What is wrong with my proof: $-1 = 1$?
Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$?
Using a new document class/template with LyX? I tried googling this elsewhere but I'm new to LaTeX and extremely confused about certain things. Basically I downloaded MikteX and I'm using the latest version of Lyx with it. Now all I basically know how to do is make pretty looking mathematics and add contents pages/titles/references/etc.  I'm looking to make a dissertation and my unviersity provides a template,  (available for download at that link).  So, what exactly do I need to do (and please be super specific as I am totally clueless to all the jargon I read).   Basically, I need to download ociamthesis.cls in the first line of that webpage. Does it matter where I save it? What exactly do I do next? While browsing on Safari and listening to iTunes I come across a video and then press the "play" button on the keyboard (to stop the music while I watch the video) it starts/stops the video instead of the music (I have no control over the music).  How can I quickly play/pause my music without affecting the video? Drop In Reputation - But No Record Of Why? In general what is the approach to switch the desktop environment or window manager on Ubuntu? The main difference between stock Ubuntu, Kubuntu and Lubuntu is supposedly the use of Unity vs. KDE vs. LXDE.  However, in order to try various desktop environments and window managers I don't want to have to reinstall the respective flavor from scratch just to get a particular desktop environments or window manager. There probably is an easier method than that.  The question: How can I easily test different desktop environments and respective compatible window managers (say KDE, LXDE, Unity, GNOME, XFCE and StumpWM)? Are there functions or packages in Mathematica to sort subsets in lex order, colex, revlex, etc.? How do I get the UTC time of "midnight" for a given timezone? The best I can come up with for now is this monstrosity:  &gt;&gt;&gt; datetime.utcnow() \ ...   .replace(tzinfo=pytz.UTC) \ ...   .astimezone(pytz.timezone("Australia/Melbourne")) \ ...   .replace(hour=0,minute=0,second=0,microsecond=0) \ ...   .astimezone(pytz.UTC) \ ...   .replace(tzinfo=None) datetime.datetime(2008, 12, 16, 13, 0)   I.e., in English, get the current time (in UTC), convert it to some other timezone, set the time to midnight, then convert back to UTC.  I'm not just using now() or localtime() as that would use the server's timezone, not the user's timezone.  I can't help feeling I'm missing something, any ideas? Sorry, I'm still a newbie to blender, but why are some edges visible only in edit mode, but not in wireframe view? is it a bug or am I missing something here?  edit mode: Edges visible. Object mode, wireframe view: Pooff, they're gone. what can i do to fix this? / thanks Remove the most popular tag from the title using JavaScript So, I see that  to fight site scrapers. But, this is harmful to User Experience. With browsers like Google Chrome, all I can see is the first few characters of the title, and it becomes annoying to try to find the right tab.  So, my request: Can you remove the tag from the &lt;title&gt; using JavaScript? Crawlers will still see the SEO-friendly &lt;title&gt;, and users won't have their user experience hampered by it. We all win. Transfer Clash of Clans account from one G+ to another G+ account? Only Fixed and Random Options are Working in Illustrator CC 2015 Callipraphy Brushes App updated but still the same. Only fixed and random options are working in Calligraphy Brushes. I have searched but can't find a solution. What's a concise way to check that environment variables are set in a Unix shell script? I've got a few Unix shell scripts where I need to check that certain environment variables are set before I start doing stuff, so I do this sort of thing:  if [ -z "$STATE" ]; then     echo "Need to set STATE"     exit 1 fi    if [ -z "$DEST" ]; then     echo "Need to set DEST"     exit 1 fi   which is a lot of typing. Is there a more elegant idiom for checking that a set of environment variables is set?  EDIT: I should mention that these variables have no meaningful default value - the script should error out if any are unset. I have a PCB with documentation. In the documentation, there is referred to specific pins of several jumpers and connectors. On the PCB however, there isn't a label or so for which pin has what number. I need to figure out which pin has which number, so that I know the function of each physical pin.  I noticed all connectors have all pins with a rounded tin pad, whilst one pin (either the first or the last when counting from right to left) has a square tin pad:     Is it some kind of convention which number this pin has? After that, how to count the other pins? Radius of circumscribed circle of triangle as function of the sides Given the length ot the sides $a , b$ and $c$  of $ \triangle ABC$.  What is the length of the radius of the circumcribed circle?  After some formula substitution I came to the monster formula:  $$ \frac {a b c}{\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}} $$  Can this formula be simplified? If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light? I know that as the velocity increases, the mass of the object also increases so it becomes tougher and tougher to move the object which ultimately leads to a requirement of infinite energy to accelerate an object to the speed of light. But I have a doubt.  As far as I know only the observable mass (Relativistic mass) increases but not the Proper Mass or Intrinsic Mass, right? The actual mass of the object will remain the same. So if the actual mass ($m_0)$ remains the same and only the observable mass increases, why is more and more energy required? I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light but is it possible to make it move faster than light?  If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?  I quote Igor Ivanov (a stackexchange user from this question ) to give an insight into my question.     The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to  popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".   EDIT  After reading the answers I have a doubt.  SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy. How do I clear the render border? In the 3d viewport I am using a border region to only render things inside the selected region. I did this with CtrlB. I cannot get rid of the region. How can I make the border disappear? Who paid for Harry's Nimbus 2000? The Nimbus 2000 was the fastest broom at the time of the Harry's first year at Hogwarts.  Professor McGonagall wanted to win the Quidditch Cup so she spoke with Dumbledore and got an exception from the rule that no first-years could have a broom for Harry.  My question is who paid for the broom?  It couldn't have been a cheap broom.  They wouldn't have taken gold out of Harry's Gringotts account without permission.  It wouldn't be right for the school to pay for it and I doubt McGonagall paid for it out of her own pocket.  So who paid for the broom? Is it possible to somehow mark a System.Array as immutable.  When put behind a public-get/private-set they can't be added to, since it requires re-allocation and re-assignment, but a consumer can still set any subscript they wish:  public class Immy {     public string[] { get; private set; } }   I thought the readonly keyword might do the trick, but no such luck. Rendering smoke in Cycles? Is "everyone" singular or plural? Which is correct?     Everyone were convinced that he would go to the game.   Everyone was convinced that he would go to the game.   I think it's "was", because "everyone" is singular, but I just wanted to check. After upgrade of any app - apps move back from SD Card to phone memory.  After that, to save phone memory, I have to move upgraded app back to SD card.   Is any possibility to avoid this because it is very annoying?  I have Android 5.0.2 on Motorola E second gen phone.