Suppose      $a,b\in \mathbb{Z}$. Is it true        $\sqrt{a}\sqrt{b}=\sqrt{ab}$. If so, then     $\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1$  But we know      $\sqrt{-1}=i$ and so $i^2=-1.$ Finally we get     $i^2=-1=1.$  Which is not true. What is the logic behind it?     Thank you in advance.
We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$  EDIT: I see this has been dealt with before but at least with  I'm not making the fundamental mistake of assuming an incorrect definition of $i^2$.
What Hogwarts classes are taught to two houses at once? I remember from my mumble about nine mumble rereads of the Harry Potter series that the following classes Harry took were taught to more than one house in the same lesson:   Potions with the Slytherins  Herbology with the Hufflepuffs  Care of Magical Creatures with the Slytherins  broomstick riding with at least the Slytherins (maybe other houses, too?)    I don't remember any student from a house that isn't Gryffindor being mentioned in the sections on Transfiguration, Charms, Defense Against Dark Arts, History of Magic, or Divination. That would mean McGonagall had to teach twice as many classes as Snape, if both subjects got the same number of lessons (which I think I recall).   Were any subjects other than Potions, Herbology, and Care of Magical Creatures taught to more than one house in the same lesson? And do we know why some subjects are taught to more students at the same time than others? Is cross-posting a question on multiple Stack Exchange sites permitted if the question is on-topic for each site? It is possible to migrate a question from one Stack Exchange site to another by closing, but if I have a question that I think is on-topic for multiple Stack Exchange sites, is it OK to post it on both (multipost)?   For example, I have a question that's earned me the tumbleweed badge on SO and I'm not sure what the best thing to do with it is. It's about a web server so it might be answerable on Server Fault but it's really more of a programming thing, hence the posting the question on Stack Overflow.    Is there any way to make the question visible on multiple sites (crosspost) and then accept the answer wherever it came from? Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods? Ethernet: how important is the magnetics topology? I have some Ethernet issues with a set of boards (6/10 are bad). They're possibly just assembly house mistakes... but I'm very concerned about my magnetics jack.  I could have sworn I saw a example schematic with my jack's topology used with my PHY before, but I can't find it. The PHY's datasheet and suggested magnetics jacks are all identical in topology, and very different than what I have.  I'm using a SMSC  PHY, with a Bel  magnetics jack.  The transformer ratios on my jack vs. a suggested jack for this PHY like the  are the same (1:1).  The inductance is the same, 350uH. However, the choke on mine is on the circuit side, not network side. Also, while the TLA has four 75 ohm resistors from taps tied to a single 1nF/2kV cap, mine has an extra set of 1nF caps between the taps and the 75 ohm resistors.  The main difference is that my jack is a POE jack, and I thought I found an example that used one exactly like it. So, the topology is obviously different just because of that feature.  At the moment, I have 4 good boards, 1 'sometimes' board, and 5 boards that just blink a LED at 11.68Hz representing some error that no datasheet explains.  How important is this? Of course there's no other jack with that footprint.  EDIT  I added some detail on the magnetics jacks above. I was just recommended the SI-60152-F, so I'll compare to that one. Of course like all the other recommendations... it also has no stock at anyone on FindChips.  LATE EDIT  It turns out my problem was the crystal layout that fed the Ethernet PLL.  On the respin, I went with a more standard non-POE jack. Java: how do I check if a Date is within a certain range? I have a series of ranges with start dates and end dates. I want to check to see if a date is within that range.  Date.before() and Date.after() seem to be a little awkward to use. What I really need is something like this pseudocode:  boolean isWithinRange(Date testDate) {     return testDate &gt;= startDate &amp;&amp; testDate &lt;= endDate; }   Not sure if it's relevant, but the dates I'm pulling from the database have timestamps. Problem with babel and tikz using \draw I'm trying the next code:  \documentclass[pdftex,spanish]{article} \usepackage{babel} \usepackage{tikz}  \begin{document}  Simple arrow:   \begin{tikzpicture}     \draw[-&gt;] (-1.5,0) -- (1.5,0); \end{tikzpicture}  \end{document}   but it doesn't work. It throws the errors:    ! Argument of \language@active@arg&gt; has an extra }. ! Paragraph ended before \language@active@arg&gt; was complete.&lt;to be read again&gt;\par \draw[-&gt;]   What can I do if it works right without the babel package? print IP address with interface name I want to show the IP address like below  lo : 127.0.0.1  eth0 : 192.168.5.123 eth1 : 192.172.0.212 wlan0 : 10.1.0.124   I'm able to print all IP address by ifconfig | awk '/inet addr/{print substr($2,6)}'. But it is only printing the IPs. Every system have their own interface names and addresses. So my script has to show their interfaces with respect to IP addresses. Why do we use the base form of verbs instead of the past form in past tense questions? Looking for story title - younger gen waters down age-slowing drink of older gen, crammed into small apartment I remember a short story in which multiple generations of people being crammed into a small apartment in a city. The magical drink that either reduced or completely stopped aging was something like a soda that came in a six pack. The story focused on visual aspects of the drink at some point. The younger family members complained that their elders were living too long and apparently decided to water down the soda, I think that conclusion was ultimately left up to the reader. Many thanks for any help tracking down this story, might be one of Kurt Vonnegut's perhaps? Different ways to prove there are infinitely many primes? Developing Unit Tests While Refactoring Legacy Code? Getting my File Manager to link to my Google Drive I have a very small amount of storage on my computer and would like to conserve it, so I would like to know if there is a way to mount my Google Drive in Thunar or Nautilus on Ubuntu 12.04.5 LTS.   Yes, I know, it's weird that I'm saying that I have a small amount of storage when I have 2 file managers installed, but I literally have 5GB out of 16GB left. What are the benefits of running chef-server instead of chef-solo? I am looking at automated deployment solutions for my team and have been playing with Chef for the past few days. I've been able to get a simple web app up an running from a base Red Hat VM using chef-solo.  Our end goal is to use Chef (or another system) to automatically deploy application topologies to the cloud as we run builds. Our process would basically run like this:   Our web app code, dependencies, and chef cookbooks are stored in SCM A build is executed and creates a single package for images to acquire and test against The build engine then deploys new cloud images that run a chef client to get packages installed. The images acquire the cookbooks from SCM or the Chef server and install everything to get up and running   What are the benefits and/or use cases for getting a Chef Server running?  Are there any major benefits to have a Chef Server hold and acquire the cookbooks from SCM vs. using chef-solo and having a script that will pull the cookbooks from SCM? How do I get list of all tables in a database using TSQL? What is the best way to get the names of all of the tables in a specific database on SQL Server? What is the most bizarre thing you have seen done with TeX I think the questions says it all! :)  By TeX, I mean any derivative of TeX as long as the code is clearly a derivative of TeX, i.e. that it could belong on TeX.SX (and not pure Lua for example).  I'll give two answers myself so you see what I mean. Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics Why doesn't "its" have an apostrophe? I know that its is the possessive and it's is the contraction, and know when to use them.  But why doesn't the possessive have an apostrophe?   "The bear's eating a fish." [contraction] "The bear's coat is brown." [possessive] "It's eating a fish." [contraction] "Its coat is brown." [possessive] "One's eating a fish." [contraction] "One's coat is brown." [possessive]    lists the etymology as "From  +‎ ", and  says that this is actually the original form:     Originally written it's, and still deliberately spelled thus by some writers until early 1800s.   So what happened to the apostrophe?  When did people stop using it, and why did they?  It seems that it's as the possessive is more natural, as most people do this until they're taught that it is wrong (or even after).  Update: Online Etymology Dictionary has been updated to include two potential explanations:     The apostrophe came to be omitted, perhaps because it's already was established as a contraction of it is, or by general habit of omitting apostrophes in personal pronouns (hers, yours, theirs, etc.).   Can anyone back up either of these arguments?  The possessive one's still has the apostrophe, despite these. Genes and Intelligence Assuming that intelligence has a genetic component,   • do we know which genes contribute to it?   and, if so,  • can we predict intelligence from genomic analysis? Any way to list similar commands? When you write the command name wrong, bash often does this:  septi@norbert:~$ good No command 'good' found, did you mean:  Command 'gold' from package 'binutils' (main)  Command 'gmod' from package 'gmod' (universe)  Command 'goo' from package 'goo' (universe)  Command 'god' from package 'god' (universe)  Command 'geod' from package 'proj-bin' (universe)  Command 'gord' from package 'scotch' (universe) good: command not found   Or sometimes it does this:  septi@norbert:~$ nftp No command 'nftp' found, but there are 23 similar ones nftp: command not found   Is there any way to ask bash to show these 23 similar commands for me? And, is there a way to show similar commands, including those that aren't yet installed, instead of running the application, ftp for example? Which "mean" to use and when? So we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mean and it's application to 'speed' related problems).  However, a question that has always intrigued me is "how do I decide which mean is the most appropriate to use in a given context?" There must be at least some rule of thumb to help understand the applicability and yet the most common answer I've come across is: "It depends" (but on what?).  This may seem to be a rather trivial question but even high-school texts failed to explain this -- they only provide mathematical definitions!  I prefer an English explanation over a mathematical one -- simple test would be "would your mom/child understand it?"