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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Imaginary number observation

What is the reason for this ? $ (i^4)^{\frac 12} = 1^{\frac 12} = 1$ $(i^4)^{\frac 12} = {({(i^2)}^2)}^{\frac 12} = i^2 = -1 $ Here, $$ i=\sqrt{-1}$$
user537100
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Problem of convexity in $ \mathbb{C}$

Consider the following set \begin{eqnarray*} S &=&\Big\{\lambda\in \mathbb{C};\;\exists (x_n,y_n)\in \mathbb{C}^2\,;\;\;\;|x_n+y_n|=1, \\ &&\phantom{++++++++++}\;\displaystyle\lim_{n\longrightarrow+\infty}|x_n|^2+x_n\overline{y_n}=…
Schüler
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What is the value of $\frac{1}{\sqrt{\pm i}}$

The above expression popped up in a problem I was solving. Using the relation $$i=\exp(i\pi/2)$$ I got $$\frac{1}{\sqrt{\pm i}}=\exp(\mp i\pi/4),$$ however, the reference I have gives the result as $$\frac{1}{\sqrt{\pm i}}=\exp(\pm i\pi/4).$$ Please…
Jonjo
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Compute complex number $(1 + i)^{14}$

Compute the complex numbers, writing your answers in the standard form $a + bi$ a) $(1 + i)^{14}$ So I am trying to use polar coordinates and De Moirre's formula to compute this but I ran into a problem. I started out by writing it in the form: $(r…
C.Math
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Equation with complex numbers and absolute value

I'm trying to solve this equation algebraically. $|z + i| < |z − i|$ The result should be: $y<0$ My result is: $y>ix$
gefavasej
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The nth roots of a complex number

Let's consider this following equation in $ \mathbb C $ : $$ (E) : z^6 + z^5 + z^4 + z^3 + z^2 + z +1=0 $$ for every $ K \in \{0,1,2,3,4,5,6\} $ : $$Z_k= e^{i \frac {2k\pi}7 } $$ 1) Show that: $ z_1,z_2 ,...$ and $z_6 $ are solutions of…
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Real part of a product?

We have: $$ \Re\{Ae^{i\phi}e^{i\omega}\}=\Re\{Ae^{i(\phi+\omega)}\}=A\cos(\phi+\omega) $$ But why does the following not give the same the…
JDoeDoe
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Is $Log(i^{1/3})=\frac{1}{3}Log(i)$?

The definition of general complex log for any non-zero complex number $z$ is $$Log(z)=\log|z|+i[\arg(z)+2m\pi], m\in \mathbb{Z}$$ With this, if $n\in \mathbb{N}$ then $Log(z^{1/n})=\frac{1}{n} Log(z)$ holds for all non-zero complex number $z$. I…
KON3
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Why isn't $\sqrt[i\pi]{-1} = e$?

My reasoning is through Euler's identity $$e^{i\pi} = -1$$ $$\sqrt[i\pi]{e^{i\pi}} = \sqrt[i\pi]{-1}$$ $$e = \sqrt[i\pi]{-1}$$ but Wolfram Alpha says otherwise, $\sqrt[i\pi]{-1} = -\pi$ So the way I see it, there are three options: Wolfram…
Zaya
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Find all complex numbers $z$ such that $z^4 \in \mathbb R $

Find all complex numbers $z$ such that $z^4 \in \mathbb R $ Here is my solution: We can use the exponential form of a complex number to say that $$z^4 = |z|^4 e^{4\theta i} \quad \mbox{We know that |z| is a real number}$$ $$z^4 \in \mathbb R…
Aemilius
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Points in the golden rectangle with complex numbers

so here I got a golden rectangle Let the point A be $z_0 = 0+0i$ G: $z_1 = 1+i$ J: $z_2 =\phi + (2-\phi)i$ L: $z_3 = 2\phi -2$ N: $z_4 = 1+(2\phi -3 )i$ P: $z_5 = (6-3\phi)+(2-a)i$ etc... I would like to find $\lim_{n\rightarrow\infty}z_n$ but I am…
davidaap
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cant every power of $i$ just equal 1?

I feel like there is a pretty critical flaw with my logic here but I cant figure out where it is. I don't have much knowledge about imaginary and complex numbers , but I've watched Khan academy's video on them earlier and saw this video here , where…
Dahen
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How to Simplify $\sqrt {-X} \times \sqrt {-Y}$

For example : $\sqrt {-18} \times \sqrt {-12}$ Would I start by multiplying the 2 numbers under a square root, In which case the double negatives cancel out? $ \sqrt {-18\times-12} = \sqrt {216} = 6\sqrt {6}$ Or get the i out of the square roots in…
Kareem Atef
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Area of region enclosed by the locus of a complex number

Find the area of region enclosed by the locus of $z$ given by $\arg(z-i) - \arg(z+i)= \frac{2\pi}{3}$ and imaginary axis (where $i= \sqrt {-1}$) What I did was I put $$\tan (\alpha) =z-i$$ and $$\tan (\beta) = z+i$$ and solving for…
Rohan Shinde
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What does $\arg\left(\frac{z-a}{z-b}\right)=\theta$ represent geometrically? ($a$ and $b \in \mathbb{C}$)

Taking $z=x+iy$ and solving the equation algebraically shows that it is a circle but I cannot understand intuitively why this is so. I would greatly appreciate a geometrical intuition behind this like there is with $|z-a|= r$ (postive numerical…