I was wondering is there any way to express $0.999999$ recurring as an actual fraction without equaling $1$? Because I tried to convert it into a fraction following the rules for normal recurring decimals like this:  $$\begin{align}n&amp;=0.999\dot9\\10n&amp;=9.999\dot9\\n&amp;=0.999\dot9\\9n&amp;=9\\\therefore n&amp;=9/9\end{align}$$  But as you can see the result is $9/9$ which ultimately is equal to $1$ . And I've even tried calculating it other ways like this:  $$\begin{align}1/3&amp;=0.333\dot3\\2/3&amp;=0.666\dot6\\\therefore3/3&amp;=0.999\dot9\end{align}$$  But it always ends up telling me that $0.9999999... = 1$. Is there any mistake in my logic? And I also realized this applied to other recurring decimals ending in $9$. E.g: $0.5999999...=5.4/9 = 0.6$ . So is there a way to write $0.999999...$ as a fraction so you can differentiate it from $1$?
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?
Can a Sorcerer protect their friend from Web by using Careful Spell? With the metamagic "Careful Spell" does the caster protect his friend from a Web spell for its full duration or just for the first save? If the friend stays in the area or re-enters the area does he continue to automatically save against the restraining effect? I'm having some trouble understanding cosets.  If I understand it correctly, cosets form a partition. So, if I have $|G:H| = 2$, then $G = H \cup xH$. Right?  In an exercise, I'm asked the following:  "For $G$ group and $H \le G$ such that $|G:H| = 2$, prove that $x^2 \in H$ for any $x \in G$"  If $x \in H$, then $x^2 \in H$ because $H$ is a group. But, if $x \notin H$, then $x \in gH$ for any $g \in G$, right? Then $x^2$ = $(gh)^2$ but I can't see why this is in $H$.  Can you give me some hint? Thanks for stopping by. If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication) Let $V,W$ be normed vector spaces, and $L(V,W)$ be the space of bounded linear operators. Usually I would only see the statement "If $W$ is Banach, then $L(V,W)$ is Banach.". But  writes that there is a converse: "If $L(V,W)$ is Banach, and if $V$ is non-trivial, then $W$ is Banach". This is pretty interesting since I never seen a converse before. I was wondering if anyone has a nice proof.  I tried reversing the proof for the usual direction, but the inequalities can't be reversed. I also tried to start with a Cauchy sequence in $W$, and construct linear operators (using Hahn-Banach to control the operator norms), but alas I cannot say much about the distance between these operators (much less say that it is Cauchy) It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know. Tree level QFT and classical fields/particles Term or phrase describing action occuring when not watching Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic. I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. Then I found logic that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is certainly cyclic if $m$ and $n$ are coprimes.  I searched on google a proof of that, to compare with mine and the answer is that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z}/mn\mathbb{Z}$ (which is clearly cyclic) $\iff$ $m$ and $n$ are coprimes.  But I got another proof and I want to know what you think about it :     I claim that $([1]_m,[1]_n)$ generates $\mathbb{Z}/m\mathbb{Z}$   $\times\mathbb{Z}/n\mathbb{Z}$, where $[a]_d$ is the canonical   projection of $a\in \mathbb{Z}$ into $\mathbb{Z}/d\mathbb{Z}$.       Let's take multiples of $([1]_m,[1]_n)$ until we get $m \cdot$   $([1]_m,[1]_n) = ([0]_m,[m]_n)$ (where I suppose $m=\min(m,n)$...   otherwise, consider $\mathbb{Z}/n\mathbb{Z}$   $\times\mathbb{Z}/m\mathbb{Z}$ and $\min(m,n)=n$).       The next step is $([1]_m,[m+1]_n)$. This pair is not equal to   $([1]_m,[1]_n)$, since $m+1 \not\equiv1$ mod $n$ ($m$ and $n$ are   supposed to be coprimes). We can now continue adding $([1]_m,[1]_n)$   to the results.      The last step is precisely reached when you got   $([mn]_m,[mn]_n)=([0]_m,[0]_n)$. At this point, the fact that $m$ and $n$ are coprimes is absolutely necessary because if they are not coprimes, $lcm(m,n)\ne m\cdot n$ and   $([lcm(m,n)]_m,[lcm(m,n)]_n)=([0]_m,[0]_n)$ but all the $m \cdot n$ possible pairs are not reached, only the $lcm(m,n)$ are ! Predicting count data with random forest? Are there any banned subjects on Area 51? I know of the following rules for creating a proposal on Area 51:  The scope must be reasonably well-defined, not too unclear (e.g. &quot;cool stuff&quot;), too opinion-based (e.g. &quot;car reviews&quot;), too broad (e.g. &quot;life on Earth&quot;), or too narrow (e.g. &quot;The 1995 academic misconduct scandal at East Oak Mews High School&quot;). One  come to Area 51 with an already existing community rather than hope to build one on Area 51. Proposals that are subsets of existing sites are not acceptable (e.g.  — deleted). These get closed as &quot;This subject is already well-covered by a live Stack Exchange site. We generally do not split off subjects simply to give them their own space.&quot; Proposals for non-English sites .  Other than the criteria above, are there any per se restrictions on site content for an Area 51 proposal? If so, what topics are currently banned? I would expect a proposal for a &quot;Let's organize a lawsuit against Stack Exchange&quot; or &quot;[Specific SE staff member] sucks, amirite?&quot; would get closed for obvious reasons, but I'm interested if there are any non-obvious topics that are not currently allowed in proposals regardless of how well-defined, well-scoped, original, and well-supported by a community they are. Having a list of banned topics would be helpful to potential proposers as well as the communities that back them. For example, I wouldn't want to go to all the effort of organizing a community, bringing them to Area 51, etc., only to have my proposal closed with a curt &quot;Sorry, due to the political climate, we are not currently accepting proposals for sites about real-life weapons.&quot; custom closure. To be clear, I am asking solely about topic restrictions (e.g. &quot;no sites about fishing, hunting, or weapons&quot;). I'm not asking about clarity requirements, the requirement to have a pre-existing community, not duplicating an existing site, or the requirement to not use sock puppets to artificially inflate proposal stats. Those requirements are already well-defined elsewhere or else are common sense. Also to be clear, I don't have a specific proposal idea in mind right now or the community to make it happen. I'm asking in general. The question could be rephrased as &quot;What are those topics where I shouldn't even bother to try to define, gather a community, etc. because any attempt to propose a site will be summarily closed due to covering a banned topic?&quot; This is not a duplicate of  because that question is about the appropriateness of expanding a proposal to cover things that are acceptable proposal topics but arguably don't make sense in combination with the original proposal scope. My question is about topics that are completely banned from proposals regardless of whether they are included in the original proposal description or tacked on by the community later. Writing someone else's master's thesis: Unethical and illegal? It is really difficult for me to decide on this situation: when someone calls me and asks for work-for-hire for his/her thesis. I don't know what to do. I want to help and earn money but I think this way of earning money is unethical.  I think that receiving the title of thesis and giving back prepared manuscript is the intention that some people have, when asking for their thesis to be written.  There are plenty of websites offering such services!   The question is that: Is it legal and or ethical to write someone else's master's thesis as work-for-hire? Synaptics driver doesn't load sometimes How to identify polygons with "flag" sliver errors Working in ArcMap, I have come across errors in a polygon layer that I will call "flag slivers" (taken from similar language in parcels call "flag lots").  These sliver polygons (see image below) are typically composed of just one additional node that user has  accidentally added.    In the images below, the "flag" and the "flagpole" are one single-part feature where the "flagpole" just overlaps itself.  The left-hand image has a total of 5 nodes.     When seen alone the errors are very obvious, but when multiple polygons are adjacent, they are nearly impossible to see because they appear to be the boundary between 2 polygons.  This likely happens because they are using a shapefile based editor, and therefore I cannot implement any topology-based editing rules to prevent this from happening in the future.  Does anyone have a way of identifying and resolving these types of errors?  I would prefer an automated method of both identification and resolution since field users are the ones who created the errors, but I am stuck cleaning up after them.  Thanks. If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$. If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.  My attempt at answering the question:  \begin{align*} x &amp;\equiv 0 \pmod{a}\\\ &amp;\Longrightarrow x\text{ is divisible by $a$}\\\ &amp;\Longrightarrow x = ma\text{ for some integer $m$}\\\ \ \\\ x &amp;\equiv 0 \pmod{b}\\\ &amp;\Longrightarrow  x\text{ is divisible by $b$}\\\ &amp;\Longrightarrow x = mb\text{ for some integer $m$}\\\ \ \\\ x^2 &amp;= (ma)(mb)\\\ x^2 &amp;= (m^2)(ab)\\\ x &amp;= \sqrt{m^2ab}\\\ x &amp;= m\sqrt{a}\sqrt{b} \end{align*} Let $m$ be $k\sqrt{a}\sqrt{b}$. Then \begin{align*} x &amp;= kab\\\ &amp;\Longrightarrow x \equiv 0 \pmod{ab} \end{align*}  Is this correct, if not can someone point me in the right direction? How to set class weights for imbalanced classes in Keras? How do websites block selecting text and how do I unblock that? There's a website () that somehow blocks selecting text. Also it blocks Ctrl+A to select everything, there is also no “Copy” in the popup menu.  What I have tried:  Some sites use JavaScript to block selection. So I disabled all JavaScript sources in no-script addon in Firefox. I tried the same site in Google Chrome with the Script Safe addon. Made sure that everything is blocked, still can’t select text. Mousepointer remains an arrow and not a text cursor also on the whole site.  Used about:config to disable JavaScript completely in Firefox. Still no effect.  I read that some sites use DIVs with the style display:block so I used Inspect to examine the styles of the site. There is not one mention of "block" on the whole website, not in embedded CSS nor in object style=-attributes.  The text is not an image or flash or some HTML5 canvas ect. Everything is within DIV and P tags but no style was found that might block the text selection.  How can the website still block any selection of text? Not to mention why browsers support such behaviours at all. I just want to use a selected word or sentence and want to search google for it using right mouse click. It’s pretty annoying when a website does such things and forces me to type technical terms and names into google by hand. It disturbs my workflow. Allow moderators to refresh a user's network profile Currently, moderators have the ability to reset a user's username and/or profile text, and propagate that to the rest of the network, resetting it everywhere. However, often, the user's stackexchange.com network profile will also need to be reset, and this is currently only possible to do by either the user in question or staff.  Instead of bothering staff every time, moderators should have something available similar to chat, where moderators can change a user's parent site and have access to a "refresh profile" button, which will fetch updated information such as username and avatar from the parent site.  Currently, the only thing that moderators can do on stackexchange.com is to see hidden accounts. It would make sense for moderators to be able to refresh profiles there, like chat, and avoid pinging CMs every time this comes up. I just want to write a script which changes my directory.  I put the below commands in the file /home/alex/pathABC  #!/bin/sh cd /home/alex/Documents/A/B/C echo HelloWorld   I did   chmod +x pathABC   In the Terminal, while in /home/alex, I ran ./pathABC, but the output is just HelloWorld and the current directory is not changed.  So what is wrong? Route SSH to host through interface Is it possible to route all outgoing requests for SSH (port 22) to mydynamicdns.afraid.org via wlan0 instead of eth0 when both are connected?  Since my IP is dynamic, I would like to use a name instead of IP. I wish to print a Stack&lt;Integer&gt; object as nicely as the Eclipse debugger does (i.e. [1,2,3...]) but printing it with out = "output:" + stack doesn't return this nice result.  Just to clarify, I'm talking about Java's built-in collection so I can't override its toString().  How can I get a nice printable version of the stack? How to keep Blender from adding frame numbers to output filename?