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.

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how to find cube roots of complex number

$c^3 = -1 + i$ How does one calculate the complex number(s) that satisfy above. Would like a general method, if there is one?
Dan
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If $z_i+z_jz_k$ are real, then $z_1z_2z_3=1$

Let $z_j=r(cosφ_j+isinφ_j), r\in R$ for $j=1,2,3$ be different complex numbers. If the numbers $w_1=z_1+z_2z_3$, $w_2=z_2+z_1z_3$, $w_3=z_3+z_1z_2$ are real, prove that $z_1z_2z_3=1$ I know one solution for this problem, but it just seems too…
user92596
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Square root and distribution

I see two ways of definining the imaginary number $i$, e.g. $i^2 = -1$ or $i = \sqrt -1$. I always thought the second one was right, yet I saw lots of websites saying that : $1=\sqrt 1=\sqrt{-1\cdot-1}=\sqrt-1\cdot\sqrt-1=i\cdot i=i^2=-1$ Which…
user178032
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How to write a complex number in polar form

Complex number given: $x = 1 + \cos \alpha + i \sin \alpha$ Desired form is something like $|x| \cdot e^{i \cdot \phi} = |x| \cdot (\cos \phi + i \sin \phi)$. I somehow got completly stuck how to convert the number to the Euler style. Maybe someone…
meinzlein
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how to find inverse point in a complex plane

How to find the inverse point of the point z=a with respect to the circle $|z-c|=r$ (where c is the origin and r the radius) ? $c+\frac{r^{2}}{a-c}$ this is the answer given in the book...how do we get it
amit
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Why $\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 \in \mathbb R \iff z_1, z_2, z_3 \text{ are along the same line}$

Prove: $$\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 \in \mathbb R \iff z_1, z_2, z_3 \text{ are along the same line}$$ My attempt: Since $z+\bar{z} =2 \operatorname{Re}z \in \mathbb R$, so we can transform like…
Zhen Zhang
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summing the powers of a complex number

Let $z=e^{\frac{2\pi i}{5}}$, then $1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9=?$ I am kind of confused since by drawing a graph, $1+z+z^2+z^3+z^4$ should be zero, but using computational softwares the result is different, and hence I do not know how…
Amber Xue
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If $\{ z_k\}_{0\leq k < n}$ are the $n$-th roots of a unity with $z_0 = 1$ prove that $n = \prod (z_0-z_k)$

Let $z_k$ be a $n$-th root of the unity, i.e. $$z_k = e^{2\pi i\frac{k}{n}}\\ k\in \{0,1,\cdots,n-1\}$$ Prove that $$(z_0 - z_1)(z_0 - z_2)\ldots (z_0 - z_{n-1})= n$$ This problem was at the end of the introduction chapter of a book about complex…
Darth Geek
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Find $n$ for which $(1+i)^{2n}=(1-i)^{2n}$

Question: Find the values of $n$ for which $$(1+i)^{2n}=(1-i)^{2n}$$ wolfram alpha tells me that the answer should be : $$n=\frac{2i\pi m}{\log(1-i)-\log(1+i)}$$ $$n=-\frac{i(2\pi m+\pi)}{\log(1-i)-\log(1+i)}$$ PS:I have no idea about the answer. I…
Hashir Omer
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Why is $ i^2 \neq (1 + i)^4$?

Today I read that you can see the number $i$ as the rotation of 90° and therefore i^2 is the rotation of 180° or -1. I also learned that $1+i$ is 45° but if this would be true I should be able to rotate 4 times with 45° and I should also get to 180°…
Maik Klein
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What is the least positive integer $n$ for which $(-\sqrt{2}+i\sqrt{6})^n$ is an integer?

Compute the least positive integer $n$ for which $(-\sqrt{2}+i\sqrt{6})^n$ will be an integer, where $i$ is the imaginary unit. I did the binomial expansion and just plugged in numbers for $n$ starting from $1$ to see any pattern. I coudn't find…
Kelly Rocks
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Finding complex solutions of an equation

How does one solve this equation. I would like to see the solution of this problem in steps. $z\cdot\bar{z}=\left|3\cdot z \right|$ EDIT: Is it possible to solve this by converting to the form $z=a+b\cdot i$ What about the solution of this…
user18960
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Help on a complex number equality problem

Find the general value of $\theta$ which satisfies the equation $\displaystyle (\cos\theta+i\sin\theta)(\cos2\theta+i\sin2\theta)...(\cos n\theta+i\sin n\theta)=1$ My thoughts: Simplest answer is $\theta= 0$ $\displaystyle (\cos…
square_one
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Express a complex number in modulus amplitude form

Express a complex number in modulus amplitude form $\displaystyle 1+\sin \alpha +i\cos \alpha $ My Attempt: $\displaystyle r\cos \theta= 1+\sin \alpha $ $\displaystyle r\sin \theta= \cos \alpha $ Squaring and adding.. $\displaystyle r^2= (1+\sin…
square_one
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Rewriting this complex square root

for some reason I can to figure out how to rewrite this square root. I have: $\sqrt{2+i}$ And I need to rewrite it into: $\frac{\sqrt{2(\sqrt{5} + 2)} + \sqrt{-2(\sqrt{5} - 2)}}{2}$ Can anybody show me I to do this? I've been trying for an hour…
user160513
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