-3
votes
1answer
31 views

When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn. For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy ...
0
votes
2answers
17 views

Function with exponent imaginary power

If we have $u=\frac{4c(e^{-is}-e^{is})}{(e^{-is}+e^{is})^2} \tag 1$ where c is a constant and s is a variable. Can we write $e^{is}$ in terms of u ? Means Can we write $e^{is}$ as $\psi(u)$ , a ...
0
votes
4answers
29 views

Find all complex and real roots of higher degree polynomials, given one root

$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!
2
votes
3answers
47 views

Roots of Unity - $x^3 = -i$

I need to find the roots of unity for the following equation: $$x^3 + i = 0$$ Thus, $x^3 = -i$. I know that $-i = \exp[i(\frac{3\pi}{2} + 2n \pi)]$ however I do not know how to get all roots. ...
4
votes
3answers
209 views

Find all values for cos(i)

In my Differential Equations class recently we have learned about Euler's Formula and Fourier Series. I am given the problem ...
1
vote
4answers
35 views

Third point of a triangle in the complex plane

I have an equilateral triangle with two points equal to $(2+2i)$ and $(5+i)$. I want to find the third point(s) (there are $2$ of these). I have that the side length of the triangle is $\sqrt{10}$.
-2
votes
1answer
33 views

Complex numbers and geometry

There exist two different complex numbers $c_1$ and $c_2$, that together with $2+2i, 5+i$ form the vertices of two equilateral triangles. Find the product $c_1c_2$.
3
votes
5answers
89 views

Complex solutions to $ x^3 + 512 = 0 $

An algebra book has the exercise $$ x^3 + 512 = 0 $$ I can find the real solution easily enough with $$ x^3 = -512 $$ $$ \sqrt[3]{x^3} = \sqrt[3]{-512} $$ $$ x = -8 $$ The book also gives the ...
0
votes
0answers
37 views

Pre-calc complex number geometry

The equation of the line joining the complex numbers $-5 + 4i$ and $7 + 2i$ can be expressed in the form $az + b \overline{z} = 38$ for some complex numbers a and b. Find the product $ab$. Maybe ...
0
votes
2answers
21 views

Find angle $\alpha$ from a complex vector

I'm trying to solve this problem from a Russian book: Find the angle which is needed to rotate the vector $3\sqrt{2} + i2\sqrt{2}$ to obtain the vector $-5+i$. EDIT: $\tan\dfrac{\pi}{6} \neq ...
0
votes
1answer
62 views

Solving system of equations with complex numbers

Equation 1$$ \frac{V_{1}}{5} + \frac{V_{1}-V_{2}}{10+j6} - 10\angle45^\circ = 0 $$ Equation 2 $$ -4V_{1} + \frac{V_{2}-V_{1}}{10+j6} + \frac{V_{3}}{-j2} + \frac{V_{3}}{8+j7} = 0 $$ Equation 3 $$ ...
0
votes
3answers
85 views

Why does this work for $ i^{2i} $?

I'm finding the principal value of $$ i^{2i} $$ And I know it's solved like this: $$ (e^{ i\pi /2})^{2i} $$ $$ e^{ i^{2} \pi} $$ $$ e^{- \pi} $$ I understand the process but I don't understand ...
1
vote
1answer
61 views

Polar form of the sum of complex numbers $\operatorname{cis} 75 + \operatorname{cis} 83 + \ldots+ \operatorname{cis} 147$

The number $\operatorname{cis} 75 + \operatorname{cis} 83 + \operatorname{cis} 91 +\dots+ \operatorname{cis} 147$ is expressed in the form $r\operatorname{cis}(\theta)$, where $0\leq \theta< ...
2
votes
4answers
54 views

Writing the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form

Now I can't finish this problem: Express the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form, where $0 < \alpha < \frac{\pi}{2}$. So the goal is to determine both ...
2
votes
2answers
53 views

Another way to solve this problem with complex expressions

The problem is this: Express $x$ and $y$ with $u$ and $v$, if $\dfrac{1}{x+iy} + \dfrac{1}{u+iv} = 1$ Where $x,y,u,v \in \mathbb{R}$, and $i^2 = -1$. I could solve it, but I used a hairy and ...
4
votes
2answers
223 views

Problem getting the real roots of this complex expression

I'm trying to get the real roots of this expression: $$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$ Where $i^2=-1$ and $z=x+iy$. I tried to simplify that with Algebra, and then separate the real ...
3
votes
3answers
102 views

Problems with trigonometry getting the power of this complex expression

I'm here because I can't finish this problem, that comes from a Russian book: Calculate $z^{40}$ where $z = \dfrac{1+i\sqrt{3}}{1-i}$ Here $i=\sqrt{-1}$. All I know right now is I need to use ...
0
votes
1answer
19 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
2
votes
2answers
65 views

Simplify $(7-2i)(7+2i)$. Found difference between mine and solution guide's and didn't know why.

I looked up the solution guide and found out: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49+4$ $=53$ Why the unknown "$i$" just disappeared$?$ I supposed it might be: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49-4i$ does it? ...
0
votes
6answers
79 views

How to find the roots of $-x^3+3x^2-7x+5 = 0$?

I would like to understand how to go about solving something like this, not just get the solution but some kind of methodology (that hopefully makes as much intuitive sense as possible); I honestly ...
0
votes
4answers
56 views

Correct this problem in complex numbers

Prove for all $|z| = 3$, $$\frac{8}{11} \leq \left | \frac{z^2 + 1}{z^2 + 2}\right | \leq \frac{10}{7}.$$ Here is what I did, $$\frac{8}{11}\leq \frac{8}{z^2 + 2}=\frac{|z^2| - 1}{z^2 ...
0
votes
3answers
45 views

For all complex numbers with $|z|=2$, inequality $2\le |z-4|\le 6$ holds

Prove for all $|z| = 2$, $$2 \leq |z - 4| \leq 6$$ I tried $|(x - 4) + iy| = |x^2 + 16 - 8x + y^2| = |20 - 8x|$ I also tried using triangle inequality $2 = |z| = |z - 4 + 4| \leq |z - 4| + 4$ ...
2
votes
1answer
45 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
1
vote
2answers
48 views

Evaluate expression in the form $a+bi$.

So, I have to evaluate $\sqrt{-3}\sqrt{-12}$ into the form $a+bi$. I know that $i^2 = -1$ so $i = \sqrt{-1}$ What I have done is: $$\begin{align}\sqrt{-3}\sqrt{-12} &= \sqrt{3(-1)}\sqrt{12(-1)}\\ ...
0
votes
3answers
46 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
1
vote
2answers
47 views

Find $c$ if $a,b, \; c$ satisfy $c = (a+bi)^3 - 107i$

Find $c$ if $a,b, \; c$ are positive integers which satisfy $c = (a+bi)^3 - 107i$ I can try expanding the cube, but that seems too direct. What other ways are there to go about this?
2
votes
3answers
120 views

Computing complex number [duplicate]

"Compute $(1 + i)^{1000}$. So far I have: $(1+i)^{4 (2^2 5^3)} $ but I am not sure how to proceed. Ideas?
1
vote
2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
2
votes
3answers
66 views

Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
1
vote
1answer
143 views

Quick complex number proof question:

How would I go about proving the following identity: $$\frac{1}{\left|z\right|} = \left|\frac{1}{z}\right|$$ I keep finding myself going in circles. I've tried using this identity: $|z|^2 = z^*z$ ...
3
votes
1answer
72 views

Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...
0
votes
3answers
42 views

Complex solutions of polynomial question

$2z^3-6z^2+mz+n = 0$ $m, n$ are real and $1+\sqrt{ 2} i$ is a solution. Find $m$ and $n$. Attempt to solve : Giving the known theorem $1-\sqrt{2}i$ is also a solution, so we can substitute each time ...
1
vote
2answers
89 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
1
vote
2answers
58 views

Solve $(2z-1)^5 - i = 0$

Solve $(2z-1)^5 - i = 0$ I started by saying that $(2z-1)^5 = i$ $(2z-1) = \sqrt[5]i$ $z =$ $(\sqrt[5]i +1) \over 2$ $z^5 =$ $(i +1) \over 32$ $z^5 =$ $1 \over32$$ *(i +1)$ From there, ...
1
vote
1answer
61 views

Find the 6th root of $-3+4i$ and plot on complex plane

So I have a rough idea on how to get the answer but I'm getting stuck on the angle or argument for the equation. The question is: Find the 6th root of $-3+4i$. I first find the $r$ value which ...
0
votes
1answer
22 views

Quadratic factor to complex numbers

How to convert this quadratic factor to complex number form? (With steps please) Reference: $Z = a + bi$, $i = \sqrt{-1}$ $$-3 + \frac{\sqrt{-12}}{2}$$ Thanks!
1
vote
1answer
43 views

complex equations question

find all solutions of the equation: $w^4 = -8(1-i\sqrt{3})$ I dont wanna be that guy, but can someone tell me what the second solution to this equation is? cuz the solution manual says it's $-1 + ...
-1
votes
1answer
49 views

Find all complex solutions to the equation

i) Find all complex solutions to the equation z^4 +1 -i*3^(1/2) = 0 I basically have no clue, any tips/advice/solutions would be great. I could also need some help with another question, this one ...
0
votes
2answers
50 views

quotient of complex numbers?

so I was wondering if you have two different equations having denominators $2+i$ and $2-i$ respectively how came the denominator of the quotient in standard form is $5$ for both equations? I tought ...
0
votes
2answers
54 views

How to solve this equation? $8z^3+1=0.$ [closed]

I would like to know how to do this question. (a) Determine all of the solutions of the equation $$8z^3+1=0.$$ Express your solutions in the form $z = re^{i\theta}$ where $-\pi<\theta \leq \pi$. ...
0
votes
1answer
58 views

How to solve the complex equation? $(x+2yi)^2 = xi.$

How to solve the following complex equation with in less than 60 seconds? $$(x+2yi)^2 = xi.$$ I know how to solve, we have to solve power first then real part equal to real part and imaginary to ...
1
vote
1answer
112 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
1
vote
2answers
32 views

Contradiction with complex gaussians…

So, I am computing something seemingly simple involving complex gaussians and constants, but I am getting a big contradiction in my calculations. The setup: Let $C$ be a complex constant, that is, ...
1
vote
2answers
22 views

Solving two varibles system equation above $\mathbb{C}$

A bit emmbarrassed to ask this newbie question: Let: $$(1+i)x + y = 2$$ $$(1-i)x + iy = 0$$ Multiplying the first equation by $(-i)$ and summing the two equations, we have: $$(2-2i)x + 2i = 0$$ ...
0
votes
2answers
24 views

Prove that function transforms $S$ into $S$

I have to prove, that function $\Phi(z):=\frac{1-\overline a z}{z-a}$ transforms $S$ into $S$ where $S:=\{|z|=1\}$ and $|a|>1$ I don't know where can I start. Any hint?
6
votes
3answers
102 views

What is the value of $\ln \left(e^{2 \pi i}\right)$

I know that $$e^{2 \pi i} = 1$$ so by taking the natural logarithm on both sides $$\ln \left(e^{2 \pi i}\right)=\ln (1)=0$$ however, why isn't this $2 \pi i$ as expected? Is it beacuse logarithms ...
3
votes
4answers
472 views

Calculating real and imaginary part of a complex number

Consider the complex numbers $a = \frac{(1+i)^5}{(1-i)^3}$ and $b = e^{3-\pi i}$. How do I calculate the real and imaginary part of these numbers? What is the general approach to calculate these ...
0
votes
5answers
175 views

How to teach newbie multiply of complex number

I want to teach a newbie the arithmetic law of complex numbers. the law of add is acceptable psychological. but multiply is not. for example, assume $$z = a+bi, w = c+di$$ He (She) may ask me: why ...
1
vote
1answer
28 views

What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?

$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
0
votes
1answer
32 views

How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$