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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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Why does this method for solving equations with complex number roots not always work?

Here's the question: $1+3i$ is a root of the cubic $z^3+6z+20=0$. Identify the other two roots. Obviously the conjugate complex number is another root so one is $1-3i$. So I use the general formula $z^2-(\alpha+\beta)+\alpha\beta=0$, where $\alpha$…
ODP
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Find the positions relative to a set.

Find positions of $\dfrac{1}z$, $\bar{z}$ and $-z$ on a set $|z|=2^2$? The position of $\dfrac{1}z$ will be $x^2+y^2=\dfrac12$ and $\bar{z}$ will be $x^2+y^2=2$ (which have no effect) and $-z=|-z|=|z|$ (which also have no effect). Can someone…
Sam123
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why sin of something is always less than that something

I was working on a proof in complex numbers which goes like this: Let $z_1=r_1\left(\cos \theta_1+i \sin \dot{\theta}_1\right)$ and $z_2=r_2\left(\cos \theta_2+i \sin \theta_2\right)$. Then $\left|z_1\right|=r_1,\left|z_2\right|=r_2, \arg…
Manohar
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For the circle $|z - 1|=1$ show that $z=1+\cos\theta-i\sin\theta$, where $-\pi<\theta\le\pi$

The question is: a). For the circle $|z - 1|=1$, show that $z=1+\cos\theta+i\sin\theta$, where $-\pi<\theta\le \pi$ b). Deduce that the point representing the complex number $\frac{1}{z}$ describes a straight line. My attempts: a). I simply thought…
Nicko
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Square root question complex

So I know how to solve $x^2= 1-i$. We would get two roots: $2^{1/4}e^{-i\pi/8}$ and $2^{1/4}e^{7i\pi/8}$. The question I am actually asked is what is $\sqrt{1-i}$. I assume both roots would be the answer since there is no "positive" root concept…
HFM
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Having $z,w,a,b\in \mathit C$ Show that the segment $zw$ is parallel to $ab$ iff: $\frac{z-w}{\bar z-\bar w}=\frac{a-b}{\bar a-\bar b}$

So the excercise says: Being $z,w,a,b\in \mathit C$ such that $z\neq w$ and $a\neq b$. Show that the segment $zw$ is parallel to $ab$ iff: $$\frac{z-w}{\bar z-\bar w}=\frac{a-b}{\bar a-\bar b}$$ My thoughts on this are that I can express any of them…
Roma_Rayado
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Finding the other solution of a quadratic equation with complex roots

I am having trouble with the following problem: The equation $ax^2 + bx + c = 0$ has real coefficients and one solution $x_1 = 2 + i$. What can be said about the other solution $x_2$? (a) $x_2 = x_1$ (b) $x_2$ is a real number (c) $x_2 = 2-i$ (d)…
Bishop_1
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Finding the smallest possible value of $$ given three values of $^{1/n}$

I have the arguments and modulus of some values of $^{1/},$ where $$ is a complex number and $n$ is a positive integer. The values of $x^{1/n}$ I have are called $x_1,x_2,$ and $x_3.$ I want to find the smallest possible value of $n,$ and also use…
avocado_pit
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If $|z-5\sqrt3-5i|=5$, then find $a+b+c+d$ such that $\left|\frac1z-\frac1{a\sqrt{b}}+\frac{i}{c}\right|=\frac1d$.

If $|z-5\sqrt3-5i|=5$, then find $a+b+c+d$ such that $\left|\frac1z-\frac1{a\sqrt{b}}+\frac{i}{c}\right|=\frac1d$. I try to make the second equation look like the first one. Multiplying both sides by $acz\sqrt{b}$, we get $$|ac\sqrt{b} - cz +…
py_math
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How to simplify the solution of $(z-1/z)^{10} = 1$ given $w^{10} = 1$ is solved

The problem requires us to solve $w^{10} = 1$ beforehand ($w=cis(\frac{2k\pi}{5})$ for $k = 0,1,2,3,\ldots, 9)$ to solve $(\frac{z-1}{z})^{10}$ = 1 and show it is z = $\frac{1}{2}$ $(1+i\cot(\frac{r\pi}{10}))$ for r = 1,2...9$. Using the prior steps…
Ophe
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Map of $z$ to $z^4$ given $\Re(z) + \Im(z) ≥0$ and $\Re(z)< 0$

I'm having trouble finding the range of $z\mapsto z^4 | \Re(z) + \Im(z) ≥0\text{ and }\Re(z)< 0 $. If I set $z=x+iy$ and let $x+y=0$, I get: $z^4= (x+iy)^2 = (x-ix)^4 = x^4(1-i)^4 = x^4(-2i)^2 = -4x^4$. Since $x$ is real, this tells me the line…
Pen and Paper
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Solving $z\bar{z}+3iz=p+9i$, $z\in\mathbb{C}$

$z\bar{z}+3iz=p+9i$, where $z$ is a complex number and p is a real constant. Given this equation has exactly one root, determine the complex number $z$. I started with $z=a+bi$…
AnthonyML
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Solution of $e^{2z}(1+z)=1-z$ is $0$ or purely imaginary

This might be a silly question, but I can't find for any solution. I want to solve $e^{2z}(1+z)=1-z$ over the complex numbers $\mathbb{C}$. It looks like that the solutions are given by $z=0$ and $ir_{n}$, where $\{r_{n}\}$ are real roots of $\tan r…
sansae
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Proving that an expression is an integrer

If $z^7 = 1$, $z \in \mathbb{C}$ and $$x = \frac{z}{1+z^2}+\frac{z^2}{1+z^4}+\frac{z^3}{1+z^6}$$ Prove that $x \in \mathbb{Z}$. I wrote $z$ as $z = \cos{\frac{2k\pi}{7}} + i\sin{\frac{2k\pi}{7}}$, $k$ having values from $0$ to $6$, and…
Reras
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Number of solutions of the equation $z^p=(\overline{z})^q$

So we have a complex number $z$ and we have to find number of solutions possible for the equation: $$z^p=(\overline{z})^q$$ So I created two cases: Case-1: $p=q$. Clearly we can see the whole equation simplify into $z-\overline{z}=0$ or…
Kshitij Kumar
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