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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If $z$ lies on the circle $|z-1|=1$ then$\frac {z-2} z$ is a purely imaginary number

If $z$ lies on the circle $|z-1|=1$ then $\frac {z-2} z$ is a purely imaginary number. This is what by book states. Did'nt understand why. Can someone help? Actually I was thinking of a more geometrical approach to the problem, as pointed out by…
user220382
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3 Dimensional Complex Plane?

We all know, that all real numbers are complex numbers also. But in real numbers, there are multi-dimensional coordinate planes. Is there such thing in complex numbers also? If yes, what is there meaning and how do we represent numbers over there?…
Aditya Agarwal
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complex no. $z$ such that $ |z|<\frac{1}{3}$ and $ \sum^{n}_{r=1}a_{r}z^{r} = 1\;,$ Where $|a_{r}|<2$

Prove that there exists no complex no. $z$ such that $\displaystyle |z|<\frac{1}{3}$ and $\displaystyle \sum^{n}_{r=1}a_{r}z^{r} = 1\;,$ Where $|a_{r}|<2$ $\bf{My\; Try::}$ We can write $\displaystyle \sum^{n}_{r=1}a_{r}z^{r} = 1$ as …
juantheron
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Root of Complex Number

I just want to clarify something here. Using elementary computation we can verify that for $x,y\in\mathbb{R}$ $$\sqrt{x+iy}=\pm\left(\sqrt{\frac{r+x}{2}}+i \sqrt{\frac{r-x}{2}}\right)$$ where $r=\sqrt{x^2+y^2}$. However, in wikipedia the algebraic…
Ajat Adriansyah
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Product of complex solutions via factorisation

I'm wondering if someone could help me out. I am asked to solve the equation: $z^6 =−1$ in part (a) of a question. I have done this and so I now have a set of solutions: $z_0,z_1,z_2,z_3,z_4.$ I'm lost in part (b): Let $z_0, z_1, z_2, z_3, z_4,…
Matty
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Solve an equation with complex numbers

The question is to solve the following equation for complex numbers $$z-i = iz +5$$ I have tried to add i to both sides which gives $$z = iz +5 + i$$ I have also tried with combinining all the terms on the LHS so i get $$z - i - iz - 5 = 0$$ Can you…
addde
  • 449
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4 answers

Complex number and exponent

If $z = -1+ i\sqrt{3}$ Is it possible that to prove by using induction $z^{2n}+2^n\cdot z^n+2^{2n}=0$ if $n$ is not multiple of $3$. I know other way of proving it.
Milan Amrut Joshi
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$z^3=w^3 \implies z=w$?

I've reached this in another problem I have to solve: $z,w \in \Bbb {C}$. $z^3=w^3 \implies z=w$? I've scratched my head quite a bit, but I completely forgot how to do this, I don't know if this is correct: $$ z^3=|z^3|e^{3ix}=|w^3|e^{3iy} $$ I…
YoTengoUnLCD
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Multiplying square roots of negative numbers

I am just learning more about complex numbers and a question popped up I can't figure out on my own, so I've posted it here. I already know $i^2=-1$ and $i=\sqrt{-1}$ (isn't it even true that $\pm i=\sqrt{-1}$?) I know $\sqrt{a} \sqrt{b} =…
DeBe
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Find all the solutions of the equation $w^3 = 1/2(1+i\sqrt{3})$ in the form $r.\operatorname{cis}(\theta)$

Could anyone please help me walk through the steps. I understand the underlying concepts through the use of deMoivre's Formula, and that it is to be written in the form of Euler's equation. I would like some guidance on the exact steps taken to…
Danika Blackburn
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Rotation around complex number

The function $$f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$$ represents a rotation around some complex number $c$. Find $c$. How would I start this? Thanks.
Rohini Sharma
  • 449
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Find vertices of equilateral triangle

There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $2 + 2i$, $5 + i$, and $c$ form the vertices of an equilateral triangle. Find the product $c_1 c_2$. So far, I have used the distance formula to get $(a−2)^2+(b−2)^2=(a−5)^2+(b−1)^2$…
Rohini Sharma
  • 449
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Complex numbers: $|\frac{1}{x}-\frac{1}{y}| = \frac{|x-y|}{|x||y|}$?

I need the result for a proof, but I can't seem to verify it. If $x,y$ are distinct nonzero complex numbers, why is it true that $|\frac{1}{x}-\frac{1}{y}| = \frac{|x-y|}{|x||y|}$? Starting with the left hand side, I've gotten to the term…
zha
  • 139
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Evaluate $(\frac{1}{-\sqrt{2}+\sqrt{2}i})^{2011}$

Evaluate $$(\frac{1}{-\sqrt{2}+\sqrt{2}i})^{2011}$$ So ... $$(\frac{1}{-\sqrt{2}+\sqrt{2}i})^{2011} = (-\sqrt{2}+\sqrt{2}i)^{-2011}$$ $$\theta=\pi - \arctan(\frac{\sqrt{2}}{\sqrt{2}}) = \frac{3\pi}{4}$$ $$-\sqrt{2}+\sqrt{2}i=\cos{\theta} + i…
Jiew Meng
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Roots of unity question.

Question Let $\omega=\cos\dfrac{4\pi}{7}+i\sin\dfrac{4\pi}{7}$. Show that $\omega-1=2\sin\dfrac{2\pi}{7}\left(\cos\dfrac{11\pi}{14}+i\sin\dfrac{11\pi}{14}\right)$. My attempt Observe that $\omega$ is a seventh root of unity. Label the roots $1,…
bibo_extreme
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