Mathematics Stack Exchange
Mathematics Stack Exchange

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 to find a real $d \ne 0$ such that the equation $(2-3i) z^2 -(d-1)z + 4+ 3i = 0$ has a real root?

I need to find a real value for d so that the equation $(2-3i)z^2 -(d-1)z + 4+3i= 0$ has real roots. I've tried using the discriminant formulae and then I got $$\frac {(d-1) \pm \sqrt{d^2 -2 d-67 +24i}}{4-6i} $$ Here I got stuck because I get…
Anonymous196
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Geometric description of set of points satisfying $\mathrm{Im}(z)>0$

How can we describe geometrically the set of points $z$ satisfying the condition $\mathrm{Im}(z)>0$ where $z$ is a complex number?
user54843
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Describe the domain in the plane $\mathbb{R}^2$

Please check if I do it correctly: Let $D=\{z\in\mathbb{C}: \left|\frac{2zi-1}{z+2i}\right|<1\}$. Describe $D$ in the complex plane. Naively, I just try to solve the equation, using the fact that it is not well-defined at…
User3231
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Order of operations with complex numbers

Which one is the correct way of solving this problem? $i\times\sqrt{-x} = i \times i \times \sqrt{x} = -\sqrt{x}$ or $i\times\sqrt{-x} = \sqrt{-(-x)} = \sqrt{x}$
Pedro
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Show $\sqrt[\sqrt{i}]{i}\approx 23$. (What are the other values?)

Let $\arg(z)=\theta$ and $|z|=r$. So, \begin{equation*} \begin{aligned} \sqrt[\alpha]{z} & = r^{1/\alpha}\cdot \left(\sin\left(\dfrac{\theta+2\pi k}{\alpha}\right)+i\cos\left(\dfrac{\theta+2\pi…
Username Unknown
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Proving that a three complex numbers form an equilateral triangle with vertices on unit circle

While solving questions from Mathematial Analysis by Apostol, I came across this question:- Q. Given three complex numbers $z_1$, $z_2$ and $z_3$ such that $\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = 1$ and $z_1 + z_2 + z_3 = 0$.…
Aniruddha Deshmukh
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How to calculate $\overline{\cos \phi}$

How do you calculate $\overline{\cos \phi}$? Where $\phi\in\mathbb{C}$. I try to proof that $\cos \phi \cdot \overline{\cos \phi} +\sin \phi \cdot \overline{\sin \phi}=1$?
Kasper
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Finding maximum value of $\arg(z_{1})-\arg(z_{2})$

If $|z|=2$ and $z_{1}$ and $z_{2}$ be two points on that circle. Then find maximum value of $\arg(z_{1})-\arg(z_{2})$. where $z,z_{1},z_{2}$ Represents complex number. $\theta=\arg(z)$ represents principle argument which means $\theta\in…
DXT
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Equality of two complex numbers with respect to argument

If two complex numbers are equal , is it necessary that their arguments are also equal ? Is the vice versa also true ? means that if the arguments of two complex numbers are equal , does it necessarily imply that they’re equal? According to me ,…
Aditi
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Locus corresponding to sum of two arguments in Argand diagram?

I am attempting to determine the locus of z for $\arg(z^2+1)$. I do know that since $z^2+1 = (z+i)(z-i)$, I could write $\arg(z^2+1) = \arg(z+i) + \arg(z-i)$ but I don’t really know how to figure out the locus of the points from here. How should I…
user107224
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When working with complex numbers, how can you solve for $x$ when it's inside $Re()$?

I'm trying to figure out the impedance of a capacitor. My textbook tells me the answer is $\frac{-i}{\omega C}$ and plugging that into the equation does work but I wanted to come up with that answer myself. So I wrote out the equation with what I…
Aust
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Square root of $-i$

I am stuck on the problem as the solutions given at two places do not either concur or or not understandable by me. The non concurring answer (here, pg #12) states: one root is $-\frac{1}{\sqrt2}-i\frac{1}{\sqrt2}$. I feel my approach will yield as…
jiten
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Why is $\log(-1) = 1.36 i$ instead of $\pi i$?

According to Google, $\log(-1) = 1.36 i$. Why is that true? Euler's identity says $e^{i\pi} = -1$. Taking the log on both sides gives $\log(-1) = i\pi (\neq 1.36i)$.
Albert Hendriks
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Prove that $\left(\frac {-1+\sqrt {-3}}{2}\right)^n + \left(\frac {-1-\sqrt {-3}}{2}\right)^n$ follows this pattern

Prove that: $$\left(\dfrac {-1+\sqrt {-3}}{2}\right)^n + \left(\dfrac {-1-\sqrt {-3}}{2}\right)^n=\begin{cases} 2, & \textrm { if } n \textrm { is a multiple of 3},\\ -1, & \textrm { if } n \textrm { is any other integer} \end{cases}$$ My…
pi-π
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How do I find the complex solution of this absolute value problem?

I have the equation $|z-i|^4=1$, and the solution needs to be in the form of $x+iy$ or $re^{i\theta}$. First I attempted this: $$|x+iy-i|^4=1$$ $$|x+i(y-1)|=1$$ $$|z|=\sqrt{x^2+y^2} \ \Rightarrow \sqrt{x^2-(y-1)^2}=1$$ $$x^2+(y-1)^2=1$$ This is a…
matryoshka
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