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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Determining $\arg(u-\sqrt{3})$ in exact form.

Let $u$ be the solution to $z^5=-9\sqrt{3}i$ so that $\frac{\pi}{2}\le arg(u) \le \pi$. Determine $\arg(u-\sqrt{3})$ in exact form. How would I go about completing this question?
Si S.
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What is the polar form of $ z = 1- \sin (\alpha) + i \cos (\alpha) $?

How do I change $ z = 1- \sin (\alpha) + i \cos (\alpha) $ to polar? I got $r = (2(1-\sin(\alpha))^{\frac{1}{2}} $. I have problems with the exponential part. What should I do now?
Kaliche
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Complex no. problem (involving Euler's identity)

Here's the problem: $$ e^{2 m i \cot ^{-1} p \cdot\left(\frac{p i+1}{p i-1}\right)^{m}}= $$ I got this from internet. I tried substituting $p=\cot(∆/2)$, managed to solve a bit of the problem but couldn't do it further. Note: There isn't any remark…
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Solve a complex equation in R

Solve $x=1+w^x + w^{2x}$ Where $w$ ,$w^2$ are cube roots of the unit and $x$ is pure real number My attempt :- let $w = \cos(120) + i \sin(120)$ Then $x = 1+\cos(120x) + \cos(240x) + [\sin(120x) + \sin(240x) ]i$ $x = 1+\cos(120x) + \cos(240x)…
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Complex number from a region

After sketching this region$$-4\sqrt{2}\le\operatorname{Re} z\le0, $$ $$\operatorname{Im} z\ge0,$$ $$|z|\ge8$$ I need the polar form and the rectangular form of the complex number that is in the region which has the smallest possible imaginary…
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Selecting the correct root of a complex number

I'm out of my league. Apologies in advance for the poorly formulated question; I'd be happy to take advice on how to express myself more clearly. I'm having a very difficult time trying to select the correct root when working with complex numbers.…
Ian
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Complex number - arctan

$$z=-2+2\sqrt{3}i\implies x=-2, y=2\sqrt3$$ $$r=\sqrt{x^2+y^2}=\sqrt{4+12}=4$$ $$\text{Angle}=\arctan\left(\frac{2\sqrt3}{-2}\right)+\pi=\frac{2\pi}{3}=120^\circ$$ 1) May I know how $\arctan\left(\dfrac{2\sqrt3}{-2}\right)+\pi$ turns into…
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How to calculate an imaginary number to high exponent?

How can I calculate something like $(i+1)^{33}$ or similar high exponent without the use of a calculator?
Straightfw
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Complex Geometry about intersection between median and circle

Point $A,B,C$, all different, lies in unit circle on complex plane. $X$ is midpoint of $BC$. $AX$ cuts the circle again at point $Y \neq A$. Express $y$ in terms of $a,b,c$. (Point $A$ is denoted by the complex number $a$, and similar notation…
Adola
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Is there an unspoken loophole in complex number theory?

I'm a high school student, who started learning about complex numbers literally today. Sorry if I say anything naive. My question is this: sqrt(-1) x sqrt(-1) = sqrt(1) = 1 [simple algebraic manipulation] however i = sqrt(-1), and is…
rohiths18
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Evaluate $\prod_{k=1}^{k=2n}[1+w^k]$, when $\omega$ is a cube root of unity

My work: I wrote it in this form $[(1+w)(1+w^2)(1+w^3)]^k$ where k is the number of repeat of this triplet in a product. Every three things consecutive factors of $(1+w) ( 1+w^2) ( 1+w^3)$ count as one power to the exponent. I wrote $ k =…
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Where did I go wrong in showing that $\left(1+\frac{ix}{n}\right)^n$ goes to $e^{ix}$ as n goes to infinity

I was curious in showing that $\left(1+\frac{ix}{n}\right)^n$ goes to $e^{ix}$ as n goes to infinity. I believe this is right as included it is included numerous times in here: Proof of Euler's formula that doesn't use differentiation?. I have:…
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Calculate Fourier Transform

I have question for which i am stuck at Calculate the Fourier Transform of $f(x) = e^-((x-1)^2)/4$ I am not sure if my answer is correct since it makes no sense...
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cos(2i) into exponential form

I need to put cos(2i) into exponential form. So far I have $$\frac {e^{i(2i)}+e^{-i(2i)}} {2} $$ $$\frac {e^{-2}+e^{2}} {2}$$ I am not sure if this is correct or where to go from here
Kat L
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$c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$

Let $c$ is complex-number satisfying : $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$ So, how could i get $(3c^{100}+\frac{2}{c^{100}}+1)(c^{100}+\frac{2}{c^{100}}+3)$ ?
freeze
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