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

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3
votes
4answers
376 views

Weird stuff happening with complex numbers on a ti-84

So, I'm trying to do some calculations for my Electrical engineering homework. This requires a bit of algebra with complex numbers. I have been finding that some of the calculations that my calculator ...
2
votes
1answer
38 views

Does $\lim_{x \to 0}({z^2\over \overline z})$ exist? $(z\in \mathbb{C})$

I am trying to figure out if $\lim_{x \to 0}({z^2\over \overline z})$ exists or not. This is a way I though to show that this does not exist but I am not entirely sure. Let $a_n={1\over n}$ and ...
0
votes
0answers
33 views

$f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
-3
votes
1answer
50 views

When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn. For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy ...
0
votes
2answers
28 views

Function with exponent imaginary power

If we have $u=\frac{4c(e^{-is}-e^{is})}{(e^{-is}+e^{is})^2} \tag 1$ where c is a constant and s is a variable. Can we write $e^{is}$ in terms of u ? Means Can we write $e^{is}$ as $\psi(u)$ , a ...
1
vote
1answer
26 views

Principal branch of the complex logarithm does not always satisfy the product formula

My book asks to prove: $\text{Ln}[i \cdot (-1+i)]$ does not equal to $\text{Ln}(i) + \text{Ln}(-1+i)$ where $\text{Ln}$ gives the principal log of the complex number. I don't see why this is true ...
0
votes
1answer
20 views

Is this log identity true?

I'm wondering if the exponent property carries forward to the complex log. In other words, for some complex numbers $z$ and $w$ does $\ln(z^w) = w\ln(z)$?
4
votes
1answer
67 views

When is $x^{2^n}$ dense in $\mathbb{S}^1$, for $|x|=1$?

Motivation: So I just saw this question: Limit when $n\rightarrow\infty$ of $\text{sgn}(\sin(2^n \pi x))$ with $x\in(0,1)$ fixed., and the answer involves diadic numbers and things of the kind. Most ...
5
votes
3answers
96 views

Can anyone tell me how $\frac{\pi}{\sqrt 2} = \frac{\pi + i\pi}{2\sqrt i}$

I was working out a problem last night and got the result $\frac{\pi + i\pi}{2\sqrt i}$ However, WolframAlpha gave the result $\frac{\pi}{\sqrt 2}$ Upon closer inspection I found out that ...
5
votes
5answers
61 views

Is $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ open or closed?

I am trying to figure out if the set $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ is open or closed or maybe none of that. I hope someone could provide a hint to solve this. Can this set be ...
0
votes
3answers
29 views

All Values of a Complex expression

I am asked to find all values to $$\left(\frac{1-i}{\sqrt2}\right)^{1+i}$$ I do not know how to approach a power with complex part. Any help would be appreciated.
2
votes
1answer
34 views

Proving $\left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin n\left(\frac{\pi }{2-x}\right)$

How to solve the following question? If $n$ is an integer, show that \begin{eqnarray} \left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin ...
0
votes
0answers
17 views

Is a complex function really just an infinite dimensional matrix?

I have recently sort of come to the understanding that integrating two functions multiplied together is a sort of infinite dimensional dot product, and I only know this from taking an undergraduate ...
14
votes
7answers
819 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 ...
0
votes
4answers
33 views

Find all complex and real roots of higher degree polynomials, given one root

$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!
295
votes
18answers
54k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
4
votes
1answer
89 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
0
votes
1answer
80 views

Imaginary Numbers [duplicate]

Can someone help me by explaining what imaginary numbers actually are, please? I know that it is defined as $i = \sqrt{-1}$ and I can find umpteen sources telling me how to calculate with them but I ...
2
votes
1answer
30 views

Variance of amplitude and phase from sin and cos regressors in polar coordinates

On a data set, I estimated the sine and cosine weights at a specific frequency, $\beta_{\sin}$ and $\beta_{\cos}$. I can extract the amplitude and phase from these regressors as follows: ...
4
votes
3answers
1k views

Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
0
votes
1answer
56 views

Why does $2x_1x_2y_1y_2 \leq x_1^2y_2^2+x_2^2y_1^2$?

When I tried to prove the triangle inequality $|z_1+z_2| \leq |z_1| + |z_2|$ algebraically for complex variables $z_1$ and $z_2$, I came across this inequality and found that this is always true no ...
1
vote
0answers
13 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
0
votes
2answers
72 views

What is the generalized definition of '<' and '>' for complex numbers? [duplicate]

I'd expect this question to be asked here before, but I've not been able to find it. The generalized definition of the multiplication operator for complex numbers is simple: The product of the ...
0
votes
1answer
34 views

$\Im\left ((a+bi)^n\right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*}$

I am trying to connect $a,b,n$ such that $$\Im\left ((a+bi)^n \right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*} \mathrm{or } \; \mathbb{R^*}$$ What I tried was write $(a+bi)^n$ as $\sqrt{a^2+b^2} ...
7
votes
1answer
190 views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
3
votes
2answers
538 views

Locus of complex numbers $z$ with restricted $(z+1-i)/(z-1-i)$

Problem Describe the locus of the following points on the Argand diagram: $$\left|\frac{(z+1-i)}{(z-1-i)}\right| = 1$$ and $$\mathrm{arg}\left[\frac{(z+1+i)}{(z-1-i)}\right] = \pm ...
0
votes
1answer
15 views

Question on the Closure Properties of $\mathbb C$

I read here: https://proofwiki.org/wiki/Properties_of_Complex_Numbers, that the Complex Numbers are closed under addition and multiplication. I'm having trouble understanding why. I realize this is a ...
0
votes
1answer
42 views

Straight Line Equation in Complex Plane

Hi there, I'm confused about the straight line equation in complex plane: how does "0 = Re((m+i)z + b)" come from "y = mx + b" ? I mean when I see "y = mx + b", I can draw a graph in my mind, but ...
1
vote
2answers
89 views

How find this $a$ such $(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$

Question: find all the complex $a$,such for every complex $z_{1},z_{2}(|z_{1}|,|z_{2}|<1,z_{1}\neq z_{2})$,such $$(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$$ My idea: ...
2
votes
3answers
104 views

Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$

State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$. By letting $z=\cos\theta+i\sin\theta$, show that $$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos ...
2
votes
1answer
84 views

About the identity $e^{i\pi}=-1$

I have a question about the famous identity of Euler $e^{i\pi}=-1$. I opened the other day this question about the number of roots of a complex number with irrational exponent. Under this light and ...
0
votes
1answer
39 views

Roots of a quintic function

I need some pointers in the right direction for this question: Three of the roots of the equation $ax^5+bx^4+cx^3+dx^2+ex+f=0$ are $-2$, $2i$ and $1+i$. Find $a$, $b$, $c$, $d$, $e$ and $f$. I ...
-1
votes
3answers
62 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
3
votes
1answer
27 views

Every compact set $S\in \mathbb{C}$ is bounded

This is my proof for every compact set $S \subseteq \mathbb{C}$ is bounded. Let $S \subseteq \mathbb{C}$ be compact and assume that it is not bounded. Then for each $z\in \mathbb{C}$ and for each ...
2
votes
3answers
77 views

Cross Ratio is positive real if four points on a circle

Given four points $a, b, c, d \in \mathbb{C}$ on a circle I want to show that the ratio $ \frac{(a-c)(b-d)}{(a-b)(c-d)}$ is a positive real number. I have shown that it is real by expanding $ ...
0
votes
1answer
62 views

Given $w=e^{2i\pi/7}$, perform algebraic manipulations with complex numbers like $w+w^2+w^4$

It is not my own homework and I forgot how to solve this kind of things. Anyway, the following are the statement of the homework and my attempts: The homework: Let $w=e^{2i\pi/7}$, $u=w+w^2+w^4$, ...
2
votes
1answer
26 views

Evaluating $|a^b|$ when $a,b$ are complex

Here, $a^b=e^{b\log a}$ for some suitable (but fixed in advance) branch of the $\log$ function. What is the most general formula for $|a^b|$ when both $a$ and $b$ are complex, and what are the ...
1
vote
1answer
40 views

Finding which base number given operations

$$ (35_a + 24_a) * 21_a = 1081_a $$ Which base is the above number? Any advice on how to solve questions like these? I tried making it in to a polynomial: $(3a+5 + 2a+4) * (2a+1) = 108a + 1$ ...
0
votes
2answers
23 views

Calculating the argument of a complex number… something tends towards infinity?

A simple question, but I like to be clinical with my choice of words: I have a complex number, $z=-i$. If I were to calculate the argument of this complex number, $arg(z) = tan^{-1}( \frac{-1}{0}) ...
0
votes
1answer
39 views

$D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact

This is the proof I wrote for $D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact. $$ \bigcup_{n=2}^{\infty}D_{1-(1/n)}(0) $$ is clearly a open covering of $D_1(0)$. Consider the finite ...
1
vote
1answer
54 views

Definition of $z^0, z=a+bi, a,b \in \mathbb{R}, z \neq 0$

Today I found in a True-False the question; Does the equality $$z^0=1, z=a+bi, a,b \in \mathbb{R}$$ hold $\forall z \in \mathbb{C^*}$? The thing is, this was never clearly defined in the book, and ...
26
votes
10answers
3k views

$i^2$ why is it $-1$ when you can show it is $1$? [duplicate]

We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$ EDIT: I see this has been dealt with before but at least with ...
1
vote
0answers
26 views

Complex trigonometric equation

Find a solution to the equation $\tan(z)=7i$ which satisfies the condition $0<\Re(z)< \pi$} We use the $\sin(z)=\frac{e^{zi}-e^{-zi}}{2i}$ and $\cos(z)=\frac{e^{xi}+e^{-xi}}{2}$. Here is ...
0
votes
0answers
88 views

Given roots on the unit circle, find the complex reciprocal polynomial

Given "all" the $m$ zeros on the unit circle of a complex reciprocal polynomial of even degree $2N > m$, can we find the polynomial? The known conditions are: We have all the $m$ zeros on the ...
1
vote
0answers
74 views

Proving Ptolemy Theorem using complex number

I am working on an assignment proving Ptolemy Theorem using complex number, and I am looking at a textbook Complex Numbers and Geometry by Hahn. Here is what I am working at this moment: THE ...
0
votes
0answers
26 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
votes
4answers
63 views

Why has my prof worked out the multiplicative inverse of complex numbers this way?

She explains how to obtain the multiplicative inverse: In what follows, z=a+bi and w=c+di are complex numbers with a,b,c,d∈R. Is there a multiplicative inverse of z? If so, what is it? Note ...
0
votes
3answers
37 views

How do you divide complex numbers in polar form?

A question in my textbook asks: Find $\frac{z_1}{z_2}$ if $z_1=2\left(\cos\left(\frac{\pi}3\right)+i\sin\left(\frac{\pi}3\right)\right)$ and ...
1
vote
1answer
25 views

The annulus $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open

I want to prove that the set $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open. This is my attempt. Let $z \in A_{r,s}(z_0)$. Then $|z-z_0|-r>0$. Let $r'=[|z-z_0|-r]/2$. Then ...
30
votes
9answers
4k views

Is “$a + 0i$” in every way equal to just “$a$”?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$). I say this is practically the case, so in every calculation you ...