I understand that the numbers that, when squared, result in 4 are ±2 because both -2 and 2 squared result in four. However, when in a radical, why is it that $\sqrt 4$ is only 2? Why isn't the sign undetermined until it the number is taken out of the square root? Why is the ± needed in front of $\sqrt 4$ to denote that the solution set includes -2?
It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.  I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified?
QGIS 3.0 and Qt 5.7 time schedule When will QGIS 3.0 with QT5.7 (possibly) be released as a stable and long term support version? I posted a question which was closed as off-topic and it seems like it could fit on another site in the network. How can I go about moving it there?   What is migration? When can a question be migrated, and who can migrate it? When should I consider migration, and when is it inappropriate? What causes a migration to be blocked and what happens after? What causes a migration to be rejected and what happens after? Anything else I should know about migration? I have installed Apache(LAMP) web server on my ubuntu machine and I want to do the above mentioned thing.I have tried creating symbolic links in the /var/www directory but it only shows files but not directories inside the directory linked.I want to redirect /var/www to my home folder so that I can access all the files and directories contained therein.Please help me. Existence of area-preserving diffeomorphism between two sets in $\mathbb{R}^2$ Suppose that we have open sets $A, B\subset \mathbb{R}^2$ such that $A$ and $B$ are simply connected and have smooth boundaries. Furthermore, let us assume that $m(A) = m(B) &gt; 0$, where $m$ is the Lebesgue measure on $\mathbb{R}^2$. My question is as follows: when does there exist an area-preserving diffeomorphism between $A$ and $B$? In this context, an area-preserving diffeomorphism between $A$ and $B$ is a is a $C^{\infty}$-diffeomorphism $\phi : A\to B$ such that $m(S) = m(\phi(S))$ for all measurable $S\subseteq A$. If an area-preserving diffeomorphism doesn't always exist between sets $A$ and $B$ as above, are there any extra conditions we can impose on $A$ and $B$ such that there does exist an area-preserving diffeomorphism?  My thoughts thus far are that we always have a $C^{\infty}$-diffeomorphism between $A$ and $B$: this is a consequence of the . It's also obvious to me that if we can deform $A$ linearly to get $B$, then that deformation will be area-preserving. Furthermore, it seems that aside from this case, there's no reason why an area-preserving diffeomorphism should exist. However, I'm having a hard time proving that there necessarily isn't an area-preserving diffeomorphism between $A$ and $B$ if $B$ is not a linear deformation of $A$. Does anybody have any ideas? "Typos" are acknowledged as "technical" errors, ones that obviously aren't caused by the writer's poor spelling. Although less frequent than in typing, they can also occur in handwritten texts, as a result of absent-mindedness or just an accidental stroke of the pen.   Since the text wasn't actually typed, it sounds funny to call them "typos". Is there a more fitting word, one that reflects the "accidental" nature of the error? I am dissecting some code and came across this,   $sql = 'SELECT page.*, author.name AS author, updator.name AS updator '      . 'FROM '.TABLE_PREFIX.'page AS page '      . 'LEFT JOIN '.TABLE_PREFIX.'user AS author ON author.id = page.created_by_id '      . 'LEFT JOIN '.TABLE_PREFIX.'user AS updator ON updator.id = page.updated_by_id '      . 'WHERE slug = ? AND parent_id = ? AND (status_id='.Page::STATUS_REVIEWED.' OR status_id='.Page::STATUS_PUBLISHED.' OR status_id='.Page::STATUS_HIDDEN.')';   I am wondering what the "?" does in the WHERE statement.  Is it some sort of parameter holder? Sum over all non-evil numbers Shuffling questions in exam wrong numbering for subcaption in scrbook When using the scrbook document class in combination with the subcaption  package, the numbering of subfigures seems broken. Consider:  \documentclass{scrbook} \usepackage{xcolor} \usepackage{caption} \usepackage{subcaption} \captionsetup{%   format=hang,%   font=small,%   labelfont={sf,bf},%   aboveskip=0.2\normalbaselineskip,%   position=top } \pagestyle{empty}  \begin{document} \begin{figure}   \caption{Black}   \label{fig1}   \centering \rule{2cm}{2cm} \end{figure}  \begin{figure}   \caption{More figures}   \label{fig2}   \centering   \begin{subfigure}{.3\textwidth}     \centering     \caption{Blue}     \label{fig2a}     \color{red}\rule{2cm}{2cm}   \end{subfigure}   \begin{subfigure}{.3\textwidth}     \centering     \caption{Green}     \label{fig2b}     \color{green}\rule{2cm}{2cm}   \end{subfigure}   \begin{subfigure}{.3\textwidth}     \centering     \caption{Blue}     \label{fig2c}     \color{blue}\rule{2cm}{2cm}   \end{subfigure} \end{figure}  Figure \ref{fig1} shows a black box. Figure \ref{fig2} shows colored boxes, where Figure \ref{fig2a} shows a red box, Figure \ref{fig2b} shows a green box, and Figure \ref{fig2c} shows a blue box. \end{document}   This results in the following display:    Clearly, the figure numbering in the text references is off by one for the subfigures. Instead, if I change the document class to article, it looks fine:     Why does this happen, and how can I prevent it? What is the difference between an "API," a "framework," an "IDE," and an "engine?" how to create a roster sheet in sharepoint 2010 This is about training management, when an employee requests for training, it goes to approval workflow and once approved, it is like the employee is enrolled.  Once it is approved I want a roster sheet to be created for that course for all the employees registered and who got approved.   The report I need to generate has to be of the enrolled users of one course and one particular session  If I give copy listitem as approved then different courses would come together in the list  The format for the report How many ounces is a British "tin" of tomatoes? If I have a British recipe that lists "1 tin chopped plum tomatoes," how many ounces is that? In British English, does "tin" usually refer to a specific size? In the US, cans generally come in 14 and 28oz sizes - which one is likely closest to what the recipe means? Concatenate multiple rows Prevent abstract environment to reset pagenumbering Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side (say $AB$) of the polygon is shorter than the circular arc with the same endpoints ($\stackrel{\frown}{AB}$). Summing all these inequalities shows the perimeter of the inscribed polygon is indeed smaller than that of the circle.  I'm wondering if there is proof that the perimeter of a circumscribed polygon always overestimates the perimeter of the circle, which is as simple as that of the inscribed polygon case. Thanks! Run VirtualBox in background, without a window? I was wondering if it's possible to run an Ubuntu virtual machine using VirtualBox in background, I mean, without any window open.  The idea is to connect via SSH to the Linux host, that would be running in background.  Has anyone ever done something like this? Is it possible? How to render lightning in Cycles How can you render a lightning bolt using Cycles? Send email using the GMail SMTP server from a PHP page How to convert a column number (e.g. 127) into an Excel column (e.g. AA) How do you convert a numerical number to an Excel column name in C# without using automation getting the value directly from Excel.  Excel 2007 has a possible range of 1 to 16384, which is the number of columns that it supports. The resulting values should be in the form of excel column names, e.g. A, AA, AAA etc. Counting the Lotto number draws which contain consecutive numbers