Why is $\sqrt{x^2 } = |x|$ ?   Squaring always produces a positive result, and you obviously square the equation before taking the root of it. So where exactly is the problem?
Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't the same equation. For example, say I put $-1$ into them:  $\begin{align} f(x) &amp;= \sqrt {x^2} \\ f(-1) &amp;= \sqrt {(-1)^2} \\ f(-1) &amp;= \sqrt {1} \\ f(-1) &amp;= 1 \end{align}$  $\begin{align} g(x) &amp;= x \\ g(-1) &amp;= -1 \end{align}$  thereby, we conclude that $f(x)$ and $g(x)$ do not produce the same results even though they are mathematically the same. This is also shown when we try to graph the functions:  :    :    :    From this, we can see that given $f(x) = \sqrt {x^2}$, when simplified is not the same as $f(x) = x$. So, is there any way to prove that $y = \sqrt {x^2}$ is not the same as $y = x$ for negative values, but is infact the same as $y = \mid x \mid$?
Convert NSDate to NSString How do I convert, NSDate to NSString so that only the year in @"yyyy" format is output to the string? As I have seen in other posts, i.stack.imgur.com is blocked by numerous companies' firewalls including my own. I have started the process to get the restriction lifted but to increase the likelihood of it being granted, have only requested that i.stack.imgur.com be unblocked.  Staying within the context of SO, imgur and the hosting arrangement SO has with them, are there any other sub-domains/domains that I should be requesting be unblocked? i.e. p.stack.imgur.com or *.stack.imgur.com?  I'm not sure if images can still be embedded/hotlinked in posts from other image hosts or not but if they can I'm not interested in unblocking those hosts, just the addresses used by SO for imgur.  Cheers :) Not booting from USB or CD (SYSLINUX Message) I am trying to install linux on my laptop, a Toshiba Satellite C6550-S5200. I did it once but something happened so I removed it then I had to destroy all data on hard drive so now I have nothing on it. Well I got a iso file burned to a CD and to a flash drive. With the flash drive I get.   SYSLINUX 4.06 EDD 4.06-pre7 Copyright (C) 1994-2012 H. Peter Anvin et al  With the CD it will start booting it but somewhere loading it up, the dots turn all orange and stay that way and my CD drive turns quiet.  Oh and some more info the images work because I tried loading them up on another pc and it worked just fine.  I manage to get the CD to boot I just had to let me pc boot up first then insert the CD and have it boot the CD then. Once I get done installing ubuntu it works fine but I have to leave the PC on 24/7 for if I turn it off the PC will freeze 5-10 seconds after booting back up no matter how I install it. Apt-get update with an @ in password error Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n \left(X_i-\bar{X}\right)^2}{n-1}?$$ How can I dual boot Windows and Ubuntu? I currently have Windows 7 on my computer and I want to install Ubuntu alongside Windows safely.  Could someone please provide clear instructions? A single word or phrase for something that necessarily causes both harm and benefit? What is a single word, or phrase, for something that necessarily causes both harm and benefit? For example, a coal-fired power plant will harm human health through air pollution, but will bring electricity and economic development to a poor area. Placing the object of an infinitive before it instead of after it Given some potential $V$, we have the eigenvalue problem   $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$   with the boundary condition   $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$   If we wish to seek solutions for   $$ E &lt; \lim_{|x|\rightarrow \infty}V(x), $$   I find textbooks like Sakurai and others assert that (without proof) set of all such eigenvalues $E$ that admits nontivial solutions is discrete, which gives us liberty to index them with natural numbers (principal quantum numbers). Can anyone help me with a little elaboration of this mathematical fact ? As a student of mathematics, I know that compact operators have discrete spectrums, so for the operator $ T : L^2 \rightarrow L^2 $given as  $$ T(\psi) = -\frac{\hbar^2}{2m}\Delta \psi + V\psi. $$   Are there methods to show that $T$ is compact? Or possibly other ways to establish discreteness of its eigenspectrum? How can I control where smoke and fire are emitted from? Finding a basis for a certain vector space of periodic polynomials I have to move a MongoDB Collection (Size: 1TB+) from Sharded MongoDB Cluster to another[5 shards on both clusters]. During this operation, it is assumed that the accompanying program will be offline, so no need to worry about inconsistent data, but I need to minimize the downtime.  I've tried testing it with mongodump and mongorestore, but looks like that is taking ages to finish.  Please share your experiences for such kind of scenarios.  Note: Cluster has only one collection, so I am open at cluster level sync as well. Does removing an application from Software Center do a remove or a purge When I remove an application already installed through the Software Center. Does it remove the app the same way as doing a purge to it (aptitude purge) or does it remove it like a "remove" (aptitude remove). Sci-Fi book female fighter pilot All I remember about the book is that the protagonist was a human female fighter pilot, she has short hair, not sure if it was blond or auburn and she was involved in making peace with the alien race.  The alien race was invading Earth, but because of her they stopped, surrendered and because of her helped Earth improve its technology. I remember that she melded with something and was starting to sort of think like them and lose a bit of her humanity.  It was set in the not too distant future. I believe I read it in the 90s. Are submodules of finitely generated free modules always free? I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free? How to access the Index Of A Generic.List By Reflection? ok, ive a class and i pass an object as property.  the object that i pass is a List&lt;X&gt;  in my class im trying to access the Object index by reflection BUT I CAN'T!!!   Example:  this class works i just wrote down the part i want to show you and i need help.  class MyClass {     private object _recordSet;     public object RecordSet     {         get { return _recordSet; }         set { _recordSet = value; }     }      public string Draw()     {         system.reflection.Assembly asem = system.reflection.Assembly.getAssembly(_dataSource.GetType());          object instance;          instance = asem.CreateInstance(_dataSource.GetType().UnderlyingSystemType.FullName);          //to access de Count of my List         int recordcount = int.Parse(_dataSource.GetType().GetProperty("Count").GetValue(_dataSource,null));          //i need to do a          for(int cont = 0; cont &lt; recordCount; cont++)         {             _dataSource[cont].Name; // &lt;-- THIS PART IS NOT WORKING!!! because i cant access the Index Directly.... WHAT TO DO!! ???         }     } } I install virtualBox on Ubuntu 16.04, but after install and create new virtual Host shows this error:     Kernel driver not installed (rc=-1908)      The VirtualBox Linux kernel driver (vboxdrv) is either not loaded or there is a permission problem with /dev/vboxdrv. Please install virtualbox-dkms package and load the kernel module by executing  modprobe vboxdrv       as root. If it is available in your distribution, you should install the     DKMS package first. This package keeps track of Linux kernel changes and   recompiles the vboxdrv kernel module if necessary.  where: suplibOsInit what: 3 VERR_VM_DRIVER_NOT_INSTALLED (-1908) - The support driver is not installed. On linux, open returned ENOENT.     How to address this modprobe vboxdrv error? If $A\subseteq\mathbb N$ and $\sum\limits_{a\in A}\frac1a$ converges then $A$ has natural density $0$ In this  to a question about a series, a theorem was stated:     If $A= \{a_i \}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the  of the set.   My background in number theory is basically zero and all my attempts to prove this have been utterly unseccessful; would anyone please help me, or at least provide me with a hint to prove it? I have a requirement for a custom app I am building to open an attachment on the parent record in a new window on create of a child record.  I have parent object Quality_Assessment__c with child object Chart_Review__c. Quality_Assessment__c has a PDF attached to it in the Notes &amp; Attachments section. When a user clicks to create a new Chart_Review__c record I need to have the PDF attachment on the Quality_Assessment__c record open in a new tab as well as the record create page for the Chart_Review__c record. Looking for help in generating the needed javascript for a custom button to accomplish this.  The 'Quality_Assessment__c' record will have a PDF file attached to it prior to the user clicking the 'New Chart Review' button.  My use case is to have users open the 'Quality_Assessment__c' record and perform what we call a Chart Review. To perform the Chart Review the user would click on a custom button called 'New Chart Review' which would do two things.   create a new 'Chart_Review__c' record displaying the new record edit page  open the PDF attachment from the related 'Quality_Assessment__c' record.    Preferably on click of the button the new record page would open in existing tab and a new tab with the PDF would open simultaneously. How would you cut a shape out of an object using another object? Coming from SketchUp one of the tools I miss most is the circle draw tool which lets you draw a circle on any face and then you could extrude it in/out. The only tool that I know of that comes close to this is the knife tool but this would be very impractical to cut out with the knife  Example:    Is there any tool in blender that can match this function? also how could you do this with squares?