Are complex numbers just vectors defined by cosine (x axis) and sine (y-axis)?
Relationship between complex number and vectors
Analyzing $\biggl\lfloor{\frac{x}{5}}\bigg\rfloor=\bigg\lfloor{\frac{x}{7}}\bigg\rfloor$ How many non negative integral values of $x$ satisfy the equation :$$\biggl\lfloor{\dfrac{x}{5}}\bigg\rfloor=\bigg\lfloor{\dfrac{x}{7}}\bigg\rfloor$$.   My try:  Writing few numbers and putting in the required equation.  $\underbrace{0,1,2,3,4}_{\implies 0=0},\underbrace{5,6}_{\displaystyle\times},\underbrace{7,8,9}_{\implies 1=1},\underbrace{10,11,12,13}_{\displaystyle\times} ,\underbrace{14}_{\implies 2=2},\underbrace{15,16,17,18,19,20}_{\text{LHS gives 3 but RHS gives 2}},\underbrace{21,22,23,24}_{\text{LHS gives 4 but RHS gives 3}}$.  The point which I wish to make here is that after $14$ we'll get $5's$ multiple before than $7's$, hence $\forall x \geq 15 \implies \biggl\lfloor{\dfrac{x}{5}}\bigg\rfloor&gt;\bigg\lfloor{\dfrac{x}{7}}\bigg\rfloor$  Hence the only solution are: $\{0,1,2,3,4,7,8,9,14\}$. Does these make sense?  I need to know other solutions that don't incorporate such analyzing, means using some properties and framing it to solve above. Doesn't Radagast care about trees? In the book The Two Towers, Treebeard refers to Gandalf as "the only wizard that really cares about trees".  But in Fellowship of the Ring, we met the wizard Radagast the Brown, whose special focus is on animals, especially birds.  Of course, animals aren't trees, but it is difficult to imagine a person loving animals without also caring deeply about trees - especially when that person loves birds most of all, and so many birds are so  dependent upon trees for their survival.  More generally, Radagast seems like an all-around nature lover, and this too suggests that he "really cares about trees".  This leads me to imagine several possibilities:   Treebeard doesn't know Radagast (this seems unlikely - they have both been around for thousands of years, so surely they have bumped into each other at some point)   Treebeard forgot about Radagast, possibly because he was annoyed at Saruman and a bit angry at the time    Radagast actually doesn't care about trees    Is there some other alternative I have overlooked, or is one of the options I mentioned correct?  Or does Tolkien never address this apparent problem? Is the set of all unitary matrices in $M_2(\mathbb{C})$ is compact? I can show that as determinant map is continuous so unitary matrices are closed but how to show they are bounded?   Please help. Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$ . Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$     My attempt  First, I know the existence of this integral since $$ \mid \int_{-a}^{a} \frac{e^{-x^2}}{1+x^2} dx \mid \;\; \leq\;\;  \mid \int_{-a}^{a} \frac{1}{1+x^2} dx \mid$$  Next, let $R &gt;0 $ be an arbitrary and $\gamma_1(t) = -R(1-t) + Rt , \;\; 0 \leq t \leq1$ and $\gamma_2(t)= Re^{i\pi (t-1)}, \;\; 1 \leq t \leq 2 $.  So, take $f(z) = \frac{e^{-z^2}}{1+z^2}$.   I tried to find the value or upper bound of norm of $$\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz$$.  However, $$ \lvert\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz\rvert \leq  \frac{\text{max}\lvert e^{-z^2}\rvert}{R^2-1}\pi R = \frac{e^{R^2}}{R^2-1}\pi R $$ So, I can't proceed this.  May I ask you how to solve this? Disallow closing of questions with a certain number of question / answer upvotes With reference to this currently open question   At the time of writing, it has 3 close votes yet has 7 question upvotes and the single answerer (11 upvotes) will have been awarded the enlightened badge (silver).  Therefore in the opinion of many (i) the question adds value and (ii) the answer adds value.  As a staring point for discussion, I suggest that 5 net question upvotes (and perhaps 5 answer net upvotes) should be enough to keep a question permanently open. Let's haggle over the numbers though. Thoughts? How to jusify {e,n} is a valid RSA public key? I have $k$ pairs of samples from different distributions. For each pair of samples I want to check if the samples are taken from a normal distribution with the same mean and variance. I assume the normal distributions for each pair is different and independent of other pairs. I do not know the mean and variance of any of the distributions.  I am doing a two sample t-test for each of the pairs. I want to combine the $k$ resulting p-values.   I could use Fisher's method but I am wondering about the distribution of the average t-value. If these were z-values rather than t-values I would use Stouffer's z-test. I could also approximate the t-distribution with a z-distribution and than use Stouffer's z-test, but I am not sure if that is a good approximation.  Is there a test like Stouffer's z-test for t-values?  Is there a better way to combine the t-values? Well, one example is this file of the game The Sims 2 for PS2: quickdat.arc. The file contains this .arc extension, but it's no one an archive file and it contains some data of the game inside. How can I do this? Put so many files in one?  How to do something like this? All the game files inside one without using a file compressor? I believe that Asimov himself recommended reading them in chronological order, however I read them in the order that they were published and think that this is the best way to do it.  What is the correct order? Individual community preferences vs. SE network policy: who wins? In the recent question , the OP asks where the following question would be appropriate:     What would be the best way to approach someone that works for me about bad odor?   CHAOS team member Abby Miller  by suggesting the proposal  &#8212; which is in commitment as I'm writing this &#8212; because     One of the up voted example questions there is about dealing with an irritating coworker. This question will be perfectly on topic for that site.   In my opinion,  qualifies for the not constructive close reason*, which applies to all sites in the network, and therefore, the question shouldn't be allowed, no matter how much it has to do with workplaces.  I know Abby isn't the only one who disagrees with me about this. I once had a talk with a mod of some beta about the presence of poll/GTKY questions on his site. His position, in short, was that his community had different rules from SO and its members enjoyed such questions. I don't remember which site it was, it might even be graduated now, but I never did end up participating there.  Anecdotal examples are nice and all, but here's the real question: to what extent can individual communities/sites override network policies? To what extend should they be able to do so?  *: not constructive comes with the description "This question is not a good fit to our Q&amp;A format. We expect answers to generally involve facts, references, or specific expertise; this question will likely solicit opinion, debate, arguments, polling, or extended discussion." Why do we have more male infants born? Why is "nodev" in /etc/fstab so important? How can character devices be used for hacking? I am learning about linux security and struggling to understand why a USB stick with a character device on it is potentially dangerous.  If I have a USB stick with a bash executable that has setuid root on it, the danger is obvious: Anybody with such a USB stick can gain root privileges on my computer if I have an entry like   /dev/sdb1 /media/usbstick auto defaults 0 0   in my /etc/fstab because defaults includes suid.  But what about character devices? How can I use a character device to gain root privileges or break stuff if a USB stick with a character device on it gets mounted with dev or defaults? du shows completely different folder size for * and folder name I'm trying to move some data over and to start with I'd like to make sure that the size is ok. I run into a problem however, if I check the folder with   du -sh  myfolder/ 780M    myfolder/   but   du -sh * .. 71M     myfolder ..   I thought it might be something to do with hidden files but this particular folder does not have any inside.  Am I missing something? General Information about Eigenvalues for an 3x3 symmetric matrix How do we find general information about the eigenvalues of an arbitrary 3x3 symmetric matrix without resorting to explicitly computing the solutions to the cubic characteristic equation?:  \begin{bmatrix}     a   &amp; b &amp; c  \\     b    &amp; d &amp; e \\     c &amp; e &amp;f \end{bmatrix}  (In this case the sum and product of the eigenvalues, and the sum of the reciprocals of the eigenvalues.) Since diagonalization preserves the trace and determinant of matrices, would it be best to just find the corresponding diagonal matrix and read off the sum and product of the eigenvalues from there? The problem is I can't think of a way to diagonalize a matrix without explicitly computing the eigenvalues. What would happen to a teaspoon of neutron star material if released on Earth? I've read on NASA's page on neutron star that one teaspoonful of that star would weigh over 20 billion tonnes on Earth. If it was somehow possible to bring it to earth would it:   Burn and disappear during Earth atmosphere entry? Assuming we have 20 billion tonnes of mass occupying the volume of a teaspoon here on Earth, would it fall through the ground under its own weight? What is the correct way to install proprietary ATI Catalyst Video Drivers (fglrx) directly from AMD? "Try Ubuntu" option is not dislayed I created a live USB using Unetbootin (distribution: ubuntu 12.04 ) from the alternate installation ISO, but while booting "try ubuntu" option is not displayed in Unetbootin menu. Is there any way to try this ubuntu distribution without installing ? Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|&lt;r\}$. How can i show that the volume of $B_r$ is equal to $br^{(n-1)}$ for $r &lt;\varepsilon$ for some $\varepsilon&gt;0$. And how can I find $b$? It is intuitively obvious. But how can one prove it? I have no idea. Solution of equation $\frac{x\cdot 2014^{\frac{1}{x}}+\frac{1}{x}\cdot 2014^x}{2} = 2014$ Solution of equation $\displaystyle \frac{x\cdot 2014^{\frac{1}{x}}+\frac{1}{x}\cdot 2014^x}{2} = 2014$   $\bf{My\; Try::}$ Clearly Here $x&gt;0$, Now Using $\bf{A.M\geq G.M}$  So Here $\displaystyle x\cdot 2014^{\frac{1}{x}}&gt;0$ and $\displaystyle \frac{1}{x}\cdot 2014^x&gt;0$ .  So $\displaystyle \left(\frac{x\cdot 2014^{\frac{1}{x}}+\frac{1}{x}\cdot 2014^x}{2}\right)\geq \left(x\cdot 2014^{\frac{1}{x}}\cdot \frac{1}{x}\cdot 2014^x\right)^{\frac{1}{2}} = \sqrt{2014^{\left(x+\frac{1}{x}\right)}} = 2014$   And equality Hold , When $\displaystyle x = \frac{1}{x}\Rightarrow x= 1$  Can we solve it  without Using $\bf{A.M\geq G.M}$  If Yes, Then please explain me.  Thanks Revolution of Koch Snowflake