As $2=\sqrt{4}$ And $4=-(-4)$ $4=-1*(-4)$ $\therefore \sqrt{4}=\sqrt{-1\times(-4)}$ $\therefore\sqrt{4}=2i^2 $ $\therefore 2=-2$. Which is absurd.  I know I have done the mistake in writing (my teacher said) :  $\sqrt{4}$=$\sqrt{-1\times(-4)}$  But I want to know how could this be a mistake? Many people say that $\sqrt{ab}$ is not equal to $\sqrt{a}$$\sqrt{b}$ , if $a,b$ &lt;$0$. But instead of saying that &quot;$\sqrt{ab}$ is not equal to $\sqrt{a}$$\sqrt{b}$ , if $a,b$ &lt;$0$&quot;. Can't this be thaught of a contradiction of accepting that root of negative number exist?
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align}
We have a fairly big MS SQL 2008R2 database that resides on a SSD drive. The drive itself only has ~110Gb of space, and the database files are the only files on the drive.  The database is in "Simple" recovery mode, and only has two files, .MDF and .LDF.   The disk is now nearly full: the MDF is currently 109Gb in size. However, SSMS tells me there's nearly 18Gb of "Space Available" (in the 'General' properties page), and if I go through the motions of Shrinking the file it also tells me there's 18Gb of space free. SSMS also tells me the database size is around 132Gb, which surprises me - that wouldn't fit on the drive!  From what I've read, shrink is a really bad idea. However, I'm starting to see replication errors (could not allocate space for object). We've previously tried shrinking the database, but within a few hours the file was back to it's original size.  How should we proceed - given there's apparently 18Gb of free space, should SQL automatically use that free space? Or is it as simple as: we really need more disk space? Make IN clause behave like AND clause I've asked the same question on , but got no answer, so that's why I'm asking it here (I've read that simple sql questions should be asked at SO, but I have no choice).  Suppose I have 3 tables: posts, post_categories and categories. I'm implementing some kind of page with filters where I can filter posts by categories. User can select multiple categories. When he selects more than one, they should be summed. I can't get it working. What I have now is simple IN() SQL clause:  SELECT posts.id, post_categories.category_id FROM posts  JOIN post_categories ON posts.id = post_categories.post_id  WHERE post_categories.category_id IN (1,2,3) LIMIT 10;   But it is not matching all ids, it is OR and I need AND. What I need here is to find all posts that have categories with id=1 AND id=2 AND id=3, but instead this query returns posts that have at least one category from the IN() list. I clicked on  to a comment in my email and it prompted a list of apps to open link. All browsers (Chrome, Firefox and etc) are in the list, but not SE app.  Most apps that can open site specific links get prompted for that app to open  the link, but not Stack Exchange links for some reason.    App Version: 1.0.95 Device Manufacturer: Asus Device Model: ASUS_Z010D   OS Version: 6.0.1 (WW_Phone-13.8.26.76-20161230) How big is "too big" for an airline seat? At what point will you be required to buy two tickets? As I said in another question, I'm bringing my parents to visit me here in Sweden next year.  My mother is around 280-300 pounds and 5'6" (130-135kg, 1.67m). I am concerned that she might be asked at the airport to buy a second seat. If it's likely I would rather just buy the second seat in advance, because I don't want her to be embarassed/humiliated by some random employee. 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