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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What is $\sqrt[4]{-1}$

So, we all know that $\sqrt{-1}=i$, that $\sqrt{-2}=\sqrt2i$, and so on. And that, for example, $\sqrt[3]{-27}=-3$. But I was wondering; what would $\sqrt[4]{-1}$ be? Since all $n^{th}$ roots of $-1$ that are odd result in $-1$ (Because…
user558017
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$e^{i\pi}$ paradox - invalid proof that $-4 \pi ^2=0$

This paradox is directly ripped off of this video by the youtube channel "Mathologer 2". There, he presents a paradox with the following steps: $$e^{i \pi}=-1$$ $$(e^{i \pi})^2=(-1)^2$$ $$(e^{i \pi})^2=1$$ $$e^{2i \pi}=1 $$ $$e^{2i \pi…
Alexander51413
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Find all complex solutions of $\sin(z)=1-i$.

For this question I tried to use this identity: $$\sin(z)= \frac{e^{iz}-e^{-iz}}{2i}$$ and let $\sin(z)= \dfrac{(e^{iz}-e^{-iz})}{2i}=1-i$, then $e^{iz}-e^{-iz}-(2i+2)=0$; multiply both sides of this equation by $e^{iz}$,and let…
Jone Will
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Why is principle value of argument generally taken as $(-\pi , \pi]$?

I have just started reading about the Modulus and Argument of Complex Numbers. In the definition, it is said that: If z is not equal to 0 and $-\pi < \theta \le \pi$ , then $\theta$ is the principal argument of z, written $\theta = \arg(z)$. My…
Vitale
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Some equation with complex numbers

Given $a,b \in \mathbb{C}$ such that $a^2+b^2=1$, it is clear that $x:=a\bar{a}+b\bar{b}$ is a real number and that $yi:=a\bar{b}-\bar{a}b$ is imaginary (i.e $y$ is real). Moreover, a direct computation shows that $x,y$ satisfy $x^2-y^2=1$. Now,…
KotelKanim
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If $|z^2-1|=|z|^2+1$, show that $z$ lies on imaginary axis

If $|z^2-1|=|z|^2+1$, how do we show that $z$ lies on imaginary axis ? I understand that I can easily do this if I substitute $z=a+ib$. How do we solve it using algebra of complex numbers without the above substitution ? My…
Sooraj S
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help with proving that the multiplication of complex roots equals 1

Let $z_1, z_2,\dots,z_n$ be all the solution in $\mathbb{C}$ of $z^n=1$. I have to prove that for odd power $n$ the multiplication of the roots equals $1$: $$z_1z_2\dots z_n=1.$$ I'm a bit stuck and would appreciate a tip how to progress. $z^n=1 …
Hector Isidor
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Do $z^{3/4}=-1 $ solutions exist?

I want to get some perspective on complex exponentiation and the best possible approach to solve them. $ z^{3/4}=-1 ;z \in \mathbb{C}$ Let $z=x+iy=\rho e^{i \theta} $ where $\rho =|z|$ and tan$\theta=\frac xy$ $w=-1=(1).e^{i(\pi+2k\pi)}…
FemtoComm
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Complex numbers question

Consider 3 nonzero complex numbers $z_1,z_2,z_3$ each satisfying $z^2=i \bar{z}$. We are supposed to find $z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$. The answers- $0$, purely imaginary , purely real respectively. I have no idea how to proceed.…
GRrocks
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Find $\sqrt{8+6i}$ in the form of $a+bi$

I need help with changing $\sqrt{8+6i}$ into complex number standard form. I know the basics of complex number such as the value of $i$ and $i^2$, equality of complex number, conjugate and rationalizing method. This is my first encounter with such…
Afiq Afandi
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$e^{i\theta} = 1$ for all $\theta$?

What is the conceptual error in the following derivation? $$ e^{i\theta} = e^{i\frac{2\pi}{2\pi}\theta} = (e^{i2\pi})^{\frac{\theta}{2\pi}} = 1^{\frac{\theta}{2\pi}} = 1 $$ It is clear to me that the second move is illegal, I just don't know why.
Joshhh
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Writing complex numbers in form $a+bi$

Can $\sqrt{i+\sqrt{2}}$ be expressed as $a+bi$ with $a,b \in \mathbb{R}$? In general, what kinds of expressions can be rewritten in that form?
rorty
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Find all $z$ such that $e^z=6i$

Am I on the right track to solving this? $$e^z=6i$$ Let…
whatwhatwhat
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Why is there only a complex conjugate, but no real conjugate?

In mathematics one often uses the complex conjugate $$ \Bbb C\to\Bbb C,\quad z=a+b\cdot\mathrm{i}\;\;\mapsto\;\; \bar z=a-b\cdot\mathrm{i} $$ This is often described as a a reflection along the real axis. But in analogy one could also define a real…
asmaier
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4 answers

Computing $(1+\cos \alpha +i\sin \alpha )^{100}$

How to prove that $$ ‎‎‎‎‎‎‎‎‎‎‎(1+‎\cos ‎‎\alpha ‎+i‎\sin ‎‎\alpha ‎)^{100} =‎ ‎2^{100}‎\left( ‎‎\cos \left(‎\frac{‎\alpha‎}{2}\right)‎\right) ‎^{100} ‎‎\left( ‎‎\cos \left(‎\frac{100‎\alpha‎}{2}\right)+i‎\sin …
6-0
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