Pi is defined as circumference/diameter, but it is an irrational number. And by definition an irrational number can't be defined by a fraction. So how is it that pi is circumference/diameter and on a side note: how are important irrational numbers like e normally found?
$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction.   Then why is $\pi$ an irrational number?
Chance of objects going against greater entropy? My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.  My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)? I don't want to watch the prisoner cart ride to the execution: is it possible to skip this?   If so, how? Can I replace the intro movie file by some empty video? If I take r as the radial vector of the moving object and v as the velocity vector of the moving object in central force field.   Then r should be perpendicular to r×v . So that depicts that r.(r×v) = 0 . So how does this say that the particle is lying on a constant plane ? " java.lang.IllegalArgumentException: uri.getScheme() == null! " error using JavaFX to play audio Enumerate non-isomorphic graphs on n vertices In the following the graphs are assumed to be undirected and simple.  1.Enumerate the number of non-isomorphic graphs on $n$ vetrices where $n$ is fixed.  Here are some ideas I had:  The number of labeled graphs is $ 2^{\frac{n(n-1)}{2}} $.  So it is enough to find the number unlabeled graphs on $n$ vertices.I have no idea for this.  2.Enumerate the number of non-isomorphic graphs on $n$ vertices and $m$ edges where $n,m$ are fixed.   Can we find a closed formula for each of this?  Any help?   Thank you! I am trying to use the AwesomeCV template () for creating a CV using TexMaker 5.0.3 with XeLaTex. However, I am not able to get this to work as I get the following error messages:    Does anyone know what the source of problem might be and how to solve it?  Thanks a lot in advance,  Cheers Adverb sentence query He is arguably the best actor of his generation   In the above sentence the adverb is placed just after "be" Can we use adverb after "be"? If we can use then please explain properly because I have studied that adverb can be used before adjective and adverb . What does "yield break;" do in C#? I have seen this syntax in MSDN: , but I don't know what it does.  Does anyone know? When a permanent magnet attracts some object, lets say a steel ball, energy is converted into for instance kinetic energy and heat when attraction happens, and they eventually collide. Does this imply that energy is drawn from the magnetic field and the magnet is depleted, making it weaker and weaker for each magnetic attraction performed?  (if the answer requires Quantum Mechanical explanations, please elaborate :)) $\lim_{h\to 0}\int \frac{f(x+h)-f(x)}{h}\,\mathrm{d}x = f(b)-f(a)$ pgfplots external with large dataset (lualatex?) I have fairly large datasets with densely strucutred data that i cant downsampe much without loosing structure. I use TexMaker on Ubuntu 14.04LTS. I use   \usepackage{tikz,pgfplots,pgfplotstable} \pgfplotsset{compat=1.9} \usepgfplotslibrary{external} \tikzexternalize   and   \tikzsetnextfilename{figs/test1}   to externalize the figures. This works good compiling with pdflatex -shell-escape. But as the data becomes to large I am out of memory. I am trying to compile using LuaLaTeX -shell-escape instead of pdfLatex but get the following error message:   ! Package tikz Error: Sorry, the system call 'pdflatex -halt-on-error -interact ion=batchmode -jobname "figs/test1" "\def\tikzexternalrealjob{article}\input{ar ticle}"' did NOT result in a usable output file 'figs/test1' (expected one of . pdf:.jpg:.jpeg:.png:). Please verify that you have enabled system calls. For pd flatex, this is 'pdflatex -shell-escape'. Sometimes it is also named 'write 18' or something like that. Or maybe the command simply failed?   so pgfPlots seem to call pdfLatex without the appropriate -shell-escape flag and without me specifying it. How can i get this to compile in LuaLatex? or can i compile only the figures in LuaLatex and the rest in pdfLatex (the opposite of what is happening now)? I found  and  example but neither helped me...  Thankfull for help and i am also thankful for any general advice that can help me resolve the problem of ploting larger data sets in pgfplots.   Cheers /J Can a set be neither open nor closed? An example would do. I cant think of any.   Thanks in advance! This is a homework problem for my real analysis class. It is problem 13.1 from Mattuck's Introdution to Analysis. The question reads:       Suppose F(x) is continuous on some open interval I and c is a maximum point inside this interval. Is it true that f(x) must be increasing immediately to the left of c and decreasing immediately to the right of c? Proof or counterexample. (Note: A constant function is considered to be both increasing and decreasing.)      My friend and I have been attempting to find a counterexample. We figured a function that begins oscillating infinitely many times near c but converges to c would work. We're having difficulty constructing such a function out of elementary functions so we think we will have to define it in another way, piece-wise using receding intervals. I have no clue how this would look. Any advice would be greatly appreciated. Detect Chrome extension first run / update "24 hours' notice" or "24 hours notice"? Which one is correct?  1: Well, you canceled on Tuesday and our policy is 24 hours' notice for all cancellations. 2: Well, you canceled on Tuesday and our policy is 24 hours notice for all cancellations.  To me personally, the first one makes more sense grammatically. But I've also heard that the second version is how they typically write it on paper forms at hospitals (that's what native speakers say). Could you please clarify my confusion? What if residuals are normally distributed, but y is not? I've got a weird question. Assume that you have a small sample where the dependent variable that you're going to analyze with a simple linear model is highly left skewed. Thus you assume that $u$ is not normally distributed, because this would result in normally distributed $y$. But when you compute the QQ-Normal plot there is evidence, that the residuals are normally distributed. Thus anyone can assume that the error term is normally distributed, although $y$ is not. So what does it mean, when the error term seems to be normally distributed, but $y$ does not? What's interesting in latus rectum? I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical applications of the curves (in engineering, physics, finance, etc.), on their geometric properties, or both.  I know there's this really easy formula to find the latus rectum of a conic from its key parameters. It's quite nice and straightforward to derive it, but...  ... What's its use? Why should anyone be interested in knowing the latus rectum of a curve? Is there any physical phenomenon, any architectural trick, or anything else where its position and/or length plays a key role?   I know just the relation between angular momentum, standard gravitational parameter and latus rectum in an astronomical orbit ($l=\frac{h^2}{GM}$, where $l$ is half the latus rectum, and $h$ is the ratio between the angular momentum associated to the orbiting body and its mass). It's cool, but a bit tricky, and probably too far-fetched for secondary school students... Is there anything simpler? Or just anything else, at any difficulty level?  Many thanks! General misconception about $\sqrt x$ How to clone ArrayList and also clone its contents? How can I clone an ArrayList and also clone its items in Java?  For example I have:  ArrayList&lt;Dog&gt; dogs = getDogs(); ArrayList&lt;Dog&gt; clonedList = ....something to do with dogs....   And I would expect that objects in clonedList are not the same as in dogs list. Crookshanks is Hermione's familiar in PoA, GoF, and OOTP. I cannot recall mention of Crookshanks following OOTP, though. What happened to Crookshanks? Does canon refer to Crookshanks in either HBP or DH?