Tagged Questions
0
votes
0answers
10 views
Does $\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - \mathtt{i}^\frac{2 k}{n+1}\right)$?
This seems to be true:
$$\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - \mathtt{i}^\frac{2 k}{n+1}\right)$$
but I don't know how to demonstrate it, and definitely not neatly. I'd like to see why it ...
2
votes
3answers
78 views
Show that $z^2=2i$ iff $z=\pm(1+i)$
I am reading Beardon's Algebra and Geometry.
Show that $z^2=2i$ iff $z=\pm(1+i)$.
For the problem in question, first I made the multiplication $(1+i)\times(1+i)$ which showed the result but I ...
1
vote
1answer
42 views
A finite sum of trigonometric functions
By taking real and imaginary parts in a suitable exponential equation, prove that
$$\begin{align*}
\frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases}
1&\text{if } k ...
0
votes
2answers
41 views
number of roots of polynomial of order n
from theorem of algebra,it is well know that polynomial of order n has exactly n roots,for exmaple quadratic equation like $ax^2+bx+c$ has three cases
let $D=b^2-4ac$ ,so we have
...
0
votes
1answer
57 views
$z= \frac{u-\overline{u}v}{1-v}$ is real is equivalent to $|v|=1$.
Let $u,v$ be complex numbers such as $u,v\notin \mathbb{R} $, and :
$$z= \frac{u-\overline{u}v}{1-v}$$
Prove that : $z\in\mathbb{R} \Longleftrightarrow |v|=1$.
2
votes
2answers
62 views
Show that $z^3 + (1+i)z - 3 + i = 0$ does not have any roots in the unit circle $|z|\leq 1$.
I need help with showing that $z^3 + (1+i)z - 3 + i = 0$ does not have any roots in the unit circle $|z|\leq 1$?
My approach so far has been to try to develop the expression further.
$$ z^3 ...
2
votes
1answer
36 views
Solution of $(E):z^2-2mz+1=0$ in $\mathbb{C}$
Suppose the equation $(E):z^2-2mz+1=0 \quad / m\in \mathbb{C}\quad z\in\mathbb{C}$ and we suppose $z_{1}$ and $z_{2}$ are the two solution of this equation.
How can I prove that $|z_1|+|z_2| = ...
2
votes
1answer
82 views
Help with particular solution to solving $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$.
I asked a question about how to solve: $$z^4−2z^3+9z^2−14z+14 = 0$$
When all you know is that there is a root with the real part of 1. I was given great answers and you can find the question ...
7
votes
3answers
129 views
How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.
I would please like some help with solving the following equation:
$$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$
All I know about the equation is that there is a root with the real part of $1$.
My approach ...
0
votes
1answer
55 views
Overdetermined system of equations
$x_{1}=u+v$
$x_{2} = \zeta u + \zeta^{2} v$
$x_{3} = \zeta^{2} u + \zeta v$
where $\zeta = -\frac{1}{2} + i \frac{\sqrt{3}}{2}$
Since $\zeta$ is a third root of unity, $\zeta^{3} =1 $.
Also, ...
1
vote
1answer
60 views
Problems with basic algebra
I'm studying for an exam in a digital communications course I'm taking, and the solution to one question has me totally lost. While finding the Inverse Fourier Transform of a function, there's one ...
0
votes
0answers
84 views
Some trigonometric equation problems
show that :
$$\left(1+\cos \frac{2\pi}{13}\right)\left(1-\cos \frac{4\pi}{13}\right)\left(1+\cos \frac{6\pi}{13}\right)\left(1+\cos \frac{8\pi}{13}\right)\left(1-\cos ...
6
votes
7answers
397 views
What is the value of $i+i^2+i^3+\cdots+i^{23}$? [duplicate]
Can anyone help me with this question and show me a step by step solution please?
The imaginary number is $i$ is defined such that $i^2=-1$. What is $i+i^2+i^3+\cdots+i^{23}$?
6
votes
1answer
84 views
determining if a complex number is a root of unity
How would you determine if $a+ib$ is a $n$th root of unity for some unknown $n$? Obviously the modulus of $a+ib$ must be $1$. But you also need to determine if the $a+ib$ is located at the vertex ...
3
votes
3answers
92 views
The distance from 1 to the other $n$th roots of unity
I want to prove that the sum of the fourth powers of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $6n$. I consider the distance from 1 to the other $n$th roots of unity ...
2
votes
2answers
65 views
The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle
I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
1
vote
1answer
33 views
Cyclotomic polynomial simplification
I try to work out what $\Phi_{12}(z)$ is:
By the fundamental theorem of arithmetic:
...
38
votes
12answers
2k views
How can I introduce complex numbers to precalculus students?
I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
1
vote
1answer
51 views
Why does this say that there are two complex roots, when they are displayed on real axes?
How do I find the real valued solutions to $3x - x^3 = \sqrt{(x + 2)}$.
http://www.wolframalpha.com/input/?i=3x+-+x%5E3+%3D+sqrt%28x+%2B+2%29
Here we see three intersections of the two graphs $f(x) ...
10
votes
3answers
198 views
What's $(-1)^{2/3}\; $?
I know that
$\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$
But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?
2
votes
2answers
108 views
How can I break up $z = \frac{5}{9+3i}$
Into its real and imaginary components? Wolfram tells me it's equivalent to $\frac{1}{2}+\frac{i}{6}$, but I don't know how to arrive there myself.
Thank you!
2
votes
1answer
95 views
Simplifying $|a+b|^2 + |a-b|^2$
I want to simplify $|a+b|^2 + |a-b|^2$ where $a, b \in \mathbb{C}$. I've used Wolfram Alpha to get
$$
|a+b|^2 + |a-b|^2 = 2\left(|a|^2 + |b|^2\right)
$$
I'm trying to understand the steps involved in ...
0
votes
1answer
197 views
Complex Numbers in Standard Form and other assorted problems
Some help on these practice questions and how they are solved would be much appreciated. Ran into some problems in Precalc.
1) Write the complex number in standard form.
$6 − \sqrt{-50}$
2) Perform ...
0
votes
2answers
685 views
Writing Complex Numbers in Standard Form
Can someone show me how to write complex numbers in standard form? I missed a few days of class and do not have the text book. Answering a simple question like the one below would help
Write the ...
2
votes
3answers
897 views
Square root of negative numbers
If:
$$a = \sqrt{ b^2 - b }$$
The problem I have is that for values of:
$0 < b < 1$
the result of:
$b^2 - b$
Is a negative number which gives rise to an error on Excel and my calculator.
...
2
votes
2answers
87 views
Solutions to $z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2 = 0, b\in \mathbb R, z \in \mathbb C$
$z_1 = 1+i$ is a given solution.
I guess what I have to find is $z_2$ and $z_3$ in
$(z - (1 + i))(z - z_2)(z-z_3) = z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2$.
I tried to divide the polynomial by $(z - (1 ...
1
vote
2answers
58 views
Is there a similar function in complex number system corresponding to logarithim in real number system?
i notice that there are $e^{i\theta}$ in math,so is there a similar function in complex number system corresponding to logarithim in real number system?
3
votes
1answer
58 views
Compute $\sum_{i=1}^{2n} \frac{x^{2i}}{x^i-1}$ where $\{ x \in \mathbb{C}$ | $x^{2n+1} = 1, x \neq 1\}$
$\{ x \in \mathbb{C}$ | $x^{2n+1} = 1$ , $x \neq 1\}$
Compute $\displaystyle{\sum_{i=1}^{2n} \frac{x^{2i}}{x^i-1}}$
0
votes
2answers
308 views
Finding the cubic root of a complex number given its relation to the sum of itself and its conjugate
I am trying to solve for $z$, given that $z^3=z+\bar{z}$.
I tried reducing this seemingly easy equation by rewriting to polar form, completing the square, and some trig manipulation but with no ...
4
votes
2answers
174 views
Complex Numbers $x,y,z$ Find $x^{2007}+y^{2007}+z^{2007}$
Let $x,y,z$ be complex numbers such that
$$x+y+z = x^{5}+y^{5}+z^{5} = 0, \hspace{10pt}
x^3+y^3+z^3=2$$
Find all possible values of
$$x^{2007}+y^{2007}+z^{2007}$$
1
vote
1answer
265 views
How do I find complex roots of a quartic polynomial using quadratic formula?
For something like:
$$
z^4 + 8z^2 + 3
$$
how can I find all the complex roots using the quadratic equation, or is there a better method?
I tried a u substitution using $u = z^2$, but then when I ...
2
votes
3answers
600 views
The square root of a variable is negative?
If the square root of a variable is negative, as shown below:
$$\sqrt x = -1$$
Then what is $x$ equal to? The closest answer I can think of is $i^4$.
$$\sqrt{i^4}=i^{\frac42}=i^2=-1$$
But if $i^4$ ...
1
vote
2answers
1k views
How can I find the roots of a quadratic function?
Bascially we are trying to find the roots of a quadratic equation, and 'apparently' there is a theorem for this, but every one that I have found so far mentions that the degree of the polynomial is ...
0
votes
1answer
117 views
Simplifying this equation (euler's formula?)
I was reading some stuff earlier today, and I wasn't sure how they changed the exponentials to trigs in this expression:
...
2
votes
1answer
147 views
Find all reals $a, b$ for which $a^b$ is also real
The title is pretty much clear, but here is a more precise formulation:
Find all pairs $(a,b)\in\mathbb{R^2}$ for which $a^b$ is also real.
I used a CAS to solve the problem and it says that the ...
1
vote
1answer
132 views
Von Mises width at half height
I'm fitting the following Von Mises type function to some data:
$(A/2\pi)e^{k\cos(\theta)}+C$ where A and k are positive.
I want to calculate the width at half height from the lowest point of the ...
4
votes
1answer
292 views
Why is the argument of $i$ equal to $\pi/2$?
So it's obvious geometrically that the argument of $z=i$ is $\pi/2$.
However the method of getting the argument is $\arctan(y/x)$. And when in the case of $z=i$, $y/x = 1/0$ which is undefined...
So ...
12
votes
6answers
735 views
How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$?
Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$?
Calculating the complex roots and then the result could be an answer I suppose, but ...
0
votes
5answers
197 views
How to find $\sqrt[3]{8i}$
How do I find the following cube root?
$$\sqrt[3]{8i} = ?$$
I tried by adding $\sqrt[3]{i^3 + 8i + i}$ but that is where my imagination quits.
2
votes
1answer
202 views
How to find logarithms of negative numbers?
Logarithms of negative numbers must be complex.
But how do you find $\ln{(-2)}$ expressed in something like $x \cdot i$ where $x \in \mathbb{R}$?
7
votes
3answers
849 views
Drawing $z^4 +16 = 0$
I need to draw $z^4 +16 = 0$ on the complex numbers plane.
By solving $z^4 +16 = 0$ I get:
$z = 2 (-1)^{3/4}$ or $z = -2 (-1)^{3/4}$ or $z = -2 (-1)^{1/4}$ or $z = 2 (-1)^{1/4}$
However, the ...
2
votes
3answers
161 views
How to calculate $\sqrt{\frac{-3}{4} - i}$ [duplicate]
Possible Duplicate:
How do I get the square root of a complex number?
I know that the answer to $\sqrt{\dfrac{-3}{4} - i}$ is $\dfrac12 - i$. But how do I calculate it mathematically if I ...
-2
votes
1answer
160 views
Smallest positive integer for equation
I am having trouble identify the smallest positive integer $n$ such that $(\frac{1+i}{1-i})^n = 1$
Can someone please throw on approach?
(Also, please correct the equation in the form of Tex/Latex ...
-1
votes
3answers
71 views
Complex Numbers Question
1) let $Z_0$ be a solution of $Z^{13}-13Z^{7}+7Z^{3}-3Z+1=0$, Is it true that $Z_0$'s conjugate is also a solution?
2) let $Z_0$ be a solution of $Z^{2}+iZ+2=0$, Is it true that $Z_0$'s conjugate is ...
2
votes
1answer
304 views
How can I solve a system of equations?
If $x, y, z$ are complex numbers, how can I solve this system of equations
\begin{cases}
x(x-y)(x-z)=3;\
\\y(y-z)(y-x)=3;\
\\z(z-x)(z-y)=3.
\end{cases}
40
votes
4answers
1k views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
1
vote
3answers
201 views
Square Roots of Complex Number $3-4i$
What I did
$z^2=3-4i$
$(a+bi)^2 = 3-4i$
$a^2-b^2+2abi = 3-4i$
Then got 2 simultaneous equations
$a^2-b^2=3$ and $2ab=-4$
Solve for $a^2$ in 1st equation: $a^2=3+b^2$
Subbed into 2nd equation ...
2
votes
1answer
134 views
Intersections of 2 circles
Let me ask a similar question to the one I did yesterday.
I got answers which said that the following problem had no general solution for x and y.
$\sqrt{(n_1-x)^2+(n_2-y)^2}=n_3$
...
3
votes
2answers
134 views
Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$
Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$ with $z \in \mathbb{C}$.
3
votes
1answer
113 views
Solve for $z$ in $4z^2+8|z|^2-3=0$
$$4z^2+8|z|^2-3=0$$
I have to find $z$.
$|z|^2 = z\cdot \bar{z}$, but I don't know if this helps in this situation.
