Why do we need to rationalize fractions?
Why rationalize the denominator?
Symbols instead of numbers as footnote markers prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit Problem:  Start page numbers from chapter one and also reflect this in the TOC.  Currently the code for TOC is:  \renewcommand{\contentsname}{Table of contents}  \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2} \tableofcontents   I would appreciate if I can be pointed in the right direction. So far I have understood that you can use \chapter* in order not to include a chapter in the TOC.  Questions:  1) How would you force Latex to start numbering from chapter 1 and forward?  2) How can you start page numbering from the first chapter? 403 Forbidden with Java but not web browser? Correlating filename with package Say, I'm given a filename like libasound.so.2. Is there a way to query the package manager to correlate the filename with its corresponding package?  I'm interested to get answers for any package manager, but my primary focus is on apt and yum. Compiling LaTeX document empties .bbl file I'm trying out biblatex for the document I'm compiling, using the biber backend on my miktex distribution. Everything, including biber, should be up to date, assuming the MiKTeX repository currently holds the latest versions of everything.  This is what the biblatex documentation (p. 101) says about how often I have to run biber to update my .bbl file:     Whenever a reference to a work which has not been cited before is   added, this procedure must be repeated. This is also the case if the   last reference to a work which has been cited before is removed   because some citation labels may change in this case.   So if I make changes to my document (which is a very common activity in the process of writing a document), but leave the citations unchanged, I should not have to re-run biber before compiling the changes. However, every time I compile my document, regardless of the compiler or documentclass, the .bbl file is emptied. Consequently, the next time I compile the document without running biber, the bibliography is removed (seeing as the .bbl file contains nothing), and all my citations are broken. For this reason, I have to run biber before every time I compile the document, which largely counters the benefit of externalizing the bibliography in the first place.  How can I make the compilers stop overwriting my .bbl file and make everything behave as described in the biblatex manual?  Upon investigating the contents of the auxiliary files, I found that the .bbl file (after running biber, before compiling) begins with the following lines:  % $ biblatex auxiliary file $ % $ biblatex bbl format version 2.3 $   Does the biblatex version 2.8 still mark the .bbl files as version 2.3? If not, could this be the problem? And if so, why does my 2.8 version of biblatex (partly) behave as version 2.3?  Any help solving this problem would be much appreciated. Why is $\infty \cdot 0$ not clearly equal to $0$? I did a bit of math at school and it seems like an easy one - what am I missing?  $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$  $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ 0}_{n\text{ times}} = 0$$  (i.e add $0$ to $0$ as many times as you like, result is $0$)  So I thought an infinite number of $0$'s cannot be anything but $0$? But someone claims different but couldn't offer a reasonable explanation why. Google results seemed a bit iffy on the subject - hopefully this question will change that. I am trying to use a file repository for updating linux/ubuntu on a non-connected machine. The architecture for this machine (uname -a):  x86_64 GNU/Linux   The results of dpkg --printarchitecture are:  amd64   It is a Trusty Tahr amd64 machine, and it specifically says this, for example, in the aptitude sources.list file. Also, the "About this Computer" in ubuntu specifically says the machine is 64-bit. So, I installed the amd64 packages on the machine. However, when I try to do apt-get update, I get errors such as:  W: Failed to fetch file:/var/spool/apt-mirror-trusty/mirror/archive.ubuntu.com/ubuntu/dists/trusty/main/binary-i386/Packages  File not found   So, in other words, even though it is a 64-bit machine, apt is inexplicably trying to read i386 packages. Why is it doing this and how can I make it use the 64-bit packages?  Note that there is nothing in sources.list that specifies i386.  The only uncommented line in the sources.list file is:  deb file:///var/spool/apt-mirror-trusty/mirror/archive.ubuntu.com/ubuntu trusty main restricted universe multiverse The whole multi-continent aspect of A Song of Ice and Fire seems to be getting out of hand. Do the other continents suffer those long winters?  Can the Others harm the other continents?  In The Lord of the Rings there were other realms than Middle-earth, but Middle-earth was still all of humanity and dwarf population and hobbits.  Sure there are stories like Romeo and Juliet where the only thing that matters is whether those 2 are going have a chance to love.  But in these fantasy / sci-fi stories there's usually danger of world domination by evil.  It's not usually just one continent is going to be a terrible (or impossible) place to live but other civilizations will flourish. Every partially defined isometry can be extended to a isometry I know that the following theorem holds true:  Let $S$ be a subset of $\mathbb R^n$, and let $f:S\to \mathbb R^n$ a map such that $d(p,q)=d(f(p),f(q))$ for every $p,q \in S$ (here $d$ is the usual euclidean distance). Suppose that $0 \in S$ and $f(0)=0$. Then $f$ can be extended to an orthogonal transformation of $\mathbb R^n$.  I have two questions:   Can someone give me a reference of a proof of this theorem? I believe that the previous theorem holds true also if we change $\mathbb R^n$ with $\mathbb C^n$ and "orthogonal" with "unitary", with nearly the same proof. Does it holds true also in the $ \ell^2$ space? With the same proof? When writing questions and answers here about the Harry Potter series, I often need a word for "someone who's not a Muggle". People often refer to "the wizarding world" as opposed to the Muggle word, so "wizard" would be one choice - but it seems slightly sexist to exclude witches in that way. On the other hand, "witch or wizard" is a bit cumbersome, much longer than the simple word "Muggle".  Is there a sensible word to use which is neither sexist nor cumbersome, that makes a good equivalent to "Muggle"? What would JK Rowling say to fill in a sentence such as this:     Muggles use technology, while [...] use magic. How does one find which files are taking up 80% of the space on a Linux webserver? The drive is constantly filling up. You've hunted down all the spare and random files you can. grep'd for coredump files and even removed some of the un-needed backups...  What would be your next move.  The actual server in question has 10GB of website files and the OS shouldn't take any more than 10GB so how do you track down what's filling a 50GB (virtual) drive? Prove the map has a fixed point Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) &lt; \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$.  The uniqueness is easy. My problem is to show that there a exist fixed point. $K$ is compact, so every sequence has convergent subsequence. Construct a sequence ${x_n}$ by $x_{n+1}=Ax_{n}$,$\{x_n\}$ has a convergent subsequence $\{ x_{n_k}\}$, but how to show there is a fixed point using $\rho (Ax,Ay) &lt; \rho(x,y)$? Formula between wire size and current Is there a mathematical formula between wire size, current, temperature and wire type(Or resistance)? For some reason my Steam just broke today after working perfectly since I installed it a few days ago.  It was originally installed via the deb from the Steam site. So I uninstalled that using the Ubuntu Software Center. Then I installed steam-launcher. When I run it from the menu it does nothing at all. When I go through terminal it gives this message, no error:  $ steam/usr/bin/steam:  line 196: /home/andrew/.local/share/Steam/steam.sh: Success   So I purge-removed it and removed the .steams directory from my home directory but on re-installation its the same story.  I am using Xubuntu 12.10 64bit.  It appears that steam .local/share/Steam/steam.sh is blank. What is it meant to have in it? I am so confused with how reinstalling it doesn't fix it. Number of total possibilities for an equation I need to find the number of possibilities for which the following equation exists:  $$x_1 + x_2 + x_3 + \cdots + x_{10} \leq 70$$  Each variable is a non-negative integer.  I tried simplifying the question to the point of finding the number of possible solutions to each equation seperatly:  $$x_1 + x_2 + x_3 + \cdots + x_{10} = 70,$$ which is ${79}\choose{10}$.  $$x_1 + x_2 + x_3 \cdots + x_{10} = 69,$$ which is ${78}\choose{10}$.  $$x_1 + x_2 + x_3 + \cdots + x_{10} = 68,$$ which is ${77}\choose{10}$, and so forth.  Then I thought about adding all of these together. This seems a bit too much work considering that the textbook solution is: ${80}\choose{10}$, and I can't seem to figure out the idiomatic way of approaching these kind of questions.   Any help is greatly appreciated. What is the T-SQL To grant read and write access to tables in a database in SQL Server? What is the exact SQL to assign db_datareader and db_datawriter roles to a user in SQL Server?  The user name is MYUSER and the database is MYDB. Where did dedicated.abac.net. come from in my rDNS and PTR records? Getting current date and time in JavaScript How to do string formatting with placeholders in Java (like in Python)?