I wish to prove that square of a number is even if and only if the number itself is even. Let's call that number $a$, we need to prove that  $$ a^2\text{ is even}\Leftrightarrow a\text{ is even} $$  We need to prove two things:   If $a$ is even, then $a^2$ is also even.   I think that I can do this part. If $a$ is even, then we can write $a=2k$ where $k$ is some integer. Now $$ a^2=(2k)^2=4k^2=2(2k^2) $$ which shows that $a^2$ is also even.   If $a^2$ is even, then $a$ is also even.   I am having difficulty here. If $a^2$ is even, then I can write $a^2=2m$ . for some integer $m$ but taking square root gives $a=\sqrt{2m}$ which can be written as $a=2\sqrt{\frac{m}{2}}$ but the problem is that will $\sqrt{\frac{m}{2}}$ always end up as some integer? I do not think so!
How can I prove that if $n$ is even then $n^2$ is even and if $n$ is odd then $n^2$ is odd?
$4$ letter words taken from the letters of CONCENTRATIONS How many words with or without meaning can be made from the letters of the word CONCENTRATIONS by taking $4$ letters at a time?  There are  $14$ letters total present in the word. $4$ letters can be picked in $14\choose 4$ ways. $4$ letters can be arranged in $4!$ ways. So ideally answer has to be ${14\choose 4} * 4!$. But I know this answer is wrong because some letters are repeating. How do I solve this then? I appreciate any help. What is the correct way to completely remove an application? I've searched the net for such information and found different command lines, like these ones:  sudo apt-get remove application sudo apt-get remove application*  sudo apt-get remove --purge application sudo apt-get remove --purge application*  sudo apt-get purge application sudo apt-get purge application*   So, what is the correct way? Is it necessary to use that "*"?  After that, I also found these commands:  sudo updatedb sudo locate application sudo rm -rf (file/folder name) How can I quickly get a player back into a session after their character dies? As a GM, what kind of techniques could I use to help a player re-join the gaming session after their character dies? I want to avoid any major interruptions to the flow of the adventure. I would also like to avoid having to end the session immediately or making them sit on the side-lines for the rest of the session. "Was" or "were" for "half a dozen" In Microsoft Word, the following sentence is flagged.  It tells me to use "was" instead of "were"     There were half a dozen books strewn about the floor.   I would think that you would use "were" since it's a quantity more than one.  You wouldn't say, "There was twelve books strewn about the floor."  Does the use of the "half" modify it somehow? Catching access violation exceptions? Example  int *ptr; *ptr = 1000;   can I catch memory access violation exception using standard C++ without using any microsoft specific. What happens if light/particles exceeded the speed of light for a particular medium? I need to run something as sudo without a password, so I used visudo and added this to my sudoers file:  MYUSERNAME ALL = NOPASSWD: /path/to/my/program   Then I tried it out:  $ sudo /path/to/my/program [sudo] password for MYUSERNAME:    Why does it ask for a password? How can I run/use commands as root with a non-root user, without asking for a password? This song by The Rolling Stones, "(I Can't Get No) Satisfaction". As a non-native speaker, I always wondered what's the exact meaning of this phrase?   Is it "I'm not getting any satisfaction" (this seems to be the most widely met translation; double negation used where single negation should have been used). Or is it "I want my satisfaction, I cannot leave without getting some" (makes much more sense to me).   Or is it something else? I've always had troubles understanding double negation as applied to English. First I was taught it's a mistake to use double negation at all, and then I see it everywhere. Why is boiling water loud, then quiet? Water in my electric kettle makes the most noise sixty to ninety seconds before the water comes to a full boil. I have been fooled many times by the noisy kettle, only to discover that the water was not yet hot enough for tea. The kettle is only at a full boil after the noise has subsided.    I have noticed the same phenomenon with many other kettles, including conventional kettles on kitchen ranges; it is not a peculiarity of this electric kettle.  Why does the boiling become quieter as the water reaches full boil? What happens to Bruce Banner when Hulk loses a tooth? In Age Of Ultron we witnessed Hulk lose a tooth.   My question is: Has Bruce Banner lost a tooth as well? Is there a PPA with the latest version of LibreOffice? There was recently another release of , but it is  a pain to have to manually go to their website and download the packages every time there is a new release. Is there a PPA that always has the latest version of LibreOffice? I would like to update a large number of C++ source files with an extra include directive before any existing #includes. For this sort of task, I normally use a small bash script with sed to re-write the file.   How do I get sed to replace just the first occurrence of a string in a file rather than replacing every occurrence?  If I use  sed s/#include/#include "newfile.h"\n#include/   it replaces all #includes.   Alternative suggestions to achieve the same thing are also welcome. How to derive the conjugate prior of an exponential family distribution I am trying to derive the conjugate prior of the univariate Gaussian distribution over both the mean and the precision. I know that the prior I'm looking for is the normal-gamma distribution, but the idea is to derive this result.  It seems that you can write down the conjugate prior of an exponential family distribution immediately if you represent it in the canonical exponential family form.  suggests a way to do just that. So far I have written the Gaussian probability function in this form and equated the following, but I don't see how the form of the conjugate prior is immediately obvious.  $$p(\mu, \lambda | D) \propto p(\mu,\lambda)p(D | \mu, \lambda)$$ Preventing itemize environment to insert initial vertical space g++ undefined reference to typeinfo The portion of the tangent to any conic intercepted between the point of contact and the directrix subtends a right angle at the focus of the conic? Hi I'm traveling to Uzbekistan and have 12 hours layover in Seoul via Korean Airlines. Do I need a visa to go out of the airport for sightseeing? I'm a Philippine passport holder. Proving orthogonality of vectors in a complex vector space I'm taking a course in Real Analysis and have come across the following question which I initially thought looked quite simple:  Let $X$ be an inner product space over $\mathbb{K}$ ($\mathbb{K} = \mathbb{C}$ or $\mathbb{R}$) Show that x and y are orthogonal if and only if $||\lambda x  + \mu y||^2 = ||\lambda x||^2 + ||\mu y||^2$ for all $\mu, \lambda \in \mathbb{K}$.  I have proven the statement in one direction (starting with orthogonality and ending up with $||\lambda x  + \mu y||^2 = ||\lambda x||^2 + ||\mu y||^2$). However, I seem to be getting stuck trying to go in the other direction. The following is what I have so far:  $||\lambda x  + \mu y||^2 = ||\lambda x||^2 + 2\bar{\lambda}\mu Re&lt;x,y&gt; + ||\mu y||^2$  Then we must have 2$\bar{\lambda}\mu Re&lt;x,y&gt; = 0$ for the desired equality to hold, and so $Re&lt;x,y&gt; = 0$.   However, for orthogonality I want $&lt;x,y&gt; = 0$ and can't see how I can make this next step.  Any help would be greatly appreciated. MAC addresses in JavaScript I know that we can get the MAC address of a user via IE (ActiveX objects).  Is there a way to obtain a user's MAC address using JavaScript? Reading a bit about recent Monica Cellio related posts, I am afraid that I may be using a language that is a lot like English, but might not be the Stack-Exchange-approved version of English. For example, my English has had a certain gender-neutral pronoun in living use for centuries. That pronoun is used throughout the literature written in this version of English. It is used in the English I was taught as a grammar school student and later as a graduate student working on a master's thesis. I believe that this English is good enough to communicate about topics relevant to the various Stack Exchange sites.  The last thing I want from Stack Exchange is to be booted out because I slip up and accidentally use the language that has served me so well for a lifetime.  The topics discussed on Stack Overflow (I almost referred to it as a "sister site;" would that have been allowed? "mother site?" Good thing I didn't?) are often centered on languages which have formal descriptions.  Where is the formal description of the language that is to be used on Stack Exchange?     Strive not with a man without cause, if he have done thee no harm.      He shall die without instruction; and in the greatness of his folly he shall go astray.