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
1
vote
2 answers

How to find $\arg(z)$ and $|z|$?

How to find $|z|$ and $\arg (z)$ $z$ is complex number and $z$ is defined by $$z=\left(\cos\frac\pi5+i\sin\frac\pi5\right)^{15}\cdot(3-3i)^{20}$$ I`ve tried to behave it like $$e^{i15\frac\pi5}\cdot e^{i20\frac\pi4}$$ and got in result =…
Ulan Duishenaliev
  • 19
1
vote
2 answers

Complex number inequality verification

If $|z| \leq 1$ and $|w| \leq 1$, show that $$ |z-w|^{2} \leq(|z|-|w|)^{2}+(\operatorname{Arg} z-\operatorname{Arg} w)^{2}. $$ I would like to know the meaning of the inequality here; does it mean that for any specific $z$ and $w$ having modulus…
Paracetamol
  • 624
1
vote
0 answers

Confusion over Complex Plane Subset Sketch

The question of concern is the following. Sketch the subset $S$ where $S={z: \frac{-\pi}{4}\le Arg(z-(-2+i)) \le \frac{3\pi}{4}} $. The provided answer is the following. This is where I am confused. Why is there a dotted boundary line here? Would…
Swiss Gnome
  • 189
1
vote
2 answers

The solution to $z^2+(1-i)z-1=0$?

I am teaching myself complex numbers, yet I can't solve this exercise: Find all complex number solutions to $z^2+(1-i)z-1=0,$ and provide them in standard $z=a+bi$ format. I've tried using the classic formula $\dfrac{-b \pm \sqrt{D} }{2a},$ and…
Arnór Bjarki Svarfdal
  • 39
1
vote
1 answer

I've solved it by finding the value of Z first. So is there any other ways that can solve it without having to find the value of Z?

Given the $z+ \frac{ 1 }{ z } =i$ Calculate the $z ^ { n } + \frac{ 1 }{ z ^ { n } }$
Ray Akabe
  • 21
1
vote
2 answers

Determine all complex number z which satistfy the equation $|z + 1| = |z - i|$

I am trying to solve the aforementioned problem but have previously only worked with complex number inequalities where the resulting set is (the inside/outside of) a circle. I am asked to determine all complex numbers z which satisfy the…
Hector Auvinen
  • 115
1
vote
3 answers

How to show $\frac{1}{\omega-3\omega^2}=\frac{4\omega^2+3}{13}$.

I was solving a cubic equation with two different methods and one of the root I got with first method is $\frac{1}{\omega-3\omega^2}$ and with second method I got $\frac{4\omega^2+3}{13}$. $\omega$ is the cube root of unity If I set one equals…
user6262
  • 1,098
  • 5
  • 18
1
vote
2 answers

Transformations of the Argand plane described in geometrical terms

So far I've learned that the transformation of $\;f(z)=\overline z\;$ is the same as a reflection in the x-axis the transformation of $f(z)=iz$ is the same as an anti-clockwise rotation of 90 deg about the origin and I'm quite happy with how to do…
user1020304
  • 11
1
vote
3 answers

How can one find the square roots of a complex number (say -7+13i, or more generally a+bi) using de Moivre's formula?

How can one find the square roots of a complex number (say $-7+13i$, or more generally $a+bi$) using de Moivre's formula? I've begun a course on complex variables recently and one of the first things we've covered is complex numbers in polar form,…
user944610
1
vote
4 answers

Using De Moivre's theorem to solve $(z−3+2i)^4 = z^4$

What are all the solutions to: $(z−3+2i)^4 = z^4$? I know I have to use De Moivre's theorem which states: $$(\cos\theta + i\sin\theta)^n=\cos\theta n + i\sin\theta n$$
Nicole
  • 53
1
vote
1 answer

is this method for solving a complex equation correct?

I need to find roots of this complex equation: $$\left(|z|^2 + 3 |z|\right)^2 \left(z^8 - 1\right)^3 = 0$$ (for the moment, let's focus only on the first parenthesis) $$\left(|z|^2 + 3 |z|\right)^2 = 0$$ My attempt: I collected $|z|$, because it's…
Gabriel Burzacchini
  • 679
1
vote
0 answers

sum of a set of complex numbers

Let $X=\{x_1,\ldots,x_n\}$ and $Y=\{y_1,\ldots,y_n\}$ be two (equal size) sets of the complex numbers. If $\sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k$, for all $k\geq 1$, then is this true that $X=Y$? How about if $x_i$'s and $y_i$'s are roots of unity?
user53093
  • 21
1
vote
3 answers

Expressing complex numbers in form $a+bi$

I know that we should express complex numbers generally in the standard form $$a+bi:a,b\in\mathbb{R}$$ Like $4+5i-2=2+5i$. But how do I express complex numbers like $e^{-i\pi/2}$ or $i+e^{2\pi i}$? Thank you!
user84905
  • 11
1
vote
0 answers

Describe geometrically sets of the following form

I have to describe geometrically sets of the form: $\{z\in\mathbb{C} : Az \bar z +Bz + \bar B\bar z + C=0\}$, $A,C\in\mathbb{R}, B\in\mathbb{C}$ . Check also for $A=0$. (I have a feeling that it is supposed to describe circles in the complex plane,…
Nikolaos Kokkalis
  • 11
1
vote
2 answers

Why will $\lim_{z\to \infty} |e^{z^3}|=0$ only if $\mathrm{Re}(z^3) < 0$?

I have come across a question in a textbook "Sketch the region in an Argand diagram where $\lim_{z\to \infty} |e^{z^3}|=0$ The solution in the book begins "This will only be satisfied if $\mathrm{Re}(z^3)$ is negative", without any justification as…
Tharaib
  • 113
1 2 3
…
99 100