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
3
votes
1 answer

Why isn't it true that $|z|\sqrt{2} \leq |Re(z)| + |Im(z)|$?

I know that $|z|\sqrt{2} \geq |Re(z)| + |Im(z)|$. When I tried to prove it I did this: $|z| = |Re(z) + iIm(z)|$ By the triangle inequality: $|Re(z) + iIm(z)| \leq |Re(z)| + |iIm(z)|$ $ |Re(z)| + |iIm(z)| = |Re(z)| + |i||Im(z)| = |Re(z)| +…
Juanma Eloy
  • 1,407
3
votes
5 answers

Prove $\ \frac{z-1}{z+1} $ is imaginary no' iff $\ |z| = 1 $

Let $\ z \not = -1$ be a complex number. Prove $\ \frac{z-1}{z+1} $ is imaginary number iff $\ |z| = 1 $ Assuming $\ |z| = 1 \Rightarrow \sqrt{a^2+b^2} = 1 \Rightarrow a^2+b^2 = 1 $ and so $$\ \frac{z-1}{z+1} = \frac{a+bi-1}{a+bi+1} =…
bm1125
  • 1,427
3
votes
3 answers

Find $z^3+bz^2+c=0$

Find $b,c\in \mathbb{R}$ where $z^3+bz^2+c=0$ And $z_1=(-\sqrt{2}-\sqrt{2}i)^5$ and $z_2=(-\sqrt{2}+\sqrt{2}i)^5$ We have $z_1=(-\sqrt{2}-\sqrt{2}i)^5=32e^{i\frac{15\pi}{4}}$ and $z_2=(-\sqrt{2}+\sqrt{2}i)^5=32e^{-i\frac{15\pi}{4}}$ Can we conclude…
newhere
  • 3,115
3
votes
2 answers

Find the maximum of the $| \left( w + 2 \right) ^3 \left( w - 3 \right)^2|$ with $|w|=1$

Let $w \in \mathbb{C}$, and $\left | w \right | = 1$. Find the maximum of the function $| \left( w + 2 \right) ^3 \left( w - 3 \right)^2|$ Since $$|(w+2)^3(w-3)^2|=|w^5-15w^3-10w^2+60w+72|$$ Let $w=\cos x+i \sin x$. Then we have an ugly form
math110
  • 93,304
3
votes
5 answers

Find all solutions $z^2\overline{z}^3=32$

$z^2\overline{z}^3=32$ I have bought about the following way: to simplify it by $z^2\overline{z}^3=z^2\overline{z}^2\overline{z}=(z\overline{z})^2\overline{z}=|z|^4\overline{z}$ to replace $z$ or by $z=x+iy$ or $z=re^{i\theta}$ which should I…
newhere
  • 3,115
3
votes
3 answers

Understanding Definition of Complex Number

In the book "Linear Algebra Done Right", a complex number is defined as: "an ordered pair $(a, b)$ , where $a, b \in \Bbb{R}$, but we will write this as $a + bi$." "The set of all complex numbers is denoted by $\Bbb{C}$: $$ \Bbb{C}=\{a+bi\ : a, b…
crayden
  • 145
3
votes
2 answers

Sum of a Complex number Z with its Conjugate equals to zero. Conclusion?

I saw an MCQ in a book that asks that sum of a complex number $Z$ with its Conjugate equals to zero if and only if Im$(Z)=0$. But my brain cannot absorb this answer. Because their sum equals to 2Re$(Z)$, therefore Re$(Z)$ must be zero. Isn't it?
Raju
  • 31
  • 6
3
votes
1 answer

Complex number rotation - Wildberger approach

I am trying to understand more about quaternions and I was watching the series of videos from Norman Wildberger, in particular I am a bit stuck on this video: https://www.youtube.com/watch?v=uRKZnFAR7yw You can see he associates to a complex number…
dd95
  • 229
3
votes
2 answers

Vectorial Analogue To Usual Multiplication of Complex Numbers

I know that a complex number can be taken as 2-dimensional vector and the dot and cross products have also been defined for two complex numbers but different from those of vectors. But my questions is "What is the vectorial analogue to the usual…
Waqar
  • 267
3
votes
3 answers

Complex numbers straight line proof

Prove that the three distinct points $z_1,z_2$, and $z_3$ lie on the same straight line iff $z_3 - z_2 = c(z_2 - z_1)$ for some real number $c$ and $z$ is complex. I know that two vectors are parallel iff one is a scalar multiple of the other,…
Q.matin
  • 2,835
3
votes
1 answer

Given a complex number $z$ that $|z-2-i|=2\sqrt{2}$. Find the maximum and minimum of $H=|z+3-2i|+|z-3+4i|$.

Given a complex number $z$ that $|z-2-i|=2\sqrt{2}$. Find the maximum and minimum of $H=|z+3-2i|+|z-3+4i|$. The minimum is easy to find: $H=|z+3-2i|+|z-3+4i| \geq|z+3-2i-z+3-4i|=6\sqrt{2}$ I am struggling with the maximum. It leads to find the…
Godgog Arsenal
  • 157
3
votes
3 answers

Finding every solution for equation of complex numbers

I need to find every solution for: $\ z^{3} + 3i \overline z = 0 $ So I tried was just to compare imaginary and complex part of $\ z^{3} $ and $\ 3i\overline z$ Ill spare you the alegbra, here is the result: $$\ a^{3} - 3ab^{2} + i(3a^{2}b-b^{3})…
bm1125
  • 1,427
3
votes
3 answers

How do I show that a complex number $z$ is a root of $z^6-1$?

This problem would be less confusing to me if one was not subtracted from $z^6$. Here is the problem: If $z=cos(\frac{\pi}{3})+jsin(\frac{\pi}{3})$, show that $z$ is a root of $z^6-1$. My approach to this problem involved using de Moivre's theorem…
AugieJavax98
  • 537
3
votes
3 answers

Need help in describing the set of complex numbers

Let set $$C=\{z\in \mathbb{C}:\sqrt{2}|z|=(i-1)z\}$$ I think C is empty , because you could put it in this way$$|z|=-\frac{(1-i)z}{\sqrt{2}}$$ but would like a second opinion.
Mykolas
  • 1,357
3
votes
5 answers

Find such a complex number $z$ so that $|z| - 2 = z +12i$

I have no idea how to approach this exercise. I've tried deducing the real component $a$ and imaginary component $bi$ by inserting $z = a + bi$ and $|z| = \sqrt{a^2 +b^2}$ into the original equation, but it just gets way longer and absurdly…
God bless
  • 2,049
1 2 3
…
99 100