I know that many people would say I don't even know this. But I know it very well that negative × neagtive = positive. But I don't know Why? So kindly give a logical answers.
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.  I went ahead and gave them a proof by contradiction like so:  Assume $(-x) \cdot (-y) = -xy$  Then divide both sides by $(-x)$ and you get $(-y) = y$  Since we have a contradiction, then our first assumption must be incorrect.  I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.  Is there a better way to explain this?  Is my proof incorrect?  Also, what would be an intuitive way to explain the negation concept, if there is one?
How do I root my Android device? This is a common question asked by those who want to root their phones. How exactly do I root my Android device? Are there risks involved? Where can I find the source code for the Ubuntu Kernel? I'd like to build my own copy of the Ubuntu kernel. Where can I get the source code? Is an integer a sum of two rational squares iff it is a sum of two integer squares? Let $a\in \mathbb Z$. Is it true that $a$ is a sum of two squares of rational numbers if and only if it is a sum of two squares of integers?  I came to face this problem while dealing with quaternion algebras. Expectation of random variables ratio What, if anything should one do if a user's "about me" profile section contains extremely inflammatory terms (racist, etc.) I don't think anyone wants to be the thought police, and I'm particularly wary of censorship on a part of the site that is a user's personal self-statement, and is a little off the beaten path.  On the other hand, I'm also concerned that we may not want a site where clicking on a user to see their other posts would expose folks to racist or other bigoted terms that almost everyone would agree are inappropriate for civil discourse.    Don't forget the "be nice" rule".  Note that I don't want a "flag user" rule, which I know has been beaten to death already, and further think that swears and semi-inflammatory opinions, etc. should generally just be ignored - I'm only asking about terms that provoke serious emotions around racial, religious, ethnic, or sexual orientation issues.  What, if anything, is the appropriate response in such cases? Combinatorics Question Asking How Many Integer Solutions Given Different Restrictions How many integer solutions of $x_1 + x_2 + x_3 + x_4 = 28$ are there with:  (a) $0 ≤ x_i ≤ 12$?  (b) $−10 ≤ x_i ≤ 20$?  (c) $0 ≤ x_i, x_1 ≤ 6, x_2 ≤ 10, x_3 ≤ 15, x_4 ≤ 21$?  \ \  My attempts (I was really only able to solve part (a) but I would appreciate if you could check my work for this one as well):  (a) First I took the total number of integer solutions with no restrictions: There would be ${28+3\choose 3}$ = ${31\choose 3}$ total solutions with no restrictions.   Then I used the inclusion exclusion principle and knew I would have to subtract [P(A) + P(B) + P(C) + P(D)] where A --> $x_1$ is greater than or equal to 13, ..., D--> $x_4$ is greater than or equal to 13. The probabilities of all these cases would equal ${15+3\choose 3}$ = ${18\choose 3}$ because once we assign 13 to one of the $x_i$ there would be $28 - 13 = 15$ remaining to distribute among the $x_i$ values. We would also need to choose which of the $x_i$ to give the 13 to so we would multiply the ${18\choose 3}$ by ${4\choose 1}$ = ${4\choose 1}{18\choose 3}$  We would then have to add back, since we "oversubtracted", the case in which there is an intersection between two of the $x_i$ or in other words two of the $x_i$ receive 13 or more. P$(A\bigcap B)$ = $28-2(13) = 2$ remaining to distribute among the $x_i$ values. So we would also have to choose which of the $x_i$ to give the 13 to so we would have ${4\choose 2}$ ways to choose them. So there would be ${4\choose 2}{2+3\choose 3}$ = ${4\choose 2}{5\choose 3}$ solutions in this case.   Thus the number of solutions would be: ${31\choose 3} - {4\choose 1}{18\choose 3} + {4\choose 2}{5\choose 3}$.  I'm not sure if this is correct and I'm really stuck on b and c so I would appreciate any help or feedback. Thank you! When I use , the results I get don't reflect recent activity. Sometimes the data I get back are two months out of date. What's wrong? This is intended to be a canonical reference for questions about SEDE not reflecting real-time results. I recently tried to use QGIS and I wondered if you can convert a draft ArcGIS map (*.mxd file) into a QGIS project (*.qgs file).   This would avoid thematize the entire project. Best way to put commas into large numbers I've started a new job and it involves looking at a bunch of big numbers. Is there an easy way to add commas to an int or decimal field to make it readable?  For example, SQL Server outputs the column on the left, but for my own sanity, I need it to look like the one on the right:  2036150 -&gt; 2,036,150     ...or would I have to write some heinous   left(right(vandalized_data),6),3) + ',' + right(left(vandalized_data),6),3)   function?  The perfect thing would be commas in the display grid, then plain integers in the output. When should you use WP_Query vs query_posts() vs get_posts()? It seems like half the tutorials in the  and around the blogosphere use  and half use . They all do similar things, so when should I use one over the others? How do I withdraw all money from my HSA account as a non-resident? I have lived in US for 5 years, and I (still) have an HSA account with BoA. Now I am living in Europe, and there's a fat chance I'll ever go back to live in US, so I would like to withdraw the money from my HSA, pay the due taxes, of course, and just be able to use them. How exactly I go about that?  Thanks in advance. Do redirection operators always open file descriptors in parallel? How to sort lines by float number I've such a file:  name: xxx --- time: 5.4 seconds name: yyy --- time: 3.2 seconds name: zzz --- time: 6.4 seconds ...   Now I want to sort this file by these float numbers to generate a new file as below:  name: yyy --- time: 3.2 seconds name: xxx --- time: 5.4 seconds name: zzz --- time: 6.4 seconds ...   I've tried the command awk '{print $5}' myfile | sort -g but this will show me ONLY the float numbers. Schema for a multilanguage database I'm developing a multilanguage software. As far as the application code goes, localizability is not an issue. We can use language specific resources and have all kinds of tools that work well with them.  But what is the best approach in defining a multilanguage database schema? Let's say we have a lot of tables (100 or more), and each table can have multiple columns that can be localized (most of nvarchar columns should be localizable). For instance one of the tables might hold product information:  CREATE TABLE T_PRODUCT (   NAME        NVARCHAR(50),   DESCRIPTION NTEXT,   PRICE       NUMBER(18, 2) )   I can think of three approaches to support multilingual text in NAME and DESCRIPTION columns:   Separate column for each language  When we add a new language to the system, we must create additional columns to store the translated text, like this:  CREATE TABLE T_PRODUCT (   NAME_EN        NVARCHAR(50),   NAME_DE        NVARCHAR(50),   NAME_SP        NVARCHAR(50),   DESCRIPTION_EN NTEXT,   DESCRIPTION_DE NTEXT,   DESCRIPTION_SP NTEXT,   PRICE          NUMBER(18,2) )  Translation table with columns for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains a column for each language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID,   TEXT_EN NTEXT,   TEXT_DE NTEXT,   TEXT_SP NTEXT )  Translation tables with rows for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains only a key, and a separate table contains a row for each translation to a language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID )  CREATE TABLE T_TRANSLATION_ENTRY (   TRANSLATION_FK,   LANGUAGE_FK,   TRANSLATED_TEXT NTEXT )  CREATE TABLE T_TRANSLATION_LANGUAGE (   LANGUAGE_ID,   LANGUAGE_CODE CHAR(2) )    There are pros and cons to each solution, and I would like to know what are your experiences with these approaches, what do you recommend and how would you go about designing a multilanguage database schema. About the localization of a UFD I was wondering, is the localization of a UFD also a UFD?    How would one go about proving this? It seems like it would be kind of messy to prove if it is true.  If it is not true, what about localizing at a prime? Or what if the UFD is Noetherian? Finding the limit $ \lim_{x\to a^+} \frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)} $ Why is the set of commutators not a subgroup? Relationship between Binomial and Beta distributions I'm more of a programmer than a statistician, so I hope this question isn't too naive.  It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.  I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is  cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1   The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.  EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia () that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know. "work for" / "work in" / "work at" I would like to know what usually follows after each proposition in sentences with the word &quot;work&quot;. These are what I understand so far: I work for my boss. ( usually for certain person ) I work in a accounting corporation. (usually for business area ) I work at Samsung. ( usually for specific name of company ) Could you please tell me if I understand rightly? Does every Abelian group admit a ring structure? Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive:  \begin{align*} &amp;a * (b * c) = (a*b) * c \\ &amp;a * (b + c) = a * b + a * c \\ &amp;(a + b) * c = a * c + b * c \\ \end{align*}  We also might have multiplicative identity $1 \in G$, with $a * 1 = 1 * a = a$ for any $a \in G$. Multiplication may or may not be commutative.  Depending on the definition, the answer could be no in the case of the group with one element: then $1 = 0$. But the trivial ring is not a very interesting case. For cyclic groups the statement is certainly true, since $(\mathbb{Z}_n, +, \cdot)$ and $(\mathbb{Z}, +, \cdot)$ are both rings. What about in general? Is there some procedure to give arbitrary abelian groups ring structure?