What does Pi equal to What is the approximation of pi in a fraction form. I am very curious to know what it is. I have been seeing pi almost everywhere.
Approximating $\pi$ with least digits Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as penalty.  For example the well known $\frac{355}{113}$ is an approximation, but it gives only 7 correct digits by using 6 digits (113355) in the approximation itself. Can you make a better digit ratio?  EDIT: to clarify the "game" let's assume that each mathematical operation (+, sqrt, power, ...) also counts as one digit. Otherwise one could of course make artifical infinitely nested structures of operations only. And preferably let's stick to basic arithmetics and powers/roots only.  EDIT: true. logarithm of imaginary numbers provides an easy way. let's not use complex numbers since that's what I had in mind. something you can present to non-mathematicians :)
I'm using a custom class file.  My main file looks like this:  \documentclass[12pt,a4paper]{thesis} \begin{document} \frontmatter     \pagenumbering{alph}      \pagenumbering{roman}     \clearpage      \cleardoublepage     \tableofcontents     \clearpage     \listoffigures     \clearpage     \listoftables   \mainmatter        \part{test}       \input{test} \end{document}   If I put this in my test.tex it does work:  \chapter{test} \label{chapter:test} test    When it becomes more text (lets say 40x the word "test") I get the error stated in my question title.  To test it I brought my class file down to this:  \ProvidesClass{thesis} \NeedsTeXFormat{LaTeX2e}  % Based on the memoir class \DeclareOption*{\PassOptionsToClass{\CurrentOption}{memoir}} \ProcessOptions \LoadClass{memoir}   I'm totally confused what is causing my error now.  This is the complete error:  ! pdfTeX error (font expansion): auto expansion is only possible with scalable  fonts. \AtBegShi@Output ...ipout \box \AtBeginShipoutBox                                                    \fi \fi The habitual past "would" versus "simple past" tense One can use the habitual past 'would' to describe past events. For example one could say,      Jim would spend weekends at the farm. He would cut weed and water the plants.   One can also use the simple past.       Jim spent weekends at the farm. He cut weed and watered the plants.   Would the use of habitual past be considered "weak" writing or prose, as opposed to the simple past?  How does the simple past in the following sentence "He would cut weed and water the plants." without any preceding time expression e.g. "spend weekends"  make a distinction between a repeated action/habit and a one-time action of cutting weeds and watering plants? As a new user without much rep, what should I do if someone else has already asked a question I have, but it has no answers? Suppose you had a question, and you really want a good answer that would solve your problem, but noticed it was already asked by someone else. However, no one has posted any answers. The question is technically considered "unanswered", but a lot of users who would otherwise bother to give a correct answer probably skip the question completely for some reason.  What should I do in this case?  One idea is to ask a new, completely identical question, in the hope users will notice the new question and post an answer there. But would that go against SE policy?     Note: This is adapted from , but that doesn't cover the case where the original question is unanswered. Also, it is not a duplicate of , as that question asks about getting attention for one's own questions, while this one and the one linked above ask about getting an answer if someone else has already posted the same thing. URL to create a new Google document What is the current URL to create a new Google Document please? Since Google last updated their Docs the old URL does not work. Labels cannot be moved although data-defined settings are active I'm trying to manually move labels but having some trouble. I'm trying to move the labels while in edit mode. I've added fields for x coordinate (all null), y coordinate (all null), rotation (all =0), and font-size (all =10). This makes the "move label" and "rotate label" buttons active, but when I click on the labels, no box around the label appears and it's impossible to move the label. What am I doing wrong? What is the expected value of the maximum of 500 IID random variables   with uniform distribution between 0 and 1?   I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction?  Thanks I would like to associate the last installed version of my Mac OS X application with a certain type of file. A little experimentation shows that the info.plist file seems to be read and interpreted by the operating system when launching the application, not when dragging the application bundle to the disk.  does not say much on that topic.  Is there any way, using a simple .dmg image as installation medium, to make sure the system associates that particular file type with my newly installed app without having to launch it first?  I should clarify that I want this to be done automatically during the installation (I don't want my users to have to do it themselves). The is sometimes pronounced &quot;tha&quot; (/ðə/) or &quot;thi&quot; (/ðiː/). Which is the correct pronunciation of this word? Are both correct and used interchangeably at specific places? If the second question is correct, please provide the rule of pronouncing at different places. Problem: A lot of the time I downvote something that clearly shows lack of research or effort on OP's part, and when they edit to improve it, I usually find out pretty late, because there are no notifications to help with that.  Proposition: add a Get notified when this post is edited button/checkbox under a post which you have downvoted (per post, not just all posts ever), so that if there is a significant improvement, you could remove your downvote or even change it to an upvote to show you recognize the improvement.  I believe this would improve public thinking about votes, and make people (who choose to participate like this on the SE sites more often) vote more freely and more often, and everybody else would not think of a &lt;0 post score as a death sentence for their contribution.  I'm not asking to enable this on SO (for a number of obvious reasons), and that's why I propose an opt-in subscription. This would be most useful on lower activity sites like the SE, in my opinion, but who knows.    I want to know if someone has improved their post after a downvote to reinforce the positive behavior and I get -4 on my good intentions? What kind of culture are you trying to cultivate? Please, explain why you disagree without silently downvoting. If you think there's a better way, yes, I want to know your thoughts. Create a tar archive split into blocks of a maximum size I need to backup a fairly large directory, but I am limited by the size of individual files. I'd like to essentially create a tar.(gz|bz2) archive which is split into 200MB maximum archives. Clonezilla does something similar to this by splitting image backups named like so:  sda1.backup.tar.gz.aa sda1.backup.tar.gz.ab sda1.backup.tar.gz.ac   Is there a way I can do this in one command? I understand how to use the split command, but I'd like to not have to create one giant archive, then split it into smaller archives, as this would double the disk space I'd need in order to initially create the archive. Enumerate paragraphs and subparagraphs like sections but with letters Electric potential is related to Electric Field by negative integral of Field with distance.If charges move away from the positive terminal (Assuming conventional flow) why is there no drop in electric potential even though the charges are moving in the direction of electric field?  Some of the answers I read for the similar question said that the electrons in wires rearrange themselves in a way to make the electric field uniform in the wire.However,this doesn’t explain the absence of potential drop in the wires (due to electric field) since the field is still there. Sentient Bomb in a city on the moon? I would like to know the title of the short story that I believe takes place on the moon.  The setting is a mall or some other very public location and the main character is a police officer.  She needs to talk a sentient bomb out of exploding.  I know there was the John Carpenter film "Dark Star" with a similar plot line but I'm trying to identify the short story and the author. How can I get back to Blender's default theme? Even though I am relatively new to Blender I am recording a tutorial for a Blender exporter. I have been advised that I should use the Default colour scheme (for the UI). I have no idea which one that is. Can anyone tell me? I read a novel where several enemy alien species were stranded together on an interdicted planet.  They had to recycle everything and keep hidden. By "keep hidden" I mean make sure there is no evidence of their presence visible from space. They recycled everything and lived in camouflaged dwellings.  The younger generation was building a mixed culture community until the older home species showed up.  One of the alien races had a body made of rings or donuts of gases that mixed to express itself. Members of this species could be taken over and controlled by rings transferred from another member of the species.  The book may be 20 years old or more. What do you call the frequent conjunction {juxtaposition} of two words? The statement that I want to make is:     "Impostor syndrome" ... why does this (conjunction of words) sneak into every text I read these days?   What is the right or best expression for "conjunction of words"? In the statement I don't want to refer to the syndrome itself, but rather to the expression. If it was one word instead of two I would want to say     ... why does this word sneak into every text I read these days.   However, writing     ________ why do these two words sneak into every text I read these days.   would miss the point, because it's not these two words that sneak into every text at various location but rather their conjunction. What size images do I need for common print sizes, such as wallet size, 8x10, and 16x20?  Is there a general formula?  How could that size change if, for example, a 16x20 was printed on canvas rather than photopaper? This is problem #6 from Section 1.2 of Ash's Real Variables With Basic Metric Space Topology.   I am asked to show that it is impossible to list the rational numbers in increasing order.   While I know it is possible to list a finite subset of the rational numbers in increasing order, I was wondering if the reason for the impossibility in this case is because there is no least element of $\mathbb{Q}$?   I mean, it's possible to impose an ordering on $\mathbb{Q}$ (correct me if I'm wrong, but it seems possible to compare any two rationals).  But the set $\mathbb{Q}$ is of course, a subset of itself, and the Well-Ordering Principle says that a set $S$ is Well-Ordered only if any subset of $S$ contains a minimal element. Since $\mathbb{Q}$ does not contain a minimal element, it does not appear that it is a Well-Ordered set.   Is this the correct idea as to why it's impossible? If not, could somebody explain to me what the correct idea is and a strategy for proving it?  Thank you ahead of time. Opacity Mask With Multiple Images (Illustrator CS5) I am trying to do an opacity mask and it is driving me crazy.  Bascially I have a layer of ice. It's four different pictures of ice overlapping each other so it looks like a whole bunch of ice. I want to have it fade in from both sides (left and right) until it is completely opaque in the middle.  I've select all of the pictures and chosen 'Make Opacity Mask' from the Transparency Panel and it disappears but then I don't know how to manipulate the opacity.   In the Transparency panel it doesn't show a picture of what I'm working with so I can't simply click next to it to create the opacity mask like I've seen in tutorials.  I've also tried to overlay a rectangle over the objects, choose them all then try 'Make Opacity Mask' but that didn't work either. Any ideas? Thanks. How to tell Google that I have changed my website URLs?