Finding nth roots of complex number If I have a complex number of the form $z = a+bi$, how would I find the complex roots? I know that each root will be equidistant from each other and will form a circle, but I'm not sure how to solve for the complex roots. I also know that for a complex number with $n$ roots, each root will be rotated $2\pi/n$ radians. It's not an issue of graphing the complex roots, but it is for solving for them.
The $n$ complex $n$th roots of a complex number $z$ Suppose $z$ is a nonzero complex number, so $z=re^{i\theta}$. Show that there are only $n$ distinct complex $n$-th roots, given by $r^{1/n}e^{i(2\pi k+\theta)/n}$ for $0\leq k\leq n-1$.   My proof: Let $z=re^{i\theta}\in\mathbb{C}$ and $a\in\mathbb{C}$ s.t. $a^{n}=z$. Let $a=\rho e^{i\varphi}$. Then, $a^{n}=(\rho e^{i\varphi})^{n}$. Setting $\rho^{n}e^{i\varphi n}=re^{i\theta}$ and rewriting in polar form, we get $$\rho^{n}(\cos(n\varphi)+i\sin(n\varphi))=r(\cos(θ)+i\sin(\theta)).$$ Thus $\rho^{n}=r$ $\rightarrow$ $\rho=r^{1/n}$ and $n\varphi=\theta\rightarrow\varphi=\theta/n$. Thus the roots of $a\in\mathbb{C}$ have the form $$r^{1/n}(\cos(\tfrac{\theta}{n})+i\sin(\tfrac{\theta}{n}))=r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))$$ for some $j\in\mathbb{Z}$ due to the periodicity of trig functions. Then, $$r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))=r^{1/n}e^{i\left(2\pi j+\tfrac{\theta}{n}\right)}=r^{1/n}e^{i(\theta+2\pi k)/n}$$ for $0\leq k\leq n-1 ,j=nk\in\mathbb{Z} $.  I feel like there could be a more concise way to put this.
The Additional Drivers settings interface is no longer available in Ubuntu 17.10  How can I install proprietary drivers in Ubuntu 17.10?  I need drivers for Wifi &amp; Bluetooth (BCM43142) and Intel Graphics.  The earlier Additional Drivers program showed a package named jockey but it no longer seem to exist in repositories. How do I make 'X-Axis Mirror' available in Pose Mode? More often that not, while posing an armature, I want to mirror the pose across the X-axis, so that when I move "brow.L", "brow.R" also moves in the same way.  You can copy and paste a mirrored pose, or select both bones prior to performing a transform, but both of these options aren't particularly good workflow wise.  In edit mode, in the tools panel, under options you can enable "X-Axis Mirror" to copy all transforms across the X-axis in real time, but this option is not available in pose mode.  I'm not very experienced with python, but edit source reveals:  col.prop(ob.data, "use_mirror_x")   My understanding is that "use_mirror_x" refers to a boolean property, but I have no idea how to find where that boolean property is defined or if it would be possible to modify it to get the same result in pose mode.  Any advice on this is appreciated. Maybe there's a better way to achieve this?  Thanks. Practical hyperparameter optimization: Random vs. grid search Increasing post view count automatically I installed Blender from official Ubuntu repository but its version is always behind the version available from the official Blender website.  I'm aware that I can launch the program within the downloaded folder, but i want to replace the install version with a new version downloaded from the website?   It must be present in the unity launcher.  I'm also aware of the available PPA's that are available, however I just want to install Blender manually.  Thanks in advance How to disable the touchpad? I want my touchpad to be disabled when I use my mouse. How can I do that? touchpad-indicator has stopped working on 11.10. It used to work on 11.04. Gnome3 is not a solution as I don't like it and find it buggy. I'm new to blender and trying to model the blade of my helicopter and can't figure out how to make an edge curved (for the red tip and selected edge). The object is a plane, with one loop cut. I attempted making a bevel Ctrl B, but nothing happens (A dotted line appears attached to the cursor, moving does nothing, left-clicking does nothing).  also I cant add a curve modifier, just says its disabled. What is the correct way to model this? How do I iterate over a range of numbers defined by variables in Bash? How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$ The following problem:  The Fibonacci number is denoted by  $f_{n}$, show that the following holds when $n$ is a positive integer:  $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$  My idea was to do an induction proof, which holds for $n$ = 1  $f_{0}f_{1} + f_{1}f_{2} = f^{2}_{2}$  So we assume it holds for $n$ and show for $n+1$  $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} + f_{2n}f_{2n+1} = f^{2}_{2n} + f_{2n}f_{2n+1}$  I fail to get  $f^{2}_{2n+1}$ out of the term  $f^{2}_{2n} + f_{2n}f_{2n+1}$  Is there a difference between "show that" and "prove that". So might be induction the wrong tool here?  Many thanks for your hints in advance :-) My reputation rating on SO took this hit an hour ago, and it is due to what appears to be the simultaneous deletion of multiple questions.  I am uncertain why it happened, am concerned, and would appreciate an explanation.  Going to  shows that it was removed due to moderation.  Was a script run deliberately, or were we hit with a question deletion attack; what happened?  -12     today -2  1 hour ago  removed     How to send a 2d Vector over UDP in c++ -2  1 hour ago  removed     How do I use the Eclipse debugger to debug a Java application? -2  1 hour ago  removed     PHP string error; inconsistency with local and server -2  1 hour ago  removed     Unable to start the MySQL service after a server migration -2  1 hour ago  removed     How to use Java to print a text file on a ticket printer? -2  1 hour ago  removed     How can all of the ZBrush Obj files be imported to make a complete model? I got my Galaxy S4 today (i337M), it's wayyy better than my old SII hehe.  I immediately removed the stock Bell firmware and put the latest snapshot available from CyanogenMod.  I'm wondering if it's possible to convert apps like S Planner, S Note, S Health to CyanogenMod.  Current stock android version available for i337M is 4.2.2. Latest Android version for CM is 4.4.  If I install CM 4.2, can I import/install the APKs and then upgrade to 4.4?  Hope you'll be able to help me :) So I had an exam today and one of the questions were: Give an example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points. At first I had no idea how to do it then I came up with this, even though I know it's wrong:  $$f(x)= \begin {cases} \ \sqrt{x}&amp;\text{if }x &lt;0\\ \sqrt{-x}\ &amp;\text{if }x &gt; 0\\ \end {cases}$$  Like I said I know it's wrong..but can someone give me an example of this? Thanks. Is  && statement() the same as if() statement()? The question is (from Topology without tears) that: Let $(X,d)$ be a metric space and $\tau$ the corresponding topology on $X$. Fix $a \in X$. Prove that the map $f:(X,\tau) \rightarrow \mathbb{R}$ defined by $f(x) = d(a,x)$ is continuous.  My first attempt was that let an open ball, $B_{\epsilon}(a)\subset \mathbb{R}$ and show that $f^{-1}(B_{\epsilon}(a)) \subset \tau$. Since $f(x) = d(a,x)$, $f^{-1}(B_{\epsilon}(a))=B_{\epsilon}(a) \in \tau$.   I am pretty sure the last sentence is wrong. I am not sure if every open ball is an open set in topology induced by metric space.  Please give me some hint and direction. Thanks "To do this" or "For doing this" to start explaining something? Many questions have been asked and answered about "to + bare infinitive" vs "for + gerund" in the middle of a sentence.  However, when one wants to start explaining something, what is better? For instance:  "Now, let us focus on baking the bread.   To do this, start by turning the oven on.", or For doing this, start by turning the oven on."?   Are there even better expressions? I have not found anything concerning this situation, and I feel that the answers about the same problem but in the middle of a sentence do not apply here. Levels or First Differences, VECM or VAR for Ultimate Impulse Response Functions? How did Dumbledore gain the loyalty of Fawkes? How did Dumbledore find Fawkes. Did Dumbledore have Fawkes even when working as a Transfiguration teacher or acquired it after having Headmaster position. They both have a very strong bond between them. When did Dumbledore gain this. Update: Dumbledore says that Ollivander wrote to him when Harry's wand was purchased; if Dumbledore did not have Fawkes before 1938, he would not know about the two feathers being in the wands, so Fawkes and Dumbledore must have met and bonded before 1938  “My wand's feather came from Fawkes?” Harry said, amazed. &quot;Yes,” said Dumbledore. “Mr. Ollivander wrote to tell me you had bought the second wand, the moment you left his shop four years ago.” -Harry Potter and the Goblet Of Fire, Chapter 36-The Parting Of the Ways.  Update 2: JKR says that Dumbledore was the only owner of Fawkes. So it is to be believed that Fawkes was a wild Phoenix before its bonding with Dumbledore.  Peter Humphreys for BBC Newsround: Who did Fawkes previously belong to and will he play a vital role in the next book? JK Rowling: I am not going to answer about the role in the next books, which probably gives you a big clue, and he has never been owned by anyone but Dumbledore. You will notice that when Harry goes back in the Pensieve in this book, Fawkes is never there, and ­­ no, I am sorry, not in this book, I take that back. When Harry has previously seen the study with a different headmaster he saw it with Dippet and Fawkes was not there then. Fawkes is Dumbledore's possession, not a Hogwarts possession. I have a Plantronics BackBeat Sense SE bluetooth headset. When Connecting it to Windows 10, it installs two profiles in Playback devices list:    PLT_BBTSENSE Hands-Free. PLT_BBTSENSE Stereo.   The Second one (Stereo) is set to be the "Default Device" and the "Default Communication Device" on the system.   At this point the sound works great and normally. The problem happens when I want to use the mic. When I start any program that uses the mic (recorder, chat, VoIP Calls, gaming, etc.) The sound suddenly stops working And I can only use the mic until I stop the recorder or the call.   The thing that kind of fixes the problem is setting the (Hands-Free) as the default device. I can hear the sound when I do that and it works with the mic but it becomes Very bad quality.  What I became to understand is that bluetooth had different profiles for sound and mic: A2DP and HSP. The first handle the voice(High Quality) but the latter handles the mic and reduces the sound quality.  Is this really the case? Is there anyway to make them work simultaneously? Is there any solution to this problem at all? What does "Change" represent in the User Leagues? What does the Change field represent on the  for SO? Refer to the highlighted areas in the image below:   Rank: the user's ranking for the selected time period (week, month, quarter, year) Change: +/- change representing... ??? Total Reputation: the user's total reputation [Time Period] Reputation: the user's reputation for the selected time period (week, month, quarter, year)  So what is the Change field measuring? Mars needs water and women I read this book in high school too many years ago for me to think about. Martian colonist are raiding earth's oceans for water and because of the conditions on Mars where only male children are conceived, they are also on the look out for young women as potential mates. Our main character while on a raid, bags not only a young woman but her brother as well. Returning to Mars they find that Earth has attacked Mars can't remember much more.