2
votes
1answer
35 views

What does taking the $n^{\text{th}}$ root of a complex number geometrically mean?

What are the geometrical implications of taking the $n^{\text{th}}$ root of a complex number, say $3+4i$. What is the geometrical implication of $\sqrt[n] {3+4i}$ in the complex plane?
0
votes
0answers
37 views

Plane geometry in the complex plane

i am asked to find the area of a triangle that has vertices $0, w_{1}, w_{2}$ in $\mathbb{C}$ by applying the transformation $z \rightarrow \bar{w_2}z.$ My attempt: since we are multiplying by the ...
4
votes
1answer
47 views

relationship between complex numbers

Consider the following: Two equilateral triangles inscribed in a circle. The vertices of the large triangle are the geometric images of the three cubic roots of $z$ (a complex number). The small ...
0
votes
1answer
78 views

How to find the roots of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ using trigonometric methods

Can all the roots (real or complex) of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ be found using trigonometric methods? Many thanks to all of answers.
0
votes
0answers
34 views

Basic complex geometry: Reflexion by a line. Where does $ \overline{z}$ go?

Where does $ \overline{z}$ go in the end? Shouldn't the formula be $f(z)= w^2 \overline{z} +2isw$? Thanks in advance. Original link: ...
2
votes
1answer
34 views

Representation of cardiod in the complex plane

I noticed that the complex function $$f(z) = \frac{2}{(z+i)^2}$$ seems to map the real line onto the cardioid given by the polar equation: $$r = 1- \cos(\theta).$$ I was wondering if there is a simple ...
1
vote
2answers
35 views

Geometric proof and extension of |a|=|b|=|c|=a+b+c=1 => a=1 or b=1 or c=1

We have $a,b,c\in\mathbb{C}$ verifying $|a|=|b|=|c|=a+b+c=1$, we have to show that $a=1$ or $b=1$ or $c=1$. That can be rather easily proved using trigonometry formulas. Is there a way to prove it ...
3
votes
3answers
50 views

Suppose $z \neq -1$ is a complex number of norm 1. Prove that $(\frac{1+z}{|1+z|})^2 =z $

Suppose $z\neq -1$ is a complex number of norm 1m $(|z| =1)$. I know that $z= n + mi$, what is the most straightforward way of solving this problem? I was also given the following sketch for a ...
0
votes
1answer
52 views

Generalised Pythagorean Theorem?

$|a+b|^2=|a|^2+|b|^2+2 Re(\overline ab)$ Can anyone explain this equality to me? How it is derived?
1
vote
2answers
40 views

the geometric explain of $t = x-\frac{a}{3}$ in the simplify of cubic equation $x^3+ax^2+bx+c=0$

Assume $$f(x) = x^3+ax^2+bx+c$$ we have $$f''(x)=2a+6x$$. we get $x = -\frac{a}{3}$ Magically, If we take the transformation: $$t = x -\left(-\frac{a}{3}\right)$$. we can transform the above ...
1
vote
1answer
37 views

Is it possible to find a formula for $d$ in terms of $a$, $b$, and $c$?

If $a$, $b$, $c$, and $d$ are complex numbers on the unit circle, and $\overline{ab}\perp\overline{cd}$, is it possible to find a formula for $d$ in terms of $a$, $b$, and $c$?
1
vote
1answer
52 views

$e^{i\theta} = \cos\theta + i \sin\theta$ for solid angle steradian , working mechanism.

$e^{i\theta} = \cos\theta + i \sin\theta$ for solid angle steradian , working mechanism. How will it work? For radian is 2D , i want for 3D.
0
votes
1answer
54 views

Shortest distance formula in complex numbers

Let $L$ be a line in $\mathbb{C}$ that makes an angle $\alpha$ with the real axis. Let $z=x+iy$ be any point on the line $L$. Let $d$ be the shortest distance from $L$ to origin. Prove geometrically ...
-2
votes
2answers
48 views

If $|z_1 - z_2| = |z_1 + z_2|$, then $\arg z_1 - \arg z_2 = \pi/4 $

Problem If $|z_1 - z_2| = |z_1 + z_2|$, show $$\arg(z_1) - \arg(z_2) = \frac{\pi}{4}.$$ Progress I have tried squaring the modulus and using double angle formula for $\tan$, but can't get to the ...
4
votes
2answers
170 views

Complex Numbers Geometry

I'm not sure where to begin on this problem - do I plug in for a and solve for z? I was also given a hint: Let z be a point on the line we're trying to describe. We have good tools in complex ...
0
votes
1answer
48 views

Locus in the complex plane given an equation

I have the question Let $a$ and $e$ be two positive real numbers, with $0 < e < 1$. Describe the locus of the points $z$ in the complex plane which satisfy $|z - ae| + |z + ae| = 2a$. I ...
4
votes
3answers
77 views

A question on plane geometry

$ABCD$ is a square with centre $O$. Let $P$ be the midpoint of $OB$ and $Q$ the midpoint of $CD$. Prove, using vectors and complex numbers, that $AP$ and $PQ$ are perpendicular segments of the same ...
1
vote
1answer
42 views

Show that a point is not contained in a region defined by two circles

Let $w_0$ a point in the complex plane, and $w,w^*$ two points on the same line that passes through $w_0$. The two points are equi-distanced from $w_0$ and on the two different sides of the line. Let ...
0
votes
1answer
154 views

Inversion of Circles

I'm studying for my exam and one of the questions I am stuck on is: Show that under inversion in the unit circle a circle with centre C and radius r inverts into a circle centre ...
1
vote
1answer
146 views

Geometry of Complex Numbers

Write down in the form ${Z}\rightarrow{AZ+B}$ the following transformations of the complex plane: (a) translation in the direction $(2,-3)$ (b) rotation about (0,1) through $\pi/4$ I know from my ...
1
vote
1answer
33 views

How find this inequality find the maximum $z_{5}$

let $z_{1},z_{2},z_{3},z_{4},z_{5}\in C$,such $$\begin{cases} |z_{1}|\le 1,|z_{2}|\le 1\\ |2z_{3}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ |2z_{4}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ ...
0
votes
2answers
443 views

Circles in Complex Planes

Points on the circle centre C and radius r are given by the equation $|Z-C|=r$ or $(Z-C)(\overline{Z}-\overline{C})=r^2$. Where $Z = x + iy$. When multiplied out, I understand that we have ...
0
votes
1answer
39 views

Why is $|z-a|=\rho$ equivalent to $|z|^2-a(z+\overline{z})=\rho^2-a^2$?

I have some problems to understand the following statement from a book about reflections in poincare half-plane modell: For $z,\overline{z} \in \mathbb{C}$ and $ a,\rho \in \mathbb{R}$ we have: ...
0
votes
1answer
52 views

Geometry using complex e powers

The question is to expresss $\cos(4\theta)$ and $\sin(4\theta)$ in terms of $\cos(\theta)$ and $\sin(\theta)$. This in itself is not that hard using geometrical rules. But my problem is that you need ...
1
vote
1answer
24 views

Collections of points of affixes $z$

Suppose a and b two different complex numbers and k a real positive number. How can determinate the collections of points z such as $ L_k=\{{z \in \mathbb{C}} / |\frac{z-a}{z-b}|=k \} $.
1
vote
1answer
24 views

Choosing vectors on a complex sphere

Consider a complex $t$ dimensional unit sphere. Say we pick $n$ points on this with inner products in the set $\{a_1,a_2,\dots,a_r\}$ (we have $n$ inner products with value $1$). Note the set ...
0
votes
1answer
67 views

Show that the equation of a line can be given as ℑm(αz+β)=0

I've just started a non-Euclidean Geometry course and the book we are using has a very brief (and not-so-helpful) section on complex numbers that we sort of went over in class. One of the questions ...
2
votes
1answer
257 views

Loci Of a Circle In The Complex Plane

I am trying to solve: $$\text{|z - 1| = 3|z + 2|, where z = x+iy}$$ workbook is asking for a sketch, but unfortunately it does not provide any answers. I seem to struggle to draw this so I am ...
2
votes
2answers
166 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
38 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
130 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 ...
9
votes
6answers
459 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
115 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
89 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
87 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 ...
0
votes
3answers
67 views

Circle in a complex plane.

Let $C$ be a circle in the complex plane, and let $x$ be a fixed, non-zero complex number. Prove that $\{xz : z \in C\}$ is also a circle. I would really appreciate any help that would get me ...
3
votes
3answers
817 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
298 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
280 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 ...
1
vote
1answer
422 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
410 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
523 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
211 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
147 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
596 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
1k 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
2k 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
303 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
204 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
331 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 ...