is an infinite sum of fractions, meaning we are dealing with rational numbers everywhere in it. If you start to approximate π (π over 4) by calculating the partial sums of the formula, you can keep using fractions - it's all just the matter of finding the least common multiple of the denominators and thus expanding the fractions.  The approximation goes like 1, 2/3, 13/15, 76/105, ...  Theoretically, you can repeat this fraction expansion infinite number of times. And no matter how long you're doing it, you will always get a rational number.   But if you never stop, you do it all the way down to infinity, the result is π over 4, which is insanely irrational. What is the point where (and the reason why) this rational pattern breaks?
Just a curiosity. I just saw a video where pi was being calculated by the series  $$1-(1/3)+(1/5)-(1/7)+...$$  My question is how can $\pi$ be irrational if we can calculate it using rational numbers?
Nested List Indices cin and getline skipping input How does an Intelligence penalty affect skill points when character-building/leveling-up? Programmatically obtaining Big-O efficiency of code I wonder whether there is any automatic way of determining (at least roughly) the Big-O time complexity of a given function?  If I graphed an O(n) function vs. an O(n lg n) function I think I would be able to visually ascertain which is which; I'm thinking there must be some heuristic solution which enables this to be done automatically.  Any ideas?  Edit: I am happy to find a semi-automated solution, just wondering whether there is some way of avoiding doing a fully manual analysis. Do fontsize environments exist? Without any doubt there are fontsize switches in the form of control sequences (described  for instance.) But, from time to time I ran into code examples where environment versions of those were used, e.g. \begin{small}...\end{small}. After testing those successfully I began to incorporate them into my TeX vocabulary.  A few days ago @GonzaloMedina commented on one of my answers that those environments do not exists and constructions relying on them work by pure coincidence. Then I tried to find out more about those ominous environments. Ominous because they don't seem to be somehow documentated. (The only thing I found that tried to explain them in a somehow systematical manner was some Wikibooks resource.)  Are these environments real or not? Distributing distinct apples and pears to three people so that each person receives at least one pear How many ways are there to distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear.   Note: The first part of the problem was the same but the word distinct was replaced with identical apples and pears. That answer is $360$ ways.   What has to be done differently when the word identical is replaced with distinct? Apologies in advance for the long question.  My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$   t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$.    Why is this so?     Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$.  Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$.  If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$.  If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers.  I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$   T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$.  Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$   I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align}   I_{ba}  &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\ &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\ &amp;= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result.  How do others feel about this argument?  I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$.  I imagine that in GR a similar argument could be made by invoking local flatness of the metric. mysqli::query(): Couldn't fetch mysqli I am on Ubuntu and I have forgotten the password for my Gnome Keyring (yes, stupid me, I know). Any way I can get it back? I have already looked at , but it says it's not possible if my home directory is encrypted and unfortunately my home directory is encrypted. About fullscreen scrolling webpages There are recently many websites using full screen scrolling pages since Apple launched their own with the iPhone 5C. (website not available anymore, but a reproduction can be found )  Some other examples are:            Is there any study about this practice from the UX point of view? More than about the scroll highjacking, about the use of full screen content?  It seems to me that it's an easy way to present information to the user when used properly and in a simple way, just like a good Power Point presentation. A way to show the user the most important things in a beautiful and smart way.  I see many developers complaining nowadays about this kind of sites, mainly because of the scrolling highjack, but it seems to me users in general, as well as clients, like them.  Any thoughts about this? Determining functions from specifications... my high school teacher asked this question today, and I am struggling to find an answer that will work. The question is outlined below:  Give an example of a cubic polynomial, defined on the open interval (-3,3), which reaches both its maximum and minimum values. How to say that the phone is held between shoulder and ear? What are common phrases an English native speaker would use for the pose of speaking on the phone where you use your shoulder to keep the receiver against your ear (and mouth), so your hands are free?  I have some ideas, but they all sound wooden, and I don't think I've ever heard anyone describe this pose in words. It's also pretty much impossible to google for a pose or look it up on leo.org or other dictionaries.  OK, since people keep asking for what phrasings I found, I'll provide two, though I was hoping I could avoid influencing people who answer here by not providing them: "balance on one's shoulder", "held/nestled/sandwiched between shoulder and ear". Review Audit failed when "late answer" answer is a duplicate I just got a review audit where the "late answer" was a duplicate of the accepted answer, but flagging it on that basis failed the review. That seems like a bug to me.  Question here:   I found that by clicking the question to view the entire Q&amp;A, and noticed that the copy was from a generic "user12345" account (probably with 1 rep as well) and the original from a real-looking account. Viewing the whole Q&amp;A is often useful when reviewing, especially late answers (which are often repeats of earlier points, disagreements with those, or random addendums better submitted as edits). I looked at the duplicate and thought "weird, rep harvesting maybe?" so I clicked "needs moderator attention" and wrote "copy and paste of accepted answer". FAIL!  I think that this review type can't ever work, because to avoid the problem I found you'd need to make up whole new "late answers" which by definition the review bot can't know anything useful about. Either that or to pass you'd just need to take any action at all, including upvoting the original of the duplicated answer. Which I didn't do, BTW, because I had no idea or opinion about the question or answer.  (this is not a duplicate of "What are review tests", it predates that and focuses on a particular defect within the process. See the comments below). Ordinal dependent variable with continuous independent variables The roots of the equation $z^n=(1+z)^n$... The complex roots of the equation      $$z^{n}=(1+z)^{n}$$   $A.$ are vertices of a regular polygon  $B.$ lie on a circle  $C.$ are collinear  $D.$ none of these  Don't know where to start.. Please help.!  NOTE:  $z$ is a complex number. As stated above, I can't connect to my home wi-fi with natty but can with Windows 7. The strange thing is I that I have gotten it to connect to other password protected networks with natty, it's just mine that it dosn't like.   I am using a WPA2 pass-code with AES Encryption and all other devises work fine on my network. Laptop used is a Dell Latitude D430. Function to determine if two numbers are nearly equal when rounded to n significant decimal digits I have been asked to test a library provided by a 3rd party. The library is known to be accurate to n significant figures. Any less-significant errors can safely be ignored. I want to write a function to help me compare the results:  def nearlyequal( a, b, sigfig=5 ):   The purpose of this function is to determine if two floating-point numbers (a and b) are approximately equal. The function will return True if a==b (exact match) or if a and b have the same value when rounded to sigfig significant-figures when written in decimal.   Can anybody suggest a good implementation? I've written a mini unit-test. Unless you can see a bug in my tests then a good implementation should pass the following:  assert nearlyequal(1, 1, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(-1e-9, 1e-9, 5)  assert nearlyequal(1e9, 1e9 + 1 , 5)  assert not nearlyequal( 1e4, 1e4 + 1, 5)  assert nearlyequal( 0.0, 1e-15, 5 )  assert not nearlyequal( 0.0, 1e-4, 6 )    Additional notes:   Values a and b might be of type int, float or numpy.float64. Values a and b will always be of the same type. It's vital that conversion does not introduce additional error into the function. Lets keep this numerical, so functions that convert to strings or use non-mathematical tricks are not ideal. This program will be audited by somebody who is a mathematician who will want to be able to prove that the function does what it is supposed to do. Speed... I've got to compare a lot of numbers so the faster the better. I've got numpy, scipy and the standard-library. Anything else will be hard for me to get, especially for such a small part of the project. I installed Ubuntu 13.04 on my Vista 64 bit machine and I thought it was going to uninstall Vista but it didn't, now I have both systems.  How do I get rid of Vista? I wanted to know if there is difference in terms of charging the battery when a MacBook Pro 16" is connected to two monitors via USB-C, both with Power Delivery capability.   In this specific case I am talking about the LG 27UK850-W monitors. The MacBook Pro 16" will be connected via USB-C to both of them and use them as external monitors. Since both can charge it as well, will the MacBook Pro  charge faster while being connected to both or will just one monitor be able to deliver the charge at a time? Do redirection operators always open file descriptors in parallel?