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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Proving that $x=\arccos(\sqrt{\sin\theta})$ is $\sin(x+iy)=\cos\theta+i\sin\theta$

Given: $$\sin(x+iy)=\cos\theta+i\sin\theta$$ To prove: $$x=\arccos (\sqrt{\sin\theta})$$ How I tried: $$\begin{align*} \sin x \cosh y &= \cos\theta \\ \cos x \sinh y &= \sin\theta \end{align*}$$ Then tried to use logarithm of hyperbolic complex…
Ashish
  • 39
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Simplifying $z^3 e^{i\pi/3} +1 = 0 $

Given $$z^3 e^{i\pi/3} +1 = 0 $$ We have, $ z^3 = e^{i2\pi/3} $ I get $$ e^{i\pi/3}z^3 = -1 $$ $$ z^3 = \frac{-1}{e^{i\pi/3}} $$ $$ z^3 = -e^{i2\pi/3} $$ instead May I know how did we arrived at $z^3$ ?
Gavin
  • 339
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Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real.

Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real. Your answer should be expressed as a set of the form $S = \{z \in\mathbb C : \text{conditions satisfied by }z\}$. I started solving for $((iz − 1 )/(z − i)) = \overline{…
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Solve the equation $((x+y i)-\frac{1}{x+y i})/{(2 i)} = 2$

Solve the equation $$\frac{\left((x+y i)-\frac{1}{x+y i}\right)}{2 i} = 2$$ So far, I got $(0, 2-\sqrt{3}i)$ and $(0, 2+ \sqrt{3}i)$ as solutions for $x$ and $y$. Do I require $2$ more solutions?
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Complex conjugate of polar form of $z \in \mathbb C$

Express in regular form the conjugates of $z \in c$ that satisfy $z^2 + 4i = 0$ Let $$ z=re^{i\theta} $$ Thus $z^2 = -4i = 4e^{i\frac{3\pi}2} $ $r^2=4, r=2$ $\theta_1 = \frac{3\pi}4, \theta_2 = \frac{7\pi}4 $ Thus, $z_1 =2e^{i\frac{3\pi}4}, z_2…
Gavin
  • 339
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Solve the equation $\lvert z\rvert^2z-3\overline z=0$

I was trying to solve the equation using the identities $z=x+iy$; $\overline z=x-iy$ and $\lvert z\rvert^2=x^2+y^2$ so as to get $(x^2+y^2)(x+iy)-3(x-iy)=0$, that is $x^3+x^2iy+xy^2+iy^3-3x+3iy=0$ Now, a complex number is null $\iff Re(z)=Im(z)=0$,…
colis
  • 93
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Factorial of $i$

With $i^2=-1$, what is $i!$ ? I'm not sure if it is $i$ or if it is an infinite product $i(i-1)(i-2)(i-3)(i-4)(i-5)\dots$ It would make sense either way. Is it not defined?
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How do i remove imaginary numbers from denominators?

I am calculating towards getting the value of an unknown variable B. I got stuck when i came to the point where i have B = 1 divided on a complex number. I guess its the correct way to show this as a whole complex number and not as a dividation $$…
bogen
  • 155
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Prove $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$ for two complex numbers

If $z_1 = r_1(\cos\theta_1+i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2+i\sin\theta_2)$ prove that $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$ and that $\arg\left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2$. I've done most of this, but I got…
hohner
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Division of complex (and algebraic) numbers (by "rationalizing" the denominator)

Ahlfors says that once the existence of the quotient $\frac{a}{b}$ has been proven, its value can be found by calculating $\frac{a}{b} \cdot \frac{\bar b}{\bar b}$. Why doesn't this manipulation show the existence of the quotient? $\frac{a}{b} =…
countunique
  • 2,449
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How to solve $(-19w + 93\overline w)^4=-1$

How to solve $(-19w + 93\overline w)^4=-1$ , if $w\in \mathbb C$ I really have no direction where to solve this question or at least a hint, can someone help?
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How to represent the solution of $z^{2}+2z+5=0$ in in Euler form?

I get the solution of $z^{2}+2z+5=0$ $z=-1+2i \;\; \overline {z}=-1-2i$
Tony
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raising a complex number to a high power.

we should decide whether the following claims are right or not, and explain our decision. let $w_1,w_2,w_3$ be three different roots for the equation $z^3=1$ a) $w_1^{1991} + w_2^{1991} + w_3^{1991}=1$ b) $w_1^{1991} + w_2^{1991} + w_3^{1991}=0$ I…
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Prove Complex Relationship

Question: Prove that if $z + \frac{1}{z}$ is real then either the magnitude of $z = 1$ or $z$ is real. I'm struggling with, I found that $2ab$ must be equal to $0$ but I don't see how that help. Thanks!
user2250537
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Describing a subset of the complex plane formed by z satisfying |z-i| + |z+i| = 3

I have been asked to describe the subset of the complex plane which is formed by the complex numbers z satisfying |z-i| + |z+i| = 3. It was easy to see that if the points z lie on the line segment joining the points i and -i, then z can only be at…