Prove that (1+j)^n + (1-j)^n is a real number for every n>0
Is there a formula for $(1+i)^n+(1-i)^n$?
The question says to solve this equation: $(z+1)^5 = z^5$  I did. Just want to find out if I did it properly and if my run-around logic makes sense.  First I begin my writing the equations as:  $$ (z+1)^5 = z^5$$ $$ \mathbf{e}^{5 \mathbf{Log}(z+1)} = \mathbf{e}^{5 \mathbf{Log}(z)} $$ So $$  \mathbf{Log}(z+1) = \ln|z+1| + \mathbf{Arg}(z+1)i $$   $$  \mathbf{Log}(z) = \ln|z| + \mathbf{Arg}(z)i $$   Now, because the natural logarithm is one-to-one, I write:  $$ \ln |z| = \ln |z+1| \Rightarrow |z| = |z+1|$$   So assign $ z = a +bi$   So that $|z| = \sqrt{a^2 +b^2} = |z+1| = \sqrt{(a+1)^2 +b^2} \Rightarrow a^2 +b^2 = (a+1)^2 +b^2 \Rightarrow a^2 = (a+1)^2 \Rightarrow a = -\frac12 $  So, $z = -\frac12 + bi$ and $z+1 = \frac12 + bi$ for some $b \in \mathbb R$  Now to find $b$  $$ \mathbf{Arg}(z+1) = \mathbf{Arg}(z)$$ $$ \tan^{-1} \frac{b}{\frac12} = \pi - \tan^{-1}\frac{b}{-\frac12}$$  I have a feeling this last part isn't quite right, so I just want to find out if I'm approaching this question properly?  Ultimately, I get $ z = -\frac12$ which upon inspection...is wrong... How does the simulation hypothesis relate to the ethics of simulated violence? Say, for the sake of this question,  is real. Are we then hurting "persons"/conscious beings in "our simulations" - video games - when we fight them? Why (not)? If so, is this unethical? Think for example about shooting an alien in the game No Man's Sky.  Edit/answer on the comment of Joseph Weissman: My reasoning is like this:   It is unethical to harm a person (with a person being everything that has ). By the simulation hypothesis (assuming it's true), everything around us (everything that we know of), is simulated. By 1 and 2, all persons we know of are simulated. So it's unethical to harm the simulated persons around us. A game is a simulation just like we are in a simulation, so the game has simulated persons. Therefore -  - it's unethical to harm the persons in a video game.   I know that the premises - besides the simulation hypothesis itself - are debatable, but that's the reason I asked the question in the first place.  I was also thinking that the characters in the video game differ because they have far less to no intelligence, that much of it is scripted. But I neither have a proof that we ourselves aren't scripted, with free will being an illusion. Also, how intelligent does a character in a simulation/video game has to be to have personhood then? I think this is a case of the .  Just to be sure, please note that this question is not about whether the simulation hypothesis is true or how likely it is to be true. It is about (one of the) consequences if the simulation hypothesis is true. It's also not a question about violence in games in general. Marketing Cloud IP Warming - is it critical that we send up to the weekly limits by domain? How can I optimise a Minecraft-esque voxel world? 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Use some algorithm to work out which blocks can currently be seen, as blocks closer to the user could obfuscate the blocks behind, making it pointless to render them. Keep the block object themselves lightweight, so it is quick to add and remove them from the trees.   I guess there is no right answer to this, but I would be interested to see peoples' opinions on the subject.  How would you improve performance in a large voxel-based world? What is the difference between the two notions of computational and statistical indistinguishability? How to Setup GIMP 2.10 on Ubuntu 16.04 (Xenial Xerus) I've created a new C++ project in Visual Studio 2008.  No code has been written yet;  Only project settings have been changed.  When I compile the project, I receive the following fatal error:     fatal error LNK1104: cannot open file 'C:\Program.obj' Cardinality of cartesian product of an infinite set with N What does it mean when Git says 'rewrite' or 'rename' in a commit message? How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number? 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? Sort and count number of occurrence of lines I have Apache logfile, access.log, how to count number of line occurrence in that file? for example the result of cut -f 7 -d ' ' | cut -d '?' -f 1 | tr '[:upper:]' '[:lower:]' is  a.php b.php a.php c.php d.php b.php a.php   the result that I want is:  3 a.php 2 b.php 1 d.php # order doesn't matter 1 c.php Is there a *simple* example showing that uncorrelated random variables need not be independent? How do I pluralize Italian foods, like pasta noodles (spaghetti, macaroni)? Possible Duplicate:             Is it proper to say spaghettis for more than one spaghetti noodle?  What about macaronis as a plural for more than one macaroni noodle? "I saw him crossing" vs. "I saw him cross" I saw him crossing the road.   I saw him cross the road.   Which one is correct and why? How to disable the "Get Windows 10" icon shown in the notification area (tray)? This icon showed up in my taskbar notification area today and I cannot seem to get rid of it:    Clicking on it displays the following screen:    So how do I disable or remove the "Get Windows 10" icon? Do we need a global test before post hoc tests? How to fix dpkg error [2]? Every time I try to install or remove a package, I get this error. I thought maybe it was the hard disk, but a disk check gave me a clean bill of health. dpkg: unrecoverable fatal error, aborting: reading files list for package 'linux-headers-3.16.0-31': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2) A package failed to install.  Trying to recover:  I tried the solution of removing the listing for this file from the dpkg status file, but to no avail. I tried purging the package but no use. I'm stuck at this point as nothing I can find on Google helps. Update: This is now what the error says: dpkg: unrecoverable fatal error, aborting: E: Sub-process /usr/bin/dpkg returned an error code (2) A package failed to install.  Trying to recover:  After trying:  Update2: Seems the drive has bad sectors, but the reallocated sector count is 0. The drive is relatively old, however, so I expect some issues. I won't be able to replace it immediately, and until then I would still like to be able to keep the system up to date (especially for security reasons). I noted that there was an issue with coping some md5 checksums related to the package mentioned in the error, but I don't know what to do with those, such as if clearing them would allow dpkg to continue. Why aren't these loops homotopic? Let $S^1 = \{z \in \mathbb{C} : |z| = 1\}$.  Take the loops $f,g : [0,1] \rightarrow S^1$, $f(t) = 1$, $g(t) = e^{2\pi it}$.  I know these represent different elements in $\pi_1(S^1, 1)$, but I don't see why $F(t,s) = e^{2\pi its}$ isn't a homotopy between $f$ and $g$. My son has been playing Minecraft on our home computer for a couple of years now.  He just got an iPad Mini 2 for his birthday. Is it possible to transfer his game and his saves onto his iPad Mini 2? Show that unit circle is not homeomorphic to the real line