I need to prove the next numbers are irrational:  $$ \sqrt{2}+\sqrt{5} $$ $$\sqrt{3}$$ $$ \log_{2}3  $$  Of course, I tried to prove it with negative evidence...
How do you prove that $\sqrt{2} + \sqrt{5}$ is irrational?    I tried to prove it by contradiction and got this equation:  $a^2/b^2 = \sqrt{40}$.
Why does prime factorization for LCM work? A method to finding the LCM of two numbers is to use prime factoring. I know how to do this, but logically why does this work in finding the lcm Last names that are English words with an extra 'e' I noticed that there are a lot of last names that have an 'e' at the end.  The pronunciation usually isn't changed from that of the base word.   Poole Steele Browne Clarke   Why do English words not have the e?  Maybe the answer to this question depends on which came first, pool or poole. Select translation: where? How to turn off Facebook notifications on Mac OS X? How to turn off Facebook notifications on Mac OS X Mountain Lion? How can I manually install a package on MiKTeX (Windows) What factors determine enemy levels or: How do enemy levels scale to the player character's? By the time, many files are still on my system and I don't need them anymore, so how to delete all files that are one year old at least? What program can be used to package software into one executable? I need to package an application into a single executable file (so that I don't have to install it, so that it will be portable).  Does anybody know what I can use to make a program portable? Reopened questions get a "modified" time stamp in the questions list just showed up on 's front page like this:    ... which makes no sense, because the question hasn't been modified since May 25th, and the person who edited it wasn't CatLord anyway:    Turns out what actually happened was that CatLord cast a fifth reopen vote to reopen the question.    So that stamp in the questions list should probably say this instead:     reopened 4m ago CatLord 10.7k   Or just:     reopened 4m ago This one-liner removes duplicate lines from text input without pre-sorting.  For example:  $ cat &gt;f q w e w r $ awk '!a[$0]++' &lt;f q w e r $    The original code I have found on the internets read:  awk '!_[$0]++'  This was even more perplexing to me as I took _ to have a special meaning in awk, like in Perl, but it turned out to be just a name of an array.  Now, I understand the logic behind the one-liner:  each input line is used as a key in a hash array, thus, upon completion, the hash contains unique lines in the order of arrival.  What I would like to learn is how exactly this notation is interpreted by awk. E.g. what the bang sign (!) means and the other elements of this code snippet.  How does it work? Are there any drawbacks to using a wish spell to increase ability scores? If not then what would stop players from just using wish repeatedly and raising their ability scores to 30?  I'm just getting back into RPGs with D&amp;D 5e after not playing for years, and I seem to remember this being covered in the editions we played back then. How can I remove a trailing newline? What is the Python equivalent of Perl's chomp function, which removes the last character of a string if it is a newline? sudo: unable to resolve host   I think I need to edit the /etc/hosts file to make an entry for 127.0.1.1 PC name.  But how can I do that if I can't access sudo? It seems to have gone since I updated to 13.10.  I have tried nano, which still doesn't work. Why are some muggles allowed to know of the magical world? Are all electrons identical? Why should two sub-atomic (or elementary particle) - say electrons need to have identical static properties - identical mass, identical charge? Why can't they differ between each other by a very slight degree? Is there a theory which proves that?  Imagine an alien of size of order of Milky-way galaxy examining our solar system with a probe of size of 10's of solar system dimension and concludes that all planets are identical. Maximal order of an element in a symmetric group Why are random walks intercorrelated? Maximum value of $\sin A+\sin B+\sin C$? I know that contents of double quotes are expanded, whereas the contents of single quotes are not, such that  echo '$1'   gives  $1   where as   echo "$1"   gives  &lt;the argument - blank in this particular example&gt;   the same as   echo $1   However, the question  led me to wonder why double quotes and not single quotes were used when declaring the alias, thus:  alias cutf="_cutf"   instead of   alias cutf='_cutf'   I would have employed single quotes as in the latter example. Most examples on the web use single too, unless there is some expansion required. However, in this case, no expansion, to my eyes, is apparent. Is it that, in this case, they are interchangeable, or are the double quotes necessary because a function definition is employed? Performance of Non Clustered Indexes on Heaps vs Clustered Indexes