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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finding all complex roots of equation

let $z = 1 +i$ Find all complex solutions such that $z^2 + \bar z^2 = 0$. My working out: $z^2 = -\bar z^2 = -(1-i)^2 = 2i$ so $z^2 = 2i$ hence $r^2 = 2 \implies r = \sqrt 2$ mod: $2\theta = \frac{\pi}{2} + 2k\pi \implies \theta = \frac{\pi}{4} +…
auriga123
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Complex number questions.

I have an exam on this kind of stuff on Monday and was wondering if anyone could help me with these questions and can you tell me what this type of question is so i can go away and revise it for example(Further trig) Question 1) The cube roots of 1…
maxmitch
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Find the value of $z_1^2 +z_2^2$?

$z_1,z_2$ belongs to $\Bbb C$ (Complex Numbers) We are given that $z_1(z_1^2-3z_2^2) = 2$ and $z_2(3z_1^2-z_2^2)=1$ If $z_1^2+z_2^2$ belongs to real numbers, then find the value of $z_1^2+z_2^2$? I tried to use the algebraic identities or taking…
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Determine real and imaginray part of $\left(\frac{i+1}{|i+1|}\right)^n$

For $n\in\mathbb{N}$, determine real and imaginary part of $\left(\frac{i+1}{|i+1|}\right)^n$. Attempt: $$\left(\frac{i+1}{|i+1|}\right)^n = \frac{(i+1)^n}{|i+1|^n} =…
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Prove that $\operatorname{Re}\left(\frac{1-z^{n+1}}{1-z}\right)=\frac{1}{2}+\frac{\sin((n+\frac{1}{2})\theta)}{2\sin(\frac{\theta}{2})}$.

My objective is to prove that: $$\operatorname{Re}\left(\frac{1-z^{n+1}}{1-z}\right)=\frac{1}{2}+\frac{\sin((n+\frac{1}{2})\theta)}{2\sin(\frac{\theta}{2})}\text{ , where $$z is a complex number }.$$ I have developed a good reasoning, but I cannot…
Manatee
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What to do after the partial progress i made using the octagon properties

If $$\frac{1}{a_1-2i} ,\frac{1}{a_2-2i}, … ,\frac{1}{a_8 - 2i}$$ represents sides of a regular polygon in complex plane , find the area of this octagon where $a_1,…,a_8$ belongs to real numbers, what I did first I took help of inversion and said…
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Simplify $N^{N(N-2)}$ with $N=(1\pm i\sqrt{3})/2$

The question Derivative and inverse is accompanied with an interesting answer by the same author and a nice video called Inverse Prime equals Prime Inverse. The video ends with the formula $C = N^{N(N-2)}$, which is simplified further in the…
Han de Bruijn
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If $a,b,c,d$ are complex number with the same modulus such that $a+b+c=d$, then $d$ is equal to one of the other three.

I've come up with a solution, but it's not so insightful, so I was wondering if anyone has a cleaner not-so-brute-force solution: Equivalently (since the case where all of them ar zero is trivial), if $\alpha,\beta,\gamma$ are complex numbers in the…
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Simplifying complex numbers in digital signal processing

An exercise in my course in digital signal processing has a problem which leads to this expression. I've been calculating it for a while but I can't quite make out how they get to this answer. I have the transfer function…
Krippkrupp
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Showing $\left(\frac{z+i}{z-i}\right)^n = -1$ implies $z$ is real

I have shown the following identity: $$ \frac{1+e^{i \theta}}{1-e^{i \theta}} = \frac{1}{2}\cot\left(\frac{\theta}{2}\right) $$ And I now need to use this to show that the the equation: $$ \left(\frac{z+i}{z-i}\right)^n = -1$$ has $\Im(z) = 0$. I'm…
Wooster
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Two maximum values for $|iz+3-4i|$ what went wrong?

I need to find the maximum value of $|iz+3-4i|$ given that $|z|\leq4$ $|iz+3-4i|\leq |iz|+|3-4i|$ $|iz+3-4i|\leq 4+5$ But I could write an inequality for $|3-4i|$ whose maximum value would be 7, not 5 so I get the maximum value to be 11 when doing…
Linkin
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Is there $z \in \mathbb{C}$ with $\mathrm{Re}(z) > 0$ such that $e^{2z}+\frac{z+1}{z-1} = 0$?

Let $z$ be a complex number with positive real part. Is it possible that $e^{2z}+\frac{z+1}{z-1} = 0$?
xen
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Solving $z^4 + 4i\bar{z} = 0$ in detail

I asked this question here Solving $z^4 + 4i\bar{z} = 0$ Though I accepted the answer, later I found out that I and WA don't agree on the result. What I did: Find the absolute value of $z$ by solving $|z^4| = |-4i\bar{z}|$ Getting $|z|^4 = 4|z|$ and…
Aladin
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If you have four complex numbers and divide each of them by the others then will one of the results have to have non negative real and imaginary parts

Say you have four distinct nonzero complex numbers and you divide each one by the other three (ie you take $\frac{z_k}{z_j}$ for each $k \neq j$). Is there a way to prove that at least one of these quotients must produce a complex number with…
k12345
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I am at a predicament at solving the radius of a circle from a complex number.

The questions are to show $|z-w|=k$ defines a circle and find the centre and radius. $|z|=x+iy$ and $|w|=a+bi$. So what I did was substitute the complex numbers in so $|x+iy-a-bi|=k$ eventually getting to $(x-a)^2+(y-b)^2=k$. Using the general…
C92wQQ
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