Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

learn more… | top users | synonyms

0
votes
2answers
506 views

Given the cartesian coordinates of four points, how to calculate the interection of two lines they form?

Given four complex numbers $A, B, C, D$ interpreted as points on the plane, how can I calculate the number that represents the intersection of the lines formed by $A, B$ and $C, D$?
2
votes
1answer
140 views

Visualizing the flat complex conic

Consider a conic $X^2 + Y^2 - Z^2 = 0$ in $\mathbb{C}P^2$. In an affine chart $Z \neq 0$ it is supposed to look like a circle (however it looks in $\mathbb{C}^2$), but the deceptiveness of imagining ...
1
vote
2answers
314 views

Fixed points of $e^z$

How would one find the fixed points of $e^z$, where $z$ is complex (if there are any)? I feel this problem probably has a really obvious answer, and for some reason, I'm just not getting it. Thanks.
4
votes
1answer
114 views

Geometric/Simpler proof for the following complex numbers problem

I wonder if there is a geometric proof or a short proof of the following: let $z_1,z_2,z_3$ be three complex numbers of modulus $r$. prove that the number $$ ...
6
votes
1answer
305 views

Why are primitive roots of unity the only solution to these equations?

I was led by this question to the following problem: Find $n$ complex numbers $\lambda_1\dots\lambda_n\in\mathbb{C}$ that satisfy $$\begin{align} \sum_i\lambda_i & =0\\ \sum_i\lambda_i^2 ...
13
votes
4answers
3k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
0
votes
2answers
2k views

Complex conjugate of function

I have a wavefunction $\psi(x,t)=Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)}$. $A$ and $B$ are complex constants. I am trying to find the probability density, so I need to find the product of $\psi$ ...
13
votes
6answers
787 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 ...
1
vote
4answers
271 views

solve complex equation

$x^8 = \frac{1+i}{\sqrt{3} - i} = \frac{\sqrt[8]{\frac{2}{\sqrt{2}}}(\cos \frac{\pi}{4} + i \sin{\frac{\pi}{4}})}{2 \cos \frac{\pi}{6} + i \sin \frac{3\pi}{2}}$ What's the way to solve this kind of ...
5
votes
1answer
967 views

Using the fifth roots of unity to find the roots of $(z+1)^5=(z-1)^5$

The question I am working on starts of with: Find the five fifth roots of unity and hence solve the following problems I have done that and solved several questions using this, however ...
4
votes
1answer
165 views

Complex number inequality?

Suppose $|z|>1$ for $z$ a complex number. I'm trying to build a certain comparison test to test convergence. I'm wondering, is it true that $$ \frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1}? $$
2
votes
2answers
110 views

How is $\mathbb{H}$ a $\mathbb{R}$-Algebra, but not a $\mathbb{C}$-algebra?

It says that $\mathbb{C}$ is not in the center of $\mathbb{H}$. Definition of $\mathbb{K}$-algebra for a ring if $Z(R)=K$. However, you can do this $(a+bI+cJ+dK)(e+fi)=(e+fi)(a+bI+cJ+dK)$. So I don't ...
6
votes
3answers
366 views

Determining $|z-1|$ when $z=\cos\theta +i\sin\theta$ and $\theta$ is acute

As the question indicates we are supposed to find the modulus of z-1. When trying to solve the problem I drew a diagram which you can see below: The book I am working in solved a similar problem ...
0
votes
5answers
274 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
394 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 $x \in \mathbb{R}$?
2
votes
2answers
328 views

How to write a complex number in polar form

Complex number given: $x = 1 + \cos \alpha + i \sin \alpha$ Desired form is something like $|x| \cdot e^{i \cdot \phi} = |x| \cdot (\cos \phi + i \sin \phi)$. I somehow got completly stuck how to ...
0
votes
1answer
108 views

Complex Numbers Equation [duplicate]

Possible Duplicate: How can you find the complex roots of i? How can I find the solutions of the equation $$(2z+1)^5-i=0,$$ over the complex numbers $z\in\mathbb{C}$?
10
votes
2answers
476 views

Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$

1) Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$. Prove: Is it true that $a \in \mathbb{Q}$? 2) Suppose $a \in \mathbb{C}$, ...
4
votes
6answers
944 views

Solving $z^4 + 2z^3 + 6z - 9 = 0$

I'm trying to solve $z^4 + 2z^3 + 6z - 9 = 0$. $z$ is a complex number. I usually can solve those equations when they are of second degree. I don't know what to do, breaking out $z$ doesn't help... ...
4
votes
3answers
315 views

Proving that $O^{51} + \bar{O}^{51}$ is real

Let $O \in \mathbb{C}$. How can I prove that $O^{51}+\bar{O}^{51}$ is a real number, or in other words: $\Im(O^{51}+\bar{O}^{51}) = 0$?
1
vote
0answers
65 views

Hyperbolic Uniformization Metrics

I have been working Euclidean Ricci Flow but have been having considerable trouble trying to implement the same discrete gradient descent functionality in hyperbolic space. I am following the ...
7
votes
3answers
2k 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
219 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 ...
13
votes
2answers
584 views

Inequality with Complex Numbers

Consider the following problem: Prove that for every set of complex numbers $\{z_i\}$, with $i$ ranging from one to $n$, there is a subset $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge ...
4
votes
1answer
239 views

How do you integrate $\cos(x^n)$, specifically for $n=-1$?

How does one integrate $\cos(x^{-1})$? I understand that the function is not defined at zero, but it is well defined, continuous, and real over the rest of $\mathbb{R}$. Nonetheless, when I put ...
2
votes
0answers
128 views

Hyperbolic Universal Covering Space

I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic ...
-2
votes
1answer
260 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
72 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 ...
0
votes
1answer
91 views

Solution of a system of second order algebraic equations in complex numbers

What is the simplest solution for this set of equations: $ \sum_{i=1,3,5,..}^{N-1} \left | x_i \right |^2=c_1,\ \sum_{i=2,4,6,..}^{N} \left | x_i \right |^2=c_2,\ \sum_{i=1,3,5,..}^{N-1} ...
4
votes
1answer
90 views

Is there a $z$ for which $z$, $1+i$, $(1+i)z$ and $e^z$ are collinear?

Is there a $z$ for which $z$, $1+i$, $(1+i)z$ and $e^z$ are collinear? There is a close call around $z = .18 + 1.09i$ but I'd like to see a mathematical solution.
0
votes
1answer
354 views

Finding a matrix that has complex Eigenvalues

I have an assignment where I need to create 2x2 matrices for each of the following Eigenvalue pairs. ...
1
vote
3answers
128 views

I don't understand this proof about Gaussian integers

Theorem: If p is a Gaussian prime and $p|zw$ for some gaussian integer $z,w \in Z[i]$ then $p|z$ or $p|w$. Suppose $p \not| z$ and lets deduce $p | w$. Let $u$ be a greatest common divisor of $p, ...
1
vote
4answers
158 views

How to analyze the modulus of $\lambda = (1-2\mu)+2\mu\cos\theta+i\nu\sin \theta$?

Consider the complex number $$ \lambda = (1-2\mu)+2\mu\cos\theta+i\nu\sin \theta $$ where $i=\sqrt{-1}$, $\mu,\nu$ are constants, and $\mu>0$. Question: How can I get that $|\lambda|\leq ...
1
vote
1answer
209 views

Why are the coefficients of the base states of a qubit complex numbers?

Why are qubits represented as $$\left|{q}\right\rangle = \alpha\left|{0}\right\rangle+\beta\left|{1}\right\rangle\equiv\alpha\left[{1 \ 0}\right]^T+\beta\left[{0 \ 1}\right]^T; ...
0
votes
1answer
84 views

Representing complex numbers with nested exponentiation of rationals

Define $L_0=Q$ $L_1=\lbrace x \in C; e^{x} \in L_0 \rbrace$ $L_{-1}=\lbrace x \in C; \ln{x} \in L_0 \rbrace$ $L_{n+1}=\lbrace x \in C; e^{x} \in L_n \rbrace$ $0$ is in $L_1$ and $L_0$. Do any ...
1
vote
1answer
80 views

Lower and upper bounds for fractional linear transformation

If we have $k(z)=\frac{z}{1-tz}$ which is convex in unit disk, then $k(\bar{z})=\overline{k(z)}$, $k(z)$ maps real axis to real axis where $|z|\leq{r}$, $t\in\mathbb{R}$. What is the upper and lower ...
2
votes
2answers
353 views

roots of complex numbers

It is known that exists formula for geting a square root of complex number without use of De Moivre formula. Will be interesting if we can find the cubic roots of complex numbers without using De ...
2
votes
1answer
309 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}
2
votes
6answers
153 views

What is the least positive integer $n$ for which $(-\sqrt{2}+i\sqrt{6})^n$ is an integer?

Compute the least positive integer $n$ for which $(-\sqrt{2}+i\sqrt{6})^n$ will be an integer, where $i$ is the imaginary unit. I did the binomial expansion and just plugged in numbers for $n$ ...
2
votes
2answers
260 views

Finding complex solutions of an equation

How does one solve this equation. I would like to see the solution of this problem in steps. $z\cdot\bar{z}=\left|3\cdot z \right|$ EDIT: Is it possible to solve this by converting to the form ...
21
votes
2answers
1k views

What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset ...
10
votes
4answers
3k views

How do you integrate imaginary numbers?

How would you find, for instance, $\int_{0}^{4} i\> x dx$? Can you just treat $i$ as a constant, or do you have to do something more sophisticated? Thanks!
12
votes
2answers
658 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
11
votes
2answers
205 views

Is it possible to construct an ordered field which is also algebraically closed?

It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
1
vote
1answer
186 views

Are numbers : $(-1)^{i} , 1^{-i} , 1^{i} $ transcendental numbers? [duplicate]

Possible Duplicate: What is the value of 1^i? According to Euler's formula : $e^{ix}=\cos x + i\cdot \sin x$ we may write : $$e^{i\cdot \frac{\pi}{2}}=i \Rightarrow \left(e^{i\cdot ...
44
votes
4answers
2k 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
2answers
135 views

Finding Complex Number $z$ in $\frac{z+2i}{z-2i}=\frac{7-6i}{5}$

What I did: Cross Multiply, try to expand out the mod and args, but they all seem to lead to dead end (probably I am not seeing something)
1
vote
3answers
114 views

Complex equation solution. How can i resolve it?

I have this complex equation $|z+2i|=| z-2 |$. How can i resolve it? Please help me
0
votes
1answer
109 views

Using De Moivre's Theorem Before & After

I was doing the following question: Show $(1+\sqrt{3}i)^9 + (1-\sqrt{3}i)^9 + 2^{10} = 0$ Hint show $(1+\sqrt{3}i)^9 = (1-\sqrt{3}i)^9 = (-2)^9$ I got $1+\sqrt{3}i = 1-\sqrt{3}i = ...
1
vote
3answers
69 views

How to get from $2^{99} \cdot (\cos{(99\times \frac{5\pi}{6}) + i\cdot \sin{(99\times \frac{5\pi}{6})}})$ to $0+2^{99}i$?

How do I get from 2nd last to last step? How did they simplify cos & sin $99\times \frac{5\pi}{6}$?