I can show that |Z|^2=|Z^2| or |Z|^3=|Z^3| where Z is a complex (a+ib) number. But is there any way of showing that for all n where n is a natural number?
A simple stupid question that I should already know the answer to:  Is this true: $\left|z^5\right| = \left|z\right|^5$ where z is a complex number with a non-zero imaginary component.  Thanks in advance.
How would one go about proving that the rationals are not the countable intersection of open sets? I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why   $$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$   isn't a counter-example. Any ideas? Thanks! I need to prove this equivalence   Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ if and only if $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$  I have already proved the reciprocal (the obvious part) but I am having difficulties with the other part  Please any hint will be appreciated.  Thanks in advance. Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$ Let $V$ be a subspace of $\mathbb R^3$ generated by the vectors $\{(1,1,1),(0,1,1)\}.$ There are $3$ vectors $u_1=(0,0,1),u_2=(1,1,0),u_3=(1,0,1)$.Then which of the following sets are connected?  $1.\left(\mathbb R^3\setminus V\right)\cup\{(0,0,0)\}$  $2.\left(\mathbb R^3\setminus V\right)\cup \{tu_1+(1-t)u_3:t\in [0,1]\}$  $3.\left(\mathbb R^3\setminus V\right)\cup\{tu_1+(1-t)u_2:t\in [0,1]\}.$  $4.\left(\mathbb R^3\setminus V\right)\cup \{(t,2t,2t):t\in \mathbb R\}$  Now $V$ is generated by two linearly independent vectors so is like $\mathbb R^2.$ So $\left(\mathbb R^3\setminus V\right)$ will be disconnected but giving the $(0,0,0)$ with them will make it connected.  No. $4$, I think will be connected also because $\{(t,2t,2t),t\in \mathbb R\}$ is in $V$ and contains the point $(0,0,0)$ Now, any neighbourhood of this point will have points from both $V$ and $\left(\mathbb R^3\setminus V\right)$ and so the union is connected.I'm not sure about no. $2$ and $3.$ How to find/grep what is between string1 and string2? I am trying to extract a value from a long string that may change over time. So for example the string could look something like this  ....../filename-1.9.0.3.tar.gz"&lt;....   And what I want to extract is the value between filename- and .tar.gz, essentially the file version (1.9.0.3 in this case). The reason I need to do it this way is because I may later run the command and the value will be 1.9.0.6 or 2.0.0.2 or something entirely different.  How can I do this? I'm currently only using grep, but I wouldn't mind using other utilities such as sed or awk or cut or whatever. To be perfectly clear, I need to extract only the file version part of the string, since it is very long (on both sides) everything else needs to be cut out somehow. As of late, and without my deliberately doing anything to make it happen, my Bash prompt has an  (i.e. @) prepended to it. This did not previously happen. Nor can I see anything in my ~/.bashrc that seems as though it ought to be making this happen.  This is on Debian Jessie GNU/Linux, using GNU Bash.  For example, my current Bash sessions look like this:  @sampablokuper@debianbox:~$ python3 Python 3.4.2 (default, Oct  8 2014, 13:14:40)  [GCC 4.9.1] on linux Type "help", "copyright", "credits" or "license" for more information. @&gt;&gt;&gt; 1+2 3 @&gt;&gt;&gt; exit() @sampablokuper@debianbox:~$    whereas originally, they would have looked like this:  sampablokuper@debianbox:~$ python3 Python 3.4.2 (default, Oct  8 2014, 13:14:40)  [GCC 4.9.1] on linux Type "help", "copyright", "credits" or "license" for more information. &gt;&gt;&gt; 1+2 3 &gt;&gt;&gt; exit() sampablokuper@debianbox:~$    Here are all the lines from my ~/.bashrc that appear to in any way relate to the Bash prompt:  # set variable identifying the chroot you work in (used in the prompt below) if [ -z "${debian_chroot:-}" ] &amp;&amp; [ -r /etc/debian_chroot ]; then     debian_chroot=$(cat /etc/debian_chroot) fi  # set a fancy prompt (non-color, unless we know we "want" color) case "$TERM" in     xterm-color) color_prompt=yes;; esac  # uncomment for a colored prompt, if the terminal has the capability; turned # off by default to not distract the user: the focus in a terminal window # should be on the output of commands, not on the prompt #force_color_prompt=yes  if [ -n "$force_color_prompt" ]; then     if [ -x /usr/bin/tput ] &amp;&amp; tput setaf 1 &gt;&amp;/dev/null; then         # We have color support; assume it's compliant with Ecma-48         # (ISO/IEC-6429). (Lack of such support is extremely rare, and such         # a case would tend to support setf rather than setaf.)         color_prompt=yes     else         color_prompt=     fi fi  if [ "$color_prompt" = yes ]; then     PS1='${debian_chroot:+($debian_chroot)}\[\033[01;32m\]\u@\h\[\033[00m\]:\[\033[01;34m\]\w\[\033[00m\]\$ ' else     PS1='${debian_chroot:+($debian_chroot)}\u@\h:\w\$ ' fi unset color_prompt force_color_prompt  # If this is an xterm set the title to user@host:dir case "$TERM" in xterm*|rxvt*)     PS1="\[\e]0;${debian_chroot:+($debian_chroot)}\u@\h: \w\a\]$PS1"     ;; *)     ;; esac   How can I get my Bash prompt to appear as it did originally? In this case what is correct?        Keep Tom and I updated.      or         Keep Tom and me updated.      I understand that me is an object pronoun. And therefore I feel the second option is correct.  Especially, because I would say "Keep him updated" and also "Keep me updated". Instead of "Keep he updated" or "Keep I updated". But something about saying "Keep him and me updated" feels wrong.  However I am not sure, because several people have told me that in this case I is the correct option because it is being used in conjunction with a name (Tom) and not a pronoun. Is that correct? How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 3? How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 3?  $$$$ I was trying to to add 2^6 (when there is no 2)+ C(6,2)2^2 (when 2 can be in two places)+C(6,3)2^3 (when 2 can be in three places)... Why is using "for...in" for array iteration a bad idea? I've been told not to use for...in with arrays in JavaScript. Why not? Visiting two Schengen countries, coming back home in between, do I apply for single-entry or multiple-entry visa? I am a resident of Turkey and I will go to Austria for 6 days beginning of June and then I will come back to Turkey. After staying 10 days I will go to Italy for 11 days at the end of June. Do I need two single-entry visa, or a multiple-entry visa, and which country do I apply to? (My first destination is Austria but I will stay more in Italy) How do I mirror the faces of a cube (edit all of them when I edit one)? Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable I am new to functions and relations, and with some concepts I am not so familiar. I have a question in an homework:     Show that $\mathbb{Q} \times\mathbb{Q}$ is denumerable.   From what I understood, denumerable means that it is infinitively countable. There are some  on the web about this topic, but I am still not understanding their explanation, maybe because they are not explained so easily (for me).  From what I understood, a set is denumerable if there's a relation with the natural numbers (but I am still not understanding what is this relation).  I have heard also about equinumerous sets (contain a bijective function = onto + one-to-one function), but I cannot relate this information with the problem to try to solve it. Positive operator is bounded For a real Banach space $X$ let $A:X\rightarrow X^*$ be a positive operator in the sense that $(Ax)(x)\geq 0$ for all $x\in X$. Show that $A$ is bounded.   I don't know how to do that, maybe it's an application of the closed graph theorem? How do I find the package that provides a file? Simple enough question: is there some shell command (or GUI method) I can use that, given the path to a file on my system, tells me what package put it there? Assuming the file did in fact come from a package, that is.  Bonus question: what if it's a file that isn't installed on my system? Is there, say, a website that will let me look up a file and see what packages, if any, provide it? literally as a hyperbole Possible Duplicate:          I have seen a lot of backlash in internet media against people using the word literally to mean something not literal. Something like "he was literally as big as a house" to mean someone was very big I think this comic sums up the anti literal movement well   Is this actually grammatically incorrect or can the word "literally" be a hyperbole strengthening a metaphor? Singular values of the data matrix and eigenvalues of the covariance matrix What is the effect where some objects are a single bright color but the rest is black and white? There is an option to filter only one color (R,G,B,Y) in my camera. What I want to know is can this effect be achieved in photoshop? And what is this called? I tried googling monochrome and came up with nothing. My camera calls it part color. Showing that $\mathbb{R}$ is connected So I know that $\mathbb{R}$ is both open and closed. But given a set, $X\subset \mathbb{R}$, $X\ne \emptyset $ that is both open and closed, how does one show that $X=\mathbb{R}$?    Here is my attempt: Using the definitions I was given, $X$ open implies that for every $x\in X$ there is a $\delta \gt 0$, $(x-\delta,~ x+\delta)\subset X$.    Also, since $X$ is closed , $B=\mathbb{R}$\  $X$ is open.    However, I don't know how to proceed. NB: This is not a homework problem. Should I use 'or' or 'nor' after a negative statement? This document does not cover the SDK interfaces nor any other reference material.   I think the above is correct, but my grammatical checker in Microsoft Word underlines nor and suggests or. Why? fatal: bad ownership or modes for chroot directory component "/" in SFTP I've been trying to configure sftp in a debian server.sshd_config:  Subsystem sftp internal-sftp  UsePAM no  Match User sftpUser ChrootDirectory /users/sftp/sftpUser ForceCommand internal-sftp AllowTcpForwarding no X11Forwarding no   Directories:   drwxr-xr-x   3 root      root      4096 Oct 20 10:59 users   drwxr-xr-x   3 root      root      4096 Oct 20 11:00 sftp   drwxr-xr-x   2 root      root      4096 Oct 20 11:00 sftpUser   cat /var/log/auth.log  Oct 20 10:58:22 w1 sshd[24634]: Accepted password for sftpUser from 201.156.103.213 port 34106 ssh2 Oct 20 10:58:22 w1 sshd[24636]: fatal: bad ownership or modes for chroot directory component "/"   sftpUser no home, bin/falseand member of user group.  In the client side I'm getting the famous Write failed: Broken pipe and then the connection is dropped.  Commenting the ChrootDirectorycommand in sshd_config makes the connection, but get's the user loose.  What am I doing wrong ?