I have found something really weird which compelled me to put up this post. I wish I could resolve this problem but was unsuccessful.   Consider a positive integer say 7 , now is it possible to prove that it is equal to 8 and 8 is equal to 9 and so on? Quite puzzled! Have a look at this picture.       \begin{align*} 7  &amp;= 7-\frac{15}2+\frac{15}2 = \sqrt{\left(7-\frac{15}2\right)^2}+\frac{15}2 \\ &amp;= \sqrt{7^2-2\times7\times\frac{15}2+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &amp;= \sqrt{49-105+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &amp;= \sqrt{-56+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &amp;= \sqrt{64-120+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &amp;= \sqrt{8^2-2\times8\times\frac{15}2+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &amp;= \sqrt{\left(8-\frac{15}2\right)^2}+\frac{15}2 \\ &amp;= \left(8-\frac{15}2\right)+\frac{15}2 = 8 \\ \end{align*}   If this is correct, we would be able to show that all positive integers are equal.   I am sure that it is not correct but I would appreciate if someone can point out what the mistake is? Or is there something to be pondered about?
$$\begin{align} 2+2 &amp;= 4 - \frac92 +\frac92\\ &amp;= \sqrt{\left(4-\frac92\right)^2} +\frac92\\ &amp;= \sqrt{16 -2\times4\times\frac92 +\left(\frac92\right)^2} + \frac92\\ &amp;= \sqrt{16 -36 + \left(\frac92\right)^2} +\frac92\\ &amp;= \sqrt {-20 +\left(\frac92\right)^2} + \frac92\\ &amp;= \sqrt{25-45 +\left(\frac92\right)^2} +\frac92\\ &amp;= \sqrt {5^2 -2\times5\times\frac92 + \left(\frac92\right) ^2} + \frac92\\ &amp;= \sqrt {\left(5-\frac92\right)^2} +\frac92\\ &amp;= 5 + \frac92 - \frac92 \\ &amp;= 5\end{align}$$  Where did I go wrong
Is there a way to view Google Chrome browsing history past three months ago? I'm trying to find a page I looked at months ago but it looks like it's disappeared off the face of the planet. I thought I would have better luck looking through my Google Chrome browsing history, but I can't see history past September. Can I force Google Chrome to display history further into the past? Can I retrieve the history manually from a directory? Is the history data gone? Add year to question and answer dates Bluetooth Broadcom 43142 isn't working Convert a list to a string in C# Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$ How can I disable the "Welcome, Name!" banner notification for Game Center? Individual Phase shift provided by Op-Amp in Closed loop configuration Too much memory getting used I have been feeling this way for a while, looking at task manager mentally ballparking how much physical memory should be in use, and noticing a large disparity. 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How could I find $x$ in this equation $x^2-x+6 \equiv 0 \pmod {9}$ This equation is like this: $$x^2-x+6 \equiv 0 \pmod{9}$$  I want to find x by using modular arithmetic. How I could do that?   My solution was:     $$\begin{align*}  x(x-1) &amp;\equiv -6 \pmod{9}\\ x(x-1) &amp;\equiv 3 \pmod{9} \end{align*}$$      So $x$ is either:   $$ x \equiv 3\pmod{9}$$      or       $$x - 1 \equiv 3 \pmod{9}$$      Which is $x \equiv 4 \pmod{9}$   I know what I have done is very dumb. So what is the correct solution? Give me the instructions also please. I'm new to data mining and I'm trying to train a decision tree against a data set which is highly unbalanced. However, I'm having problems with poor predictive accuracy.  The data consists of students studying courses, and the class variable is the course status which has two values - Withdrawn or Current.   Age Ethnicity Gender Course  ... Course Status   In the data set there are many more instances which are Current than Withdrawn. Withdrawn instances only accounting for 2% of the total instances.  I want to be able to build a model which can predict the probability that a person will withdraw in the future. However when testing the model against the training data, the accuracy of the model is terrible.  I've had similar issues with decision trees where the data is dominated by one or two classes.  What approach can I use to solve this problem and build a more accurate classifier? Other users have been leaving me angry and hostile comments accusing me of plagiarism and saying that I'm a liar and I should stop stealing answers. But I have only been copying content from other sites, not taking answers from other users.   I don't understand what is wrong here. Shouldn't we be focused on building all the content and knowledge in a single place, much like Wikipedia does?  What does it matter that I copy and paste content I've found elsewhere so long as the question gets an answer? Mathematical notation around the world Should I use "who" or "which" when referring to a specific animal species? I'd like to know what option to use, specifically when referring to a species by its scientific name. How do I remove a file named "NUL" on Windows? I have a Windows XP box (NTFS filesystem) on which I found a file named NUL. I have not been able to remove this file in any usual way. The file appears to be owned by Administrator in the SYSTEM group, unlike any other file in the same directory (the other files are owned by my user id).  How do I get rid of this file? Where did it come from? Natbib -- change author conjunction in citation call-out from 'and' to '&' I'm using natbib, german babel and natbib creates output of the form   Foo and Bar (1337)   I would like to get   Foo &amp; Bar (1337) for citet and (Foo &amp; Bar 1337) respectively for citep. JavaScript YAML Parser I have a custom script and I want to send a desktop notification (the one that appears in the top right corner of the screen) with a custom message. How do I do that? How to install networkx python in QGIS? How to install and how to compile networkx python on QGIS?  I always get an error that no module named networkx in the console of QGIS I copied the file networkx in C:\Program Files\Quantum GIS Lisboa\apps\Python27\Lib\site-packages and there are not change. What are the alternatives to the FHS? I'm a long time Linux user for over 15 years but one thing I hate with a passion is the mandated directory structure. I don't like that /usr/bin is the dumping ground for binaries or libs in /usr/lib, /usr/lib32, /usr/libx32, /lib, /lib32 etc... Random stuff in /usr/share etc. It's dumb and confusing. But some like it and tastes differ.  I want a directory structure where each package is isolated. Imagine instead if the media player dragon had it's own structure:  /software/dragon /software/dragon/bin/x86/dragon /software/dragon/doc/README /software/dragon/doc/copyright /software/dragon/lib/x86/libdragon.so   Or:  /software/zlib/include/zlib.h /software/zlib/lib/1.2.8/x86/libz.so /software/zlib/lib/1.2.8/x64/libz.so /software/zlib/doc/examples/... /software/zlib/man/...   You get the point. What are my options? Is there any Linux distro that uses something like my scheme? Can some distro be modified to work like I want it (Gentoo??) or do I need LFS? Is there any prior art in this area? Like publications on if the scheme is feasible or unfeasible?  Not looking for OS X. :) But OS X-inspired is totally ok.  Edit: I have no idea how PATH, LD_LIBRARY_PATH and other environment variables that depend on a small set of paths should work out. I'm thinking that if I have the KDE editor  installed in /software/kate/bin/x86/bin/kate then I'm ok with having to type the full path to the binary to start it. How it should work for dynamic libraries and dlopen calls, I don't know but it can't be an unsolvable engineering problem.