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
5 answers

Let $x-\frac 1x=\sqrt 2 i$. Then find the value of $x^{2187}-\frac{1}{x^{2187}}$

Solving the quadratic equation, we find x to be $$x=\frac{i \pm 1}{\sqrt 2}$$ Moving back to the original expression $$\frac{(x^2)^{2187}-1}{x^{2187}}$$ $$=\frac{-i-1}{x^{2187}}$$ I don’t know how to solve further The answer is $\sqrt 2 i$
Aditya
  • 6,191
3
votes
3 answers

Maximum value of $Z$

How to find maximum value of $| Z| $ if: $$ \Big| Z-\dfrac{4}{Z} \Big|=2; $$ Where $Z$ is a complex mumber
ABC
  • 6,034
3
votes
1 answer

Show that there are certain properties for (z-i)/(z+i)

I have to do the following: Show that the following set: $D=\lbrace\frac{z-i}{z+i}|z \in C, \Im(z)>0\rbrace$ has a $l \in R^+_0$ and $\phi_1,\phi_2\in[0,2\pi]$ with $\phi_1\le \phi_2$ that $D$ can be described as: $D=\lbrace…
3
votes
2 answers

Finding the locus of a complex number

Find the locus of $\arg\left(\frac{z-3}{z}\right) = \frac{\pi}{4}$ where $z$ represent complex number. Working: $\arg\left(\frac{z-3}{z}\right) $ can be written as $\arg(z-3)-\arg(z) = \frac{\pi}{4}$, or $\arg\left((x-3)+iy\right) -…
3
votes
3 answers

Prove that if $z$ is uni modular then $\frac{1+z}{1 + \bar z}$ is equal to $z$.

The expression can be written as $$\frac{1+z}{\overline {1+z}}$$ Since $z \cdot \overline z=|z|^2$ $$\overline{1+z}= \frac{1}{1+z}$$ As $|z|=1$ So it will become $(1+z)^2$ But the answer is $z$. What am I doing wrong?
Aditya
  • 6,191
3
votes
2 answers

How do I simplify the squareroot of -14i?

I was thinking that I should separate the term $(-14i)^{0.5}$ into $(-14)^{0.5}$ and $i^{0.5}$ then into $i$ and $14^{0.5}$ and $i^{0.5}$ But Wolframalpha says that the answer is $(1-i)7^{0.5}$, and I don't think that's the same as…
helpme
  • 661
3
votes
2 answers

Solve $z^3-3z^2+3z+7=0$ and sketch solution set

Solve the equation $$z^3-3z^2+3z+7=0$$ and sketch solution set. My work: Since $z=-1$ is a root of the equation then we proceed by doing Ruffini's Rule we observe that $$z^3-3z^2+3z+7=(z+1)(z^2-4z+7)=(z+1)(z-(2+\sqrt{3}i))(z-(2-\sqrt{3}i))=0.$$…
manooooh
  • 2,269
3
votes
3 answers

Suppose z is any root of $11z^8 + 20 iz^7 + 10iz –22 = 0$. Then $S = |z|^2+| z|+ 1$ satisfies?

Suppose z is any root of $11z^8 + 20 iz^7 + 10iz –22 = 0$. Then $S = |z|^2+| z|+ 1$ satisfies ?(A) $S \leq 3$ (B) $3 < S < 7$ (C) $7 \leq S < 13$ (D) $S \geq 13$ Where do I start? I cannot simplify $z^7=\frac {22-10iz}{11z+20i}$. This could be…
Tapi
  • 1,688
3
votes
3 answers

Find a condition on real numbers $a$ and $b$ such that $\left(\frac{1+iz}{1-iz}\right)^n = a+ib$ has only real solutions

I´m new on this. I need to find a condition that relates two real numbers $a$ and $b$ such that $$\left(\frac{1+iz}{1-iz}\right)^n = a+ib$$ has only real solutions This is what I got till now. $$\left(\frac{1+i(a+ib)}{1-i(a+ib)}\right)^n =…
Ro_Mac
  • 51
3
votes
2 answers

How to find the real part of this fraction?

I don't understand how to get the real part of the following fraction. $\quad\dfrac{1}{a + jw}$ Here, $j$ is the imaginary unit. What's the process of retrieving the real part? Thanks.
amorimluc
  • 183
3
votes
2 answers

How to show $\sin(-iy)=i\sinh(y)$?

How to show $\sin(-iy)=i \sinh(y)$? I get: $\sin(-iy)=\frac{1}{2i}(e^{-iy}-e^{iy})=\frac{1}{2i}(\cos(y)-i\sin(y)-\cos(y)-i\sin(y))=...=-\sin(y)$. I don't get it. $-sin(y) \neq i sinh(y)$ - look at here
user2723
3
votes
2 answers

ellipse in complex numbers

Argue geometrically that: $$|z-4i| + |z+4i|=10$$ is an ellipse. I've thought about it but have no good idea. I understand these are two distances, one to $z_o=4i$ and the conjugate, but what then? Would you give me a hand?
user436603
3
votes
2 answers

Why is $i\cdot \sin(x)$ not $\cos(x)$?

I recently repeated some math basics of the Fourier transform and of course stumbled across Euler's formula. When reading the term $\cos(x) + i\sin(x)$ I wondered why it could not be written as $2\cos(x)$. Since all professors always emphasize that…
3
votes
2 answers

If $\mid z_{0} \mid = 1, z_{0} \in \mathbb{C} $ prove that then $\forall z\in \mathbb{C}$ such that $z \neq z_{0}$...

If $\mid z_{0} \mid = 1, z_{0} \in \mathbb{C} $ prove that then $\forall z\in \mathbb{C} , z \neq z_{0}$ $$\left| \frac{z-z_{0}}{1- \bar{z}z_{0}} \right|= 1$$ P.S. if $z = x+y i$, $\bar{z} = x - y i$. I tried by multiplying given fraction with $…
user560461
  • 1,735
  • 8
  • 16
3
votes
4 answers

How does $e^{5ix} = -1 $ have 5 solutions?

Part of the solution to a question in my book says $e^{5ix} = -1 $ has five solutions for x. There is no further explanation. How do I arrive at this result?
Hema
  • 1,329