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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find the complex number that satisfies the following conditions

Find all values of $z \in \Bbb C$ such that: $z + \bar{z} = 18$ and $z.\bar{z} = 84$. I don't know how to get that values, someone can help me to solve this?
Tomi
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Prove that for any complex numbers: $a,b,c$ , $|a|=|b|=|c|=1$ and $abc \neq -1$ number $\frac{a+b+c+ab+ac+bc}{1+abc}$ is a real number.

Prove that for any complex numbers: $a,b,c$ , $|a|=|b|=|c|=1$ and $abc \neq -1$ number $$x = \frac{a+b+c+ab+ac+bc}{1+abc}$$ is a real number. I wanted to calculate $2 \cdot Im(x) = x- \overline x$ and show that it's equal to…
Vojtie
  • 143
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For $z \in \mathbb{C}\setminus \{0\}$, is $|1/z| = 1/|z|$?

I have been trying to prove that for $z \in \mathbb{C}$ and $z \neq 0$ that $|1/z| = 1/|z|$ but with no success. Not all the properties of real numbers can be applied to complex numbers, of course.
El antonio banderas
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Maximum value of modulus in exponential form

I am trying to find the maximum value of $\left|e^{i\theta}-2\right|+\left|e^{i\theta}+2\right|\mbox{ for }0\le\theta\le2\pi$. I can replace $e^{i\theta}$ with $a+ib$ and then proceed to get a function of $a$; Namely; $$\sqrt{5+4a}+\sqrt{5-4a}$$ I…
Anthony Macks
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Algebra exercise

If $a^4+a^3+a^2+a+1=0$ find the value of $a^{2000}+a^{2010}+1$ I got this problem in a book and tried to solve it.I multiplied with suitable powers of a and added and subtracted alternatively to get…
Math enthusiast
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Find $\displaystyle\sum_{r=0}^n\binom nr\cos(r\theta)$using$1+\cos(\theta)+i\sin(\theta)=2\cos(\frac\theta2)(\cos(\frac\theta2)+i\sin(\frac\theta2))$

I have proven the identity shown as the textbook requests, and made attempts on manipulating the identity. The most I have procured is $$(1+(\cos(\theta)+i\sin(\theta))^n=2^n\cos^n(\frac\theta2)(\cos(n\frac \theta2))$$Recognising that…
Sam
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Complex number equivalency

I'm a bit confused over the solution to a complex ode: $i\alpha y = \beta y''$ The solution to the characteristic polynomial is $r = \pm \sqrt{i\alpha/\beta}$. Somehow my book is getting the solution: $r = \pm…
tau-neutrino
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Do roots of $z=1$ add up to 0?

This problem has been treated several times on the site, but there's a minor (and possibly stupid) aspect I fail to understand. The general statement is "The sum of n-th degree complex-roots of 1 is 0". How I fail to understand it: (1) The equation…
Cure
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A method for solving cubic equation

So I'm reading Beardon's Algebra and Geometry, and in chapter on complex numbers, author gives the following method for solving cubic equation: Suppose we want to solve cubic equation $p_1(z)=0$, where $p_1(z)=z^3+az^2+bz+c$. Now…
Sarunas
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Why does it make sense to talk about the 'set of complex numbers'?

In my complex analysis course we've discussed quite a few times the idea that $\mathbb{C}$ is really 'the same thing' as $\mathbb{R}^2$ with the added complex multiplication operation. I've also read a number of the popular posts here including this…
masiewpao
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If $\frac{z-\alpha}{z+\alpha},(\alpha \in R)$ is a purely imaginary number and $|z|=2$, can we find value of $\alpha$ geometrically?

If $\dfrac{z-\alpha}{z+\alpha},(\alpha \in R)$ is a purely imaginary number and $|z|=2$, then find value of $\alpha$. Now I took $\dfrac{z-\alpha}{z+\alpha}=t$ and as t is purely imaginary, and use the fact that $t+ \bar{t}=0$ and obtained the…
Mathematics
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Proving that a complex disc $D$ is the unit disc if $D=D^2$

Let $D= \{z \mid\vert z-z_O\vert \leq r \}$ be a closed disc in the complex plane and $D^2=\{z_1z_2 \mid z_1,z_2 \in D \}$ . Prove that if $D=D^2$, $D$ is a unit disc and $z_O=0$. I really have no idea how to start this problem, $z_1z_2$ might…
John
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A general proof of $f\left(\bar{z}\right)=\overline{f\left(z\right)}$

As a school student I have seen a striking property of functions . $$f\left(\bar{z}\right)=\overline{f\left(z\right)}$$ Where $z$ is a complex number and $\bar{z}$ it's complex conjugate. For eg: $z=x+iy$ then $\bar{z}=x-iy$ where $x,y \in…
hrkrshnn
  • 6,287
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What is $\frac{d}{dx}x^i$?

What is $\frac{d}{dx}x^i$ where $i=\sqrt{-1}$? Does the proof as for real indices apply in the same way here?
Amin Parvaresh
  • 439
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Prove $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}=2$ if $\frac{1}{a+w}+\frac{1}{b+w}+\frac{1}{c+w}+\frac{1}{d+w}=\frac{2}{w}$

If $\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}+\frac{1}{d+\omega}=\frac{2}{\omega}$, where $a,b,c,d\in\mathcal{R}$ and $\omega$ is a non-real cube root of unity, then prove that…
Sooraj S
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