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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.

19229 questions
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One root of a complex number is right so why isn't the other?

I had the equation: $( 1 + 2 i ) w^2 + 4 w - ( 1 - 2 i ) = 0$ To find $w$ I did the following: $( 1 + 2 i ) w^2 + 4 w - ( 1 - 2 i ) = 0$ $[( 1 + 2 i ) w - 1 ][ w + ( 1 - 2 i )] = 0$ Which gives: $ w = 1/(1 + 2 i) = (1/5) - (2/5) i$ and $ w = -(1…
Mirte
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Show that $|z - w| \geq \big||z|-|w|\big|$

Show that $|z - w| \geq \big||z|-|w|\big|$ given $z = x + iy$ and $w = u + iv$. So far I have, $$|z - w| \geq \big||z|-|w|\big|$$ $$\sqrt{(x - u)^2 + (y - v)^2} \geq |\sqrt{x^2 + y^2} -\sqrt{u^2 + v^2}|$$ At this point I get a little 'undone'. I…
Retty
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$\lim_{z \to \exp(i\pi/3)} \frac{z^3+8}{z^4+4z+16}$

Find $$\lim_{z \to \exp(i \pi/3)} \dfrac{z^3+8}{z^4+4z+16}$$ Note that $$z=\exp(\pi i/3)=\cos(\pi/3)+i\sin(\pi/3)=\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}$$ $$z^2=\exp(2\pi…
Username Unknown
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How to find the $\max|z|$ and $\min|z|$ for $|z+(4/z)|=2$?

Let $z$ be a complex number that satisfies $|z+(4/z)|=2$ then find $\max|z|$ and $\min|z|$ . I just need to know that how I should approach this problem.
user383555
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Solve $z^2+|z|=0$

Solve for $z$: $$z^2+|z|=0$$ I found this question on my textbook but I unable to derive an answer to it as a modulus of $z$ is present. I am unable to take an approach please guide me ..thanks
Enlightened
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Complex locus of circle

In my book it is given $\left\lvert \frac{z-z_1}{z-z_2}\right\rvert = k$ where k is not equal to 1 &0. Here z represents the locus of circle . But I could not understand here what is $z_1$ & $z_2$
Koolman
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Find the absolute value of $z$ for the condition

$z$ is a complex number, such that $$\frac{z-1}{z+1}$$ is purely imaginary. Then what would be the absolute value of $z$? Options were given as follows: $|z|=0$ $|z|=1$ $|z|>1$ $|z|<1$
Bhavya
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Proving an interesting identity

Let $Q(z)=(z-\alpha_1)\cdots(z-\alpha_n)$ be a polynomial of degree $>1$ with distinct roots outside the real line. We have $$\sum_{j=1}^n \frac{1}{Q'(\alpha_j)}=0.$$ I know an interesting but indirect proof using the continuity of the Fourier…
HyJu
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Complex $2$ to the exponential form

I am trying to find the exponential form of the complex number $2$. There is no imaginary part given, just the real part $2$. Because of this I'm sort of confused how to treat this as a 'complex number'. What value does the imaginary part have? The…
TheAlPaca02
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Difficulty using Euler's relations

I am trying to solve a problem using Euler's relations, I am not sure where to start with it, here is the question : Using Euler's relations, simplify $z = e^{2+i\pi/2}$ If someone could explain to me how to do this, including the methodology from…
Aztec warrior
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Which one of $|z|, |z|^2$ is the complex modulus?

Some texts appear to define the complex modulus as the magnitude, that is $\sqrt{x^2 + y^2}$ of any complex number $z = x + iy$. Other texts seem to define the modulus as the square of that number, in other words the magnitude squared, or the…
mcandre
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Complex Numbers Questions...

I am having trouble with these problems: Find all complex numbers $z$ satisfying the equation $$\frac{z+1}{z-1} = i.$$ The value $$\left(\frac{1+\sqrt 3}{2\sqrt 2}+\frac{\sqrt 3-1}{2\sqrt 2}i\right)^{72}$$ is a positive real number. What real…
Oran2013
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Is it true that $e^{\frac{i}{2}\pi}+e^{-\frac{i}{2}\pi}=0$?

As in the title. Is it true that: $$e^{\frac{i}{2}\pi}+e^{-\frac{i}{2}\pi}=0$$ And if it indeed is, how could one prove it? Maybe it's a silly question, but - frankly I'm kinda new to complex exponentials.
user263286
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Prove identity involving powers and trigonometric functions

Need help proving that: $$(1+\cos\alpha+i\sin\alpha)^{n}= 2^{n}\cos^{n}\frac{\alpha}{2}\left(\cos\frac{n\alpha}{2}+i\sin\frac{n\alpha}{2}\right)$$
Mykolas
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Find all $z\in\mathbb{C}$ satisfying $z^2 = |z|^2$

Let $z=x+yi$, I get $y=xi$ finally, but what's the next step?
Linda dadad
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