A rather simple question I suppose, however I have not seen a discussion on this topic on stackexchange as of yet. In fact, when I was in high school I once used the following 'proof' to prove to my teacher that $1.9999... = 2$, where the dots indicate an 'infinite' number of decimal $9$'s:  $$x = 1.99999...$$  $$10x = 19.9999...$$  $$9x = 19.9999... - 1.9999... = 18$$  $$x = 2$$  $$1.9999... = 2$$  My teacher, who has a Masters degree in Mathematics in university, then told me that infinity is a tricky concept in mathematics and that this needn't work the way it seems to. Now, many many years later, being an econometrics student myself, I have found myself thinking about this 'proof'. To me, it seems that claiming a number to have an 'infinite' number of decimal $9$'s is faulty in itself; it would be much more logical to state that the number of $9$'s approaches infinity and as a result, that $1.9999...$ is asymptotically equal to $2$, but not equal to $2$.   However, when Googling this question I am overwhelmed with the number of results where people claim that it is in fact exactly, literally, equal: that $1.9999... = 2$ is undisputably true. As a result I am now unsure whether it is or is not true.   Thus, is $1.9999... = 2$?
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?
Cancelling special & general relativistic effects I found Minecraft's marvelous large worlds extremely slow to navigate, even with a quad core and meaty graphics card.  I assume Minecraft's slowness comes from:   Java, as spatial partitioning and memory management are faster in native C++. Weak world partitioning.   I could be wrong on both assumptions. However, this got me thinking about the best way to manage large voxel worlds. As it is a true 3D world, where a block can exist in any part of the world, it is basically a big 3D array [x][y][z], where each block in the world has a type (i.e BlockType.Empty = 0, BlockType.Dirt = 1, etc.)  I assume that to make this sort of world perform well, you would need to:   Use a tree of some variety (//) to split all the cubes out; it seems like an oct/kd would be the better option, as you can just partition on a per cube level not a per triangle level. Use some algorithm to work out which blocks can currently be seen, as blocks closer to the user could obfuscate the blocks behind, making it pointless to render them. Keep the block object themselves lightweight, so it is quick to add and remove them from the trees.   I guess there is no right answer to this, but I would be interested to see peoples' opinions on the subject.  How would you improve performance in a large voxel-based world? error: Unable to find vcvarsall.bat Draw power from a portable mobile charger to power circuits Was voldemort's body going to die? Voldemort has gained "immortality" via his horcruxes, but was his body immortal to age? Looking at all the questions about Voldemort and his horcruxes what was his plan to keep his body alive. We know his first Horcux was created at age 16, yet we see him age still so it seems like eventually his body would die of old age and his soul would be left in the half dead state he was in after his spell rebounded off Harry. What was his plan here, why would he not want something such as the elixir of life to keep his vitality?     While the Elixir of Life does indeed extend life, it must be drunk regularly, for all eternity, if the drinker is to maintain his immortality. Therefore, Voldemort would be entirely dependent on the Elixir, and if it ran out, or was contaminated, or if the Stone was stolen, he would die just like any other man. Thereafter, I am convinced, he intended to continue to rely on his Horcruxes: he would need nothing more, if only he could regain a human form. He was already immortal, you see ... or as close to immortal as any man can be.' (Albus Dumbledore)   Horcruxes seem to fail at something important, keeping you alive, they stop you from dying truly yes, but they don't keep you alive. Harry is shown over and over again to be able to beat better wizards because hes has faster reflexes both because of youth and his genetics, but presumably by 50 harry will not be as fast as his 16 year old self even considering their longer life expectancy. Why operate in a body of a 50 year old? Is the sorting algorithm used by .NET's `Array.Sort()` method a stable algorithm? Is the sorting algorithm used by .NET's Array.Sort() method a  algorithm? Internet is working, but Firefox 67 browser cannot connect after update How do I upgrade an iPhone 4 from iOS 7.1.2 to iOS 8 or 9 I am considering buying a Dell XPS 15 laptop. The laptop has a Nvidia 420m card which should work with Linux but I keep reading that the Nvidia Optimus technology isn't supported on Linux. I am not really interested in switching from Nvidia to Intel to save power but need to know that the Nvidia card will in fact work if I installed Ubuntu.  If anyone has experience using a Nvidia card with Optimus technology or even better the exact laptop in question (Dell XPS 15 with 1GB NVIDIA® GeForce®  GT 420M) it would be great.  A major problem holding people back from adopting Linux is this sort of hardware issue. I am a long term Ubuntu user and supporter who can't afford to make a mistake with a purchase like this. I don't want to spend £500+ to find I have no graphics acceleration because Windows 7 is not an option. How to start a GUI software on a remote Linux PC via SSH Sometimes I need to start XMBC media player or other GUI software a one of my PC (small Xubuntu PC used as a media center).  Usually I do this starting a X11vnc server on the remote PC via SSH and than connecting with an Xvnc client to the Xfce desktop.  Is there a way to start a GUI software on a remote Linux PC via SSH?  Thanks! I am developing a set of classes that implement a common interface. A consumer of my library shall expect each of these classes to implement a certain set of static functions. Is there anyway that I can decorate these class so that the compiler will catch the case where one of the functions is not implemented.  I know it will eventually be caught when building the consuming code. And I also know how to get around this problem using a kind of factory class.  Just curious to know if there is any syntax/attributes out there for requiring static functions on a class.  Ed Removed the word 'interface' to avoid confusion. How do I copy a property from an active object to selected objects? I have been using the Copy Attributes add-on to copy some properties from active object to selected objects. This add-on is not enabled by default and must be explicitly enabled. But its a very useful workflow enhancing tool.    There is a limitation though. Not all properties can be copied and even those that can be copied are copied in groups (Its hard-coded in to the add-on).  My question is if there is a built-in feature or any another other way to copy any property from the active object to the selected objects? Force JavaScript exception/error when reading an undefined object property? Live in Florida & work remote for a New York company. Do I owe NY state income tax? I have been down in Florida for the past 3 1/2 years and continue to pay New York state taxes out of the money I earn for a company I work remote for that is in New York.  Should I be paying New York state taxes on this income?  Or do I not need to pay New York state tax because I live in Florida?  Our NY accountant says I need to continue to pay since company is out of NY, however others say I don't need to pay.  Pls advise which is correct. Thank you. What's the best way to call a generic method when the type parameter isn't known at compile time, but instead is obtained dynamically at runtime?  Consider the following sample code - inside the Example() method, what's the most concise way to invoke GenericMethod&lt;T&gt;() using the Type stored in the myType variable?  public class Sample {     public void Example(string typeName)     {         Type myType = FindType(typeName);          // What goes here to call GenericMethod&lt;T&gt;()?         GenericMethod&lt;myType&gt;(); // This doesn't work          // What changes to call StaticMethod&lt;T&gt;()?         Sample.StaticMethod&lt;myType&gt;(); // This also doesn't work     }      public void GenericMethod&lt;T&gt;()     {         // ...     }      public static void StaticMethod&lt;T&gt;()     {         //...     } } Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$? If a product of relatively prime integers is an $n$th power, then each is an $n$th power I have unistalled and reinstalled the Ubuntu Software Center as per info I found in a similar thread and I got the same response about line 91 or something like that.  I just tried to upload a screen shot but since I'm new it won't allow me to.  I also can not figure out how to cut and paste anything so I have to hand type what the error screen says, both when I attempt to open the software center and nothing happens, when I try to enter commands into the terminal to uninstall, reinstall, whatever I get the same following:  COULD NOT INTITIALIZE THE PACKAGE INFORMATION An unresolvable problem occured while initializing the package information Please report t:his bug against the 'update-manager' package and include the following error message: 'E: Malformed line 91 in source list/etc/apt/sources.list (dist parse)  E: The list of sources could not be read.,  E: The package list of status file could not be parsed or opened.   How do I report bugs?  What can be done about this.  I have searched and everything everyone says to do leads me back to the same line error message.  So, I don't know how to get to line 91 in the source list; to tell you what it says. Sorry, I'm really new to this. That is what I need is to find out how to get there and fix what it says. I would really like to NOT have to re partition my hard drive and start from scratch, so I'm really looking forward to getting this problem solved. I need to be able to install new software. Fitting text to a shape in TikZ I am designing a poster, which has some geometrical shapes (TikZ paths) surrounded by some text. I would like the text to flow neatly around these shapes. I bet there's some clever TikZ magic that could do this.  EDIT: This is the sort of effect I want, sorry to not be clear:    \documentclass[a4paper]{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \draw (0,0) to (2,-2) to (2,-4) to (4,-4) to [out=up, in=down] (6.5,0); \node [text width=6cm, anchor=north west] at (0,0) { \hspace{5pt}  This is the sort of effect I want to \\ \hspace{23pt} achieve, with text automatically \\ \hspace{36pt} constrained by some \\ \hspace{46pt} geometrical form. Here \\ \hspace{55pt} it's just a curve, \\ \hspace{57pt} but this could \\ \hspace{57pt} also be the \\ \hspace{57pt} text label \\ \hspace{57pt} of a node.}; \end{tikzpicture} \end{document} In the standard PrintDialog there are four values associated with a selected printer: Status, Type, Where, and Comment.  If I know a printer's name, how can I get these values in C# 2.0?