Why does the minimal polynomial in $\mathbb{R}[X]$ have to have degree less than or equal to $2$?
Can all real polynomials be factored into quadratic and linear factors?
Prove that the square root of a positive integer is either an integer or irrational Is my proof that the square root of a positive integer is either an integer or an irrational number correct?  The proof goes like this:  Suppose an arbitrary number n, where n is non-negative. If $\sqrt{n}$ is an integer, then $\sqrt{n}$ must be rational. Since $\sqrt{n}$ is an integer, we can conclude that n is a square number, that is  for some integer a.  Therefore, if n is a square number, then $\sqrt{n}$ is rational.  Suppose now that n is not a square number, we want to show that the square root of any non-square number is irrational.  We prove by contradiction. That is, we suppose that the square root of any non-square number is rational. So $\sqrt{n} = \frac{a}{b}$, where $a,b \in Z^+, b \neq 0$. We also suppose that $a \neq 0$, otherwise $\frac ab = 0$ , and n will be a square number, which is rational.   Hence $n = \frac {a^2}{b^2}$, so $nb^2 = a^2$.    Suppose $b=1$. Then $\sqrt n = a$ , which shows that n is a square number. So $b \neq 1$. Since $\sqrt n &gt; 1$, then $a&gt;b&gt;1$.  By the unique factorization of integers theorem, every positive integer greater than $1$ can be expressed as the product of its primes.  Therefore, we can write $a$ as a product of primes and for every prime number that exists in $a$, there will be an even number of primes in $a^2$. Similarly, we can express $b$ as a product of primes and for every prime number that exists in $b$, there will be an even number of primes in $b^2$.  However, we can also express $n$ as a product of primes. Since $n$ is not a square number, then there exist at least one prime number that has an odd number of primes.  Therefore, there exists at least one prime in the product of $nb^2$  that has an odd number of primes. Since $nb^2=a^2$ , then this contradicts the fact that there is an even number of primes in $a^2$ since a number can neither be even and odd.  Therefore, this contradicts the fact that $\sqrt n$ is rational. Therefore, $\sqrt n$ must be irrational.  Is this sufficient? Or is there any parts I did not explain well? Difference between $\sum_{i=1}^{k}{\frac{s_i}{kn_i}}$ and $\frac{\sum_{i=1}^{k}{s_i}}{\sum_{i=1}^{k}{n_i}}$ I am getting confused at how to calculate the average probability.  Suppose we repeat a kind of binary survey $k$ times each of which was done on a completely separate sample group.  For each $i^{th}$ group, $i=1,2,3,...,k$, Let $n_i$ be the number of samples and $s_i$ be the number of positive results. With this we know that the probability $p_i$ of the positive result for the $i^{th}$ group is $s_i/n_i$  With my ignorance, I happened to use two ways to calculate the average probability in my work wrongly assuming that they are the same:  $$\frac{p_1+p_2+p_3+...+p_k}{k}$$ and $$\frac{s_1+s_2+s_3+...+s_k}{n_1+n_2+n_3+...+n_k}$$  I don't know which is the correct way to calculate the average probability. So could you please explain the difference of these two and when to use which? Is this correct for Rudin exercise 3.7? Prove the series is convergent This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \ge 0$.  Proof:  I will attempt to show that the cauchy criterion is met so $${\sum_{k=n}^m {\sqrt {a_k} \over k}} \le \epsilon $$ for all $n,m \le N_1$. Im disregarding the absolute value since its all positive terms.   Since $\sum{a_n}$ converges, we know that the $\lim \limits_{n \to \infty} a_n =0$ So there exists some $N_2$ such that for all $n \ge N_2$, $$   a_n \lt \ {{(m-n)}^2k^2(\epsilon)^2} $$ I am again disregarding the absolute values because its always positive Let N=max{$N_1,N_2$} then for all $n,m \ge N$ $$\sum_{k=n}^m {\sqrt {a_k} \over k} \le \sum_{k=n}^m {\sqrt {{(m-n)}^2k^2(\epsilon)^2} \over k} \le \sum_{k=n}^m{(m-n)(\epsilon)} = \epsilon $$  Maybe this is just crazy, should I just use Cauchy-Schwarz inequality and forget about this or could I possibly make this method work? Alternative to Photoshop or Paint.NET What's the best way of accessing field in the enclosing class from the nested class? Are there any differences between insmod and modprobe? Images as Planes in Eevee having weird transparency Shell programming temp=$1 vs temp=1 What's the difference between temp=$1 and temp=1? Why do I need the dollar sign?  For example:   #!/bin/bash  temp=$1 cell=$((($temp-32)*5/9))  echo $cell Story identification: Old man has hacker neighbor who mysteriously dies Does anyone know this story? It was a novelette published with another one (I don't remember what it was) in . The premise was that the narrator is an old man whose neighbor, a hacker, dies under mysterious circumstances. A younger woman who is an expert in computer security is hired by the police to investigate what happened, and she has a fling with the narrator. After a while she dies as well, and the narrator realizes that it is an evil force that comes through electric lines is responsible for the killings. The title has interesting formatting in that it ends with a square (much like a command prompt cursor: [_] ) How to create script with auto-complete? I recently found out about Ubuntu for Android Multi-Core Smartphones and was digging around on the Ubuntu website as well as the Canonical website but have yet to find any information on if it available to the public yet ?  I am very interested to try it our on my Android and if it is available for the public am wondering where and how I can get a copy of "Ubuntu for Android" ?  If it is not available then perhaps someone can shed some lite on what sort of timeframe it will be until it is available to the general public? What would be the symbol for 'multiplication on division'? The symbol I would like to obtain ought to be like the symbol '\pm'. Let's say we have a Python script my_script.py that does some data processing with Blender. This script accepts some arguments (e.g. arg1 arg2). How can I forward arguments to the script when using it with the CLI?  blender --background --python my_script.py   The only trick I could find was :  echo "arg1 arg2" &gt; my_input.txt &amp;&amp; blender --background --python my_script.py   Is this really the only way to do that?  Update  I found a  on Stack Overflow. An example of a regular function over an open set Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$ Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$ What is the meaning of "stop to do"? My teacher said that "stop to do" had two meanings: quit doing present work to do next work (which is following after the word "do") and pause for a while then continue. I have searched a lot about its meanings but what i got just about the first meaning. Is the second one true?  For examples:   I stop to play games.   First meaning: i stop doing something in order to play games.  Second meaning: i pause playing games for a while then i continue. How do you clone a Git repository into a specific folder? Executing the command git clone git@github.com:whatever creates a directory in my current folder named whatever, and drops the contents of the Git repository into that folder:  /httpdocs/whatever/public   My problem is that I need the contents of the Git repository cloned into my current directory so that they appear in the proper location for the web server:  /httpdocs/public   I know how to move the files after I've cloned the repository, but this seems to break Git, and I'd like to be able to update just by calling git pull. How can I do this? Defaulting to the sign-up form? Recently I've noticed more and more sign-in / sign-up workflows defaulting to the sign-up option first and providing a minimal "Sign-In / Login" action somewhere else on the page. What's the reason behind this practice?  Examples where I've seen this recently: I just installed Call of Duty 4 and when I try to run the single player mode nothing happens. I just see a quick flicker on the desktop but the game doesn't launch. When I try to run it again, it gives me this message:      It appears that cod4 did not quit properly the last time it ran. Do you want to run the game in safe mode?    Multiplayer seems to work fine and when I try to switch to single player from the multiplayer menu the same thing happens, it just goes back to my desktop and that message shows up again What are pros and cons for MacPorts, Fink and Homebrew? I'm just migrating from Ubuntu Linux to Mac, and everything is new and I'm re-learning a lot of stuff.  On Linux I had the excellent apt-get to manage software packages. I googled for an alternative on Mac and found about MacPorts, Fink and Homebrew.  I will use this computer primarily to develop Ruby on Rails applications.  So, what are the differences between them? Which are the upsides and downsides? Which one is best maintained and has more packages?