Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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1
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2answers
24 views

Tangent and Circle in Complex Plane

Question:- Three points represented by the complex numbers $a,b$ and $c$ lie on a circle with center $O$ and radius $r$. The tangent at $c$ cuts the chord joining the points $a$ and $b$ at $z$. ...
2
votes
2answers
56 views

On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
-2
votes
0answers
15 views

find the principal part of the function at its isolated singular point

find the principal part of the function at its isolated singular point, and determine whether that point is a pole, a removable singular point, or an essential singular point. (a) f(z) = z exp(1/z) (...
-2
votes
0answers
14 views

Start and endpoint of line, creating arrow heads [on hold]

I have a start point(5.6,4) and an endpoint (6.1,3.15) I want to make an arrow head at the start point that is an equilateral triangle(60 degrees) with a length of .1. How can I accomplish this? ...
5
votes
1answer
61 views

Circles in complex plane.

Find the real value of a for which there is at least one complex number satisfying $|z+4i|=\sqrt{a^2-12a+28}$ and $|z-4\sqrt{3}|\lt a$. My solutions:- Graphical solution:- $|z+4i|=\sqrt{a^2-...
0
votes
2answers
27 views

Squareroot of complex number to the square $\sqrt{z^2}$

I have to calculate $\sqrt{z^2}$ an I am confused about how to procede. I thought about introducing $z=|z|\exp(i\phi+2\pi k) \implies z^2=|z|^2\exp(2i\phi+4\pi k)$. Hence, $$\sqrt{z^2}=\sqrt{|z|^2\...
-1
votes
1answer
137 views

Complex transcendentals not known in component form?

Are there any transcendentals whose real or imaginary components have not been found in exact form?
8
votes
3answers
1k views

Example of a complex transcendental number?

Researching transcendental numbers I have only come across ones with a transcendental real part. I can't think of any which are pure imaginary or are not based on a real transendental number, t, of ...
2
votes
1answer
17 views

Mutually orthogonal vectors in a complex vector space?

Consider a Matrix $A \in \mathbb C^{m \times n}$, $m<n$ which is build by vectors like $$ A = \begin{pmatrix} | & | & & | \\ \vec a_1 & \vec a_2 & \cdots & \vec a_n \\ | &...
0
votes
0answers
55 views

Spec$(R)$ a scheme of finite type over $\mathbb{C} \implies R$ is a finitely generated algebra over $\mathbb{C}$.

Suppose $(\text{Spec}(R), \tilde{R})$ is a scheme locally of finite type over $\mathbb{C}$. We want to show that $R$ is a finitely generated $\mathbb{C}$-algebra. Since $(\text{Spec}(R), \tilde{R})$ ...
1
vote
1answer
30 views

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
1
vote
2answers
49 views

Complex Numbers: A Basic Manipulation

$$ z = e^{i \phi} \tan \Bigg (\frac{\theta}{2} \Bigg). $$ What is, $$ \frac{|z|}{1 + |z|^2} $$
0
votes
2answers
36 views

How to separate real and imaginary parts of an expression

i am taking a course on complex numbers and I need to know how to separate the real and imaginary parts of a trigonometric expression like 1) $$\cos^{-1}(ix)$$ 2) $$\sin^{-1}(e^{i\theta})$$
11
votes
2answers
976 views

Clarification on “Every polynomial function of degree $\ge1$ has at least $1$ zero in the complex number system.”

The Fundamental Theorem of Algebra says "Every polynomial function of degree $\ge1$ has at least $1$ zero in the complex number system." My question is, where do the rest of the zeroes of the ...
1
vote
3answers
65 views

If $|z|=\sqrt{a^2+b^2}$, then what is $z$?

Perhaps I’m having some difficulty understanding the complex plane. Say you have a complex number $z=a+bi$, where $a$ is the real part and $b$ is the imaginary part. Why do you plot the real part on ...
1
vote
1answer
26 views

multiplying 3 variable a ,bi, cj

I know how to do this $A × B = ( a + bi ) × ( c + di )$, but I don't understand how to do this $ A × B =( a + bi +cj) × ( d + ei +fj) =? $ I'm not sure how to group them or what to do with the i ...
1
vote
2answers
62 views

Complex equation $z^2 + i\overline{z} = 0$

I need to calculate the trigonometric form of the complex solution and then show the algebric form. $$z^2 + i\overline{z} = 0$$ As far as I know when I have in the equation the $\overline{z}$ the ...
0
votes
1answer
65 views

On the inequality $|z_1-z_2|^2 \lt (1+c)|z_1|^2+(1+\frac{1}{c})|z_2|^2$

Now, I know this question has been asked here but my question doesn't deal with finding a solution, my question deals with checking the validity of the question. Question:- If $z_1, z_2$ are ...
-6
votes
0answers
26 views

Proof for $|z_1+z_2| \le |z_1|+|z_2|$ and $|z_1-z_2|\ge |z_1|-|z_2|$ [closed]

I need proof for $$|z_1+z_2| \le |z_1|+|z_2|$$ and $$|z_1-z_2|\ge |z_1|-|z_2|$$
0
votes
2answers
28 views

How is $\left| \exp(iaRe^{i\theta}) \right|\le e^{-aR\sin\theta}$?

In one book on complex variables, in the proof of Jordan's Lemma, For any constant $a>0$, and any radius $R>0$, it is stated that $\left| \exp(iaRe^{i\theta}) \right|\le e^{-aR\sin\theta}$. I ...
1
vote
1answer
48 views

Is this a pure imaginary number?

I've met this formula and I need to demonstrate that it is purely imaginary (it has no real part). $\frac{1}{2}\log(-\exp(i2\pi q))$, //for a real "input" q. As I don't know much about maths, what I'...
-8
votes
0answers
41 views

Find real and imaginary part of the following complex number [closed]

Sqrt(17)divided by 2+ 12 divided by sqrt(70)
2
votes
1answer
58 views

Sum of series $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$

I need to find sum of the series involving cube roots of unity $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$. Found it in an old test paper. I applied Geometric Progression Sum Formula....
0
votes
4answers
114 views

If $i^4 = 1$ then isn't $i = 1$ [closed]

The imaginary unit "$i$" is equal to the square root of $-1$ by definition. If we take $i$ to the forth power then we get $1$ But if $i^4 = 1$ then solving for i we get 1 instead of $\sqrt{-1}$ Can ...
8
votes
4answers
170 views

Proving $\frac{1}{\cos^2\frac{\pi}{7}}+ \frac {1}{\cos^2\frac {2\pi}{7}}+\frac {1}{\cos^2\frac {3\pi}{7}} = 24$

Someone gave me the following problem, and using a calculator I managed to find the answer: $$\frac {1}{\cos^2\frac{\pi}{7}}+ \frac{1}{\cos^2\frac{2\pi}{7}}+\frac {1}{\cos^2\frac{3\pi}{7}} = 24$$ ...
0
votes
1answer
35 views

Problem involving modulus of complex numbers [closed]

Let $a, b \in \Bbb C$, such that either $|a| = 1$ or $|b| = 1$. Show that $|a-b|\le|a-\bar a b|$.
2
votes
1answer
70 views

Why must $|z|\gt 1$ be the necessary condition

Question:- If $\left|z+\dfrac{1}{z} \right|=a$ where $z$ is a complex number and $a\gt 0$, find the greatest value of $|z|$. My solution:- From triangle inequality we have $$|z|-\left|\dfrac{1}{...
4
votes
1answer
95 views

An inequality involving two complex numbers

Let $z_1, z_2 \in \mathbb C$ and $a,b \in \mathbb{R} \setminus \{0\}$. Prove that $$|z_1|^2+|z_2|^2-|z_1^2+z_2^2|\le 2\dfrac{|az_1+bz_2|^2}{a^2+b^2}\le |z_1|^2+|z_2|^2+|z_1^2+z_2^2|$$ ...
1
vote
1answer
50 views

Finding the orthocentre of a trinagle.

Now, I know this has been asked here but my question is something else so please bear with me. Question:- If the vertices of a triangle are represented by $z_1, z_2, z_3$ respectively then show ...
-1
votes
0answers
8 views

Find locus of $z$ satisfying given condition [closed]

$$|z|^2 + (z+ \bar{z})-2=0$$ In this question I've put $z= a+bi$ and then solved but I have a problem in plotting values of $a$ and $b$, please help me out.
6
votes
2answers
236 views

Complex numbers as exponents [duplicate]

Is there any formula to calculate $2^i$ for example? What about $x^z$? I was surfing through different pages and I couldn't seem to find a formula like de Moivre's with $z^x$.
1
vote
3answers
72 views

Why is $-1>0$ not enough?

Theorem: Prove that no order can be defined in the complex field that turns it into an ordered field. Proof: Suppose complex field is an ordered field. So, either $i$ or $-i$ must be positive. ...
4
votes
1answer
217 views

Infinite tetration of $-2.5$

Let $a_n$ be the sequence $z, z^z, z^{z^z} ...$ for $z \in \mathbb{C}$. This is sometimes called the iterated exponential with base $z$. I am investigating the above sequence for $z = -2.5$. After ...
1
vote
1answer
27 views

Path of Complex Numbers [closed]

Find the image $S'$ of the square $S$ with vertices at $1+i$ ,$2 + i$ ,$2+2i$ and $1+2i$ under the linear mapping $T(Z) = Z +2 -i$. Write down the geometric description.
-2
votes
2answers
41 views

Complex variables help [closed]

Find the value of $(1-i)^i$..
-1
votes
3answers
32 views

complex variables problem [closed]

By using the polar form of the complex number prove that, $|z_1 z_2| = |z_1| |z_2|$ and $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$
0
votes
2answers
35 views

Re-defining the complex unit for teaching purposes

I often come across students who are confused by the idea that the complex unit, $i$, is defined as $i^2 = -1$. Since we are using the complex numbers in an engineering course, we use the complex ...
-2
votes
1answer
52 views

Is this matrix going to be real or complex?

I hope that this is the right forum where to post this question (and not here). I have a Chi-Square Kernel Matrix (using the second version, which is positive-definite) ...
-2
votes
2answers
38 views

Find modulus of $\frac{|z_1-z_2|}{|1-(z_1)(\overline{z_2})|}$ [closed]

If $z_1$ and $z_2$ are two different complex numbers and $\lvert z_1\rvert=1 $ then find $$ \frac{\lvert z_1-z_2 \rvert}{\lvert 1-z_1\bar{z_2} \rvert} $$
0
votes
1answer
44 views

Prove Basic Complex Number Inequalities

Let $$z_1 = a_1 + b_1i$$ $$z_2 = a_2 + b_2i$$ where $$|z_j| = \sqrt{a_j^2 + b_j^2}$$ Prove $$|z_1 + z_2| \le |z_1| + |z_2|$$ $$|z_1 + z_2| \ge |z_1| - |z_2|$$ $$|z_1 - z_2| \ge |z_1| - |z_2|$$ $$...
0
votes
4answers
45 views

Prove $|z1/z2| = |z1|/|z2|$ without using polar

Prove $|z1/z2| = |z1|/|z2|$ where $$z_1 = a_1+b_1i$$ $$z_2 = a_2+b_2i$$ $$|z_1| = \sqrt{a_1^2+b_1^2}$$ $$|z_2| = \sqrt{a_2^2+b_2^2}$$ $$RHS = \frac{|z_1|}{|z_2|} = \frac{\sqrt{a_1^2+b_1^2}}{\sqrt{...
0
votes
1answer
50 views

Find |z| if the given expression is purely imaginary [closed]

Find $|z|$ if $\dfrac{z-2}{z+2}$ is entirely imaginary. I know that if a number is purely imaginary, then $z-\overline{z}=2i$(some integer)
2
votes
2answers
40 views

Find the modulus of $|z-5|/|1-3z|$ when z is given

If $z = 3-2i$ then find $$\frac { \left| z-5 \right| }{ \left| 1-3z \right| } $$ I've substituted z by $|z|^2/z$ conjugate but still cant figure out what to do, Thanks in advance
3
votes
2answers
53 views

Roots of Unity with Rational Real Parts

All of the $4^{\text{th}}$ and $6^{\text{th}}$ roots of unity have real parts that are rational numbers. Are these the only roots of unity $z$ such that $\text{Re}(z)\in \mathbb{Q}$ ?
1
vote
3answers
33 views

Verify $\frac{(z_1+z_2)^2}{z_1\times z_2} \geq 0$

I have two complex numbers, $z_1$ and $z_2$, that both have the modules equal to 1 and their arguments are $\theta_1$ and $\theta_2$, respectively. I'd like to verify that $$\frac{(z_1+z_2)^2}{z_1\...
0
votes
4answers
103 views

$ z = 1 + 2i $ - Prove that $ z^n \notin \mathbb{R} $ [duplicate]

$$ z = 1 + 2i \ (complex \ number) \\ z^n = a_n + b_ni \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) \\ We \ know \ that: \ b_{n+2} - 2b_{n+1} + 5b_n = 0 \\ a_{n+1}=a_n-2b_n \\ b_{n+1}=b_n+2a_n $$ ...
1
vote
0answers
21 views

$|\mathcal{R}((2a+ib)^{2n+1})|\neq b$ for coprime $2a,b$ and $n>1$

Assume $n>1$ is natural and set $f_n(a,b):=\mathcal{R}((2a+ib)^{2n+1})$ Prove that for every coprime pair $2a,b\in\mathbb{Z}$: $|f_n(a,b)|>b$. Note that we have $b|f_n(a,b)$ so the only thing ...
-1
votes
2answers
29 views

Example of non-algebraic field extension of $\mathbb{C}$ [closed]

Can you give me an example of some non-algebraic field extension of $\mathbb{C}$? In case there is any of course. I've been thinking about it for a while but can't find one single example, or one ...
0
votes
0answers
31 views

What is the value of $x^{n}$ when $x\in\Bbb N$ and $n\in\Bbb C$? [duplicate]

How can I calculate $x^{n}$ when $x\in\Bbb N$ and $n\in\Bbb C$ ? Respectively $x^{ni}$
1
vote
3answers
50 views

$ z^n = a_n + b_ni $ Show that $ b_{n+2} - 2b_{n+1} + 5b_n = 0 $ (complex numbers)

$$ z = 1+2i \ (complex \ number) \\ z^n = a_n + b_ni \ \ \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) $$ Prove that $ b_{n+2} - 2b_{n+1} + 5b_n = 0$ How can I solve this? Thank you! EDIT: Or ...