How can I find the continued fraction expansion for the square root of 5. Do this without the use of a calculator and show all the steps.
If I want to find the continued fraction of $\sqrt{n}$ how do I know which number to use for $a_0$? Is there a way to do it without using a calculator or anything like that? What's the general algorithm for computing it? I tried to read the wiki article but was overwhelmed and lost. I tried Googling but couldn't find a website that actually explained this question.  If anyone has a good site that answers these questions either, please let me know. Thanks!
Post apocalyptic journey in search of knowledge from fallen society (ours) I'm racking my memory for a post-apocalyptic novel that I borrowed from a library maybe 15 years ago, but I'm sure it was published a few years before that though (maybe early 90s?).  I remember a party of people travelling across the flooded USA after some kind of apocalypse (climate change/flooding?). Towards the end of the book the group makes their way to New York(?) and finds a partially flooded library, discovering a set of books that gives hope for the return/rebuilding of society. I used sculpting on a plane, and things looks great. However, when I press P to play, the whole ground become a mess. I can see through it in various places, and there are also nasty looking black edges everywhere.  another problem, I made a UV Sphere to display the sky, but when I press , P to play it disappears! sudo apt-offline install not updating working with procedure  to update Ubuntu 18.04 offline machine, the sudo apt-offline install /path_to_downloaded_package_folder do not apply any updates/upgrades. The beginning of the massage I get is :   sudo apt-offline install ./updates_path/ No changelog available (Y) Yes. Proceed with installation (N) No, Abort. (C) Display changelog (?) Display this help message. What would you like to do next:  (y, N, ?)y Proceeding with installation ERROR: Path for keyring is invalid: /etc/apt/trusted.gpg gpgv: Signature made Mon 06 Jan 2020 08:17:27 AM MST gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" gpgv: Signature made Mon 06 Jan 2020 08:18:52 AM MST gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" gpgv: Signature made Mon 06 Jan 2020 08:18:25 AM MST gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" gpgv: Signature made Thu 26 Apr 2018 05:38:40 PM MDT gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" security.ubuntu.com_ubuntu_dists_bionic-security_InRelease synced. security.ubuntu.com_ubuntu_dists_bionic-security_Release synced. security.ubuntu.com_ubuntu_dists_bionic-security_Release.gpg synced.   What am I doing wrong? Every time I open a folder from the desktop, a prompt is shown that the said folder is opened. How do I stop that? Why is current a scalar quantity? Cannot change stroke width in Inkscape I am trying to change the stroke width of an icon in Inkscape but, unlike other times I have done this, this time the stroke width appears to be 0 px and I am not allowed to make it thinner:    The stroke width is actually quite large already, is there another setting I should edit to make it thinner?  You can download the file I am working on .  BTW: sorry for the potato quality of the screenshot, this is what you get using Inkscape on a retina screen :/ What does f-stop mean? Is it the same thing when people say "2 stops" for example? Calculating $\lim_{x\to0} \left\lfloor\frac{x^2}{\sin x \tan x}\right\rfloor$ Find $$\lim_{x\to0}  \left\lfloor\frac{x^2}{\sin x \tan x}\right\rfloor$$ where $\lfloor\cdot\rfloor$ is greatest integer function   I am a high school teacher. One of my students came up to ask this limit.  For $\lfloor\frac{\sin x}{x}\rfloor$, I have used $\sin x &gt; x$ using increasing decreasing functions.  I tried to prove $x^2 &gt; \sin x \tan x$ using increasing /decreasing  function but I am not getting it. I found this animal, which I suspect it's an insect, on the floor of a kitchen in Spain some days ago. It's about 8-10 mm long. I don't think I've ever seen it before. What is it? The unique style of the diagrams at  has an informative but nice hand-drawn touch. I guess they are actually drawn by hand but just recently on our partner site for Mathematica someone asked how to draw a similar diagram such as this one with Mathematica's plot functions ().    Now without Mathematica this should be possible as well but I have a hard time getting the details right. Can such a diagram be created in TeX with a similar design? My first attempt does not look particularly great but here it is (the font is from ):      \documentclass{standalone}     \usepackage{tikz,color}     \usepackage{pgfplots}     \usepackage{fontspec}     \setmainfont{Humor-Sans}      \begin{document}     \begin{tikzpicture}     \pgfplotsset{every axis/.append style={line width=1pt}}      \begin{axis}[%         axis x line=bottom,         axis y line=left,         xtick={1.2, 2.9, 4.5},         xticklabels={-3 days, -1 day, 5 minutes},         yticklabels={}     ]     % 1st plot     \addplot [cyan!80!white, samples=200, domain=0:6] {0.05*rnd+3+(sin(deg(x))^2)/sqrt(x)*exp(-(x-2))};     % white background for 2nd plot     \addplot [white, samples=200, domain=0:6,line width=4pt] {0.05*rnd+0.4*x+2+x^2*sin(deg(x))^2*exp(-x)};      % 2nd plot     \addplot [red!80!white, samples=200, domain=0:6] {0.05*rnd+0.4*x+2+x^2*sin(deg(x))^2*exp(-x)};      \end{axis}       \node at (2.7,4.6) {downhill};     \draw  plot[smooth, tension=.7,line width=2pt] coordinates {(1.2679,5.5206) (1.9454,5.493) (2.4017,4.9261)};      \end{tikzpicture}      \end{document} sudo credential caching on by default Why does autoboxing make some calls ambiguous in Java? I noticed today that auto-boxing can sometimes cause ambiguity in method overload resolution. The simplest example appears to be this:  public class Test {     static void f(Object a, boolean b) {}     static void f(Object a, Object b) {}      static void m(int a, boolean b) { f(a,b); } }   When compiled, it causes the following error:  Test.java:5: reference to f is ambiguous, both method     f(java.lang.Object,boolean) in Test and method     f(java.lang.Object,java.lang.Object) in Test match  static void m(int a, boolean b) { f(a, b); }                                   ^   The fix to this error is trivial: just use explicit auto-boxing:  static void m(int a, boolean b) { f((Object)a, b); }   Which correctly calls the first overload as expected.  So why did the overload resolution fail? Why didn't the compiler auto-box the first argument, and accept the second argument normally? Why did I have to request auto-boxing explicitly? how to prove that $\ln(1+x)< x$ Note: the eventual outcome of this discussion involved dropping this particular off-topic reason, along with other changes. See:  &nbsp; &mdash;Shog     I was using the reason "demonstrate a minimal understanding" myself quite frequently, but the more I review, the more I have doubts about it and I think this reason can either be dropped entirely or at least rephrased, so I started to use "problems with code you've written must describe the specific problem" instead:     Questions asking for code must demonstrate a minimal understanding of the problem being solved.   So the poster has posted some description, maybe even a snippet of code and in his mind that already proves that he has that minimal understanding required. After all he described the problem and what he tried to solve. So telling him that he lacks that understanding is rather rude, and from the posters point of view doesn't even apply. What both version of the reason not emphasize is that a description of a problem is not various versions of     it doesn't work   but that we rather mean really specific details (see below). I can often see in the reopen review queue that the "improved" version just includes a more verbatim description, while what usually would be expected is some specific error message, a trace or a log, or whatever applies.  Then it continues with:     Include attempted solutions, why they didn't work, and the expected results. See also: Stack Overflow question checklist   comparing it to:     problems with code you've written must describe the specific problem — and include valid code to reproduce it — in the question itself. See SSCCE.org for guidance.   Which says essentially the same, only in more detail.   So the "minimum understanding" reason is IMO a more rude version of the reason telling the user that an SSCCE is missing.  The intent of giving this reason is to tell the user that he needs to give exact details to be able to help him, so IMO a checklist would be much better. Something like:     Your question was closed because it may not provide enough details. Things you can do to improve your question to get an answer include:         Format the code in a way that makes it readable and easily understandable.   Provide the smallest working sample of code demonstrating the problem (link to SSCCE)   Include your input and output.   If applicable provide logs (i.e. Android debugging log, stack trace, etc.).   If applicable provide the exact error messages (compiler, etc.).      And something like this should be best included already in the banner that appears when a question is closed instead of being hidden in some pages you often enough have to actively looking for.  It might be helpful to adjust this, based on the tags or close reason, as I realize that I'm more focused on the programming area, and the above list may not apply to other areas of SO.  Update  Another thing I forgot to mention (but wanted to) is this:     Questions asking for code   Is IMO essentially an invitation of the type of questions     I have a problem, can you provide the solution? (with an optional please)   while     Questions concerning problems with code you've written   puts at least some emphasis on the fact that the poster has to put some effort himself in trying to solve the problem by providing the code which he used to attempt the solution. Is the convex hull of closed set in $\mathbb R^{n}$ is closed? Is convex hull of closed set in $\mathbb R^{n}$ closed? I accidentally discovered () that a closing square bracket within an optional argument delimited by [ ] can cause problems.  Here is example code illustrating the issue:  \documentclass{article}   \begin{document}  \tableofcontents  \section[Square brackets ([ and ])]{Square brackets: [ and ]} % bad: the TOC now has "Square brackets ([ and", and the actual title is a mere ")".  Some text.  \section[{Square brackets ([ and ])}]{Square brackets: [ and ]} % good (works as originally intended)  Some text.  \end{document}   The text to be used as the optional (as well as the non-optional) argument contains a ], and this character is not supposed to function as the optional argument terminator but as a matching square bracket/parenthesis. Perhaps this is as it should be, but the question is how to best circumvent any potential problems. Enclosing this in a group works, but I don't know whether doing this has other side effects (perhaps not here in the case of \section but hopefully someone else has a better example of where grouping could lead to a problem). There is no (obvious) way other than writing [{ }] because [ and ] are not normally escaped. That [ and ] are normal characters and bear syntactic function (as optional argument delimiters) creates this potentially problematic situation.  Is simply enclosing the entire optional argument in a group ({ }) a good solution? Are there other/better ways? I was running my OS off a USB stick fine and the wifi was working. I installed it to my SSD and setup again but not the WIFI adaptor isn't working.  I followed the first time  and it worked but since installing onto the SSD and doing the same its not working.  No wlan0 interface found Please check drivers, try lsusb to check USB device is present root@goldminer:~# lsusb Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 002: ID 0bda:0811 Realtek Semiconductor Corp.  Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub   On the working device  root@coalminer:~# dkms status amdgpu, 18.10-572953, 4.13.16-hiveos, x86_64: installed amdgpu, 18.10-572953, 4.4.0-127-generic, x86_64: installed nvidia, 390.59, 4.13.16-hiveos, x86_64: installed nvidia, 390.59, 4.4.0-127-generic, x86_64: installed rtl8812au, 4.2.2, 4.13.16-hiveos, x86_64: installed rtl8812au, 4.2.2, 4.4.0-127-generic, x86_64: installed   On the SSD  root@goldminer:~# dkms status amdgpu, 18.10-572953, 4.13.16-hiveos, x86_64: installed amdgpu, 18.10-572953, 4.4.0-127-generic, x86_64: installed nvidia, 390.59, 4.13.16-hiveos, x86_64: installed nvidia, 390.59, 4.4.0-127-generic, x86_64: installed rtl8812au, 4.2.2, 4.13.16-hiveos, x86_64: installed (WARNING! Diff between built and installed module!) rtl8812au, 4.3.8.12175.20140902+dfsg: added rtl8812au, 5.2.9: added   Any advise? I think from trying to fix it ive been making it worse. What happens if you do sudo passwd without specifying a user? I wanted to change my own password (as an admin) so I did the following command:  sudo passwd   and then it prompted for a password but when I tryed to relog, my password was not changed. I'm now wondering what password did I changed since it did prompted me to enter a password but I'm not sure what password did it changed exactly.  I know that if I do sudo passwd &lt;MY_USER&gt; it works, I'm just wondering without a user what happens.  Thank you How to start a GUI software on a remote Linux PC via SSH Sometimes I need to start XMBC media player or other GUI software a one of my PC (small Xubuntu PC used as a media center).  Usually I do this starting a X11vnc server on the remote PC via SSH and than connecting with an Xvnc client to the Xfce desktop.  Is there a way to start a GUI software on a remote Linux PC via SSH?  Thanks! I need an adjustment layer which will invert in the same way as when you select some text in a text layer