This is a real basic question.  Why is $e$ the base for polar form of complex numbers?  In high school maths I learned that e is useful in derivatives etc. And it's conventional to use it for exponential growth/decay functions, although really we could use any base number for them if we tweak the exponent appropriately (eg. $Ae^t = Ax^{kt})$ for appropriate values of x and k).  So could any number be used as the base for polar form of complex numbers? Is $e$ just convention or is there some special relationship to $i$ that makes this work?
From Euler we've learned that $z=re^{i\theta}$.  And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$.  Why must we use e to represent these numbers correctly? It seems that I could arbitrarily choose a different exponent $z=r\pi^{i\theta}$ and get the same size for $z$ as I did before: $|z|^2=r\pi^{i\theta}\times r\pi^{-i\theta}=r^2$  What did I miss?
Like most Python developers, I typically keep a console window open with the Python interpreter running to test commands, dir() stuff, help() stuff, etc.  Like any console, after a while the visible backlog of past commands and prints gets to be cluttered, and sometimes confusing when re-running the same command several times.  I'm wondering if, and how, to clear the Python interpreter console.  I've heard about doing a system call and either calling cls on Windows or clear on Linux, but I was hoping there was something I could command the interpreter itself to do.  Note: I'm running on Windows, so Ctrl+L doesn't work. Switching window manager/desktop environments? In general what is the approach to switch the desktop environment or window manager on Ubuntu? The main difference between stock Ubuntu, Kubuntu and Lubuntu is supposedly the use of Unity vs. KDE vs. LXDE.  However, in order to try various desktop environments and window managers I don't want to have to reinstall the respective flavor from scratch just to get a particular desktop environments or window manager. There probably is an easier method than that.  The question: How can I easily test different desktop environments and respective compatible window managers (say KDE, LXDE, Unity, GNOME, XFCE and StumpWM)? I entered chmod -R 777 /usr/bin and now sudo is not working.  It says sudo must be setuid root.  Some advice online said to run chown root:root /usr/bin/sudo chmod 4755 /usr/bin/sudo.  On entering chown root:root /usr/bin/sudo it shows an opened in readonly mode error. HTTP GET request in JavaScript? I need to do an  request in JavaScript. What's the best way to do that?  I need to do this in a Mac OS X dashcode widget. A while back I read about the super Kamiokande detector detected a large neutrino flux and then several hours later a supernova was seen. Anyone know of this with sources? I don't recall the source at the moment. prevent the deletion of account records if they have oppurtunity trigger preventAccountDeletionifopp on Account (before delete) {     for(account accdel:trigger.old){        Opportunity oppprevent =[select Name,(select Name,Owner from Opportunitys)from ACCOUNT =: accdel.id ];         if(oppprevent.size() &gt; 0){           accdel.adderror('Opportunity record is refering this record.So, you cannot delete it...!');         }    } } What is the secret of making a really juicy burger? As the title says; What is the secret of making a really juicy burger? What is the expected value of the maximum of 500 IID random variables   with uniform distribution between 0 and 1?   I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction?  Thanks Uninstalling Homebrew messed up git I uninstalled Homebrew using .  After running the above commands, I can't find .bashrc and .bash_profle. All files are gone from Documents , Desktop and Downloads :'(  When I do git status on /Users/ava it shows:  # On branch master # Untracked files: #   (use "git add &lt;file&gt;..." to include in what will be committed)  ... all the files which are in /Users/ava   history   113  cd `brew --prefix`   114  rm -rf Cellar   115  brew prune   116  rm `git ls-files`   117  rm -r Library/Homebrew Library/Aliases Library/Formula Library/Contributions   118  rm -rf .git   119  rm -rf ~/Library/Caches/Homebrew   120  sudo chown -R $USER:admin /usr/local   121  ruby -e "$(curl -fsSL https://raw.github.com/mxcl/homebrew/go)"   122  brew doctor   What just happened? "EOL while scanning single-quoted string"? (backslash in string) import os xp1 = "\Documents and Settings\" xp2 = os.getenv("USERNAME") print xp1+xp2   Gives me error   File "1.py", line 2  xp1 = "\Documents and Settings\"                                ^ SyntaxError: EOL while scannning single-quoted string   Can you help me please, do you see the problem? I can't seem to find a story related to mine so here it is.  I am 25 and a German citizen. In 1998 when I was 7 years old my family moved to the states and we stayed until early 2002 and left voluntarily. Basically we overstayed. I am aware that my parents had the ten year ban and since I was a minor at the time of our overstay I did not receive such ban.  Ever since 2002 I have not attempted to go back to the USA for any reason. However now that I am older I would like to go back for vacation and stay for a short period of time (one month) during my christmas break.  I've lived in Germany since leaving, have a great job, enough money in the bank account. My whole life is here and I have no intention on moving anywhere else.  After reading a bunch of stories online I have decided to apply for a B2 visa to be on the safe side but I was curious about the ESTA approval. For the last question. 8) Have you ever stayed in the United States longer than the admission period granted to you by the U.S. government?   I should put yes but is it a no if I stayed when I was a minor? And what are my chances on getting the B2 Visa? Do I need a Turkey transit visa for a layover? I have an Indian passport and am planning to visit Germany with a valid visa. To book a flight for Germany (Mumbai – Dusseldorf), I came across some connected flights that stop at Istanbul.  Do I need to have visa for Turkey?  This is the first time I am travelling outside my country, and any information on this will be helpful. Arrangements of MISSISSIPPI with all S's and P's separated If the all S,P are separated in the word MISSISSIPPI then the total possible arrangements are $k\frac {10!}{4!.4!} $ then value of k is? Now all S,P are separated so we can have slash method. Thus they can be placed alternatively in $6! $ ways and rest 5 can be placed in $5! $ ways. Thus we equate to get $$6!.5!=k\frac {10!}{4!.4!} $$  to get $k=13.7$ while $k=4/3$ .Where is my mistake Thanks Why don't gksu/gksudo or launching a graphical application with sudo work with Wayland? Why does "kill -9 0" end my console session? I just wonder why kill -9 0 ends my tty console session?   What exactly is PID 0, it is not listed on a ps aux? Is it ethical to omit the PhD credential from my resume? I am applying for a Statistical Programmer position at a commercial research organization (CRO), and I am wondering if it is unethical to omit my PhD in Statistics from my resume?  All of my degrees (Bachelor's, Master's, PhD) are in Statistics, and I am just wondering if it would be ethical to just list my Bachelor's and Master's degree on my resume, and indicate my experience as a RA while I was a PhD student as my "work experience"? (Since this Statistical Programmer position only requires Master's degree plus some years of experience)  I am asking this question because I do not want to do anything unethical/cause me troubles later on. When no model comparison, should I use REML vs ML? I'm running LMM, and I will make no comparison of models. Could I ask which one should I use between REML and ML? Why are American-style options worth more than European-style options? Why are American-style options worth more than European-style options?   I realize I can exercise American-style options anytime before the  expire, but I can only exercise European-style options during their  "exercise period" (usually right when they expire, but no earlier).   So it makes sense that an American option is worth at least as much as  a European option.   But why is it worth more? If I exercise my American option early, I  may make more money than waiting until expiration, but I also may make  less. Mathematically, there's no advantage, since I'm equally likely  to make as much money by waiting for expiration.   Mathematically speaking, is there ever a good reason to exercise an  American option early?   I realize people don't always behave logically, but even the formulas  that valuate options show that American options are worth more.   EDIT (answering comments):    Many of the comments people made would apply to lookback options  (), but not to American  options.  American options can do everything European options can and more. I  understand that, and that means American options can't be worth any  LESS than European options, but I'm still not clear on how you would  mathematically calculate this extra value.  @Aaronaught: "The difference between an American and European  option is the difference between getting N chances to get it right (N  being the number of days 'til expiration) and getting just one  chance. It should be easy to see why you're more likely to profit  with the former, even if you can't accurately predict price  movement."    REPLY: I don't think you really get N chances. Once you exercise the  option, that's it, no more chances. And, if you decide not to exercise  the option today, and the underlying's price falls, you can't go back  in time and exercise it yesterday.    @jdsweet: Think of it this way, if you traveled back through time  one month - with perfect knowledge of AAPL's stock price over that  period - which happens to peak viciously then return to its old price  at the end of the period - wouldn't you pay more for an American  option?    REPLY: Well, no. I'd simply buy a cheaper European option that expires  right at the spike. Or, I'd buy a European option that expires later,  and sell it when the spike occurs. American options don't give you  precognition, so you're still guessing as to when to sell/exercise.    @jdsweet: "one of the more common reasons people exercise (as  opposed to sell) an American option before expiration is if an  unexpected dividend (larger than remaining time value of the option)  was just announced that's going to be paid before the option contract  expires"    REPLY: This makes sense. I'm actually trading FOREX options, so there  is no dividend. In that scenario, are European and American options  worth the same? Why is variance squared? Which is correct: "what if there was" or "what if there were"? Possible Duplicate:          Is this correct grammar?     What if there was a Stack Overflow on…    Shouldn't it be "what if there were a Stack Overflow on…"?