Direct Irrationality Proof for $\sqrt{3}$ and $\sqrt{6}$
How can you prove that the square root of two is irrational?
How to change Gnome-Terminal title? Add a note about downvoting on Meta Suppose $G$ is a group and $K\lhd H\lhd G$ are normal subgroups of $G$. Is $K$ a normal subgroup of $G$, i.e. $K\lhd G$? If not, what extra conditions on $G$ or $H$ make this possible?  Applying the definitions, we know $\{ghg^{-1}\mid h\in H\}=H$ and $\{hkh^{-1}\mid k\in K\}=K$, and want $\{gkg^{-1}\mid k\in K\}=K$. Clearly, the best avenue for a counterexample is if $gkg^{-1}\not\in K$ for some $k\in K$ and $g\in G-H$.  If no such element exists, $\{gkg^{-1}\mid k\in K\}\subseteq K$ implies $\{gkg^{-1}\mid k\in K\}=K$ because if $k'\in K$, $gk'g^{-1}=k\in K\Rightarrow k'=g^{-1}kg\in\{gkg^{-1}\mid k\in K\}$. Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$ Padding trailing whitespaces in a string with another character I'd like to output hello world over 20 characters.  printf "%-20s :\n\n" 'hello world!!'  # Actual output hello world!!        :  # Wanted output hello world!!========:   However, I don't want to complete with spaces but with "=" instead. How do I do that? postfix fails to send mail with "fatal: unknown service: smtp/tcp" Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$ Is there a case where including the same header twice is actually helpful? How can I form a word like "quadruple" for any number I want? I'm not sure what these are called, but how can I form a word like "quadruple" for any number I want? Like 5× as much is quintuple, what is 31× as much or 147× as much? I want to know how they are formed so I can make my own.  Similarly, how do I construct the prefixes, such as unicycle, bicycle, tricycle, quadricycle? I figured out 12 is duodeci-, but how can I form any prefix like this? Best answer versus best solution anyone tell how is this effect created which change lips color to orange and little orange on face without brush etc tool how to change the only lips color to orange without brush etc tool and how this is created and little orange on face Inverse of a Positive Definite Let K be nonsingular symmetric matrix, prove that if K is a positive   definite so is $K^{-1}$ .   My attempt:  I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know what to do next. R raster: masking a raster by polygon also remove cells partially covered? I want to mask out a raster based on the values of a polygon: i.e., set to NA all values of the raster that are not covered by the polygons. However, it seems that raster::mask() masks out cells (i.e. set them to NA) even if they are partially covered?  This is similar to the question: , but unlike crop(), mask() does not have the snap="out" option.   Check: here the right point of the triangle touches a cell, yet that cells is masked out.   library(raster) library(sp)  r &lt;- raster(xmn=1, xmx=5, ymn=1, ymx=5, nrows=4, ncols=4) r[] &lt;- 1:length(r)  Sr1 = Polygon(cbind(c(2,4,4,1,2),c(2,3,5,4,2))) Sr2 = Polygon(cbind(c(4.1,4,2),c(2,3,2)))  SpP = SpatialPolygons(list(Polygons(list(Sr1), "s1"), Polygons(list(Sr2), "s2")), 1:2)  plot(mask(r, SpP)) plot(SpP,  add=TRUE) Can I get tourist visa on arrival at Dubai airport? Removed Python 3 and now Ubuntu Software Center, terminal and other applications don't work I am a new Ubuntu user running Ubuntu 13.04.  I've been trying to install and use a DeDRM tool, but I've been having problems doing so. I was following some directions that said I needed a version of Python 2.7, but not anything Python 3.0 or later (because 3.0 versions are missing some necessary libraries?). I tried to install Python 2.7.5.6, but it didn't work. I thought it was probably because I had the later version of Python, so I went through terminal and removed Python 3.3, so I could install the earlier version of Python.  Now that I've uninstalled Python 3.3, a lot of applications no longer work, including the terminal and the Ubuntu Software Center. I have no idea how to fix this problem now. Let H={e,(12)(34)} and K ={e, (12)(34), (13)(24), (14)(23)} be subgroups of $ S_4$ where e is identity element. Then which of following is true   H and K are normal subgroups of $S_4$ H is normal in K and K is normal in$ A_4$ H is normal in $A_4$, but not normal in $ S_4$ K is normal in $S_4$ but H is not   How should this question be approached, checking left and right cosets of these subgroup is very time consuming. Checking gh$ g^{-1}$ $\in $ H for every g in $ S_4$ is also time consuming. I understand how Insight and Perception can be used passively: for the DM to determine if an enemy Deception or Stealth attempt is successful without alerting the players that someone is deceiving or sneaking.    But the Observant feat in the Player's Handbook states:      [...] You have a +5 bonus to your passive Wisdom (Perception) and passive Intelligence (Investigation) scores.     So in what situation would a DM be rolling something against the player's Investigation without letting them know? Computing $\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$ I was asked to evaluate the following sum: $$\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$$  I'm trying to use $$\int_{0}^{\infty} \frac{\sin(x)}{x}dx=\frac{\pi}{2}$$ However, it doesn't seem to work. Any help is greatly appreciated. Where did the name "Stack Exchange" come from? I get Stack Overflow (which I'm guessing was the first site?) and Super User, Code Review, and the other sibling site names make sense to me.  But I've never heard of a stack exchange before.  Is a stack exchange a real thing, or is this site's name a combination of Stack Overflow and an exchange of ideas? Black hole "no hair" theorem The  (or conjecture), suggests that black holes can be entirely described by their mass, angular momentum and charge. All other details of the BH formation are lost.  Is there a simple way of understanding why you would be able to tell the difference between a BH made from protons and one made from electrons, but not between one made from matter rather than anti-matter?  i.e. What is special about charge, and why aren't lepton number, baryon number, strangeness etc, also black hole properties?