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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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How can you find the complex roots of i?

A variation of the Root of Unity problem. I want to find all possible answers to this: $$z^n = i$$ Where $$i^2 = -1$$
Talvi Watia
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real and imaginary part in $\sin z$ where z is complex

I wanted to know, how can I determine the real and imaginary part in $\sin z$ where $z \in \Bbb{C}$? Well, this is a part of a series of questions comprising the same in $\log z$ and $\tan^{-1} z$ I was able to solve this but no idea on how to…
Shobhit
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Solve $\operatorname{Arg} (z-2) - \operatorname{Arg} (z+2) = \frac{\pi}{6}$

I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$. I know that $$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} \frac{z_1}{z_2},$$ but that's only valid when…
Alraxite
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If "$i^2 = -1$" is an imaginary number, $i$ why is there no imaginary number for "$|x| = -1$"?

The equation "$i^2 = -1$" has no real solution, so there's an entire number system of imaginary numbers that satisfy this. So why is there not imaginary numbers that satisfy "$|x| = -1$"? There aren't any numbers, real or imaginary, that satisfy…
e4494s
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Is $\sqrt{-1}$ positive or negative?

Do the concept of positive or negative make sense in this case? I remember that $\mathbb{R}^2$ has four quadrants thus ordered pairs of numbers could be $(+,+),(+,-),(-,-)(-,+)$, I presume that complex numbers have positivity or negativity for both…
Red Banana
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Solving $z^4=(2+3i)^4$

To solve the equation, I calculated right side: $z^4=(2+3i)^4=(-5+12i)^2=-119-120 i$ And then I get the correct answer: $z_k=\underbrace{\sqrt[8]{119^2+120^2}}_{\sqrt{13}} \times Cis(\cfrac{\pi+\tan^{-1}(\frac{120}{119})}{4}+\cfrac{k \pi}{2}),…
Aligator
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Sum of powers of complex numbers equals zero implies the numbers themselves are zero?

Suppose we have the complex numbers $x_1,x_2,\dots,x_N$ and that we know that $$x_1+\dots+x_N =0,$$ $$x_1^2+\dots+x_N^2 =0,$$ $$\dots$$ $$x_1^N+\dots+x_N^N =0.$$ Does this imply that all $x_m=0$? There should be enough equations to get a unique…
ECA18
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What's the modulus of $(\sqrt{11}-i)^{1000}$?

The modulus of $\sqrt{11}-i$ is $\sqrt{11+1} = \sqrt{12}$ and the modulus of that squared is $\sqrt{144}$ so is the answer $12^{500}$? Or does the pattern change in some kind of way?
Jaemin Kim
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Simple Complex Number Problem: $1 = -1$

Possible Duplicate: -1 is not 1, so where is the mistake? I'm trying to understand the exact point of failure in the following reasoning: \begin{equation*} 1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{\sqrt{-1}^2\sqrt{-1}^2} =…
Sami
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Showing that for any complex number $z$, either $ |z + 1| \geq 1/\sqrt{2} $ or $ |z^2 + 1| \geq 1 $

I was reading complex Number from A to Z There was an example which asks to proof that for any complex Number $z$ either $$ |z + 1| \geq 1/\sqrt{2} $$ or $$ |z^2 + 1| \geq 1 $$ In the book proof was by contradiction and I had no problem in…
pde
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Resolution exercises on complex numbers

How do I solve this equation in the field of complex numbers?: $$|z|^2 - z|z| + z = 0 $$ My solutions are: $$z_1 = 0$$ $$z_2 = -1$$
Vincenzo Iannucci
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Sum of a complex, finite geometric series and its identity

I have the formula for summing a finite geometric series as $$1+z+z^2\cdots +z^n = \frac{1-z^{n+1}}{1-z},$$ where $z\in\mathbb{C}$ and $n=0,1,...$. I am asked to infer the identity $$1+\cos\theta+\cos 2\theta+\cdots+\cos n\theta =…
madisonfly
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Prove that $\frac{(z_1 + z_2)(z_2 + z_3)...(z_{n-1} + z_n)(z_n + z_1)}{z_1 \cdot z_2 \cdot ... \cdot z_n}$ is real

$z_1, z_2, ... z_n$ are complex numbers such that $|z_1| = |z_2| = ... = |z_n|$. How to prove that $\frac{(z_1 + z_2)(z_2 + z_3)...(z_{n-1} + z_n)(z_n + z_1)}{z_1 \cdot z_2 \cdot ... \cdot z_n}$ is real? I've tried writing $z_1, z_2, ..., z_n$ in…
G. Abraham
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Prove that all three roots of $f(x)=0$ are zero. Also prove that $a+b+c=0$.

Let $f(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients and all real roots, also $|f(i)|=1$ where $i=\sqrt{-1}$. Prove that all three roots of $f(x)=0$ are zero. Also prove that $a+b+c=0$. As $f(i)=-i-a+ib+c=1$ and…
learner_avid
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Proving complex numbers

Let $z_1,z_2$ be two complex numbers such that $z_1 + z_2$ and $z_1\dot\ z_2$ are each negative real numbers. Prove that $z_1$ and $z_2$ must be real numbers. My attempt at a solution follows, Let $z_1 = a+bi$ and $z_2=c+di$. The hypotheses…
Q.matin
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