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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Is there a formula to extract the real part of a complex number?

I was pondering the topic of complex numbers and how we just look at the them and extract the real and imaginary parts because it's easy given that we write it like 4+2i. I was wondering if there was some mathematical formula, in which we could put…
Nick the coder
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What's the magnitude of a real number?

As a student of mathematics (first year master degree) I have to admit that I'm somewhat ashamed to ask this. We know that if $z=x+iy$ is a complex number then we can identify it as $z=r\cdot\exp(i\theta)$. But what if $z$ is real - in other words…
Cynic Yahya
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find the sum to $n$ terms of the series $1+4w+9w^2+...+n^2w^{n-1}$ where $w$ is $n$th root of unity

I want to find the sum to $n$ terms of the series $$1+4w+9w^2+...+n^2w^{n-1}$$ where$w$ is $n$th root of unity. Let $$S_n = 1+4w+9w^2+...+n^2w^{n-1}$$ then $$ wS_n=w+4w^2+....+(n-1)^2w^{n-1}+n^2$$ therefore…
Rayees Ahmad
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Find all complex numbers $z$ such that $|z|=\frac{1}{|z|}=|1-z|$

Problem: Find all complex numbers $z$ such that $|z|=\frac{1}{|z|}=|1-z|$. Basically I have an idea how to solve this and I get $x=\frac12$ but how should I express it mathematically? Should I go and find $y$ also?
tot11
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Vertices of equilateral triangle inscribed in the unit circle

Prove that if $z_{1}+z_{2}+z_{3}=0$ and $|z_{1}|=|z_{2}|=|z_{3}|=1$ then the points $z_{1},z_{2},z_{3}$ are the vertices of an equilateral triangle inscribed in the unit circle $|z|=1$. My idea was the following: since the sum of the numbers has to…
Mykolas
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Solve $|z|=\arg z$

Im trying to find all set of points on the complex plane for which $|z|=\arg z$. I rewrote $|z|= \sqrt{x^2 + y^2}$ and $\arg z$ as $\tan^{-1}(y/x)$.I set them equal But im not sure what to do next.
Eamonn
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How to prove that $|z^2| = |z|^2$ where $z = a+bi$?

I just started my topic on complex numbers and I'm stuck on this question. What I have managed to get (I go wrong here, I don't know where though): $z^2 = (a+bi)^2 = a^2 + b^2$, so $|z^2| = \sqrt{(a^2 + b^2)^2 + 0^2} =…
user373679
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A Complex Inequality

I just need to show that : $$\int_0^{2\pi}\left|{\frac{i(Re^{i\theta})^\lambda}{1+Re^{i\theta}}}\right| d\theta \le \int_0^{2\pi} \frac{R^\lambda}{R-1}d\theta : 0 < \lambda <1 , R>1$$ Is there some trivial geometrical argument I don't see?
Squirtle
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How to solve this complex equation for the modulus of z?

The question is as follows: All the roots of the equation $11z^{10}+10iz^9+10iz-11=0$ lie: $\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (i=\sqrt{-1})$ (a) inside $|z|=1$ (b) on $|z|=1$ …
FreezingFire
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If $\left | \frac{z_1 + z_2}{z_1 - z_2}\right | = 1$, then what is $z_1\over z_2$

Question: $z_1,z_2$ are two complex numbers with $z_2 \neq 0$ and $z_1 \neq z_2$ and satisfying: $$\left | \frac{z_1 + z_2}{z_1 - z_2}\right | = 1$$ Then $z_1\over z_2$ is: A) Real and negative B) Real and Positive C) Purely imaginary D) None of…
Gummy bears
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Real and imaginary part functions

How can real and imaginary part of $e^{ix}$ be used in complex tasks? I am aware that sine is expressed as imaginary part and cosine as real part, but I am confused when there is some sort of operation that is applied on these, how should I proceed?…
BTestQ
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How to get the complex number out of the polar form

How does one get the complex number out of this equation? $$\Large{c = M e^{j \phi}}$$ I would like to write a function for this in C but I don't see how I can get the real and imaginary parts out of this equation to store it in a C structure.
some_id
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Find $\Im((\cos 12^\circ +i \sin 12^\circ +\cos 48^\circ+i\sin 48 ^\circ )^6)$

Find $\Im((\cos 12^\circ +i \sin 12^\circ +\cos 48^\circ+i\sin 48 ^\circ )^6)$. I've solved this problem but I think I've taken the long way to do this, so I am asking if there's some slick way to solve this. That's how I solved it: I've applied…
Mr. Y
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Condition that all 3 roots of $az^3+bz^2+cz+d=0$ have negative real part

The problem is - 'find the condition that all 3 roots of $f(z)=az^3+bz^2+cz+d=0$ have negative real part, where $z$ is a complex number'. The answer - '$a,b$, and $d$ have the same sign.' Honestly, I have no clue about how to proceed. Here is what…
GRrocks
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Modulus of complex number less then equal to 1

It $|z+1|\le 1 \text{ and } |z^2+1|\le 1$, then we have $$ |z|\le 1.$$ I wrote $z=x+iy, x,y\in \mathbb{R}$ and the inequalities from hypothesis become \begin{equation} (x+1)^2+y^2\le 1 \text{ and } (x^2-y^2+1)^2+4x^2y^2\le 1 \end{equation}...and I…
Delia
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