How is 0.9... equal to 1 How is $0.9999\ldots$ equal to $1$? I researched it a bit on , and I find a lot of different ways, but I'd just like to clear upon them. Intuitive: I understand that this explanation is saying that $1$ is the highest number greater than $0.\bar{9}$, but that doesn't make sense why they are equal, for example, $0$ wouldn't be equal to $0.00000(1)$. Discussion on completeness: I honestly didn't understand what it meant, but in the next paragraph it says the previous paragraph isn't proof. Formal proof: I followed it for a while until to the $0\le 1-x\ldots$ line, then I got lost in how it was trying to prove. Algebraic arguments: I don't follow how this one works, because, say $x = 0.99$ (To simplify things, it could be $0.999$ if wanted), $10x = 9.9$ $10x = 9+0.9$ while in the argument it stated $10x = 9+0.99$ (I believe $0.99$ should be $0.9$) $10x = 9+x$ $9x = 9$ $x = 1$ Analytical Proof: This time I haven't got the faintest clue how this relates, and what it's saying. I know I didn't cover most of the arguments, and there's a lot of math concepts I haven't scratch the surface of yet, but I was just curious how this is true. I didn't understand if possible for the questions, show a summary of what it was trying to say. Lastly, I just wanted to say I have a very elementary knowledge of mathematics. I still have an extremely long journey to go, so if don't understand a lot of the complicated, or even simple concepts that may help explain the problem, please be patient.
Is it true that $0.999999999\ldots=1$? I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is?
As the title suggests, I’m simply looking for a method to compute the number of Eulerian Paths in the complete graph $K_5$, where all paths start from the same node. Since there are 5 nodes, all Eulerian paths must be Eulerian circuits as well.  Thanks. Is this a correct demonstration for why elements above untriseptium cannot exist? With the confirmation that elements 113, 115, 117, and 118 are indeed fundamental elements that are now to be named on the periodic table, the next question is: what is the highest atomic number possible for an element? Feynman had a go at this years ago, and he derived (according to my limited understanding) that above element-137 (informally denoted 'Feynmanium') the electrons in the nearest orbit about the nucleus would be traveling at a velocity greater than the speed of light. (Note I am considering the Bohr model of the atom in this question.) I wished to demonstrate to my friends why this is and I came up with the following explanation.  According to Bohr's quantum condition, the angular momentum of an electron about the nucleus is given by $$ L=m_evr_n=n\hslash \implies r_n=\frac{n\hslash}{m_ev},$$ where $m_e$ is the mass of the electron, $v$ its velocity, $n$ an integer, and $r_n$ the radius of the $n$th possible orbit. Because we are concerned with the nearest orbit to the nucleus, $n=1$; and thus $$r=\frac{\hslash}{m_ev}.$$   Now, according to Coulomb's law, if the electron orbits about the nucleus the centripetal motion can be described by $$\frac{Ze^2}{4\pi\epsilon_or}=m_ev^2, $$ where $Z$ denotes the number of protons in the nucleus (the atomic number), and $e$ the elementary charge. Solving for $Z$ and substituting in $r$ from above yields $$Z=\frac{4\pi\epsilon_o \hslash v}{e^2}. $$ But what is $v$? Well, the maximum velocity an electron could ever have is the speed of light, and we wish to find the atomic number associated with an orbiting electron traveling at this speed, so we set $v=c$ and obtain our final result of $$Z=\frac{4\pi\epsilon_o\hslash c}{e^2}\approx 137.521,$$ which implies that for $Z&gt;137$, the electrons at a position of $n=1$ in the Bohr model would have a velocity $&gt;c$; and thus the highest atomic number achievable on the periodic table is 137.  Again, I just want to make sure this is a correct method for deriving element-137 before presenting. Perhaps one could explain how relativity plays a role here. I know Feynman used the Dirac equation to get this result...so could anyone (subsequently of course) expatiate on this in a simplistic manner? Thanks! Convert UTC/GMT time to local time We are developing a C# application for a web-service client. This will run on Windows XP PC's.  One of the fields returned by the web service is a DateTime field. The server returns a field in GMT format i.e. with a "Z" at the end.  However, we found that .NET seems to do some kind of implicit conversion and the time was always 12 hours out.  The following code sample resolves this to some extent in that the 12 hour difference has gone but it makes no allowance for NZ daylight saving.  CultureInfo ci = new CultureInfo("en-NZ"); string date = "Web service date".ToString("R", ci); DateTime convertedDate = DateTime.Parse(date);               As per :     UTC/GMT Offset       Standard time zone: UTC/GMT +12 hours   Daylight saving time: +1 hour   Current time zone offset: UTC/GMT +13 hours     How do we adjust for the extra hour? Can this be done programmatically or is this some kind of setting on the PC's? I am running Ubuntu 12.04.1 LTS i686 on VirtulBox. Every time I check the updates, there's a new Linux kernel. If I install the new kernel, I have to install the Virtulbox Guest Additions again and reboot the server. I don't want to do this every week. I know I can manually uncheck the kernel packages from update manager, but is there a way to skip the kernel update automatically? I found an answer here, but it's for Ubuntu 10. Thanks. How to access the road graph plugin in python? I am developing a few scripts in Python for QGIS 1.8 to automate some geo-tasks and one of them would be to calculate the shortest distance from a hospital to each building in a town.  I would like to take advantage of the already existing Qgis plugin but, as it is a C++ plugin, I am not sure whether it is possible to access it from Python? Is there a binding or a trick for that?  If not, any further advice would be appreciated! As the base data is OpenStreetMap data stored in a PostGIS DB, I may also try with pgRouting. But the QGIS plugin would be my favourite choice if possible.  PS: this question is close to this one:  but I believe it's still different enough to be worth a new thred. Dismiss Windows 8 Volume Box (top left corner) When adjusting the volume in Windows 8.1, a box appears at the top left corner of the screen. This box sticks around for ~13 seconds before it fades away. If you attempt to dismiss it by click on it, it switches to the music app.  How do you disable or dismiss this annoying box? At the very least, can the time visible be adjusted? More times than not, it's in the way. Wireless USB Adapter 0bda:c811 Realtek Semiconductor Corp How to Connect & Share your Internet Connection (Wired & Wireless) I have Ubuntu 12.04 on a HP 430 notebook and this has a single wired internet connection and I would like to share this with a LG Optimus One with Android 2.3.3. I followed  instructions and created a hotspot as shown below, but UbuntuHost is not visible on the LG Optimus One when I enable the Wifi on it.  To summarize, I want my Laptop with an internet connection to act as a WiFi router and access the internet from other devices (LG Optimus One with Android in this case).  What additional steps have to be done? Why does the sizeof operator return a size larger for a structure than the total sizes of the structure's members? The  specifically says not to ask  for career advice but people do it anyway.  Should I consider the Help Center outdated or only loosely enforced?    And what about a career advice site on the network?  There are 70+ sites, and most of them are for relatively narrow interests (which is not a criticism, I enjoy looking through the many sections).  A career section seems like it would be very popular.  Also, a careers section would help keep career questions off the other sites. In C++, sizeof('a') == sizeof(char) == 1. This makes intuitive sense, since 'a' is a character literal, and sizeof(char) == 1 as defined by the standard.  In C however, sizeof('a') == sizeof(int). That is, it appears that C character literals are actually integers. Does anyone know why? I can find plenty of mentions of this C quirk but no explanation for why it exists. Killing form on $\mathfrak{sp}(2n)$ I have the same question as  from a long time ago. Is there an easy way to see that the Killing form on $\mathfrak{sp}(2n)$ is $\kappa(x,y) = (4n+2) \mathrm{tr}(xy)$?  For example, the Killing form for $\mathfrak{sl}(n)$ can be found from the easier to calculate case of $\mathfrak{gl}(n)$ using the decomposition $$\mathfrak{gl}(n) = \mathfrak{sl}(n) \oplus \mathbb{C}.$$ I don't know a similar argument for $\mathfrak{sp}(2n)$ though. Is there an ingame way to change volume? I like the music, but I still prefer my own music.  Is there an ingame way to change the volume?   I can't find it. It is being surprisingly hard to achieve. I've tried so far:   Download file manager apps from Google Play. Unfortunately, "share" shares the files, not the file names. None of them had a option like copy file names Download backup apps. Unfortunately, none of them backup the file names of a folder.   Any idea to copy to the clipboard or share the file names of all the files of a folder? Protect .NET code from reverse engineering? Why is the direction of cross products of two vectors perpendicular to the plane? Why is the direction of cross products of two vectors perpendicular to the plane? How is that possible? How to create an interactive bash script to collect arguments for a command in linux? I'm trying to create an interactive bash script where I can collect user input variables to be used as arguments in another command. Can someone provide me an example of this?  Currently i want to do:   what type of content you want to delete? what is the status?   then feed it into a mysql delete query command. Where do I find out how a command/environment is defined? Say I want to redefine the itemize environment, or make a modified version of it. I want to know how it is currently defined. Where do I find this information? If it's some particular package command, I can just look at the .sty file for that package, but if it's something more basic, I don't know where to look...  Is there a reference manual of where all this stuff is explained? Or is there a file in the depths of my texmf tree I should poke around in? Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$ For days I tried several ways to install Windows 10 . With Windows Update I have always an error and with the media creation tool microsoft I have this message: " The installation failed in SAFE_OS phase with an error when BOOT operator " with this code : 0xC1900101 - 0x20017 Having tried everything I do not know what to do ... Can you help me please ? Thank you  PS: I migrated my OS with the software "Paragon migrate OS to SSD" from my hard disk to my SSD. Maybe I have this errors because I did this....