}}\operatorname{Re}(1-e^{iny})$?

I am studying Fourier analysis and had a question involving the following equality: $$\operatorname{Im}{\frac{1-e^{iny}}{-2i\sin{\frac{y}{2}}}}=\frac{1}{2\sin{\frac{y}{2}}}\operatorname{Re}(1-e^{iny})$$ I can see that…
mathreads
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$\arg(r_+) = 0$ and $\arg(r_-) = \pi$

$r_+$ = positive real number $r_-$ = negative real number I am confused $\arg(r_+) = 0$ and $\arg(r_-) = \pi$ for $\arg(r-) = \pi$ : for example, $\arg(-1)$ can be represented as $-1 + 0i $ $\cos(\arg(-1)) = \frac{-1}{1} = -1$; this…
EM4
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Boundary of the absolute value of a sum of complex numbers

I have this question that I have no idea how to approach! Let $w_{1},w_{2},...,w_{n}$ be complex numbers such that there exists a constant $C$ such that $|w_{1}^k+ w_{2}^k + ... + w_ {n}^k| \leq C$ for all whole numbers $k \geq 0$. Show that…
pencil321
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Are $\mathbb{C}-\mathbb{R}$ imaginary numbers?

Background I am teaching senior high school students about the structure of numbers. Start from defining $\mathbb{Q}$ and $\mathbb{R}$ as the rational and real numbers respectively, we can define $\mathbb{R}-\mathbb{Q}$ as the irrational numbers. I…
Display Name
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Polar form of $\frac{2(1+i)(1-i\sqrt{3})}{(-\sqrt{3}+i)^{30}}$

$\frac{2(1+i)(1-i\sqrt{3})}{(-\sqrt{3}+i)^{30}}$ So I solved this by using polar form and then turned it to cis for (1+i) the argument is $\sqrt{2}$exp($\frac{\pi}{4}$i) for (1-i$\sqrt{3}$) the argument is 2exp($\frac{-\pi}{3}$i) for…
EM4
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Evaluate $|z_1 + z_2 + z_3|$ on unit circle given $ \frac{ z_1^2}{z_2 z_3} + \frac{ z_2^2}{z_1 z_3} + \frac{ z_3^2}{z_1 z_2} +1 =0$

given three complex numbers $(z_1 , z_2 , z_3)$ lying on the unit circle and related by the equation $ \frac{ z_1^2}{z_2 z_3} + \frac{ z_2^2}{z_1 z_3} + \frac{ z_3^2}{z_1 z_2} +1 =0$, find the sum of all possible values of $|z_1 + z_2 + z_3|$ I…
tryst with freedom
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Find real numbers r and s so that $a_{n+2}+ra_{n+1}+sa_n = 0$ and $b_{n+2}+rb_{n+1}+sb_n = 0$

I already know that $b_{n+1} = a_n +3b_n$ and $a_{n+1} = 3a_n - b_n$. So $a_{n+2} = 3(3a_n-b_n)-(3b_n+a_n) = 9a_n-3b_n-3b_n-a_n = 8a_n-6b_n$ and $b_{n+2} = 8b_n+6a_n$. So we can rewrite the whole thing as $8a_n-6b_n+r(3a_n-b_n)+sa_n =…
KTF
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Complex numbers limits

$\lim_\limits{z\to\infty} \sqrt{z-2i} - \sqrt{z-i} ,$ where z is complex no. How to evaluate this? I tried by assuming $z = x+iy$ and evaluated $z-2i = x+ i(y-2)$ and $z-1 = x + i(y-1)$ and after putting the value in the given question , I couldn't…
user268439
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How to show that $|z_1+z_2|+|z_1-z_2| = |z_1+\sqrt{z^2_1-z^2_2} |+|z_1-\sqrt{z^2_1-z^2_2} | $?

How to show that $$|z_1+z_2|+|z_1-z_2| = |z_1+\sqrt{z^2_1-z^2_2} |+|z_1-\sqrt{z^2_1-z^2_2} | $$ Is this question correct? I tried my best to prove it But got nowhere near, Any help
Angelo Mark
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Rotation of angle $\frac{\pi}{4}$ about the point $i$

Need to find an isometry which would rotate about the point $i$ by $\frac{\pi}{4}$. So I was thinking that first I return the given point to orign, make the rotation and then translate back, right? That would look…
Sarunas
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Finding The Complex Roots Of $4z^5 + \overline z^3= 0$?

Hello everyone how can I find all the complex roots of: $4z^5 + \overline z^3= 0$? I tried to mark $a+bi = z , a-bi = \overline z$ and $4(a+bi)^5 +(a-bi)^3 = 0$ But I don't know how to continue.
xxx
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natural logarithm of a complex number $\ln(a+bi)$

I am trying to find a formula for $\ln(a+bi)$, is my working correct? $$a+bi=re^{i\theta},\,\,r=\sqrt{a^2+b^2},\,\,\theta=\frac{|b|}{b}\arctan\left(\frac ba\right)$$ and so: $$\ln(a+bi)=\ln(re^{i\theta})=\frac{\ln(r^2)}{2}+i\theta$$ $$\therefore…
Henry Lee
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Complex polarization identity proof getting stuck towards the end w.r.t. the imaginary part

I'm working on a homework problem regarding the proof for the polarization identity for complex scalars. I've taken a look at another question on this community (Polarization Identity for Complex Scalars) and have tried working it out on my own, but…
Sean
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