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
2
votes
2 answers

How to compute $i^i$?

My question is a bit straightforward. How can I solve $i^i$? Do I have to work it out based on polar form of complex number? Even that doesn't seem to help!!
Avery
  • 127
2
votes
1 answer

What is the principal argument of $-5-5i$?

When i calculated it by $\tan^{-1}\dfrac{y}{x}$, I got $\dfrac{\pi}{4}$, then i added $\pi$ to make it in the right quadrant, so my final answer is $\dfrac{5\pi}{4}$ However, the correct answer is $-\dfrac{3\pi}{4}$...why is that?
Mohdak
  • 121
2
votes
4 answers

Where is the world of Imaginary numbers?

Complex numbers have two parts, Real and Imaginary parts. Real world is base of Real numbers. but where is (or what is) the world of Imaginary numbers?
Ahmad
  • 39
2
votes
4 answers

find two numbers that add up to 8 and multiply to 20

Find two numbers that add up to 8 and multiply to 20. Only the complex number "i" (imaginary number) is allowed other than the real numbers. But you do not necessarily have to use "i" if it is unnecessary. Just putting out the possibilities I've…
user112533
  • 155
  • 2
  • 3
  • 10
2
votes
3 answers

Find all the solutions of $e^z=i$

Find all the solutions of $e^z=i$ I don't know where to start. I want to do $z=\ln(i)$, but have no idea where that would lead me. Thanks in advance.
Mr Croutini
  • 1,158
2
votes
2 answers

Is there an impossible to solve equality in $\mathbb{C}$?

exactly like how ${x^2=-1}$ is impossible in $\mathbb{R}$ is there any equation that is impossible in $\mathbb{C}$ and how to deal with ?
Mostafa 36a2
  • 153
  • 1
  • 9
2
votes
6 answers

Is there a number that ${x^2=i}$ where ${i^2=-1}$

Imagine that there is a number ${x^2=i}$ where ${i^2=-1}$ where should it be as it is not in ${C}$ , and how to draw it , and where ?
Mostafa 36a2
  • 153
  • 1
  • 9
2
votes
1 answer

Sketch the set of points in the complex plane satisfying

$|z+1|-|z-1|=4$ Solution: Using reverse triangle inequality: $||z+1|-|z-1||\le|(z+1)-(z-1)|=|2|=2$ Thus, there are no points, because $|z+1|-|z-1|$ can't be 4. Is this correct?
user102417
  • 111
2
votes
3 answers

solving equations with complex numbers?

How would I find all of the complex roots to $(x + yi)^5 + 16(x - yi) = 0$? I am lost as to where to start. Binomial expansion seems like it would take too long. I'm guessing it has to do with conversion to polar form, but without being given the…
ben
  • 31
  • 1
  • 3
2
votes
1 answer

Complex roots on a parallel to a bisector

This is from a collection book of problems on complex variables (Volkovyskii, Lunts, Aramanovich). I don't know how to tackle it without involving heavy unpromising calculations: Prove that both values of $\sqrt{z^2-1}$ lie on the straight line…
Weltschmerz
  • 6,875
2
votes
1 answer

$ |1-z \bar w|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$

Given $z,w \in \mathbb C$, show that $$ |1-z \bar w|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$$ I think I need to use the equation $|z|^2 = z \bar z$ Thanks for any help.
user110441
2
votes
3 answers

Imaginary numbers and polynomials question

I have a task which I do not understand: Consider $w = \frac{z}{z^2+1}$ where $z = x + iy$, $y \not= 0$ and $z^2 + 1 \not= 0$. Given that Im $w = 0$, show that $| z | = 1$. Partial solution (thanks to @ABC and @aranya): If I substitute $z$ with $x +…
ScientistAnja
  • 21
2
votes
2 answers

Find solutions for $z^3 + 4\sqrt{3} -4i = 0$ ($n$th roots of complex numbers)

I'm not sure if I have done the last parts of this right. $z^3=-4\sqrt3+4i$ Let $z^3=w$ So $w=-4\sqrt3+4i = 0$ $|w| = \sqrt{-4\sqrt3^2+4^2}$ $=8$ I have drawn the points $4$ and $-4\sqrt3$ on a complex plane and found out the angle $θ$ (which is…
user88720
  • 497
2
votes
4 answers

Why do we write complex numbers as $e^{i \theta} $?

Why can't we write complex numbers as $2^{i \theta} $ or $-40^{i \theta} $? Why does it have to be $e$?
usual me
  • 783
  • 5
  • 15
2
votes
2 answers

Solve $(z+ \bar{z}=|z^2+1|)$

Solve this equation: $$z+ \bar{z}=|z^2+1|$$ I tried the following. $$x+iy+x-iy=|z^2+1|$$ $$2x=|z^2+1|$$ $$x=(|z^2+1|)/2$$ and I came to a dead end. How can I proceed?
malloc
  • 407
1 2 3
…
99 100