How do we deal with irrationals in Physics?
Is it possible for a physical object to have an irrational length?
Does there exists an Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.  Attempt In my opinion It does not exists.  If there exists such an isomorphism say $\phi :\mathbb{C}^* \to \mathbb{C}$ then, from the fact that Isomorphism takes identity to identity we have $$\phi \left( 1 \right)=0$$  From the fact      Given $\phi:G \to H$ be an isomorphism, then For a fixed integer $k$ and a fixed group element $b$ in $G$, the   equation $x^k=b$ has the same number of solutions in $G$ as does   the equation $x^k=\phi\left(b\right)$ in $H$.   We have $x^4=1$ and $x^4=0$ must have same number of solutions, but the solutions to these equations are 4 and 1 only. I am a student who has taken the basic calculus courses but I am working through Calculus (the fourth edition) by Spivak in my spare time in order to both review the material and to gain a more rigorous understanding of the concepts. I am currently in Chapter 8 where he supplies a proof of the basis of the Intermediate Value Theorem, Theorem 7-1 (pp. 135-136 of the fourth edition). It seems mostly straightforward but I think there is some nuance of it that I don't grasp.  I am including in this post the statement Theorem 6-3 since he references it in the proof of Theorem 7-1 as well as part of Theorem 7-1 itself.     Theorem 6-3 Suppose $f$ is continuous at $a$, and $f(a) &gt; 0$. Then $f(x) &gt; 0$ for all $x$ in some interval containing $a$; more precisely, there is a number $\delta &gt; 0$ such that $f(x) &gt; 0$ for all $x$ satisfying $|x-a| &lt; \delta$. Similarly, if $f(a) &lt; 0$, then there is a number $\delta &gt; 0$ such that $f(x) &lt; 0$ for all $x$ satisfying $|x-a| &lt; \delta$.     Problem 6-16 is also referenced. However,it is the same thing as Theorem 6-3 for one-sided limits. The following proof is the Theorem 7-1 up through the paragraph which I don't fully understand.     Theorem 7-1: If $f$ is continuous on $[a,b]$ and $f(a) &lt; 0 &lt; f(b)$, then there is some number $z$ in $[a,b]$ such that $f(x) = 0$.  Proof: Define the set $A$ as follows:  $$A = \{x : a \le x\le b, \mbox{ and } f \mbox{ is negative on the interval } [a,x] \}.$$  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Clearly $A \ne \emptyset$, since $a$ is in $A$; in fact, there is some $\delta &gt; 0$ such that $A$ contains all points $x$ satisfying $a \le x &lt; a + \delta$; this follows from Problem 6-16, since $f$ is continuous on $[a,b]$ and $f(a)&lt;0$. Similarly, $b$ is an upper bound for $A$ and, in fact, there is a $\delta &gt; 0$ such that all points $x$ satisfying $b-\delta &lt; x \le b$ are upper bounds for $A$; this also follows from Problem 6-16, since $f(b) &gt; 0$. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; From these remarks it follows that $A$ has a least upper bound $\alpha$ and that $a &lt; \alpha &lt; b$. We now wish to show that $f(\alpha) = 0$, by eliminating the possibilities $f(\alpha) &lt; 0$ and $f(\alpha) &gt; 0$. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Suppose first that $f(\alpha) &lt; 0$. By Theorem 6-3, there is a $\delta &gt; 0$ such that $f(x) &lt; 0$ for $\alpha - \delta &lt; x &lt; \alpha + \delta$. Now there is some number $x_0$ in $A$ which satisfies $\alpha - \delta &lt; x_0 &lt; \alpha$ (because otherwise $\alpha$ would not be the least upper bound of $A$). This means that $f$ is negative on the whole interval $[a,x_0]$. But if $x_1$ is a number between $\alpha$ and $\alpha+\delta$, then $f$ is also negative on the whole interval $[x_0,x_1]$. Therefore $f$ is negative on the interval $[a,x_1]$, so $x_1$ is in $A$. But this contradicts the fact that $\alpha$ is an upper bound for $A$; our original assumption that $f(\alpha) &lt; 0$ must be false.    The point I don't understand is in the final paragraph of the excerpt. Why does Spivak split the interval $[a,b]$ up into the subintervals $\lbrack a,x_0 \rbrack$ and $[x_0,x_1]$? Isn't it true that, because $\alpha$ is a least upper bound of $A$ and $f$ is continuous on $[a,b]$, it must be true that if $a \le x &lt; \alpha$ then $f(x)&lt;0$? I understand the need to pick out the $x_1$ but not the $x_0$. Especially perplexing is the statement "there is some number $x_0$ in $A$ which satisfies $\alpha - \delta &lt; x_0 &lt; \alpha$ (because otherwise $\alpha$ would not be the least upper bound of $A$." Wouldn't it be true that if there were some $x_0$ such that $a \le x_0 &lt; \alpha$ and $f(x_0) \ge 0$ that $x_0$ would be an upper bound for $A$, contradicting that $\alpha$ is the least upper bound of $A$?  Thanks for any help. Lost supplementary user groups for GUI login session Ubuntu 16.04.5  Very weird situation - after Xorg crash, now every time when I login into Unity, my user account doesn't have it's system groups like adm, sudo, plugdev, fuse etc. But when I login on virtual console (Alt+F1) or over SSH - all my additional groups are there, as reported by id command.  Neither of /etc/group* or /etc/gshadow* files have changed, my user is still listed there in many system groups.  It looks like some systemd session parameters for GUI logins or lightdm cache files got screwed. But I have no idea where to start looking..  A lightdm log:  $ journalctl -au lightdm  -- Reboot -- systemd[1]: Starting Light Display Manager... systemd[1]: Started Light Display Manager. lightdm[3223]: pam_kwallet5(lightdm-greeter:setcred): (null): pam_sm_setcred lightdm[3223]: pam_unix(lightdm-greeter:session): session opened for user lightdm by (uid=0) lightdm[3345]: pam_succeed_if(lightdm:auth): requirement "user ingroup nopasswdlogin" not met by user "chm" lightdm[4867]: pam_ecryptfs: Passphrase file wrapped lightdm[3345]: pam_kwallet5(lightdm:auth): (null): pam_sm_authenticate lightdm[3345]: pam_kwallet5(lightdm:setcred): pam_kwallet5: pam_sm_setcred lightdm[3345]: pam_unix(lightdm:session): session opened for user chm by (uid=0) lightdm[3046]: Error writing to daemon: Bad file descriptor lightdm[3046]: Error writing to daemon: Bad file descriptor lightdm[3046]: Error writing to daemon: Bad file descriptor lightdm[3046]: Failed to write utmpx: Permission denied Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ I'm trying to prove using induction that $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have:   Base case: $P(1) = (1) + (8) + (27) = 36, 36$ can be divided by $9$ so the base case is valid Inductive step: let $P(n)$ be the statement $9$ divides $n^3 + (n+1)^3 + (n+2)^3$. Assume $P(k)$ is true, so $9$ divides $k^3 + (k+1)^3 + (k+2)^3$.   And this is where I'm stuck. I'm not sure how to demonstrate that $9$ divides $P(n)$ when $n = k+1$. If someone could step me in the right direction that would be awesome. Thanks! Why is my find not recursive? I am running the following command, but it is not performed recursively:  find . -name *.java   I know there are java files further down in the current directory but it is performing the find on the current directory only. I am using OS X, 10.9. Programming pattern to generate VGA signal with micro-controller? SSH connection problem with "Host key verification failed..." error I can connect to another Ubuntu machine in my LAN via SSH. On both of then PC's I installed   but from another Ubuntu computer I can not connect to my PC via SSH and I got this error:     Host key verification failed... How to understand pipes When I just used pipe in bash, I didn't think more about this. But when I read some C code example using system call pipe() together with fork(), I wonder how to understand pipes, including both anonymous pipes and named pipes.  It is often heard that "everything in Linux/Unix is a file". I wonder if a pipe is actually a file so that one part it connects writes to the pipe file, and the other part reads from the pipe file? If yes, where is the pipe file  for an anonymous pipe created? In /tmp, /dev, or ...?   However, from examples of named pipes, I also learned that using pipes has space and time performance advantage over explicitly using temporary files, probably because there are no files involved in implementation of pipes. Also pipes seem not store data as files do. So I doubt a pipe is actually a file. Dictionaries on iOS 8 not working Group algebra ever a field? Let $G$ be a group. Let $K$ be any field and consider its group algebra $K[G]$. It is known that if $G$ is a finite group then $K[G]$ is never a field. Or even if $G$ is not finite: if $G$ has at least one non-trivial torsion element, then $K[G]$ is not a field.  Question: Do there exist infinite groups $G$ such that $K[G]$ is a field or a division ring? (In, On or At) GitHub? What's the difference between say:     "The project will be on GitHub",      "The project will be in GitHub" and      "The project will be at GitHub"? Edits that shadow a suggested edit inconsistently reject but include the suggestion I have the edit privilege. If I edit a post which has a pending suggested edit, then:  My edit starts from the version resulting from the suggested edit. I'm not notified of that, or even of the existence of the suggested edit. (There's actually a clue, but it's easy to overlook — the edit summary field is pre-filled with the summary from the suggested edit.) The suggested edit is marked as rejected by Community with the message “This edit conflicted with a subsequent edit.” The suggester is not credited in the edit history, my revision combines the suggested edit with my modifications.  This can happen if an edit is suggested while I have the page open: I click on “edit”, and think I'm editing the version that I was just reading, but in fact I'm editing somebody's suggested edits. It also happens if I deliberately visit /posts/12345/edit, which . Examples: a  that I approved and then wanted to ; a  where I . There are two bugs here, but I'm making a single report because they're very closely linked.  I'm not told about the suggestion, so my edit includes content that I'm not aware of, yet is attributed to me. The suggestion is marked as rejected and the suggester is not credited in the edit history, yet the suggested changes are made. Why does the weighing balance restore when tilted and released I'm talking about a Weighing Balance shown in the figure:    Press &amp; Hold on onside of the horizontal beam and then release it. It makes some oscillations and comes back to equilibrium like shown in the figure.  Both the pans are of equal equal masses. When the horizontal beam is tilted by an angle using external force, the torque due to these pan weights are equal in magnitude &amp; opposite in direction. Then why does it come back to it position? What's making it to come back? Simplest or nicest proof that $1+x \le e^x$ 50 question per month limit? I get a message:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Why is there a question limit? 50/month is just not enough :) Explaining to laypeople why bootstrapping works I recently used bootstrapping to estimate confidence intervals for a project. Someone who doesn't know much about statistics recently asked me to explain why bootstrapping works, i.e., why is it that resampling the same sample over and over gives good results. I realized that although I'd spent a lot of time understanding how to use it, I don't really understand why bootstrapping works.  Specifically: if we are resampling from our sample, how is it that we are learning something about the population rather than only about the sample? There seems to be a leap there which is somewhat counter-intuitive.  I have found a few answers to this question here which I half-understand. Particularly . I am a "consumer" of statistics, not a statistician, and I work with people who know much less about statistics than I do. So, can someone explain, with a minimum of references to theorems, etc., the basic reasoning behind the bootstrap? That is, if you had to explain it to your neighbor, what would you say? "What have you tried" epidemic Statistics  () In Q1 of 2012 - 0.0134 such comments per question (~1.33%) In Q2 of 2012 - 0.0167 such comments per question (~1.66%) In Q3 of 2012 - 0.0169 such comments per question (~1.68%) In Q4 of 2012 - 0.0193 such comments per question (~1.92%) In Q1 of 2013 - 0.0223 such comments per question (~2.22%)  The vast majority of these questions need serious help.  Unfortunately, there is a growing trend towards abuse Along with the good, comes some bad -- as usual in life. These comments are bad when  Posting &quot;what have you tried&quot; multiple times per question is severely obnoxious Posting &quot;what have you tried&quot; when the user has clearly shown that they have tried to solve it on their own  It is becoming increasingly common to see &quot;what have you tried?&quot; comments in these scenarios. Some examples     Workarounds Flagging Comments - It is considered a good thing to be liberal with comment flagging -- that helps us clean up after the fact. &quot;Too Chatty&quot; is a no-brainer here. Unfortunately, that only cleans up after the damage has been done. All signs point toward the continued increase of the trend of using &quot;what have you tried&quot; comments. I spend a lot of time editing and cleaning up bad questions. This trend came up on my radar toward the end of last year and it is getting worse. So, I thought at the very least that I'd raise some awareness. I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?  Secondly, the book wants me to use the fact that if $n$ is an integer that is not a perfect square, then $\sqrt n$ is irrational. This means that $\sqrt 6$ is irrational. How are we to use this fact? Can we reason as follows:  $\sqrt 6$ is irrational  $\Rightarrow \sqrt{2 \cdot 3}$ is irrational.  $\Rightarrow \sqrt 2 \cdot \sqrt 3$ is irrational  $\Rightarrow \sqrt 2$ or $\sqrt 3$ or both are irrational.  $\Rightarrow \sqrt 2 + \sqrt 3$ is irrational.  Is this way of reasoning correct? How much electronics and other valuables can I bring duty-free when going to India? I am planning to travel to India (COK airport, Kerala) from United States. I want to take my laptop as well as my tablet (iPad). Am I allowed to take both in my backpack with my documents related my work. I think I am allowed to take only one computer as per the rule. Will the customs consider my iPad as a computer?  What all I can take?   One of my friends took 2 laptops (1 work and 1 personal) and had trouble at the airport.  Also I am planning to take some wristwatches (7 of them, each avg price is 45 $). I read in a website that I can take stuff up to max 35000 INR. Is that true? In that case, how will I manage my stuff? In my experience, input type="text" onchange event usually occurs only after you leave (blur) the control.  Is there a way to force browser to trigger onchange every time textfield content changes? If not, what is the most elegant way to track this “manually”?  Using onkey* events is not reliable, since you can right-click the field and choose Paste, and this will change the field without any keyboard input.  Is setTimeout the only way?.. Ugly :-)