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

learn more… | top users | synonyms

6
votes
1answer
129 views

determining if a complex number is a root of unity

How would you determine if $a+ib$ is a $n$th root of unity for some unknown $n$? Obviously the modulus of $a+ib$ must be $1$. But you also need to determine if the $a+ib$ is located at the vertex ...
6
votes
1answer
398 views

How to choose a proper contour for a contour integral?

When analyzing real integrals with contour integrals, how does one choose a proper contour integral? Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
6
votes
2answers
272 views

Does it make sense to compare complex numbers in certain circumstances?

I know that $\mathbb{C}$ is an unordered field and that (strictly non-real) complex numbers cannot be 'compared' in the sense that one is less than/greater than another. However, we can compare real ...
6
votes
8answers
223 views

Am I wrong in thinking that $e^{i \pi} = -1$ is hardly remarkable?

I believe my trouble is that the identity, $e^{i \pi} = -1$, comes down to the definition of the exponentiation of $i$, which seems rather obscure to me. This is my current understanding of ...
6
votes
2answers
236 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
6
votes
3answers
383 views

Determining $|z-1|$ when $z=\cos\theta +i\sin\theta$ and $\theta$ is acute

As the question indicates we are supposed to find the modulus of z-1. When trying to solve the problem I drew a diagram which you can see below: The book I am working in solved a similar problem ...
6
votes
2answers
172 views

A Complex Inequality

I just need to show that : $$\int_0^{2\pi}\left|{\frac{i(Re^{i\theta})^\lambda}{1+Re^{i\theta}}}\right| d\theta \le \int_0^{2\pi} \frac{R^\lambda}{R-1}d\theta : 0 < \lambda <1 , R>1$$ Is ...
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 ...
6
votes
2answers
117 views

Evaluate $\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}\ \mathrm dx$ by complex methods

find integral $$\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}dx$$ what I had in mind is to use Euler formula, to turn it into a complex integral and change the limits of integration from $ -\pi$ to ...
6
votes
2answers
239 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
114 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
206 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
115 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
1answer
219 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
639 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
466 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
229 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
175 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
473 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
391 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
466 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
6answers
641 views

Is the Complex Conjugate the Only Way to Get a Real Number?

Is the complex conjugate of a number (or a real multiple of it) the only complex number which, when multiplied with the original number, gives a real number?
5
votes
6answers
148 views

What's the result? $1/i=?$, where $i=\sqrt{-1}$ [duplicate]

I just had my first math class in the university, and I understood everything pretty well, but I think I have misread this one because I read that the result is $-1$. Thanks for your answers!
5
votes
5answers
190 views

Picture/intuitive proof of $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$?

Is there a nice geometric, intuitive or picture proof as to why the easily algebraically provable identity $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$ is true? Note I'm not looking for a ...
5
votes
5answers
52 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 ...
5
votes
4answers
187 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
809 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
1k 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
247 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
254 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
173 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
191 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
377 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
1k 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
456 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
184 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
2k 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
227 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
432 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
3answers
94 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
1answer
116 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
3answers
752 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 ...