How do we know that the number $1$ is not equal to the number $-1$? (I am not talking about the multiplicative inverse of an arbitrary field, but the integer/rational, real or complex number $1$.) Is that an axiom? Since someone in the comments asked for more background/motivation: I'm just a naive bachelor student and I wanted to prove that for any number $x$, $x=-x$ implies $x=0$ (which is true if and only if $1\neq -1$, see ). Why is this question different to the question in my last post? In my last post, I wanted to know if you can prove that $x=-x$ implies $x=0$ for an element $x$ of an arbitary field. It turned out that you can not (there are fields where this is not the case). In addition, it turned out that this is true if and only if $1\neq -1$. ($1$ stands for the multiplicative identity, not the number one.) Since we normally assume that $x=-x$ implies $x=0$ if $x$ for some number $x$ (at least in my lectures), I wanted to know wheter you can prove that $1\neq -1$ or not. (To be honest, I didn't think that you have to distinguish e.g. between the integer $1$ and the natural number $1$...) To put it into a nutshell: My first question was wheter you can prove $1\neq-1$ where $1$ stands for the multiplicative identity of an arbitrary field (the answer is no.) But we know that the answer is yes if $1$ stands for the integer/real number/complex number one, which led to this post.
Let $F$ be a field and $x\in F$. If $x=-x$ and $1\neq -1$, then $$0=\frac{x+x}{1+1}=\frac{1+1}{1+1}x=x.$$ This means that the statement in the title is true if and only if $1\neq -1$. But how do we know that $1\neq -1$ for an arbitary field?
I have Solaris 8 OS installed in French.  Is it possible to switch from French to English without reinstalling the OS ? If yes how ?  When locale -a i have this :  POSIX common en_US.UTF-8 C iso_8859_1 en_CA en_CA.ISO8859-1 en_US en_US.ISO8859-1 en_US.ISO8859-15 en_US.ISO8859-15@euro fr_CA fr_CA.ISO8859-1 iso_8859_15 th th_TH th_TH.TIS620   Even when I try to edit /etc/default/init with :  LANG=en_US.ISO8859-15 LC_COLLATE=en_US.ISO8859-15 LC_CTYPE=en_US.ISO8859-15 LC_MESSAGES=en_US.ISO8859-15 LC_MONETARY=en_US.ISO8859-15 LC_NUMERIC=en_US.ISO8859-15 LC_TIME=en_US.ISO8859-15 LC_ALL=en_US.ISO8859-15   and reboot : nothing happens.  To verify the result i always use date and when i reboot : i also have the date in french.  P.S : I have to do this configuration in a computer which isn't here in my office, so i have to this manipulation in a an another computer here in my office and if it works, i'll do it to the other computer.  So the computer here is configured in English and i have to switch it to French. Here is the content of my ~./dt :  C                     errorlog              sessionlogs Desktop               errorlog.old          sessions Trash                 errorlog.older        simulateur.dtwmrc appconfig             help                  startlog appmanager            icons                 startlog.old asr.dtwmrc            outils.dtwmrc         startlog.older bin                   palettes              tmp bookmarks             sdtfonts              types dtwmrc                sdtpcaction.conf      wsmenu en_US.ISO8859-15      sdtpcaction.conf.bak   Thanks Probability of two uniform random numbers being more than $\frac{1}{2}$ apart Two numbers are chosen randomly and independently from the unit interval $(0,1)$.  What is the probability that they differ by more than $\frac{1}{2}$? Hint: Think of the $2$ numbers chosen as coordinates in the Cartesian plane. Draw a square with vertices at $(0,0)$,$(1,0)$,$(1,1)$, and $(0,1)$. Allowable choices for the $2$ points lie below the line $y=x-\frac{1}{2}$ or the above $y=x+\frac{1}{2}$. Find the combined area of these regions.  I need help on this. I know what $y=x-\frac{1}{2}$ and $y=x+\frac{1}{2}$ looks like but I am not understanding the problem as a whole given a polynomial $p(x,y)$ from $\mathbb Z[x,y]$. I want to substitute $x=1$ leaving $y$ as it is. The command Value(p,[1,y]) does not work. Can you give me a hint?  After that it gets a little more complicated. Let  $$p(x,y)=x^6+y^6$$ After substituting $x=1$ we have $p(1,y)=1+y^6$. Given a natural number $j$. I want to find all expressions $y^j$ in $p$ and substitute these $y^j$ by $1$ and the others by $0$. In this case (and for $j=6$) we would get $1+1=2$. If $$p(x,y)=x^4y^2+x^2y^6$$ we should get $0+1=1$. Is there any way to do this?  Thanks for reading! edit: Well, it turns out I used the function Value wrong. The right code for my problem would be Value(p,[x,y],[1,y]). So my first question is solved. How do I pass this array of c strings correctly? Lets say I have a  creature that says "Whenever another creature enters the battlefield under your control, you may gain 1 life." and I currently have all 4 of them in the battlefield. Then I cast another creature and : Do I get 1 or 4 life? Does that effect trigger for every priest or just once? thanks for help How to flag a typo question I came across  today. In reading the answer and the comments it turns out that the issue was just a simple typo that the OP made. So that being the case I decided that this question probably needs to be closed since this will not be helpful to any future vistors. However, in looking at the flag reasons (I don't have the close ability yet), I don't see a flag that would be appropriate.   I know some of these options might still be influx because of the . However, until a different off-topic flag description is created, how should this be flagged? Or should this not be flagged at all and just allowed to live on or wait for a user with close votes to use a custom off-topic close option? Use of var keyword in C# After discussion with colleagues regarding the use of the 'var' keyword in C# 3 I wondered what people's opinions were on the appropriate uses of type inference via var?  For example I rather lazily used var in questionable circumstances, e.g.:-  foreach(var item in someList) { // ... } // Type of 'item' not clear. var something = someObject.SomeProperty; // Type of 'something' not clear. var something = someMethod(); // Type of 'something' not clear.   More legitimate uses of var are as follows:-  var l = new List&lt;string&gt;(); // Obvious what l will be. var s = new SomeClass(); // Obvious what s will be.   Interestingly LINQ seems to be a bit of a grey area, e.g.:-  var results = from r in dataContext.SomeTable               select r; // Not *entirely clear* what results will be here.   It's clear what results will be in that it will be a type which implements IEnumerable, however it isn't entirely obvious in the same way a var declaring a new object is.  It's even worse when it comes to LINQ to objects, e.g.:-  var results = from item in someList               where item != 3               select item;   This is no better than the equivilent foreach(var item in someList) { // ... } equivilent.  There is a real concern about type safety here - for example if we were to place the results of that query into an overloaded method that accepted IEnumerable&lt;int&gt; and IEnumerable&lt;double&gt; the caller might inadvertently pass in the wrong type.  var does maintain strong typing but the question is really whether it's dangerous for the type to not be immediately apparent on definition, something which is magnified when overloads mean compiler errors might not be issued when you unintentionally pass the wrong type to a method. 32 bit or 64 bit OS on a 512 mb VPS? Initialize class fields in constructor or at declaration? I've been programming in C# and Java recently and I am curious where the best place is to initialize my class fields.  Should I do it at declaration?:  public class Dice {     private int topFace = 1;     private Random myRand = new Random();      public void Roll()     {        // ......     } }   or in a constructor?:  public class Dice {     private int topFace;     private Random myRand;      public Dice()     {         topFace = 1;         myRand = new Random();     }      public void Roll()     {         // .....     } }   I'm really curious what some of you veterans think is the best practice. I want to be consistent and stick to one approach. After two days of installation, strangely my Ubuntu machine is restarting instead of shutdown from both desktop GUI and command line.  sudo shutdown -h now sudo halt  # (doesn't shutdown, instead freezes on boot screen Plymouth)   I had force shutdown by powering off the machine.  This is not happening all the times.  I have been using Ubuntu for 6 months and never experienced this kind of problem. Find the general solution of $4u_{xx}+5u_{xy}+u_{yy}+u_x+u_y=2$ How do I solve the following second order partial differential equation?  $4u_{xx}+5u_{xy}+u_{yy}+u_x+u_y=2$  I have classified the equation to be hyperbolic and changed variables to obtain the canonical form as $u_{\epsilon\nu}={1\over{3u_\nu}}-{8\over9}$ which I believe is correct but I am struggling to find the general solution? A step by step solution would be appreciated. Fast way of copying IMAP to IMAP on CentOS I'm trying to copy of IMAP emails from one server to my new server but after Googling for hours and trying several tools, nothing is working. Is there a tool that actually works?  I am copying hundreds of thousands of emails so using a email client is out of the question, I tried it, and at the rate of a email per 2 seconds, it would take a few weeks or months!  www.linux-france.org/prj/imapsync/ is commonly mentioned, however, the tool is no longer available.  I tried www.athensfbc.com/imap_tools/ which just hangs and does nothing.  Any migration tools that work? How to get reflections from 3D to show up on real images How can I take shallow depth of field photos with a point-and-shoot camera? This is about that old chestnut, $\newcommand{\d}{\mathrm{d}} \int \sin^2 x\,\d x$.   OK, I know that ordinarily you're supposed to use the identity $\sin^2 x = (1 - \cos 2x)/2$ and integrating that is easy. But just for the heck of it, I tried using the $u$-$v$ substitution method (otherwise known as integration by parts).  $$ \int \sin^2 x\,\d x = \int \sin x \sin x\,\d x $$  We can say $u=\sin x$ and $\d u=\cos x\,\d x$ while $\d v = \sin x\,\d x$ and $v = -\cos x$. When we put it all together:   $$\int \sin^2 x\,\d x = u v - \int v\,\d u = -\sin x \cos x - \int -\cos^2 x\,\d x$$  and doing the same routine with $\cos^2 x$ we get $u = \cos x$, $\d u = -\sin x\,\d x$, $\d v = \cos x\,\d x $ and $v = -\sin x$, leading to:  $$\begin{align}   \int \sin^2 x\,\d x = uv - \int v\,\d u     &amp;= -\sin x \cos x - (-\cos x \sin x - \int -\sin^2 x\,\d x) \\     &amp;= -\sin x \cos x + \sin x \cos x - \int \sin^2 x\,\d x \end{align}$$  which eventually works out to  $$2\int \sin^2 x\,\d x = 0$$  So I wanted to get an idea why this didn't work. Maybe it's higher math and the why will be beyond me (I would think that might be the case), or maybe it's one of those proofs that looks absurdly simple when shown that I am just unaware of. How do I include lines in resolv.conf that won't get lost on reboot? Text gets stretched or disappears when scrolling When scrolling in the Stack Exchange Android app, the text gets stretched vertically, code disappears, and sometimes the entire post content disappears.  This doesn't make the app completely useless, but it's pretty close.   Motorola Razr MAXX, Android 4.0.4. In my past experiences, I have almost never typeset my manuscripts according to the formats required by the journals to which I would like to submit. I leave my manuscripts as produced by the LaTeX article documentclass.  Recently I am wondering: Would such a behavior generally give handling editors a negative first impression? What does "it" refer to in "it's raining"? How can I 'join' an array adding to the beginning of the resulting string the first character to join?