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
13
votes
2 answers

Complex numbers, polynomials

Let $a$ be complex number such that $a^5 + a + 1 = 0$. What are possible values of $a^2(a - 1)$? I have tried to find $a$. Is there any way to find it?
chaos
  • 599
12
votes
3 answers

Prove that the complex expression is real

Let $|z_1|=\dots=|z_n|=1$ on the complex plane. Prove that: $$ \left(1+\frac{z_2}{z_1}\right) \left(1+\frac{z_3}{z_2}\right) \dots \left(1+\frac{z_n}{z_{n-1}}\right) \left(1+\frac{z_1}{z_n}\right) \in\mathbb{R} $$ I have tried induction and writing…
12
votes
2 answers

How to prove sum of powers property of roots of unity?

We know that $1+\alpha_1+\alpha_2+...+\alpha_{n-1}=0$ where $\alpha_i$ are the roots of $z^n=1$. How can I prove that: $$1+\sum_{i=1}^{n-1}\alpha_i^m=\begin{cases}0\quad m\in Z,m\not\equiv0\pmod n\\n\quad m\in Z,m\equiv0\pmod n\\\end{cases}$$
RE60K
  • 17,716
11
votes
3 answers

Equation of a circle in a complex plane

In the book I'm reading, one of the exercises starts with mentioning that $\left|\frac{z-z_1}{z-z_2}\right|=c$, where $c\neq 1$ is a constant, is an equation of the circle. It seems clear to me that $|z-z_1|=c$ is an equation of the circle in a…
11
votes
9 answers

Solve the equation $z^3=z+\overline{z}$

I have been trying to solve an equation $z^3=z+\overline{z}$, where $\overline{z}=a-bi$ if $z=a+bi$. But I cant find any clues on how to move forward on that one. Please help.
Viktoria
  • 119
11
votes
2 answers

Principal $n$th root of a complex number

This is really two questions. Is there a definition of the principal $n$th root of a complex number? I can't find it anywhere. Presumably, the usual definition is $[r\exp(i\theta)]^{1/n} = r^{1/n}\exp(i\theta/n)$ for $\theta \in [0,2\pi)$, but I…
goblin GONE
  • 67,744
11
votes
6 answers

Definition of multiplication of complex numbers

I know that given two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, the multiplication of these two numbers is defined as $$ z_1*z_2 = (ac - bd) + i(ad + cb) $$ I also know that I can easily derive this formula by applying the distributive…
user222753
10
votes
3 answers

Solve complex equation $z^3 = i$

I have this $z^3 = i$ complex equation to solve. I begin with rewriting the complex equation to $a+bi$ format. 1 $z^3 = i = 0 + i$ 2 Calculate the distance $r = \sqrt{0^2 + 1^2} = 1$ 3 The angle is $\cos \frac{0}{1}$ and $\sin \frac{1}{1}$, that…
S4M1R
  • 701
10
votes
1 answer

Interpret to a complex plane!

$\newcommand{\Re}{\operatorname{Re}}\newcommand{\Im}{\operatorname{Im}}$The question is: Interpret $$ \Re z + \Im z = 1 $$ geometrically in the complex plane. Writing $z = x + yi$, the condition $\Re z + \Im z = 1$ becomes $x + y = 1$. Now should…
Ara
  • 461
10
votes
3 answers

Is this a valid method of finding magnitude of complex fraction

If I have a complex fraction $\dfrac{a+bi}{c+di}$ and I want the magnitude, then will it be $\left|\dfrac{a+bi}{c+di}\right|=\dfrac{|a+bi|}{|c+di|}$? Scratch that ... I just found the answer on another page; however, I'm still unclear why it's true?
Daniel B.
  • 601
10
votes
1 answer

i^i^i^i^... Is there a pattern?

I was messing around with $i$ and I (haha) noticed that certain progressions arise when I keep on raising $i$ to $i$ to $i$ and so forth. Though, I am not really quite sure what is going on (and I don't have time to explore further). In other words,…
Just_a_fool
  • 2,256
10
votes
5 answers

Alternative to imaginary numbers?

In this video, starting at 3:45 the professor says There are some superb papers written that discount the idea that we should ever use j (imaginary unit) on the grounds that it conceals some structure that we can explain by other means. What is…
picakhu
  • 4,906
10
votes
2 answers

Is taking modulus on both sides of an equation valid?

This might look like a copy of another question, but what I'm about to propose here is new. There's this question, Find the least positive integral value of n for which $(\frac{1+i}{1-i})^n = 1$ While solving, if we multiply what is within the…
10
votes
2 answers

Why should I use the binomial theorem to solve $(1+i)^8$?

I have recently started on Edexcel AS and A Level Modular Mathematics FP$1$. We are tasked to solve $(1+i)^8$ and beside the question they have given the hint to use the binomial theorem. However I solved it in a much easier way: …
10
votes
3 answers

Is there a problem in defining a complex number by $ z = x+iy$?

The field $\mathbb{C} $ of complex numbers is well-defined by the Hamilton axioms of addition and product between complex numbers, i.e., a complex number $z$ is a ordered pair of real numbers $(x,y)$, which satisfy the following operations $+$ and…
Pierre
  • 349