Tagged Questions
2
votes
2answers
133 views
$i^{th}$ root(s) of unity
If we define $S:=\{z\in\mathbb{C}:z^n=1\}$ (i.e. the $n^{th}$ roots of unity), then $|S|=n$ (i.e. we have $n$ of them). We can even go as far as to say: $$S=\{z_k:k\in\mathbb{N}\cap[1,n]\}=\{e^{2\pi ...
1
vote
1answer
30 views
similarity : $z'=(1-i)z+1+i$ with the curve of $e^x-1-x$.
Let $S$ be the similarity defined by : $S(z)=(1-i)z+1+i$, for a complex number $z$ in the complex plane.
What is the image of the curve : $y=e^x-x-1$ by the similarity $S$.
My work : Let $z=x+iy$ ...
2
votes
2answers
51 views
Axis of glide reflection
Need to show that if $f$ is a glide reflection then there is only one line $L$
such that $f(L) = L$
What I know is that a glide reflection is an isometry
$$f(z)=a\bar{z}+b,$$ such that $|a|=1$ and ...
7
votes
5answers
159 views
If $A,B,C,D$ are complex numbers on the unit circle with $A+B+C+D=0$, then they form a rectangle
Let $A, B, C, D$ be points on a unit circle. Prove that if $A+B+C+D=0$, then $A,B,C,D$ make a rectangle. (Use complex numbers.)
How do I prove this? I tried to use the dot product of 2 adjacent ...
1
vote
1answer
96 views
Complex Numbers geometry question
on each edge of a quadrilateral ABCD you build a square such that the points H, G, F, E are the centers of these squares (the intersection of the diagonals).
I need to use complex numbers to prove ...
0
votes
1answer
67 views
Geometric meaning of $Arg(\frac{z_{1}}{z_{2}})=2Arg(\frac{z_{3}-z_{1}}{z_{3}-z_{2}})+2\pi k$ for $z_{i}$ on the unit circle
I was given the following question:
Let $z_{1},z_{2},z_{3}\in\mathbb{C}$ s.t $|z_{1}|=|z_{2}|=|z_{3}|=1$,
it is known that
$Arg(\frac{z_{1}}{z_{2}})=2Arg(\frac{z_{3}-z_{1}}{z_{3}-z_{2}})+2\pi
...
2
votes
1answer
66 views
Find a complex number geometrically
Consider the triangle $\Delta ABC$, which $D$ is the midpoint of segment $BC$, and let the point G be defined such that $(GD)= \frac{1}{3}(AD)$. Assuming that $z_A, z_B, z_C$ are the complex ...
2
votes
3answers
436 views
Radius of the spherical image of a circle
This is question 5 on page 20 of the book Complex Analysis by Lars Ahlfors. I have no idea how to answer that problem:
Find the radius of the spherical image of the circle in the plane whose ...
0
votes
2answers
131 views
What is the image of $|z-4i|+|z+4i|=10$?
What is the image of $|z-4i|+|z+4i|=10$?
I tried to simplify this equation but it is too difficult for me. I tried squaring both side but it was too long and I can't get equation like any curve.
...
1
vote
1answer
199 views
Complex numbers as coordinates
Is it possible to have an n-dimensional geometry where each coordinate can be a complex number, or would it make no sense, i.e. lead to contradictions?
Spacetime, can be described as having 4 ...
0
votes
1answer
174 views
Reflect a complex number about an arbitrary axis
This should be really obvious, but I can't quite get my head round it:
I have a complex number $z$. I want to reflect it about an axis specified by an angle $\theta$.
I thought, this should simply ...
4
votes
1answer
272 views
Using the complex logarithm to find the sum of angles in a triangle.
Suppose you have a triangle with vertices $a$, $b$, and $c$. I asked earlier how you can define the angles in a triangle based on the $\log$ function. I received the answer that, for instance, the ...
0
votes
2answers
239 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$?
1
vote
1answer
153 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; ...
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.
$\sqrt{(n_1-x)^2+(n_2-y)^2}=n_3$
...
1
vote
3answers
371 views
Find opposite vertices of a rhombus, given the other 2
I am stuck with this problem. I posted an earlier problem with a square, where rotation with i of 90 degrees was possible. This one is a rhombus, how should I proceed?
Given ABCD is a rhombus with ...
6
votes
2answers
814 views
How do I calculate the equation of a circle given 3 complex numbers?
Given three complex values (for example, $2i, 4, i+3$), how would you calculate the equation of the circle that contains those three points? I know it has something to do with the cross ratio of the ...
11
votes
4answers
738 views
Equation of the complex locus: $|z-1|=2|z +1|$
This question requires finding the Cartesian equation for the locus:
$|z-1| = 2|z+1|$
that is, where the modulus of $z -1$ is twice the modulus of $z+1$
I've solved this problem algebraically ...
8
votes
2answers
249 views
If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear
If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear.
I really can't seem to get anywhere on this problem, ...
5
votes
1answer
159 views
For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero
Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
6
votes
2answers
256 views
Two points on circle resulting in 5 equal regions
What values of $Z_1$ and $Z_2$ make the five regions of the unit circle, shown below, equal in area? $\overline{Z_1}$ and $\overline{Z_2}$ are conjugates of $Z_1$ and $Z_2$; in other words they lie ...
