What is a simple proof that $\sin 1^\circ$ is irrational?    I have no idea how to solve this problem.
In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational,  and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but I am curious as to whether anyone has a method for proving this without making use of the Taylor Series Expansion. I feel as though doing so must be possible using some number theory, but am low on ideas as to an alternative approach to the question.  Note:  If anyone is interested in my solution using Taylor Series Expansion (although it is not the focus of my question), here it is :      From    $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} -  \dots$$   We see that    $$ \alpha = \sin 1 = 1 - \frac{1^3}{3!} + \frac{1^5}{5!} -  \dots $$   Given integers $a$ and $b$, if $\alpha = \frac{a}{b}$ then it follows that $b!\alpha \in \mathbb{Z}$, and $b!\alpha = C + D$ where $C \in \mathbb{Z}$ and we have :   $$ D = \begin{cases} \pm(\frac{1}{b+1} - \frac{1}{(b+1)(b+2)(b+3)} + \dots) \text{    if $b$ is even}\\ \pm(\frac{1}{(b+1)(b+2)} - \frac{1}{(b+1)\dots(b+4)} + \dots ) \text{ if $b$ is odd}\\ \end{cases} $$   In each case we can see that $0 &lt; D &lt; 1$, giving us a contradiction.    Thus, we have that $\sin 1 $ is irrational.    $$ \blacksquare $$
Is it possible to write a Ping class in C# that will run within Windows 8 Metro environment? Passing list to viewparams variables in GeoServer SQL views? I have a sql view which take only one parameter, but this parameter is a list, and a need to know how can i pass this parameter in the url  Here is my sql view  SELECT * FROM shop WHERE id IN %list%   When I put the default value for %list% as (1, 2 ,3) as example, the layer preview works and show the shopa with the default ids,  but when I try this in the url  viewparams=list:(1,2,3)   does not work.   I already tried other things like   viewparams=list:"1,2,3"   but nothing works.  Does anyone know how to do this? how can I add an overlapped anchor in both overlapped paths automatically The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \&amp; \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$   Proof by Strong Induction : $\bbox[5px,border:1px solid green]{\color{green}{n = 1 }}$ $2 \mid F_1$ is false. Also, $3 \mid 1$ is false. The implication [False $\iff$ False] is vacuously true.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Induction Hypothesis}}}$ Assume that $2 \mid F_i \iff 3 \mid n$ for every integer $i$ with $1 ≤ i ≤ k$.  $\bbox[5px,border:1px solid green]{\color{green}{k + 1 \text{th Case}}} \;$  To prove:  $\quad 2 \mid F_{k + 1} \iff 3 \mid k + 1.$   $\bbox[5px,border:1px solid green]{\color{green}{n = k + 1 = 2}} \;$ $2 \mid F_2$ is false. Also, $3 \mid 2$ is false. So [False $\iff$ False] is vacuously true.  Hence assume that $k + 1 ≥ 3.$ We now consider three cases:  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 1: } k + 1 = 3q}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 \require{cancel}\cancel{\mid} (k − 1)$. By the ind hyp, $3 \require{cancel}\cancel{\mid} k    \iff F_k$ odd &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Since $F_{k+1} = F_k + F_{k−1}$, thus $F_{k+1}$ = odd + odd = even.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 2: } k + 1 = 3q + 1}}$  Thus $3 | k$ and $3 \require{cancel}\cancel{\mid} (k − 1).$ By the ind hyp, $3 | k \iff F_k$ even &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Thus $F_{k+1}$ odd.  $\bbox[5px,border:1px solid green]{\color{green}{{\text{Case 3: }} k + 1 = 3q + 2}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 | (k −1).$ By the ind hyp, $3 \require{cancel}\cancel{\mid} k \iff F_k$ odd and $3 \mid (k − 1) \iff F_{k - 1}$ even. Thus $F_{k+1}$ odd. $\blacksquare$  $\Large{1.}$ Does the proof clinch the $(\Leftarrow)$ of the $(k + 1)$th case?   $\Large{2.}$ Since the recursion contains $n, n - 1, n - 2$, thus the recursion "time lag" is $3$ here. So shouldn't $3$ base cases be checked?   $\Large{3.}$ Further to #2, shouldn't "assume $k + 1 \geq \cancel{3} 4$" instead?   $\Large{4.}$ Shouldn't the $n = k + 1 = 2$ case precede the induction hypothesis?  I referenced . Source: Exercise 6.35, P152 of Mathematical Proofs, 2nd ed. by Chartrand et al    Supplement to peterwhy's Answer:  $\Large{1.1.}$ I wrongly believed that all 3 Cases proved the $\Leftarrow$. I now see that Case 1 is $\Leftarrow$ via a Direct Proof. Cases 2 and 3 are $\Rightarrow$ via a Proof by Contraposition. Nonetheless, how would one foreknow/prevision to start from $3 \mid n$  for both directions of the proof? Derive Fourier transform of sinc function We know that the Fourier transform of the sinc function is the rectangular function (or top hat). However, I'm at a loss as to how to prove it. Most textbooks and online sources start with the rectangular function, show that   $$\int_{-\infty}^\infty \text{rect}(x)e^{i\omega x}dx=\int_{-1/2}^{1/2}e^{i\omega x}dx=\left.\frac{e^{i\omega x}}{i\omega}\right\vert_{-1/2}^{1/2}=\text{sinc}(\omega/2)$$  and then just invoke duality and claim that the Fourier transform of the sinc function is the rectangular function. Is there any way of deriving this directly? i.e., starting with the sinc function?   I've tried, but I'm not sure as to how to proceed. I know that the sinc is not Lebesgue integrable and only improper Riemann integrable. Some vague recollection of these being important in the Fourier transform hinders my thought process. Can someone clear things up for me? Is it possible to add non-number characters on the left of a code block?  Specifically, I want to add ... to correspond with some source code (see image).    I am currently using commands to stop / start the line numbers. Here is the code:  \documentclass[12pt]{article}  \usepackage[a4paper, margin = 20mm, top = 30mm, bottom = 25mm]{geometry} \usepackage{xcolor, soul} \usepackage{listings}  \definecolor{codebackground}{rgb}{0.95, 0.95, 0.92}  \lstset {     frame            = tb, % hrule above and below     keepspaces       = true,     columns          = flexible,     basicstyle       = {\normalsize\ttfamily},     escapeinside     = {(*@}{@*)}, % for escaping     backgroundcolor  = \color{codebackground},     showstringspaces = false,     language         = C,     keywordstyle     = \color{blue},     stringstyle      = \color{red},     commentstyle     = \color{teal},     numbers          = left, % {none, left, right}     firstnumber      = 1,     numberstyle      = \scriptsize\color{black},     numbersep        = 5pt }  \let\origthelstnumber\thelstnumber \makeatletter % for stopping line numbers \newcommand*\stopnumber {     \lst@AddToHook{OnNewLine}     {         \let\thelstnumber\relax         \advance\c@lstnumber-\@ne\relax     } } % for starting line numbers \newcommand*\startnumber[1] {     \setcounter{lstnumber}{\numexpr#1-1\relax}     \lst@AddToHook{OnNewLine}     {         \let\thelstnumber\origthelstnumber         \refstepcounter{lstnumber}     } } \makeatother  \begin{document} \begin{lstlisting}[firstnumber = 456] int third_function(const int c) {     /* error checking */     if(c &lt; 0)     {         return;     } (*@ \stopnumber @*)      ...     (*@ \startnumber{512} @*)     /* other stuff */     int d = 4;     char* word = "third function"; (*@ \stopnumber @*)      ...     (*@ \startnumber{678} @*)     return (c * d); } \end{lstlisting} \end{document} I live in an apartment which provides Internet service included in the rent. They use a company called Pavlov for the Internet    wireless seems to be working fine but wired connection is at least %30 faster.  Ethernet, Cat5 outlet is built in the wall, and there is only 1 outlet in each room.  I would like to take this 1 outlet coming out the wall and multiply it into 4 wires, for desktop, Playstation, TV and laptop, without losing any Internet bandwidth.   I have absolutely no idea whether this line is coming from a switch or a router but I have been researching Ethernet splitter, routers, switches, hubs and haven't found a solid answer.   Is there a way I can do this?  This picture cleared up few things:     It seems that an Ethernet switch is to Ethernet as a USB hub is to USB, but with packet inspection.  What is a 10/100Mbps Network Switch, actually, and what is the cap? How can i check if $remote_addr IP is NOT in CIDR range in nginx? I am using nginx latest web server. How can i check if the $remote_addr ip is NOT in specified CIDR range and return appropriate error code.  i would like to return specific error codes if IP is NOT in specific CIDR IP ranges.  how can i do this for CIDR IP range ?  e.g.    If remote_addr is not in 123.123.123.123/24 then return 444 code . What component to use for audio signal switching? I'm looking to build a guitar pedal switcher based on a Raspberry Pi or Arduino (have not yet decided, but I have an unused Raspberry around and I lean towards that). Anyway, digital logic.  Now, I need to switch audio signal obviously and I need to do it quietly (both physical silence and electrical silence). Please note that the signal is line level throughout most of the build, except for the input which is much lower. I will probably have 2 or 3 loops before the preamp so those may not encounter line level signal.  From what I read and know, so far, mechanical relays are not really an option due to the clicking noise they make. I also read something about FETs, but that's a bit out of my league and I would appreciate any info on these.  PS: I will probably use different power rails for the digital and analog paths.  PS2: The whole thing will be mounted in a 1U rack enclosure, so please tell me if I should take into consideration any thermal issues. What do the different colors mean in ls? What do the different colours in Ubuntu's ls command mean? For example, when I type the ls command in one of my folders, I get one of the files in light green, the other (which is a folder) in blue with green highlighting.   What do those colours mean, and there is any manual about all the colours? Graphics file extensions and their order of inclusion when not specified When using 's \includegraphics you can specify the file extension or not. For generic use and perhaps portability, extensions are commonly dropped. The types of "approved" inclusions and their order of precedence depends on the driver/compiler being used. What are these driver-defaults?  Related: I need to label the four corners of a dataframe as shown in the image. Can this be done with dynamic text scripts? Why do shields in Star Trek "wear out"? how to get line spacing greater than one and half spacing and less than double space Does anyone know how to get line spacing in a paragraph to be greater than one and half spacing and less than double spacing? I tried \renewcommand{\baselinestretch}{value} but it did not work. Factorial of 0 - a convenience? Selenium/PhantomJS raises error Does anyone know what this probability question is asking? We, as high school students have been taught that-because Bohr's model of an atom assigns specific orbits for electrons-that it is better than Rutherford's model. But what Rutherford failed to explain was why electrons don't emit EM waves &amp; fall into the nucleus. I don't see how the introduction of 'atomic orbitals' overcame this defect. Can't it still radiate EM waves? I would like to delete, from the following figure:  the top circle, but without removing the black edge that separate the orange part from the gray part. The figure above has been obtained by the following code: \documentclass[border=0.2cm]{standalone}  % Required packages \usepackage[dvipsnames]{xcolor} \usepackage{tikz}   \begin{document} \begin{tikzpicture}[thick, set/.style = {circle,      minimum size = 3cm,      fill=black!30}]  % Set A \node[set,label={135:$A$}] (A) at (0,0) {};  % Set B \node[set,label={45:$B$}] (B) at (1.8,0) {};   % Intersection \begin{scope} \clip (0,0) circle(1.5cm); \clip (1.8,0) circle(1.5cm); \clip (0.9,1.5) circle(1.5cm); \fill[orange!60](0,0) circle(1.5cm); \end{scope}  % Circles outline \draw (0,0) circle(1.5cm); \draw (1.8,0) circle(1.5cm); \draw (0.9,1.5) circle(1.5cm);  \end{tikzpicture} \end{document}  A rough example of the desired result is: Where can I find other RPG players? I'd really like to learn how to play pen-and-paper RPGs.  I know that it takes a few players to actually sit down and play, but I'm not at all sure where to find more gamers.  How do I find existing groups to join? Or, are there any resources online that may help me find existing RPG groups or players?