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
2 answers

Finding point of intersection of tangents through complex numbers

If $P$ and $Q$ be two points on the circle $|w|=r$ represented by $w_1$ and $w_2$ respectively, then find the complex number representing the point of intersection of the tangents at $P$ and $Q$. My attempt: Well frankly i didnt even know how to…
rohit_r
  • 215
3
votes
3 answers

Prove $a_0=a_1=\dots =a_{p-1}$

If $w=\cos \frac{2\pi}{p}+i\sin \frac{2\pi}{p}$ and $p$ is a prime and $a_0,a_1,\dots ,a_{p-1}$ are non zero integers and $a_{p-1}w^{p-1}+\dots +a_1w+a_0=0$ Prove $a_0=a_1=\dots =a_{p-1}$ I got a solution somewhere but don't know how it works: "The…
Taha Akbari
  • 3,559
3
votes
2 answers

Solving complex equation $z^2 + (1+i) \overline{z} + 4i = 0$

Consider the following equation, where $z \in \mathbb{C}$, $i$ is the imaginary unit and $\overline{z}$ is the conjugate of $z$: $$ z^2 + (1+i) \overline{z} + 4i = 0 $$ What is the method to deal with equations such as this? I have tried various…
bp99
  • 1,107
3
votes
1 answer

Finding $z \in \mathbb{C}$ such that $|z^{6}+z^{3}+2|= 4+|4z^{2}+4|$

everyone. I don't know how to start to solve this exercise that asks for the $z \in \mathbb{C}$ such that $$|z^6+z^3+2|= 4 +|4z^2+4|$$ And $$|z|=1$$ I tried finding the roots of $z^6+z^3+2$ but that didn't help me to find the region that describes…
3
votes
3 answers

Find all complex numbers $z$ satisfying the equation

I need some help on this question. How do I approach this question? Find all complex numbers $z$ satisfying the equation $$ (2z - 1)^4 = -16. $$ Should I remove the power of $4$ of $(2z-1)$ and also do the same for $-16$?
3
votes
2 answers

Equation of locus of points satisfied by $\frac{\left|z+3i\right|}{\left|z-6i\right|}=1$

Equation of locus of points satisfied by $\frac{\left|z+3i\right|}{\left|z-6i\right|}=1$ The answer I got is $y=\frac{3}{2}$, but the answer given in my book is $y=-0.5x+2.25$ Can anyone please confirm which is the right answer Edit: Sorry, the…
3
votes
2 answers

Solve $\sin z=0.75i$

Find all complex solutions or $\sin(z)=0.75i$ I started with $$\frac{e^{iz}-e^{-iz}}{2i}=0.75i$$ $${e^{iz}-e^{-iz}}=-1.5$$ $${e^{iz}-e^{-iz}}=e^{ln(-1.5)}$$ But it is keeping get complicated
newhere
  • 3,115
3
votes
1 answer

Difference of two complex polynomials

Prove that for any integer $m>1$, $(z+a)^{2m}-(z-a)^{2m}=4maz\prod_{k=1}^{m-1}z^2+a^2\cot^2{\frac{k\pi}{2m}}$ I started off by expanding the left hand side using the binomial therem and noticed that some terms cancelled out and I ended up with the…
3
votes
2 answers

Solutions for a Complex Numbers Equation

I have the equation $z^{8}=\bar{z}$ I had to solve it, and to find the sum and product of the solutions. I did all that and found that the solutions are (in degrees): cis of: 0, 40, 80, 120, ..., 320. The sum was 0 and the product 1 (I used…
3
votes
3 answers

Proving that $z$ is Purely Imaginary

I'm stuck on a question in relation with complex numbers: If $z\neq0$, and that $$\left\vert{\frac{z+1}{z-1}}\right\vert=1,$$ prove that $z$ is purely imaginary. I tried breaking the modulus up into two separate parts, and then multiplying both…
3
votes
1 answer

Finding the length of a complex vector

Let us say the vector $u = (1 - i, 2 + 3i, 5)$. How would I find $\|u\|$? I have tried solving this through: $\sqrt{(1-i)^2 + (2+3i)^2 + 5^2}$ But I ended up needing to find the square root of a complex number, which from what I understand is…
3
votes
2 answers

In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?

Given dual numbers, what would be the value of $0^\varepsilon$ and $\varepsilon^\varepsilon$?
Anixx
  • 9,119
3
votes
3 answers

Real and imaginary parts of $(x + iy)^n$

For $z = x + iy$, we have, for example: $\Re (z)=x$ and $\Im (z)=y$ $\Re (z^2)= x^2-y^2$ and $\Im (z^2)=2xy$ $\Re(z^3)=x^3-3xy^2$ and $\Im (z^3)=3yx^2-y^3$ I know that I have seen the formulas for $\Re (z^n)$ and $\Im (z^n)$ as explicit functions of…
3
votes
2 answers

Find the real and imaginary parts of $ln(z)$

This is on my homework on differentials and partial differentiation, so I'm not sure what application these could have on the natural log of z
stack ex
  • 350
3
votes
1 answer

Finding foot of altitude of triangle with known vertices

I've encountered this problem recently: Let $z_1,z_2,z_3\in\mathbb C$ be three distinct points and L be the line segment joining $z_1,z_3$. Let the straight line passing through $z_2$ meet the line L perpendicularly at w. Find w in terms of…
user441558