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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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Confusion in least argument of z

$a$ and $b$ are ordered pairs of a point on the Argand plane. For tan $\theta = b/a$. It’s $z = a+ib$ My sir taught me that unique value of theta = $\tan ^{-1}|b/a|$ where theta belongs to $0,\pi/2$. Since then only it is present in the first…
Srijan
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Confusion in Argand plane

When I say a argand plane , in my textbook it says the plane having a complex number assigned to each of its point. I interpreted a meaning for this statement down below , let me know if it’s correct Second is does it mean that when I talk about a…
Srijan
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Why is the sqrt. of negative one relation important for imaginary numbers?

Why is the square root of negative one an important relation for imaginary numbers and quaternions, etc.? I mean, if we are going to have imaginary numbers, couldn't they be imaginary without needing to be defined in terms of the square root of…
CommaToast
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Argument of complex number: what am I doing wrong?

Find $\arg z$ where $$z=-\frac{i\omega-w_0}{i\omega+w_0}$$ where $\omega$ and $\omega_0$ are positive numbers. My attempt was this: $$\arg z=\pi +\arctan\left(\frac{\omega}{-\omega_0}\right)+\pi-\arctan\left(\frac{\omega}{\omega_0}\right)$$$$=2\pi…
Granger Obliviate
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For $z+\frac{1}{z}=2\cos\alpha$ calculate $z^n+\frac{1}{z^n}$ If $n$ is a positive integer

For $z+\frac{1}{z}=2\cos\alpha$ calculate $z^n+\frac{1}{z^n}$ If $n$ is a positive integer here's my first attempt $z^2-2\cos\alpha+1=0$ so we have $z=\cos\alpha\pm i\sin\alpha= e^{\pm i\alpha}$ but I thought there is a simple approach What other…
user62498
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How is $i^3$ = $\sqrt{-i}$.

I new to complex numbers.So please don’t mind. We know $i^2$ = -1. Then $i^3$ is written as $= -i$ Then does it’s steps of writing follow this: $-1^2$= $i^3$= 1 So in terms i = It says in my book = - $\sqrt{-i}$. How did we get this ? I have done…
user864449
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Prove that if $z$ in $\mathbb{C}$ is a solution of $z^5= 1$ then the conjugate of z is also a solution

I have to show that $(z^*)^5=1$, if $z^5=1$, but how do I do that? I tried to show it by inserting $x-yi$ into the equation, so $(x-yi)^5=1$. After using the binomial formula for a power of $5$, I thought I might be able to see similarities between…
Jon
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What does $2|z+3i|=|z-i|$ represent?

$z$ is a complex number. What does $2|z+3i|=|z-i|$ represent? Putting $z=x+iy$, I get a circle with radius $\frac83$. But the equation of circle is $|z-z_1|=r$. How to intuitively see that $2|z+3i|=|z-i|$ represents a circle?
aarbee
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Complex conjugate doubts

$$ \frac{\bar{z}+i}{1-i\bar{z}} $$ I wanted to know what is the conjugate to simplify the equation from above, i mean: Would it be like this: $$ \overline{(1-i\bar{z})}=1+i\bar{z} $$ Or: $$ \overline{(1-i\bar{z})}=1+iz $$ since…
James
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How to solve $\arg z^3 =0$ where $z\in\mathbb{C}$?

Can someone show me how to solve $$\arg z^3 = 0$$ I don't know even how to start because of the zero.
Konrel
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$z_1,z_2,z_3\in \mathbb{S} (1)$ form an equilateral triangle iff the sum is zero

Let $z_1,z_2,z_3$ be complex numbers of module 1. Prove that they form an equilateral triangle if, and only if, $z_1+z_2+z_3=0$. I know that if they form am equilateral triangle, then $$ \frac{z_3-z_2}{z_2-z_1}= \frac{z_1-z_3}{z_3-z_2}. $$ I…
Senna
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If $z=\cos\theta+i\sin\theta$, where $\theta$ is real, show that $\frac{1}{1-z}= \frac{1}{2}\left(1 + i\cot(\theta/2)\right)$

Question.(source) If $z=\cos\theta+i\sin\theta$, where $\theta$ is real, show that $$\frac{1}{1-z}= \frac{1}{2}\left( 1+ i\cot\frac{\theta}{2}\right)$$ I have been trying to rearrange the right-hand-side in order to make it look like the required…
newm
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Is this property of complex numbers true: $(a^* \times b)^*=(a\times b^*) $

I was wondering if this property of complex numbers is true: $(a^* \times b)^*=(a\times b^*) $ $a^*$ is the complex conjugate of a If anyone could tell me where I can find more about the properties of complex numbers, it would be helpful. Do complex…
PhysicsQuestion
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Is there a geometric proof of the de Moivre's formula?

I am trying to understand (intuitively) this formula here: $$e^{ic} = cos (c) + i.sin (c)$$ I understand the infinite sum (traditional) approach 1, but I am looking for something more geometric, maybe because of the involvement of the trig…
Bipasha
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complex numbers exponential form

I wish to show that $\cos^2(\frac{\pi}{5})+\cos^2(\frac{3\pi}{5})=\frac{3}{4}$ I know the solutions to $z^5+1=0$ are $-1$, $e^{i\frac{\pi}{5}}$, $e^{-i\frac{\pi}{5}}$, $e^{i\frac{3\pi}{5}}$, $e^{i\frac{-3\pi}{5}}$ and…
Anthony Macks
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