I have learnt that a fraction with a surd in its most simplest form should have the surd in the numerator and not the denominator? Why is it convention not to leave the surd on the denominator? Is it just arbitrary convention or does it have an actual purpose?
In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$.  says that "there is a bias against roots in the denominator of a fraction". But such fractions are well-defined and I'm failing to see anything wrong with $\frac{1}{\sqrt{2}}$ - in fact, IMO it is simpler than $\frac{\sqrt{2}}{2}$ because 1 is simpler than 2 (or similarly, because the former can trivially be rewritten without a fraction).  So why does this bias against roots in the denominator exist and what is its justification? The only reason I can think of is that the bias is a relic of a time before the reals were understood well enough for mathematicians to be comfortable dividing by irrationals, but I have been unable to find a source to corroborate or contradict this guess.
What’s the point in having “www” in a URL? Other than for historical reasons, is there is reason to have “www” in a URL?  Should I create a permanent redirect from www.xyz.com to xyz.com, or from xyz.com to www.xyz.com? Which one would you suggest and why? Can my child attend a private school whilst I'm on USA ESTA? I am British, currently staying in the USA for 6 weeks on an ESTA Visa Waiver. I am living with a local family and whilst here my child (9 years old) has been attending a private (non-state) school with the children of the local family.  Today somebody told me that by having my child attend a private school I may be in violation of the terms of the ESTA Visa Waiver.  I have found nothing about this at   Where can I find official documentation / online advice about this? Uploading video to youtube in right format I have a Sony CX115 and I have a little footage I'd like to put on youtube. I've downloaded Arista from the Ubuntu Software Center and then I've downloaded its Youtube HD plugin. I've converted my video and uploaded it to youtube... The result is a flashy video with a lot of grey in it:    What could I do to publish it in high quality so it'll be viewable? What's the difference between a temp table and table variable in SQL Server? Chat messages should use consistent icon in iOS app It'd be nice if the chat notifications in the feed used the default Stack Exchange icon like the ones in the inbox do*:     Since items in the feed are grouped so that multiple items from the same site , the site icon helps delimit grouped items a bit less subtlely than the grey divider lines, and chat replies' lack of one causes them to sort of blend in to the previous group.  *Ideally it'd use the icon of the site the room was associated with, but I don't think that information is readily available I've been using the $K$-fold cross-validation a few times now to evaluate performance of some learning algorithms, but I've always been puzzled as to how I should choose the value of $K$.  I've often seen and used a value of $K = 10$, but this seems totally arbitrary to me, and I now just use $10$ by habit instead of thinking it over. To me it seems that you're getting a better granularity as you improve the value of $K$, so ideally you should make your $K$ very large, but there is also a risk to be biased.  I'd like to know what the value of $K$ should depend on, and how I should be thinking about this when I evaluate my algorithm. Does it change something if I use the stratified version of the cross-validation or not? Find the center of the symmetry group $S_n$.   Attempt:  By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\  g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is always the trivial permutation. Suppose $a$ is in $S_n$, but not equal to identity. Now we can imagine the permutation as bijective function that maps from $\{1,2,\dotsc,n\}$ to $\{1,2,\dotsc,n\}$.  So suppose $p$ is a permutation map. Then $p$ maps from a location $i$ to a location $j$. Take $p(i) = j$ where $i\neq j$. Let $k$ be in $\{1,2,\dotsc,n\}$, where $k$, $i$ and $j$ are all different elements. The cycle $r = (jk)$, then we will see if this commutes.  $rp(i) = rj$    Can someone please help me, I am stuck?  Thank you. Are there ways to determine if a wall is load bearing? Other than looking up blue prints, which many homeowners may not have, are there ways to determine if a wall is load bearing?  Methods I can think of might include:   Going up in the attic to check if ceiling trusses run perpendicular to the wall If the wall is an exterior wall   Any others? Simulate Multiple Monitors I'm trying to find out if there is a product out there to simulate Multiple Monitors on one large monitor/tv?  So take a screen and turn it into something like below...  -------------------------------- |               |              | |               |              | |               |              | |               |              | -------------------------------- |               |              | |               |              | |               |              | |               |              | --------------------------------   Something like this...even though in reality it's a single screen.  A software solution is perfectly fine as long as it functions like the built in Windows 8 functionality (Start screen only consumes the screen it was started on/etc).  If anyone is curious as to the reason.  I'm looking to get a treadmill desk since I'm a developer and don't like being sedentary all day.  So the two options that I see are having 4 monitors and mirroring them (I don't want to have to worry about forgetting to move the window that I need) or getting a large monitor/tv and putting it on an arm that I can simply swivel into place.  There are dual-monitor swivel mounts, but they are EXTREMELY pricey and the ones I've found don't appear to have the range of motion that I'm anticipating needing.  EDIT:  I found ... but I can't find the answer to the start menu question (I've sent the company a message on facebook).  Hoping I hear good news as this appears to offer many of the features I'm wanting.  So Fingers crossed there. Question: If $(1+x)^n=\binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+\dotsm+\binom{n}{n}x^n$, prove that $\binom{n}{1}-2\binom{n}{2}+3\binom{n}{3}-\dotsm+(-1)^{n-1}n\binom{n}{n}=0$  My attempt:  I wrote the expression for $\binom{n}{k}$ for each term in the L.H.S and then tried to simplify but could not reach the R.H.S.  Please help. I have no idea on how to begin. I am not allowed to use calculus. I have a Linux instance that I set up some time ago.  When I fire it up and log in as root there are some environment variables that I set up but I can't remember or find where they came from.     I've checked ~/.bash_profile, /etc/.bash_rc, and all the startup scripts.  I've run find and grep to no avail.   I feel like I must be forgetting to look in some place obvious. Is there a trick for figuring this out? I looked up several tutorials on how to install and they dont work. I do everything correctly but then i get this:     The file   '/home/mike/Downloads/minecraft.jar'   is not marked as executable.  If this   was downloaded or copied from an   untrusted source, it may be dangerous   to run.  For more details, read about   the executable bit.   I run the minecraft.jar with the Sun java runtime 6 but it doesnt work:/. Any help would be greatly appreciated! PGFPlots: Understanding how to draw plots on different layers and arrange them If $A \in M_{n}(\Bbb{R})$ is an Upper triangular matrix with diagonal entries $1$ such that  $A \neq I$, then what can we say about the diagonalizability of $A$ ?  I know that if the matrix has distinct eigenvalues or the set of eigenvectors are linearly independent then the matrix is diagonalizable.  And that the eigenvalues of the triangular matrices are given by the diagonal elements like  and , but they work nocely if we had distinct elements on the main diagonal.  But in my case I have same value 1 on the main diagonal, how can I approach about the diagonalizability of the matrix? calculator's angle answer for trig ratios that can work in more than 1 quadrant on the unit circle Why does the calculator do a cc (counterclockwise) rotation for positive trig ratios instead of clockwise,  and a clockwise rotation for negative sine &amp; tan instead of cc and a counterclockwise rotation for negative cos ratios instead of a clockwise   ie. in degree mode  $\cos^{-1}(-5/12)=114.62$  $\sin^{-1}(-5/12)=-24.62$  $\tan^{-1}(-5/12)=-22.61$  Is it maybe picking the value that involves the least amount of computing power? or is it a matter of convention? or am I overlooking something? Deleting Individual SE Accounts Sheet Specific Scripts I'm currently using the below script to timestamp when an edit is made to column 7 on a sheet named "Sheet 1".  function onEdit(e) {    if ([7].indexOf(e.range.columnStart) != -1) {    e.range.offset(0, 7).setValue(new Date()).setNumberFormat("dd.MM.yyyy at HH:mm");    }   }     I have a second sheet "Sheet 2" which I need to run the same script on but against column 5.  function onEdit(e) {    if ([5].indexOf(e.range.columnStart) != -1) {    e.range.offset(0, 7).setValue(new Date()).setNumberFormat("dd.MM.yyyy at HH:mm");    }   }     I've tried .getSheetByName("Sheet1") but just cant seem to get it working correctly. Elsarticle - Abstract two columns wide "Was going to" versus "would be" as the past tense of "will" Do you think that "was going to be" or "would be" could be used as past tense of will?     I thought this book was going to be your choice.        I thought this book would be your choice.   Do these two sentences basically mean the same thing or are they slightly different? I think the title says it all. I can't see any reason employers have to incentivize employees to contribute to their 401(k) evenly throughout the year. Instead couldn't they use a relatively simple formula that is based on how much of their income the employee has contributed to their 401(k) thus far for the year? If that or their current contribution is over the threshold, they get the match this pay period, even if they didn't contribute anything.  EDIT: To clarify, my employer matches 100% of the first 3% of my contribution and 50% of the next 2%. The match is added every pay period along with my contribution. I am pretty sure there is no "true-up" feature, so if I were to hit the $17.5K limit early, and not be able to contribute at least 5% of my salary, I would not get the full match. So I have to be careful to make my contributions evenly throughout the year to get the maximum match. Just trying to figure out my employer's motivation for requiring this.