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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sum of complex number of different magnitude

Is there a systematic way to express the sum of two complex numbers of different magnitude (given in the exponential form), i.e find its magnitude and its argument expressed in terms of those of the initial numbers?
aflous
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Locus of Complex Number

Would be great to get your help in finding the locus of this complex number $z$: $|z-z_1|+\sin \alpha|z-z_2|=\sin \theta$ From this question I proceed to a refined one- What would $$|z-z_1|+2|z-z_2|=k$$ represent?
DeepK
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How to expand out negative powers for complex numbers

I have the following expansions but I don't know how my teacher gets them. Apparently there is a formula for it (though the guy who told me didn't know it well), but I cannot find it in my notes. For example, let $z$ be a complex number.…
s1047857
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Proof using complex numbers

Prove that $\left|\dfrac{z-w}{1-\bar{z}w}\right| = 1$ where $\bar{z}$ is conjugate of $z$ and $\bar{z}w\ne 1$ if either $|z| = 1$ or $|w| = 1$. I used $|c_1/c_2| = |c_1|/|c_2|$ and multiply out with $z = x + iy$ and $ = a+ib$ but I am getting stuck…
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Explain why there are two complex numbers z such that $|z| = 1$ and that satisfy the equation $|z| = |z-1|.$

I must find both such complex solutions and express them in Euler form and usual form. So it's been a while since I've touched the imaginary/real plane. However, from what I remember, $z = a + bi$. If $|z| = 1$ we can assign specific values to a…
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locus of a complex number in the form |z-a|=k|z-b|

how does one find the locus of a set of complex numbers defined in the form |z-a|=k|z-b| for example in the question (CIE ALEVELS MATHS/9709/May-June 2013/Paper 33/Question 7) below we have to find the locus of the set of complex numbers…
phantom.omaga
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Why is $\Re(e^{(\lambda e^{i\theta}-\lambda)})\neq e^{(\lambda(\Re (e^{i\theta}-1)))}$?

Why is $\Re(e^{(\lambda e^{i\theta}-\lambda)})\neq e^{(\lambda(\Re (e^{i\theta}-1)))}$ ? I always thought, that $\Re$ is linear, but if I compute LHS; $\Re(e^{(\lambda…
OBDA
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How to solve $2 - \sqrt{3}i + z^{3}=0 $?

I have tried to solve this "problem" so that i have $z$ switched with $a+bi$. Then after some hours of solving this riddle i have gain an enormous numbers of lines but with no solution. $2 - \sqrt{3}i + z^{3}=0 $ $z=a+bi $ I want to know how much…
Primus
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Complex number with real part as 0

it is kinda of awkward, but is Equation: 0+3i=0? Or it simply means that it is imaginary number?
user146148
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Finding an expression for the complex number Z^-1

So I want to find out an expression to express: $$z^{-1}$$ I know the answer is: $$z^{-1} = \frac{x-iy}{x^2+y^2}$$ But how would I go about proving this/the steps to this?
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Every imaginary number is also a complex number?

How is it possible that every imaginary number (multiple of i ) is also a complex number?
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Explanation on how this solves for the real part of a complex fraction

I'm trying to solve the following fraction to find out what omega $\omega$ will leave me with only the real parts, assuming I know the values L, C, and R. $z = \dfrac{ \dfrac{L}{C} + \jmath\omega RL}{R + \jmath(\omega L - \dfrac{\omega}{\omega…
user1675074
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What is $\tilde{\Bbb{C}}$

$\tilde{\Bbb{C}}$ was defined in the following manner $\tilde{\Bbb{C}} = \Bbb{R} \cdot 1 + \Bbb{R} \cdot e$ with $1 \cdot 1 = 1, 1 \cdot e = e \cdot 1, e \cdot e = 1$ Could you elaborate more on this please? Or simply what is its name?
george
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Determining all complex Z in the equation

Let $n \in \mathbb N$. Determine all complex numbers $z \in \mathbb C $ such that $ |z| ^{n-2} = 1.$ How would i begin this question, thanks!
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$z$ is a complex number, what is the solution of $z^n=-1$ for $n$ an interger and $\geq 2$

$z$ is a complex number, what is the solution of $z^n=-1$? For $n$ an interger and $\geq 2$. How can we expand $z^n$? Thanks in advance.
Ian
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