I am just curious if 0 is a real number. The definition of a real number is all rational and irrational numbers. And the def definition for rational number is that &quot;$\mathbb{Q}={a\div b|a,b\in\mathbb{Z}}$&quot;. But in done say that $0$ is a whole number and some day that it is not. In some websites some say that $0$ is a real number, and in some not. Please help me. Thanks in advance
There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-real? For what reason is zero real, and how is it that a value such as infinity isn't when zero is?  Also, what purpose does zero even have? It's obvious that it's used as a place holder, but what value does it have alone? What makes it any different from other numbers? I realize that it is typically interchanged with nothing, but that would technically mean that zero is nothing—and that's not true. People have told me already so many times that zero does not necessarily mean nothing, and I figured it out a long while ago anyway. So, what I want to know is the actual value that can be associated with zero.  However, my question brings up another topic that has been argued and disputed about and disagreed on for years. Zero divided by zero, or 0/0. There are a lot of varying answers about this, but what's been constant throughout everything is that people claim 0/0 is not a real number. So, if I may ask, why is it that zero is considered a real number, and not 0/0? And why is it considered non-real in the first place? True, there is no defined value, but the expression is anything but meaningless; if it was, then there wouldn't be so many people trying to understand it. I guess what I'm trying to figure out is the reason behind the expression's classification as non-real—especially when the sole number of zero is real, despite not having an evident value.  This covers a few different things, so I was uncertain about what to make the title of these questions. I apologize for that. If anyone can answer, though, I'd be extremely grateful!
Regulatory bodies and authoritative dictionaries for English Some languages have a "regulatory body" issuing recommendations and guidelines regarding the use of that language.  For example in the case of Spanish it's the  whose status is recognised in all Spanish-speaking countries. The Academy, among other things, publishes a dictionary (""), in print and , which is usually given a lot of prestige (but is not without , of course).  Are there any such authorative—or at least influential—institution(s) or publication(s) for the English language? Say person A and person B are having a discussion/argument.   Person A tells person B that his views are arrogant, or closed-minded, or some other derogatory adjective. But in fact it is person A's views that are arrogant, closed-minded, etc. Person B's views are usually the exact opposite of what person A is accusing him of.  This sort of situation happens a lot when discussing politics or religion, for example.  What do you call person A, for calling person B something that he himself is?  I guess, for the most part, person A doesn't actually realize that he is the one that the adjective applies to, and not person B. Maybe there is a different name if person A knows the adjective also applies to him. Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero? SQL Server default collation vs database with different collation - potential problems? I'm struggling with the design of new database that has to use a different collation (Y Collation) than the SQL Server Instance default collation (X Collation). I'm looking for potential problems.  Temp Tables by default will be created with X Collation so any joins on text data types will fail unless I create temp table with Y collations on columns.  What about SQL Server string functions: REPLACE, SUBSTRING, etc.? Will they always be returned in X collation? What about when I use REPLACE(...) COLLATE Y? Do you see any potential data loss or maybe any different problems? How do I compare 2 rows from the same table (SQL Server)? I need to create a background job that processes a table looking for rows matching on a particular id with different statuses. It will store the row data in a string to compare the data against a row with a matching id. I know the syntax to get the row data, but I have never tried comparing 2 rows from the same table before. How is it done? Would I need to use variables to store the data from each? Or some other way? (Using SQL Server 2008) What if the universe is rotating as a whole? Suppose in the milliseconds after the big bang the cosmic egg had aquired some large angular momentum. As it expanded, keeping the momentum constant (not external forces) the rate of rotation would have slowed down, but it would never reach zero.  What implications would that have to measurments to distant supernova and CMB radiation? Do we have any experimental data that definitely rules out such as scenario? And to what confidence level?  Edit A  suggests that the universe might indeed be spinning as a whole. Anyone care to poke holes at it?  Edit 2 Even more  places limits on possible rotation. This is, how I download various master branches from GitHub, and I aim to have a prettier script (and maybe more reliable?).  wget -P ~/ https://github.com/user/repository/archive/master.zip unzip ~/master.zip mv ~/*-master ~/dir-name   Can this be shorten to one line somehow, maybe with tar and pipe?  Please address issues of downloading directly to the home directory ~/ and having a certain name for the directory (mv really needed?). Is $K[a,b]=K(a,b)$ for algebraic $a, b$? A lady and a monster A famous problem:  A lady is in the center of the circular lake and a monster is on the boundary of the lake. The speed of the monster is $v_m$, and the speed of the swimming lady is $v_l$. The goal of the lady is to come to the ground without meeting the monster, and the goal of the monster is to meet the lady.  Under some conditions on $v_m,v_l$ the lady can always win. What if these conditions are not satisfied?  Edited: the monster cannot swim. If the conditions are not satisfied, then monster can always perform a strategy such that the lady will not escape the lake. On the other hand this strategy is not desirable for both of them because they do not reach their goals. As there was mentioned, this deals with undecidability of the problem. On the other hand, if you imagine yourself to be this lady/monster, you can be interested in the strategy which is not optimal. What is it? If there are such strategies in the game theory?  Edited2: My question is more general in fact. If we have a game with one parameter $v$ when two players $P_1, P_2$ are enemies and if $v&gt;0$ then for any strategy of $P_2$ the player $P_1$ wins. If $v\leq 0$ then for any strategy of $P_2$ there is a strategy of $P_1$ such that $P_2$ does not win and vice versa. I am interested in this case. From the mathematical point of view as I have understood the problem is undecidable since there is no an ultimate strategy neither for $P_1$ nor for $P_2$. But we are solving somehow these problem IRL. Imagine that you are a lady in this game - then you would like to win anyway even while knowing that your strategy can be covered by the strategy of the monster. On the other hand, the monster knows that if he will cover all strategies of the lady she will never reach the shore and he will never catch her. I mean they have to develop some non-optimal strategies. I hope now it's more clear. A person who cannot understand that there may be other perspectives I'm searching for a word to describe a person who cannot understand that there may be different perspectives. I initially thought of single-minded, but that pertains to determination.  One example of several perspectives could be the difference between a process and an action. Please assume that a process is composed of several actions. A process may be composed of other processes, but these processes would be actions in this greater process.  If you accept my definitions, then depending on the holistic or granularity level, we may consider something a process or an action, depending on whether we are considering its parts or how it's part of a greater thing.  A person may have trouble understanding that it's not true or false that something is a process or an action, it depends on how it relates with everything else from the perspective that we take. From one perspective it's a process, from another it's an action. For some people, things seem to be in one and only one absolute way, which is wrong, IMHO.   Is there a word to describe this kind of problem that some people seem to have?  PD: It's like . But I don't think most people know the word monist.  Here are some other, superficially similar terms that actually mean something quite different. A racist is (approximately) someone who considers other races inferior. A narrow-minded person considers other perspectives wrong or irrelevant, or may not consider them, but is aware that they exist. A monist would not even understand that there is more than one race or multiple perspectives; there is simply one reality, and that's it. Desktop doesn't remember brightness settings after a reboot Every time I reboot my machine the brightness goes back to 100% in Gnome. I wish it would keep the last setting. Is there anyway? I have several identical elements with different attributes that I'm accessing with SimpleXML:  &lt;data&gt;     &lt;seg id="A1"/&gt;     &lt;seg id="A5"/&gt;     &lt;seg id="A12"/&gt;     &lt;seg id="A29"/&gt;     &lt;seg id="A30"/&gt; &lt;/data&gt;   I need to remove a specific seg element, with an id of "A12", how can I do this?  I've tried looping through the seg elements and unsetting the specific one, but this doesn't work, the elements remain.  foreach($doc-&gt;seg as $seg) {     if($seg['id'] == 'A12')     {         unset($seg);     } } Deleting a channel on slack.com Can I move the Unity launcher? Where has all the RAM gone? Java Generics: List, List, List Can someone explained, as detailed as possible, the differences between the following types?  List List&lt;Object&gt; List&lt;?&gt;   Let me make this more specific. When would I want to use  // 1  public void CanYouGiveMeAnAnswer(List l) { }  // 2 public void CanYouGiveMeAnAnswer(List&lt;Object&gt; l) { }  // 3 public void CanYouGiveMeAnAnswer(List&lt;?&gt; l) { } Is there any way to check whether a file is locked without using a try/catch block?    Right now, the only way I know of is to just open the file and catch any System.IO.IOException. On every use profile page the search bar on the top of the page already contains user:&lt;id&gt; text to simplify search for post of the currently visible user. But this doesn’t work correctly for the  user:        Instead of searching for posts owned by Community user I’ve got Jeff Atwood posts:        In this case minus sign was just ignored and instead of (-1, i.e. Community user), the (+1, i.e. Jeff Atwood) is used. Same happened for any other negative id for the “user:” option in the search box. Update Migration Path List for Non-Moderators Infinite Series $‎\sum\limits_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$ ‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since if we set $C:=\sum_{m=2}^{\infty}|a_m|&lt;\infty$‎, ‎then‎ $$\sum_{n=1}^{\infty}\sum_{m=2}^{\infty}|a_m\frac{1}{n^m}|\leq\sum_{n=1}^{\infty}\sum_{m=2}^{\infty}|a_m|\frac{1}{n^2}\leq C\sum_{n=1}^{\infty}\frac{1}{n^2}&lt;\infty‎$$ and ‎by Cauchy's double series theorem‎, ‎we can switch the order of summation‎: $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=1}^{\infty}\sum_{m=2}^{\infty}a_m\frac{1}{n^m}=\sum_{m=2}^{\infty}a_m\sum_{n=1}^{\infty}\frac{1}{n^m}=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ This shows that $‎\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}=\sum_{n=1}^{\infty}\frac{1}{kn(kn-1)}$.  My Questions:  1) It's obvious that $\sum_{n=1}^{\infty}\frac{1}{2n(2n-1)}=\log(2)$, but how can I evaluate $\sum_{n=1}^{\infty}\frac{1}{3n(3n-1)}$?  2) Is there another method to evaluate $\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$?