# Tagged Questions

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?
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 ...
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 ...
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.
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: ...
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 ...
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 ...
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 ...
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?
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 ...
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 $c$?
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.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
33 views

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 $z=x+iy$ ...
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 ...
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 ...