I recognize that a rational number a/b means that b does not divide a and also that a and b are relatively prime. Does that mean rational numbers are always expressed in their simplest form? Or is my statement about coprime wrong? I have tried using Euclid's Algorithm and Bezout's Identity, since that is the section that the question comes from but I have not gotten it to work out. Please Help.
It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing 'even' with 'divisible by $3$'), one can prove that $\sqrt{3}$ is irrational, as well.  On the other hand, clearly $\sqrt{n^2} = n$ for any positive integer $n$.  It seems that any positive integer has a square root that is either an integer or irrational number.        How do we prove that if $a \in  \mathbb N$, then $\sqrt a$ is an integer or an irrational number?      I also notice that I can modify the proof that $\sqrt{2}$ is irrational to prove that $\sqrt[3]{2}, \sqrt[4]{2}, \cdots$ are all irrational.  This suggests we can extend the previous result to other radicals.        Can we extend 1? That is, can we show that for any $a, b \in \mathbb{N}$, $a^{1/b}$ is either an integer or irrational?
What does it do exactly if I click in the window of cmd? Suppose I run cmd.exe and ping localhost. After that I ping localhost again but I click into the window and the ping stops until I press enter.  What happens exactly when I click into the window? Does it suspend the process or something else? I attached a screenshot to clarify this issue.  Note: I do not actually click but drag using the mouse which creates a white box in the window as you can see on the screenshot.  Edit: further clarification: What I really wish to know is whether dragging the window halts the cmd process or not?  Edit: even more clarification: I know that I'm using Quick Edit and pressing enter copies the contents I selected. What I do not know is what happens in the background. Load jQuery from Stack's CDN if Google's CDN is blocked (or always) Support for OpenID ended on July 25, 2018 Can't login with Stack Exchange OpenID when HTTPS-Everywhere plugin is active - apparent protocol mismatch I can't seem to log in with my Stack Exchange OpenID. When I try, I get the error message:     Unable to log in with your OpenID provider:      The openid.return_to parameter (.) does not match the actual URL (.) the request was made with.   It works if I disable the HTTPS-Everywhere plugin in Firefox, but if I then re-enable the plugin and go to a site on which I am not logged in (in which case login is normally nearly seamless), it fails again with the same error message, or at least something very similar.  I'm not aware of having done anything that might have caused this, although I will admit I don't recall seeing https:// in the address bar on Stack Exchange before (I do now, and that's a good thing IMO!). I was using the site  and it worked fine then, but not now... How do I connect to a MySQL database using a python program? LaTeX on Stack Overflow? in questions, answers, and comments. Can we get something like this going on Stack Overflow?  I think it’d be appropriate as I at least pretty often want to write something like n^2 and would benefit greatly from prettier markup. Best Place to put Feature Requests? How to align prices in columns when both 2 digit and 4 digit variations of a price exist? I am trying to display data similar to coinmarketcap with prices for coins, some above 1$ in which case they are rounded to 2 zeroes, some below 1$ in which case they have 4 digits following the decimal. I saw a bunch of posts here where people have suggested to use right alignment for numeric applications/prices  Approach 1 Decimal aligned   1. $13,700.05  2.      $0.0048   Approach 2 Right aligned   1. $13700.05  2.   $0.0048   Which of these approaches do you suggest? Thank you for your time One-byte-off pointer still valid in C? Use of "save" or "save for" in sentences What does "javascript:void(0)" mean? I'm training a neural network for regression. The input vector consists of $140$ entries.   For each feature vector entry, I calculate both min and max over the training set. The difference max-min of some entries are smaller than 1e-5. I understand I can normalize such feature entries by say MinMaxScalar. However, if the difference max-min is this small, can I discard such entries since they seem to be so insensitive? How to create an interactive bash script to collect arguments for a command in linux? I'm trying to create an interactive bash script where I can collect user input variables to be used as arguments in another command. Can someone provide me an example of this?  Currently i want to do:   what type of content you want to delete? what is the status?   then feed it into a mysql delete query command. Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s or two consecutive 1s. Note: Problem from "Kenneth Rosen's DM and it's applications" and solution from "Students solution guide for use with ... applications"  Let P(n) be the number of strings not containg two containing two consecutive zeros or two consecutive ones.  I got the answer  P(n)=P(n-1)+2P(n-2)+2P(n-3)+...+2P(0) +2  but then the book says the sequence also satisfies the recurrence relation  P(n)=2P(n-1)+P(n-2). I am not able to understand this part. Chatroom owners can star their own messages I was able to star my own message by pinning it and unpinning it.  Pinning a message adds a star, even to your own messages, but when the pin is removed, the star remains. $f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki How to resolve conflicting distributions in apt-get When doing an apt-get update/upgrade I see that warning  W: Conflicting distribution: https://downloads.plex.tv/repo/deb public InRelease (expected public but got )   I was upgrading to Ubuntu Zesty 17.04 before and do not know how to fix the above. Already have seen ? but there are no suggested fixes for that.  Maybe one of you can give some hints? Thanks! How do I join several objects into one? I'm creating side skirts for a truck and I am duplicating the frames I want. The problem is it's making the duplicated object into another object. How do I put it into the same one? Do I need to pay in full for hostel reservation to show proof of accommodation for a Schengen visa? I am applying for a Schengen visa to travel to Germany and one of the requirements is showing proof of accommodation. I booked all my hostels via hostelworld.com, but only paid the non-refundable deposit (a couple of dollars for each place) instead of the entire amount. I intend to pay the rest of it on arrival at the hostel (or cancel it and go CouchSurfing, thus saving money).  But for the visa application, will the non-refundable deposit be enough to show proof of accommodation?  The reason I ask is, I emailed one of the hostels requesting a document showing my name and accommodation confirmation, and they replied saying they could only give me such a document once I pay in FULL.  Is this how I'm supposed to book for all places?  Have I been doing it wrong all this while (by only paying the tiny non-refundable deposit instead of full hostel fare)? Computing dominant area in polygon using ArcGIS Desktop?