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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Are all interior points limit points in complex analysis?

The definition of limit point z for a set S in complex analysis states that there exists at least one point of the set inside the deleted neighbourhood of z.Does this imply that all interior points of the set are limit points? Edit: I had written…
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Question about asymptotic behaviour of argument of complex number

Let $r\in\mathbb{R}^{+}$, $\theta\in\mathbb{R}$ and $z_{0}\in\mathbb{C}$. Does $\arg{(r\text{e}^{i\theta}+z_{0})}\longrightarrow\theta$ as $r\longrightarrow\infty$?
BasicUser
  • 907
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Factorisation over $\Bbb C$ of $z^2 -10z+30$

I haven't done these questions in a long time, so I am just wondering if my approach and answer is correct. When asked to $z^2-10z+30$ over $\Bbb C$, My approach: I complete the square of the equation, and would get $(z-5+\sqrt 5)(z-5-\sqrt 5)$ My…
user256922
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Prove that for any integer $m>1$, $\ \ (z+a)^{2m}-(z-a)^{2m}=4maz\prod_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)]$.

Prove that for any integer $m>1$, $$(z+a)^{2m}-(z-a)^{2m}=4maz\prod\limits_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)].$$ This how tried to do it: Expand the two brackets on the right hand side and end up with, after cancelling the $a^{2n}$ terms…
Jean
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Definitions for complex numbers

I could not find this question anywhere else. But why are addition, subtraction, division, and other operations defines they are in complex numbers? Could they defined as something else?
Azby9021
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solving difficult complex number proving

if $z= x+iy$ where $y \neq 0$ and $1+z^2 \neq 0$, show that the number $w= z/(1+z^2)$ is real only if $|z|=1$ solution : $$1+z^2 = 1+ x^2 - y^2 +2xyi$$ $$(1+ x^2 - y^2 +2xyi)(1+ x^2 - y^2 -2xyi)=(1+ x^2 - y^2)^2 - (2xyi)^2$$ real component $$(1+…
lolisme
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Show that $\cos(6x)= 32\cos^6x -48\cos^4x +18\cos^2x -1$

After writing down $\cos6x$= $Re (\cos x + i\sin x)^6$, I used the binomial theorem to expand the expression. Very soon it got really tedious and after trying $5$ times, fruitlessly, to arrive at the given expression, I gave up. Is there a shorter…
user140161
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How does uncertainty propagate through an equation with complex variables?

I am trying to understand how uncertainty propagates through systems with complex variables. Given the general error propagation formula $$ \sigma^2_u = \left(\frac{\partial u}{\partial x}\right)^2\sigma_x^2 + \left(\frac{\partial u}{\partial…
Will
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Why is there only one type of imaginary number?

We've defined the square root of -1 as an imaginary number i (or j, if you're a physicist). Is there any reason why we can't/haven't made other systems of imaginary numbers for other "impossible" operations, like log base 1 or dividing by zero?
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how can I simplify this $\sqrt{i}+\sqrt{2i}+\sqrt{3i}$

Is there an easy way to simplify the $$\sqrt{i}+\sqrt{2i}+\sqrt{3i}$$
E.H.E
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Find all values of $\sqrt[4]{-1+i}$

Okay. I know how to solve for all values of $\sqrt{-1} $ but $\sqrt{-1+\iota} $ confuses me a bit. I got the value of r to be $\sqrt 2 $ I ended up with this: $ z_k = \sqrt[8]{2}\;cos(\frac{2k+1}{4}\pi)+ i\sqrt[8]{2}\;cos(\frac{2k+1}{4}\pi)$ I used…
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The Fundamental Theorem of Algebra and Complex Numbers

We had a quiz recently in a linear algebra course, and one of the true/false question states that The Fundamental Theorem of Algebra asserts that addition, subtraction, multiplication and division for real numbers can be carried over to complex…
Ella
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Find the root of a complex number

Find all complex numbers $z$ such that $$z^2=12−16i,$$ and give your answer in the form $a+bi$. We set $$z= a+bi,$$ thus, $$z^2 = (a^2 - b^2) + (2ab)i.$$ Equating both $z^2$ we have $$ a^2 - b^2 = 12\text{ and }ab = -8.$$ I am told that I can…
amundi12
  • 509
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Complex number weird sum.

Question given in red. My working in black. $$\color{red}{\sum_{r=0}^{50}z^r=0}\iff z_k=\exp\underbrace{\left(\frac{{\cal i}2\pi k}{51}\right)}_{\theta_k},k\in\{n\mid n\le50,n\in\mathbb N\},i:=\sqrt{-1}$$ I proceed:…
RE60K
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Complex number, how to solve

Calculate i)$(1+i)^i$ ii)$(-1)^{\frac{1}{\pi}}$ I did i)$(1+i)=\sqrt{2}e^{i\frac{\pi}{4}}$. Knowing that if $z$ and $c$ are complex numbers $z^c=e^{c\log…
Roland
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