Is there a method by which we can prove that $$\sqrt{3}+\sqrt{7}$$ is irrational. It's obviously an irrational number, but I want to prove that mathematically.
Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}, \ \ a \in \mathbb{Q}$, which I don't think is possible, but how to show it?
Can we exempt downvoted accepted answers from getting the top spot? virtualbox crash on kernel 4.13.0-26 Ubuntu 16.04 LTS Virtualbox 5.0.40 ubuntu r 115130  Today after upgrading to kernel 4.13.0-26 guest vbox machines started to crash whole ubuntu. just hard poweroff is working.  If I startup with kernel 4.10.0-42 everything works fine.  what may be the problem ? How to move a window if titlebar is overlapped by Ubuntu menu Somehow the title-bar of chrome was overlapped by Ubuntu menu-bar like attached image.  In this case how to move this chrome window? Can a single sentence be used to distinguish between isomorphic classes of finite structures? The answer seems to be yes, judging from the following exercise I found in the book Mathematical Logic by H.D. Ebbinghaus, J. Flum, and W. Thomas:     Let $S$ be a finite symbol set and let $\mathfrak{U}$ be a finite $S$-structure. Show that there is an $S$-sentence $\varphi _{\mathfrak{U}}$ the models of which are precisely the $S$-structures isomorphic to $\mathfrak{U}$.   I think I have an idea of how to solve this exercise, but I seem to be unable to materialize it. Thanks. How do I display a decimal value to 2 decimal places? Copy file geodatabase using Python? I am trying to copy fgdb from one path to another using Python.  copy(base.gdb,dest)   And got the message:  Error Info:  [Errno 13] Permission denied:   How to fix it? How to fix apt: Signature by key uses weak digest algorithm (SHA1)? Unbounded continuous function on non-compact metric space Suppose $(S,d)$ is a non-compact metric space. Is it possible to construct an unbounded continuous function from $S$ to $\mathbb{R}$? If it is possible, please show the construction method :)  Here's my attempt to solve this problem: Because $S$ is not compact, we can find a neighborhood assignment function $N(x)$ so that a finite cover is not possible. Then each time we can find an uncovered point from the union of neighborhoods before, and change the function by $f(x):=max\{f(x),\text{a function which equals to the indice of this point at that point and gradually vanishes at further points from the selected point}\}$. However I'm concerned that while this function is unbounded, there may be some invalid points for this function (goes to infinity)...  EDIT: As I found out there is a duplicate question, however the elementary solution is not presented in that question, so I still wish to know about this problem. :) What is the easiest way to strip a desktop edition to a server edition? We have installed Ubuntu desktop edition on our development server. Now that we have it in a data center we would like to strip it down to a server edition.  Is there an easy way of doing so rather than just going in and uninstalling packages by hand? About two weeks ago I had a problem with my PC that my brother wrote a question about at .  I still haven't solved the problem, so my PC is unusable now.  Now I think that it would be simpler to re-install both Win7 &amp; Ubuntu than repair them, but there're some data on their partitions. I want to recover this data.  These partitions are not mountable for some reason. I don't know why.  sudo fdisk -l about the problematic partitions:     Device    Boot      Start         End      Blocks   Id  System /dev/sda1            2048     147912703    73955328   83 Linux /dev/sdb2            206848   174079999    86936576    7  HPFS/NTFS/exFAT   And in GParted:      And the error of the mount command:  Failed to mount '/dev/sdb2': Invalid argument The device '/dev/sdb2' doesn't seem to have a valid NTFS. Maybe the wrong device is used? Or the whole disk instead of a partition (e.g. /dev/sda, not /dev/sda1)? Or the other way around?   And the exact same message for /dev/sda1  How can I recover my data from those HDDs?  EDIT     I have tried to specify the type of the drive:  $ sudo mount -t ext4 /dev/sda1 /olddisk mount: wrong fs type, bad option, bad superblock on /dev/sda1,        missing codepage or helper program, or other error        In some cases useful info is found in syslog - try        dmesg | tail  or so $ sudo mount -t ntfs /dev/sdb2 /olddisk NTFS signature is missing. Failed to mount '/dev/sdb2': Invalid argument The device '/dev/sdb2' doesn't seem to have a valid NTFS. Maybe the wrong device is used? Or the whole disk instead of a partition (e.g. /dev/sda, not /dev/sda1)? Or the other way around?   EDIT     sudo ntfsfix /dev/sdb2 returns:  Mounting volume... NTFS signature is missing. FAILED Attempting to correct errors... NTFS signature is missing. FAILED Failed to startup volume: Invalid argument NTFS signature is missing. Trying the alternate boot sector Unrecoverable error Volume is corrupt. You should run chkdsk. Consider a cement block of the same density as sea-water resting on the (perfectly polished) seafloor. I also assume the side-faces of the cement block are highly polished, so that no vertical hydrostatic forces can attack on them. A cable is attached to the top of the block.  My question ist: How much force must one apply to the cable before the cement block begins to lift off the seafloor?  The normal response would be to say that only a negligible force is needed - as the block is practically suspended in the water.  However, as the bottom surface of the cement block is not in contact with the water, it is NOT experiencing the hydrostatic pressure from below - only from above.  Before the crane starts lifting, the (zero-sum) forces on the cement block are:  $$F_{Seawater Pressure} +F_{Gravity on Cement Block} - F_{Seafloor Resistance} = 0$$  as the crane starts applying Force, the Seafloor will reduce its Resistance in the same measure, leaving the forces on the cement block zero:  $$F_{Seawater Pressure} +F_{Gravity on Cement Block} - (F_{Seafloor Resistance}+F_{Crane}) = 0$$  Just before the Cement block lifts, the Seafloor will stop applying Force and the Crane will be applying the only upward Force:  $$F_{Crane} = F_{Seawater Pressure} +F_{Gravity on Cement Block}$$  Once the Cement block lifts off the sea-floor, hydrostatic pressure kicks in on the lower cement surface, suspending the block, but before that - boy does it take a lot of force to lift it!  What is wrong with this reasoning? "Will + be + verb -ing)" vs "Will + verb" ? Can someone explain me the constructions (or however they are called) with the "Will + be + verb -ing)" and the "Will + verb" ? Mass Lead Conversion - Programmatically or Using Data Loader TikZ: turning edges into arrows using "child" notation I'm using TikZ for the first time to draw a simple Kripke model, and I'm using the child notation to produce new nodes. How can I turn my edges into arrows?  \begin{tikzpicture}   \node [circle,draw] (q0) {}   child {node [circle,draw] (q1) {$p$}     child {node [circle,draw] (q4) {$p$}       child {node [circle,draw] (q7) {$p\land q$}}       child {node [circle,draw] (q8) {$p\land q$}}       child {node (q9) {} edge from parent[-&gt;, dashed]}}     child {node [circle,draw] (q5) {$q$}}     child {node (q6) {} edge from parent[dashed]}}   child {node [circle,draw] (q2) {$p$}}   child {node (q3) {} edge from parent[dashed]}; \end{tikzpicture}   I know how to change the color or thickness of all edges by passing an option at the beginning of \begin{tikzpicture}, but I can't turn them into arrows.   Also, while we're at it, is there a way to label these edges? Or am I left with simply using node notation for all my nodes?  Thanks for all your help, Piotr Since the new top bar has been rolled out - a number of different people, established and new users alike, have asked questions on where a specific link has disappeared to with the new top bar design. Where has the link to &lt;X&gt; gone? Ok, after seeing , I have decided to skip to the end and go with .  So what I need is a beginner's practical guide to Git.  "Beginner" being defined as someone who knows how to handle their compiler, understands to some level what a  is, and has touched source control without understanding it very well.  "Practical" being defined as this person doesn't want to get into great detail regarding what Git is doing in the background, and doesn't even care (or know) that it's distributed.  Your answers might hint at the possibilities, but try to aim for the beginner that wants to keep a 'main' repository on a 'server' which is backed up and secure, and treat their local repository as merely a 'client' resource.  So:  Installation/Setup    How do you set up Git?  Try to cover Linux, Windows, Mac, think 'client/server' mindset.         Working with the code          How do you handle binary files (visio docs, for instance, or compiler environments)? How do you merge files changed at the "same time"?    Tagging, branching, releases, baselines    How do you pull a particular 'release'?   How do you resolve conflicts and complete the merge? How do you merge parts of one branch into another branch?        Other   Describe and link to a good GUI, IDE plugin, etc. that makes Git a non-command line resource, but please list its limitations as well as its good.    - Cross platform, included with Git  - Cross platform history viewer, included with Git  - Mac OS X  - Mac OS X history viewer  - Cross platform, commercial, beta  - console GUI for Linux  - GUI for Windows, Linux  - package for Windows, includes friendly GUI  Any other common tasks a beginner should know?     How do I work effectively with a subversion repository set as my source control source?   Other Git beginner's references                 Delving into Git     (and )    I will go through the entries from time to time and 'tidy' them up so they have a consistent look/feel and it's easy to scan the list - feel free to follow a simple "header - brief explanation - list of instructions - gotchas and extra info" template.  I'll also link to the entries from the bullet list above so it's easy to find them later. Is there an IDictionary implementation that, on missing key, returns the default value instead of throwing? The indexer into Dictionary throws an exception if the key is missing.  Is there an implementation of IDictionary that instead will return default(T)? I know about the TryGetValue() method, but that's impossible to use with LINQ. Would this efficiently do what I need?: myDict.FirstOrDefault(a =&gt; a.Key == someKeyKalue);  I don't think it will as I think it will iterate the keys instead of using a Hash lookup. Help us test the shiny new "User Activity" page! (Plus a bunch of new features.) Why can't I buy the Windhelm house (Hjerim)? I've finished the Stormcloak storyline, and now cannot become a Thane of Windhelm without buying Hjerim. The Steward keeps saying something has happened and he can't sell it right now.  What the heck am I missing? I've been doing every random little quest from people in town I can find, hoping to unlock the house. I'm well over the 5/5 requirement of helping people.  Any ideas? Taking a skyline photo with a person closer to the camera during night time How can I take a photo which shows the skyline and the person who is standing around six feet away from the camera with proper lighting at night?   For taking just the skyline I've used long shutter speed and it turned out great, but when including a person on it, the person becomes  blurry because it's impossible for a person to remain static for that long.   Ps.  I do have two flashes, 35mm prime lens as well as 18-300mm zoom lenses as well. The camera I use is Nikon D7200.