Logical proof that .9999...=1 I know that  mathematics says that $0.99999....=1$. but some say that logic says that $0.99999....&lt;1$. what I want to know, is weather or not there is logical proof (proof without formulas, and that relies solidly on common sense ) that $0.99999....$ must be equal to $1$.  I have heard great arguments for both sides, but they all seem to end with logic and math disagreeing. I want answers that are insightful, and make math and logic agree with each other.   OK a lot of people are asking what the logical proof is, well a few of the ones that I have heard of so far are: If you move a ninth of distance to your destination then a ninth of whats left, then a ninth of whats left after that, and so on, and so on, you will never get there. Another logical question that comes up is: How can an infinite sum be equal to any non fractional whole number, isn't it never ending? can you help me explain why these ideas are flawed without using math?  Also if you'r wondering what I mean when say logical proof, I mean proof by logical deduction with as little use of mathematical formulas as possible, in other words I want proof using common sense. all of the other proofs I have seen are great and fallow the laws of mathematics perfectly, but their are some people that agree with their own common sense more than math. I want an answer I can give to them.  so the less formulas the better, try to rely on firm logic.
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?
If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$ Let $A$ be a commutative Noetherian ring and let $I$ and $J$ be ideals of $A.$  Suppose that $I\subseteq J$ and that $A/I \cong A/J$ as rings.  I want to prove that $I=J.$    Observations so far:  1) If we drop the hypothesis $I\subseteq J,$ then the result is false:  $$\mathbb{Q}[X,Y]/(X)\cong \mathbb{Q}[X,Y]/(Y) \text{ but } (X)\neq (Y).$$  2) If $A/I\cong A/J$ as $A$-modules, then $I=J$:  $$I=\mathrm{Ann}_A(A/I)=\mathrm{Ann}_A(A/J)=J.$$  I'm now a bit stuck as to how to proceed. Any hints would be most helpful!  Many thanks :) Non-uniform distribution of p-values when simulating binomial tests under the null hypothesis I heard that under the null hypothesis the p-value distribution should be uniform. However, simulations of binomial test in MATLAB return very different-from-uniform distributions with mean larger than 0.5 (0.518 in this case):   coin = [0 1]; success_vec = nan(20000,1);  for i = 1:20000     success = 0;     for j = 1:200         success = success + coin(randperm(2,1));     end     success_vec(i) = success; end      p_vec = binocdf(success_vec,200,0.5);     hist(p_vec);   Trying to change the way in which I generate random numbers didn't help.  I would really appreciate any explanation here. VB6 IDE cannot load MSCOMCTL.OCX after update KB 2687323 So we have something like:  \documentclass{article} \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2} \usepackage{amsmath}  \begin{document}  \tableofcontents  \section{Brushless Motor Fundamentals} \subsection{Brushless Motor Operation} \subsubsection{DC Motor Operation} Torque is generated in DC motors from the magnetic force, also known as the Lorentz force, which is produced when an electric current is passed through a coil in a magnetic field. This force is given by $F=q[E+(v\times B)]$, where $F$ is the force perpendicular to the coil, $E$ is the electric field in the coil, $B$ is the magnetic field, and $v$ is the velocity of the charged particles in the coil. From mechanics, torque is $\tau=F\times r$.  \section{Brushless Motor Fundamentals 2} \subsection{Brushless Motor Operation} \subsubsection{DC Motor Operation 2} If the electrical force is ignored and the remaining magnetic force is used, with the assumption that $v$ is perpendicular to $B$, we find that $\tau=qvBrsin\theta$.  \end{document}   How to show subsubsections and paragraphs in TOC? What laptop specs are important for intensive image editing work? How important is the following in purchasing a laptop that will be used for image editing:   processor speed amount of RAM integrated vs. dedicated graphics card HD speed (SSD; or 5400/7200/10000 rpm) revtex4-1 and colortbl doesn't fill cells The MWE below shows that the \cellcolor command doesn't properly fill a table cell in revtex4-1 where it would do so in a simple article. Can anyone help me get fills that extend to the boundaries of my cells in revtex4-1?    \documentclass{revtex4-1} \usepackage{colortbl} \begin{document} \begin{tabular}{ |c|l|l| } \hline \rowcolor{red} A    &amp; B     &amp; C     \\ \hline Hello World         &amp; other     &amp; \cellcolor{blue}stuff\\ \hline \end{tabular} \end{document} For example, when I type ping 8.8.8.8 &amp;, why can we see ping process working? And when I type find / -name '*test*' &amp; , it's also seen on monitor.  Why is that? Is it not really in background? I am an operator (OP) on my friends Realm. Today I tried to play and noticed that no commands would work for me, or anyone else granted OP. They only worked for the owner of said realm.   He has de-oped us and re-oped us with no cure to the problem. What's causing this?   Also another problem that when he tries to reload the realm it fails every single time, any options that would fix this problem? Regaining control of a group after being removed by another admin I created a Facebook group and was later removed by another admin.   How do I get my group back? Why is the cone measurable? Problem Denote by $\lambda_n$ the Lebesgue measure on $\mathbb {R}^n$. Let $h \in \mathbb {R}_{&gt;0}$. Let $A \subset \mathbb {R}^{n-1}$ be Lebesgue-measurable with finite measure. Define the cone $C(A,h)$ as the union of all straight lines connecting a point in $A \times\{0\}$ with the point $(0, h) \in \mathbb {R}^{n-1} \times \mathbb{R}$. Prove that $\lambda_n (C(A,h))=\frac{h}{n}\cdot \lambda_{n-1}(A)$. Where I struggle  Assuming that $C(A,h)$ is (Lebesgue-)measurable, this is a simple application of Fubini. How Fubini is used is all clear to me.  However, why is $C(A,h)$ measurable? It seems one could roughly argue as follows: As $A$ and $[0,h]$  are measurable so is $A \times [0,h]$. As $A \times [0,h]$ is measurable it is the union of a null set and a Borel set. $C(A,h)$ is the image of $A \times [0,h]$ under a certain diffeomorphism (c.f. ). Diffeomorphisms map null sets to null sets, and Borel sets to Borel sets. Hence, $C(A,h)$ is (as a finite union of measurable sets) measurable.  Unfortunately, I am not allowed to use the above properties of diffeomorphisms. What are other ways to prove the claim? Please only give a hint to set me thinking. Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$ I need help proving the following statement:  For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$  The statement is true, I just need to know the thought process, or a lead in the right direction. I think I might have to use a contradiction, but I don't know where to begin.  Any help would be much appreciated. STI vs STD vs Sexually Transmitted Virus Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? How can i set different wallpapers for Ubuntu 15.04 login and Desktop This has been bothering me very much. i have memory that Ubuntu 12.04 had this configuration? am i wrong? finally, is it possible for the login screen to have different wallpaper from the desktop? is this possible in Ubuntu 15.04? i would appreciate a stable understandable response. thanks :)  -ben How can I get back a system file after deleting it from my Mac? I've deleted a system file from my Mac and need to get it back.  System files can include:   hidden files in the root of / - especially unix files in /private  /System files /Library files /Applications files that ship with the core OS   System files are not any app, user files or files that can be regenerated by a known sequence such as cache files, Spotlight index files, etc…    Canonical question and answer for . I am able to parse strings containing date/time with time.strptime  &gt;&gt;&gt; import time &gt;&gt;&gt; time.strptime('30/03/09 16:31:32', '%d/%m/%y %H:%M:%S') (2009, 3, 30, 16, 31, 32, 0, 89, -1)   How can I parse a time string that contains milliseconds?  &gt;&gt;&gt; time.strptime('30/03/09 16:31:32.123', '%d/%m/%y %H:%M:%S') Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "/usr/lib/python2.5/_strptime.py", line 333, in strptime     data_string[found.end():]) ValueError: unconverted data remains: .123 How to run updmap-sys after installing cm-super? My original question was:   The answer that seems suitable to my cause is to install cm-super. I did that (using package manager [winxp 32 bit, MikTex installation 2.9 recently updated]). But still the command  \usepackage{cm-super}    yields an error.  User H Cavendish suggested to run updmap-sys, but I am not sure about how to do that on WinXP.   2011 November 04 edit: I made it sure that I have installed 'cm-super' - but still get the same error:   I have also tried updating the file name databases - but no avail. Example of a recursive set $S$ and a total recursive function $f$ such that $f(S)$ is not recursive? Browsing wikipedia, I stumbled on the following: "The image of a computable set under a total computable bijection is computable." Given the form of the theorem, there must be some example of a example of a recursive set $S$ and a total recursive function $f$ such that $f(S)$ is not recursive (otherwise the statement would have presumably been stronger). But I'm having difficulty dreaming up a case where this happens. Check tag classes with watir? How to disable "time travel" entirely? Every time I open a question I get this oh so funny 90s feel with scrollers and whatnots. In every question I need to click it off. The button seems to only work for the specific page I load; loading a different page shows the theme again.  Is this an attempt to keep people from using the site for a few days or how do I tell you "please never ever do anything like this if you want me to keep contributing here"? Yes I have no sense of humor if this is an attempt at that.  How do I turn off "time travel" (also known as "back to the future") mode off permanently, on all questions, and preferably network-wide?