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

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6
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2answers
233 views

Discovery of complex numbers

A popular story about the discovery of the complex numbers goes as follows. Once the formula for the solution of the cubic equation has been discovered its application to the equation $x^3=15x+4$ ...
6
votes
3answers
113 views

Solving equation $(z-i)^3=(4-i\sqrt{48})z^3$ in $\mathbb C$.

I need to solve the following equation in $\mathbb C$. $$(z-i)^3=(4-i\sqrt{48})z^3.$$ I tried with trigonometric form , but having $z-i$ on the LHS is confusing me, since I get ...
6
votes
4answers
203 views

Solve $\operatorname{Arg} (z-2) - \operatorname{Arg} (z+2) = \frac{\pi}{6}$

I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$. I know that $$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} ...
6
votes
2answers
95 views

Some equation with complex numbers

Given $a,b \in \mathbb{C}$ such that $a^2+b^2=1$, it is clear that $x:=a\bar{a}+b\bar{b}$ is a real number and that $yi:=a\bar{b}-\bar{a}b$ is imaginary (i.e $y$ is real). Moreover, a direct ...
6
votes
1answer
110 views

Axiomatic definition of complex numbers

Trying to build axiomatically the set $\mathbb C$ of complex numbers, my first attempt was to define $\mathbb C$ with three structures: addition, multiplication and conjugate: $\langle\mathbb ...
6
votes
2answers
159 views

Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
6
votes
1answer
214 views

Why is the bailout value of the Mandelbrot set 2?

For the past few days I've been studying the Mandelbrot set, and many say that if the iterations of a point stay within a magnitude of 2, the point converges. A very natural question of "why is the ...
6
votes
1answer
632 views

What is the formula for the first Riemann zeta zero?

I found this approximation of which an earlier version I posted in the chat room: $$7 \pi -\text{Log}\left[\frac{7}{2} e^{-7 \pi /2}+\frac{5}{2} e^{-5 \pi /2}+\frac{3}{2} e^{-3 \pi /2}+e^{5 \pi /2}+2 ...
6
votes
1answer
436 views

What are the uses of split-complex numbers?

The set of Complex numbers can be used in lots of domains like geometry, vectorial calculations, solving equation with no real solution etc. But what are the uses of split-complex number that can't be ...
6
votes
1answer
225 views

Complex Logs and Roots of Unity

I need to find all the solutions to the following using logarithms: $(e^z-1)^3=1$ where z is a complex number. I am told that using roots of unity I can break this equation down but I must be missing ...
6
votes
3answers
125 views

Towards a formula for the Euler $\phi$ function?

$\Phi_n(1)$ and $\Phi_n(-1)$ for the cyclotomic polynomials are well-known. I am now looking for $$\Phi_n(i)$$ and/or $$\Phi_n(-i)$$ with $i$ the complex unit. The reason is : I suppose it is ...
6
votes
1answer
114 views

An Inequality question

I have the following question. I have to find a $\delta>0$ such that for all complex numbers $x,y$ the following holds true - \begin{equation} \frac{1}{2\pi}\int_0^{2\pi}|x+e^{it}y|\,dt \ge ...
5
votes
7answers
470 views

What is the value of $i+i^2+i^3+\cdots+i^{23}$? [duplicate]

Can anyone help me with this question and show me a step by step solution please? The imaginary number is $i$ is defined such that $i^2=-1$. What is $i+i^2+i^3+\cdots+i^{23}$?
5
votes
6answers
389 views

Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?

In my ongoing strugle to understand $e^{\pi i}$ I managed to narrow down my conceptual difficulty. I'm having intuitive trouble understanding why $(1 + iX/n)^{n}$ is conceptually the same as a ...
5
votes
7answers
464 views

Is $0^0=1$ postulate independent of all other axioms of complex numbers?

This question is inspired by the other question which asked for a proof that $i^i$ is a real number. Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
5
votes
4answers
185 views

Why is it impossible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that $|z_1- z_2|=|z_1-z_3|=|z_2-z_3|=|z_1-z_4|=|z_2-z_4|=|z_3-z_4|$?

A. It is possible to find distinct $z_1,z_2,z_3\in \mathbb C$ such that $|z_1-z_2|=|z_1-z_3|=|z_2-z_3|$. Answer: True B. It is possible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that ...
5
votes
5answers
2k views

What is the norm of a complex number?

I'm in a number theory class and I'm trying to understand what the norm is... For some complex number $Z = a +bi$, $Z$ times the conjugate of z is equal to $(a^2)+(b^2)$. Most of what I've read about ...
5
votes
1answer
108 views

A case where $z^z = 0$ where $z$ is complex number

Is there any case where $z^z = 0$ where $z$ is complex number? The case excludes the case where $z=0$.
5
votes
3answers
803 views

Complex conjugate of $z$ without knowing $z=x+i y$

Is it possible to determine (and if so, how) the complex conjugate $\bar{z}$ of $z$, if you don't already know that $z = x + i y$? I think you can use $\log(z)$ to get the angle, and therefore the ...
5
votes
3answers
986 views

Comparing real and complex numbers

If I'm correct, a complex number can be interpreted as a set in the following manner: $$ \forall x, y \in \mathbb{R}, x + yi = \{(x,\ y)\}.\ \mathbf{(1)} $$ My question is, is it technically ...
5
votes
3answers
226 views

$n$th derivative of $e^x \sin x$

Can someone check this for me, please? The exercise is just to find a expression to the nth derivative of $f(x) = e^x \cdot \sin x$. I have done the following: Write $\sin x = \dfrac{e^{ix} - ...
5
votes
2answers
4k views

Multiplying complex numbers in polar form?

Could someone explain why you multiply the lengths and add the angles when multiplying polar coordinates? I tried multiplying the polar forms ($r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot ...
5
votes
2answers
251 views

Can a Gaussian integer matrix have an inverse with Gaussian integer entries?

Is there any way to characterize the set of complex matrices with Gaussian integer entries whose inverses also have Gaussian integer entries? I'm aware of the numerous examples of integer matrices ...
5
votes
4answers
157 views

$e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?

My question is simply whether the well-known formula $e^{i \theta}$ $=$ $\cos \theta$ $+$ $i \sin \theta$ a definition or there is some proof of the result. It seems to me that the formula is a ...
5
votes
5answers
189 views

Find all real values $a$ and $b$ such that $a+ib=i^{i^{i}}$?

Find all real values $a$ and $b$ such that $a+ib=i^{i^{i}}$ ? My Try : using the fact that $z^w=e^{w \log z}$, First I compute $i^i$ $$i^i=e^{i \log i}= e^{i (\log |i| + i arg (i))}=e^{- ...
5
votes
5answers
373 views

Complex numbers and trig identities. I've heard this question is easy but I don't know how. Help?

Using the equally rule $a + bi = c + di$ and trigonometric identities how do I make... $$\cos^3(\theta) - 3\sin^2(\theta)\ \cos(\theta) + 3i\ \sin(\theta)\ \cos^2(\theta) - i\ \sin^3(\theta)= ...
5
votes
1answer
999 views

Using the fifth roots of unity to find the roots of $(z+1)^5=(z-1)^5$

The question I am working on starts of with: Find the five fifth roots of unity and hence solve the following problems I have done that and solved several questions using this, however ...
5
votes
3answers
1k views

Polynomial of degree 4 with real coefficients, two complex roots given.m

Write in the form f(z) = 0, where f(z) is a polynomial of degree 4 with real coefficients, the equation having (3 + i) and (1 + 3i) as two of its roots. Can anyone help me? I'm guessing the two ...
5
votes
4answers
449 views

solve $ z^2 -6z + 25 $ into complex conjugate

I need to solve this :$$ z^2 -6z + 25 = 0$$ My book says 'complete the square' so : 1.$$ (z - 6/2)^2 -36/4 + 25 $$ 2.$$ (z - 3)^2 -9 + 25 $$ 3.$$ (z - 3)^2 + 16 $$ Now how exactly does the above ...
5
votes
1answer
183 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
5
votes
2answers
100 views

Could anyone explain this curious solution to $7^{2x}= 2^x$

I typed the following on wolfram alpha today : $7^{2x} = 2^x$ and found this as a solution besides $x=0$: $x = \dfrac{2\pi i n}{\log2 - 2\log7}$ where $log$ has a base of $e$ and $n$ is any integer. ...
5
votes
3answers
1k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
5
votes
2answers
5k views

Complex Conjugate of Complex function

I am currently reading Hamming's Numerical Methods for Scientists and Engineers. On pg. 79 he discusses the topic of finding the zeros of a complex analytic function. He then proceeds to discuss ...
5
votes
2answers
226 views

Number of Complex Roots of a Complex Polynomial

This is related to the question I asked regarding finding the complex roots of $z^3+\bar{z}=0$. It turned out that there were 5 complex roots, but because the equation was of degree 3 I was only ...
5
votes
3answers
422 views

Determine the set $\{w:w=\exp(1/z), 0<|z|<r\}$

I can't do this exercise of Conway's Book: For $r>0$ let $A=\{w:w=\exp(1/z), 0<|z|<r\}$, determine the set $A$. Any hints?
5
votes
1answer
115 views

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ .

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ ? How to find if it is convergent or not? Thanks!
5
votes
2answers
83 views

Is $\mathrm{GL}_n(\mathbb C)$ divisible?

A group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Is the group $\mathrm{GL}_n(\mathbb C)$ divisible? Or more precisely, ...
5
votes
2answers
196 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
5
votes
3answers
731 views

Additive quotient group $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the multiplicative group of roots of unity

I would like to prove that the additive quotient group $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the multiplicative group of roots of unity. Now every $X \in \mathbb{Q}/\mathbb{Z}$ is of the form ...
5
votes
2answers
225 views

Complex analysis : If $z =re^{i\theta}$, then prove that $|e^{iz}| =e^{-r\sin\theta}$

Problem : If $z =re^{i\theta}$, then prove that $|e^{iz} | =e^{-r\sin\theta}$ My working : $z = re^{i\theta} = r(\cos\theta + i\sin\theta)$ $\Rightarrow iz = ir(\cos\theta +\sin\theta) $ = ...
5
votes
3answers
100 views

$w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$

I am stuck on the following problem: Let $w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$.Then the triangle in the complex plane with $w_1,w_2,-1$ as vertices ...
5
votes
2answers
93 views

If $\left |z-3\right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$

If $\left |z-3 \right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$. I have got to the point where I have $\left |z \right |= \sqrt{x^2+(y+1)^2} = \sqrt{(x-3)^2+y^2}$ But really ...
5
votes
2answers
206 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
5
votes
2answers
263 views

Primitive roots of unity

I am trying to show that, If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then $$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} ...
5
votes
2answers
891 views

How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate?

How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate? I started with $$(z^2 + 3)^2 + 16 = 0$$ $$(z^2 + 3)^2 = - 16$$ $$z^2 + 3 = \pm 4i$$ Is this the way to start solving the equation or am ...
5
votes
1answer
66 views

Confused about exponents and imaginary/real answers

I am confused about some exponent behavior. $$(-2)^{7.6} = (-2)^{\frac{76}{10}} = ((-2)^{76})^{\frac{1}{10}} = ((-2)^{\frac{1}{10}})^{76}$$ Is there something wrong in this logic? When I plug the ...
5
votes
1answer
75 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
5
votes
5answers
307 views

Prove that $(x^2-x^3)(x^4-x) = \sqrt{5}$, where $x= \cos(2\pi/5)+i\sin(2\pi/5)$

Prove $(x^2-x^3)(x^4-x) = \sqrt{5}$ if $x= \cos(2\pi/5)+i\sin(2\pi/5)$. I have tried it by substituting $x = \exp(2i\pi/5)$ but it is getting complicated.
5
votes
2answers
129 views

Expand $(4+i)(5+3i)$ and show $\pi/4=\arctan{1/4}+\arctan{3/5}$

I can't remember ever having done this before so if someone could help me out that would be great. The question is expand $(4+i)(5+3i)$ and hence show that $\pi/4=\arctan{1/4}+\arctan{3/5}$. ...
5
votes
2answers
1k views

Prove that the zeros of an analytic function are finite and isolated

Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$. The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...