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"I could go to Vegas. Learn to play Black Jack. Memorize four hundred fifty two consecutive digits of pi, a few hundred measly cards are easy."
—Fred, Angel

A quick way to show that a character is a genius is to have him recite pi to an absurd number of places. Pi, with its endless parade of decimal digits, has both mystique and geek cred  it has its own day, it's splashed across mugs and Tshirts, calculating 110 millions of digits of π is a very common way of testing your CPU's power, and it's honored by movies and songs. Most of us never memorize it past a few places, so anyone who can fire off a hundred surely must be a genius, right?
For the record, odds are that your computer "knows" pi to the nearest multiple of 2^62, about 2.168 * 10^19, so about eighteen reliable digits. Ditto for e. You can compute it much more precisely, but you'll need to make your own storage.
The truth is, only a handful of digits are needed for most applications only 11 decimal places are needed to calculate the circumference of the Earth to a millimeter^{[2]}, while only 39 are needed to find the circumference of a circle the size of the visible universe to a precision of a hydrogen atom. There's not much point in memorizing a hundred places other than to show off. For reference, here's a handy guide.
Perhaps the reason Reed Richards Is Useless is because he spends his time memorizing numbers he'll never use. It's also a common attribute of the AbsentMinded Professor and the TV Genius, as the contrast is a wonderful thing to show or rant about.
Inversely, a quick way to show that a character is stupid is have him not know what pi is at all: "Apple or cherry?"
Contents
Anime and Manga
 In the Cowboy Bebop movie, Teen Genius Ed sings a little song that involves reciting pi. (With a few wrong numbers!)
 Sayonara Zetsubou Sensei had a kid recite the first few digits of pi...and get excessively hyped for this feat.
 Kyon of Suzumiya Haruhi supposes that the Intergrated Data Entity is smart enough that it "could calculate pi to the hundred millionth decimal point in just two seconds and do all sorts of advanced tricks."
 The first episode of Hell Girl: The Cauldron of Three had Enma Ai ask the person she was taking to hell (a teacher) to recite pi while ferrying him there.
 In The Prince of Tennis, one of Sadaharu Inui's Image Songs has him reciting pi.
 In the third episode of Angel Beats, after Noda questions Takeyama's usefulness, Takeyama literally drives him into submission by reciting hundreds of digits of pi.
 Subverted in a Fourth Wall Mail Slot omake of Eyeshield 21, a reader asked how many digits of pi Hiruma can memorize. He replied, "3.14. If you need more than that, go check it from a book or something." The team mascot commented that he's the kind of guy who doesn't remember useless trivial stuffs.
 In Tantei Opera Milky Holmes, Queen Claris (as Sherlock) only needs to get as far as "3.14" to convince the others that she's not Sherlock.
 Ichika from Infinite Stratos recites Pi in his mind to force him to get his mind off the naked Charlotte who's changing right next to him after she dragged him into a changing room both to hide from some girls and to show him her outfit. It largely works as he enters a mantra and seems to be determined to recite the numbers.
Fanfics
 Mentioned as a subversion in Harry Potter and The Methods of Rationality, chapter nine, to lampshade Harry's broader intellect as compared to Hermione's Book Smarts. Harry knows pi down to six decimals because that's accurate enough for most practical purposes. Hermione knows a hundred decimals because that was what was printed in her math textbook.
Film  LiveAction
 This is the motif of Darren Aronofsky's film Pi. The protagonist is a Mad Mathematician who seems to discover a 216digit number that holds the Meaning of Life. In the climax of the film, he recites the numbers while having a breakdown. Ironically, the actual number pi appears nowhere in the movie.
 In Never Been Kissed, the Denominators put on a bake sale with a banner reading: "Pie = $1.00. Pi = 3.1457869986". Only the first two digits after that decimal point are correct, the rest is gibberish.
 In Night At the Museum II the protagonist and his 'girlfriend' have to solve a riddle to which the answer is Pi (something about the heart of the tomb of the pharaoh). They were helped by Bobblehead Einsteins, who proceed to tell them the correct combination. The protagonist constantly mixes them up, but the girlfriend remembers the numbers correctly.
Literature
 In The Ancestral Trail, Richard defeats Madelbiot by asking him "What is the complete value of pi?"
 Discworld:
 The numbersobsessed Mr. Bent from Making Money mentions that he'd mastered all the numbers except pi, but was working on it in his spare time and was sure he'd crack it soon.
 Inverted in Moist's previous adventure, Going Postal. The Post Office's old sorting engine was designed by the infamous Bergholt Stuttley "Bloody Stupid" Johnson, who decided that pi was untidy, what with all that "never ending" nonsense, and therefore designed the crucial cog of the engine to have a pi that was exactly 3. This was all well and good until they started getting letters that hadn't been written yet, and then letters that would have gotten written if the writer hadn't written a different letter instead, and then letters that would never have been written in this particular leg of the Trousers of Time but were starting to bleed over from other legs because the engine was hurting for things to sort.
 In one of the books in Bryan Davis' Dragons in Our Midst series, partdragon genius Ashley has to verbally calculate the square root of pi out to a certain number of digits as a passcode for her own (artificially intelligent, talking, almostassmartasher) computer.
 In the novel version of Carl Sagan's Contact it is discovered that at a particular distant point of the base 11 expression of pi a long sequence of 0 and 1's occurs that forms a bitmap of a circle. (Incidentially, pi is believed to be a normal number, which means that any sequence of numbers will eventually occur, with a frequency that corresponds to its random probability.)
 Cadaeic Cadenza, a pi mnemonic taken to 14 chapters and 3835 digits, containing "Pilish" imitations of The Raven, Hamlet, etc. And then the author went even further with Not A Wake, which encodes over ten thousand digits.
 Colin from An Abundance of Katherines by John Green memorized the first 99 digits of pi when he was ten by making up a 99word sentence in which then first letter of each word corresponds to a digit of pi (a=1, b=2, etc.).
 Bree from JPod did a gradeschool report on pi and wound up becoming slightly freakishly obsessed with it.
 In Anathem, chapter five of "The Book" consists of a large number of digits of pi, which avout are expected to memorize as a punishment.
Live Action TV
 On Stargate Atlantis, Rodney tracked his memory loss by reciting pi to twelve places every so often.
 In an episode of Sliders, Quinn Mallory recites pi to 13 places while playing a fullcontact trivia/ball game on a giant Othello board.
 Doctor Who:
 The Doctor reels off the square root of pi to thirty decimal places in the Doctor Who episode "Midnight". And the woman who's copying everything he says recites it too. At the same time. Out of sync by several digits. By now everyone's realized that something is very wrong with her, but it's at this point when they start getting suspicious of him, too.
 In the original series episode "The Five Doctors" the correct path across a trapped chessboard is based on pi ... somehow. The Master has it memorized well enough to dance across the chessboard, and the First Doctor knows enough to get himself and Tegan across.
 Walter from Fringe recited pi to 101 places to help him sleep. And the code to a combination lock on his old garage is 314159. And in The Plateau, video of a mentally impaired man who's been made intelligent by an experimental drug trial show him reciting the digits of pi to 1000 decimal places while solving differential equations in his head.
 Happens in Cutey Honey the Live to show that Honey Kisaragi/Cutey Honey and Miki Saotome/Sister Miki are not normal.
 An episode of Clarissa Explains It All featured the title character and her brother on a game show, where he blows the question "What is pi?" with the standard flavorsofpi mistake. Since he prides himself on being a child prodigy, and hates when people prove he's not, this sends him into a period of stunned shock...from which he resurfaces in time to interrupt his sister and prevent her from winning them the game.
 Grant has 3.14 as his "prisoner number" in Myth Busters's "Prison Escape" myth.
 In the Star Trek the Original Series episode "Wolf in the Fold", Spock Logic Bombs a villain who's taken over the Enterprise's computer by instructing it to "compute to the last decimal place the value of pi".
 As noted in the page quote, Fred from Angel has memorized 452 digits of pi. In the same scene the quote comes from, she scares away a homeless guy by reciting it.
Magazines
 In Mad Magazine's "Potrzebie System of Weights and Measures", halavah (used to define the unit of mass) is described as a form of pie whose specific gravity is 3.1416 and specific heat is .31416.
Music
 Weird Al Yankovic's "White & Nerdy"  "My myspace page is so totally pimped out/ Got people beggin' for my top 8 spaces/ Yo, I know pi to a thousand places/ Ain't got no grills, but I still wear braces..."
 Kate Bush wrote a song called "π", about a mathematician who's fascinated with the number, and some verses consist of her singing a few hundred digits.
 The Hard N Phirm song linked above is really more a parody of this.
 Songs to Wear Pants To once wrote a song to serve as a mnemonic for the first 50 digits of pi. Just count the letters in each word:
Man, I can't  I shan't  formulate an anthem where the words comprise mnemonics, dreaded mnemonics for pi. The numerals just bother me always, even the dry anterior. Try to request something lower (zero) in numerary aptitude. Even I, pantaloon gallant, I cannot actualize the requested mnemonics, the leading fifty, I... 
 No points for guessing what famous song Public Enema used as the framework for their musical mnemonic unofficially titled "Mathematical Pi." Memorize the chorus and play along with Mouthful of Pi at home today!
 The rhyme "Secant, tangent, cosine, sine/Three point one four one five nine" appears in football chants/cheers at particularly nerdy colleges.
 This cover of "Ren'ai Circulation" is basically just a Kana Hanazawa soundalike singing the first 400 or so digits of pi.
Newspaper Comics
 FoxTrot:
 One strip had Jason Fox reciting enough digits of pi to fill several panels.
 Another strip had a scene at a nerd camp that Jason and Marcus were attending. When a kid is late for Roll Call, the counselor tells him to do a pushup and recite pi to thirty places.
 In Monty, when Robotman disappears, he is replaced by an alien human hybrid whose full name is 3.1415..., but his friends call him Mr Pi. Because the name was given to him as an insult he later changes it to Dave.
 In Pearls Before Swine, Goat tries to recite pi as a pickup line. It doesn't work.
Tabletop Games
 In the Net Runner collectible card game there exists a card "PI in the 'Face"; its cover art includes a decimal expansion of the number Pi.
Like so many concepts of Netrunner, this card's design can be traced back to R. Talsorian's Cyberpunk 2.0.2.0. Roleplaying Game. Interestingly, in Cyberpunk, PI in the 'Face is not a defense program for data forts, but an attack program used by Runners to fry a data fort's CPU. [...] If the program makes a successful attack, the CPU will be trapped calculating Pi. 
Video Games
 Moe the Clown does this in the second Phoenix Wright Ace Attorney game. It isn't to show intelligence, it's just one of his many puns.
 Minamimoto from The World Ends With You can not only recite Pi to 156 places, but use it as a massive attack so strong that Crystal Dragon Jesus had to retreat to another universe. "PiFace" is also his nickname, and his HP from the two different times you fight are 3141 and 5926.
 In Ratchet and Clank Going Commando, Clank gets repaired by Ratchet, and nearly speaks the final digit of Pi when he wakes up.
 In Metal Gear Solid 4, Sunny can be heard singing the digits of pi before a Mission Briefing.
 In Tales of the Abyss, Lorelei's fonon frequency is Pi.
 In Mountain of Faith, Marisa, getting bored of going through the stages, grumbles that she could recite pi to a thousand places by the time she got to her destination. Cirno, in "A Genius Comes When You Least Expect!", oneups her by claiming to know pi's EXACT value (the last digit, is, of course, 9; see Richard Feynman in Real Life below).
 If you play as Ami/Sailor Mercury in the Sega Genesis Pretty Guardian Sailor Moon game, she starts off the second level beginning to recite pi before Luna interrupts her.
 A character begins mouthing off the Pi sequence in Persona 2 after being challenged by a Jerkass professor. Only the professor doesn't know if the student's right, as he's an English teacher.
Web Comics
 This Toothpaste For Dinner cartoon.
 According to this Nothing Comes Naturally strip, Cerine regularly uses pi as a password.
 Subverted This Saturday Morning Breakfast Cereal Comic.
 Legostar Galactica: Security Chief 46 takes offense to being asked to help with a mission on a world with a mathbased culture:
46: What? Because I'm a robot I'm automatically good at math? That's just stupid! 
 Variation in Homestuck: the Autoresponder doesn't show off just by reciting pi to an excessive number of digits, which isn't really all that impressive for a computer program, but by claiming to have calculated the final digit of pi (four, apparently). This is obviously impossible (pi is an irrational number that has no end) and he's only saying it to screw with Dirk, as Dirk rightly points out.
 Mike Bookseller: "Lark to cash/wrap 3.141592653589."
Web Original
 Pi was the basis of one hideously difficult puzzle in the Alternate Reality Game Lockjaw, as chronicled in The Unholy Saga of Where.gif.
"Day 7: ...Despite the fact that we've spent an entire friggin week on this problem, it should be simple high school mathematics from here on in. Day 8: 300 grown men and women suddenly simultaneously realize that they learned nothing in high school math.". 
Western Animation
 The Simpsons:
 Some girls on a playground of a school for the gifted play hopscotch, singing, "Cross my heart and hope to die / Here's the digits that make pi / 3.1415926535897932384...".
 Apu mentions he can recite Pi to 40,000 places as proof of his excellent memory, correctly stating that the last digit is a 1. Homer throws in an obligatory "mmmmm, pi" here.
 Professor Frink shouts out "Pi is exactly 3!" to silence an audience of rowdy Scientists, apologizing afterward for taking such extreme measures.
 In an episode of Count Duckula, there was a character who had memorised pi to 15,000 places. Someone asked him about this and he started demonstrating. Every time the scene cut back to him for the remainder of the episode, he was still reciting.
 One episode of The Batman involved a bitter exchild genius forcing the people who fixed the game where he lost to play a mock game show. Rules were simple everyone gets one question, on any topic. They win and he lets them go free if they can ask him a question he doesn't know the answer to. One of them tries asking him the exact value of pi he dismisses the question as being impossible to answer, but in the name of being a fair sport, he recites the first one hundred digits just to prove he's not weaseling out of it. Batman ruins the scheme by asking him "Who is the Batman?"
 Reversed in a Pinky and The Brain short on Animaniacs. Brain tells Pinky to ask him any question:
Pinky: All right, what is Pie? 
 Futurama:
 One episode mentions the last digit of Pi as a game show question.
 There is a Πth Avenue in New New York and ΠKEA stores.
 In a special feature of the first Futurama movie, there is a math lecture and one of the topics is about pi. The lecturer asks the audience (that is, the Futurama staff which is chock full of math PhDs) how many digits of pi they know. Eventually, Matt Groening starts rattling off the correct digits...then some nonsense digits before ending with "333...and then just a whole bunch of 3's!"
 "Donald In Mathmagic Land" has a circular bird who recites digits of pi. And gets them wrong.
 In The Adventures of Jimmy Neutron Boy Genius, Jimmy stops his Nanobots' second reign of terror by tricking them into deciphering the value of Pi. After a stressful readthrough of all the digits, they seizeup and overload on "five". He got the idea from seeing his Bumbling Dad with a lot of pie.
Real Life
 Daniel Tammet, an Asperger's savant, holds the current European record of reciting 22,514 digits from memory. He describes the experience in his autobiography, Born on a Blue Day.
 Richard Feynman stated that he wanted to memorize the first 761 digits of pi because at that point, there is a sequence of six 9's so he could recite the numbers to there, then go "999999...and so on." "That point" was named in his honor afterwards. Douglas Hofstadter told the same anecdote. This may just be a genius thing.
 ↑ Please don't actually top that.
 ↑ And due to the effects of General Relativity, you would get the wrong answer anyway.