"Even when Cop and a Half half-assed it, it was still three-quarters cop! —wait, I gotta check my math on that... (carry the nine... times pi... plus two...) Yeah, I'm right."
As You Know, some writers didn't exactly do well in math class. So, it seems that, with the passing of time, all their knowledge of math was condensed into a pile of information that they now paste on every chalkboard in school scenes or write in class notes, regardless of the knowledge necessary to understand those equations.
For example, a first-grade class will sometimes have lots of Gratuitous Greek such as "pi" (π) written on the board, which would probably only confuse the students who would only be learning arithmetic. Other popular examples are integrals, and their well-known "large S" ( ∫ ) symbol. Very often, however, there won't be any variable of integration. Elementary algebra will be represented as something like "x + y = z", which is entirely meaningless without a description of what the variables represent or how they relate to each other.
Let's not forget about E=mc^{2}, the famous equation which has been degraded into nothing but a complex-looking decoration in everything from kids' cartoons to science shows. (E=mc^{2} is actually Albert Einstein's formula for mass-energy equivalence; far more people have heard of it than have any idea of what it actually means).
Also related to relativity^{note }but this time general relativity instead of special relativity, the Einstein field equations (R_{ab} - (1/2)g_{ab}R = (8πG/c^{4})T_{ab}) are rising in prominence as a decorative formula, as E=mc^{2} has started to become a cliché. Loosely, the terms of the left describe the curvature of space-time (which is perceived as gravity) while the terms on the right refer to the distribution of matter.
And then there's the big sigma (∑), the summation symbol. Nothing says smart like a big ol' sigma. There's also big pi (Π) notation (the symbol for a direct product).
Trigonometric relations and the Pythagorean theorem are also popular. But don't ask to see words like "sine", "cosine", and "tangent".
Furthermore, there're only two mathematical mistakes anyone ever makes, no matter how advanced their knowledge of the field; they either "misplaced the decimal point" or "forgot to carry the one". (Sign errors are conveniently absent.)
And math isn't the only subject that gets this treatment. Blackboards full of chemical formulae, sentence diagrams or plot/theme/character diagrams that stretch all over the whiteboard, or genealogies and timelines that look like a tangle of yarn are less common, but serve the same purpose.
This happens for three reasons: First, to ensure that there is some teaching going on, as the show itself thankfully never needs to show the actual classes. Second, to scare the young viewers into believing that they're going to see this stuff when they get older. Third, as a way of showing someone is really smart, often combining it with Room Full of Crazy.
If you were looking for the regular MC Hammer, go here.
Examples
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Anime & Manga
A rather infamous example-turned-meme comes from a particular scene in Futari wa Pretty Cure. The original scene is actually intentional, intended to give Honoka an opportunity to show off how smart she is by pointing out the error, but it fails to do anything but make the teacher look stupid.
One scene in Neon Genesis Evangelion has some of this panned through before we see the characters in school. It makes effort to show more than just disjointed numbers, but, as an AMV Hell segment showed, it got several equations horribly wrong.
An episode of Air Gear shows the main character skating on a fence. The camera freezes when he jumps, and it overlays a Cartesian plane (height over time) with a parabola tracing his jump so far. y=x^{2} was written on the bottom. Never mind that the curve was concave down, so the formula should be negated. Never mind that he jumped from under the x-axis to cross the y-axis, which means he leapt from underground into negative time.
Surprisingly averted in an episode of Ouran High School Host Club, where a brief glimpse of Kyoya's homework shows real mathematical notation that actually makes some sense, even if the context in which it is found is unlikely.
In Kore Wa Zombie Desuka, the protagonist makes the mistake of asking the resident "genius" to explain a math problem from the beginning. A few hours later: "...Spiral galaxies spin like this, kind of like a top. You need a string to spin it, and that's where string theory comes from. Oh, right. I want to touch on super string theory too..." Later on, after asking how soup could be used to destroy monsters (it makes sense in context), they are given a lengthy off-screen explanation concluding with "I can't answer any of your questions though. I don't understand the super string theory stuff either."
Some of the math questions that appear on the board in Puella Magi Madoka Magica for an eighth grade math class are taken from, among other places, the entrance exam for Tokyo University. More information here.
Could be justified: the setting seems futuristic, and it's possible that, as technology becomes more advanced, our knowledge will as well.
Played with in a Sgt. Frog episode where the platoon disguise themselves are children at the Fuyuki siblings' school and Giroro is given a problem in the form of a chalkboard full of incomprehensible fomulae. He stares at it all days and only hours after class had conclude does he notices a little "x 0" in the corner which apparently applied to everything, making the answer simply zero.
Comics
In one Carl Barks comic book, the blackboard in a kindergarten classroom has a slightly illegible, possibly nonsensical mathematical expression written on it, including an integral sign (which, incidentally, does appear to have a variable of integration).
An old Sidney Harris cartoonlampshades this one: two scientists are standing in front of a blackboard full of equations. Toward the bottom right of the board, the chaos of integrals, summations, and other mathematical gobbledygook suddenly ends with "and then a miracle occurs" and an answer. The caption reads "You need to be more explicit here in step two."
Léonard le Génie does this often. Once he accidentally scribbled part of a long equation on the side of Raoul the cat.
Fan Works
One segment^{note }It's at 43:28 in the AMV Hell 4 for AMV Hell focused on a slow crawl across a blackboard covered with math equations in Neon Genesis Evangelion. Halfway through the piece, we hear the Kevin Spacey in Superman Returns "WRONG!", and as the music is swapped out for King Missile's "Equivalencies", all the errors on the board are pointed out with red marker. Humorously, the segment also points out that although one solution does several things wrong, it ends up at the right answer anyway.
Films — Live-Action
The Day the Earth Stood Still:
The 2008 remake includes a scene where Klaatu et al. visit someone who won Nobel Prize for his work on biological altruism. He has a blackboard covered in maths, which may or may not be correct, including the words "Event Horizon". An event horizon is simply a boundary beyond which no events can be detected with the most commonly referenced one being that which marks the point of no return around a black hole. If there was ever a biological singularity, (evolving into god?), the word could come into play. Also, it sounds dark and foreboding.
In the 1951 original, Klaatu visits a leading scientist, but the man is not home. Klaatu makes an addition to a blackboard-covering equation, then leaves his contact information with the scientist's housekeeper. The addition to the equation was apparently intended to convince the scientist not to write off his unknown caller as a joke.
In School of Rock, Jack Black's character writes E=mc^{2} on the board while pretending to teach the children something. Played with slightly in that he is totally clueless about teaching and this was presumably the only vaguely mathematical formula he could remember, and the school principal doesn't bat an eyelid when she walks into the room, even though the children are preteens.
When he's actually teaching the class the only subject he knows (rock, of course) he fills the board with a complicated diagram of rock bands.
In the 2000 remake of Bedazzled, according to a blackboard in the background, a schoolteacher (Elizabeth Hurley) has assigned her students to prove Fermat's Last Theorem as homework. Of course, the schoolteacher was actually the devil and all of these scenes in-between wishes were of her intentionally screwing up the lives of other people.
A minor example in Matilda, where the kindergarten students are learning the multiplication tables. Most kindergarten teachers are happy when their students can add one-digit numbers. This is possibly justified due to the Sadist Principal, Miss Trunchbull.
Justified in the fact Trunchbull is English, and English school children start formal education earlier than American children.
In Good Will Hunting, the protagonist does some Super Epic Hardcore Math that had previously stumped even the professors. Turns out it was one of the more basic proofs from linear algebra.
A Serious Man has a Dream Sequence featuring an implausibly large blackboard covered with enough of the stuff to fill a book.
Admittedly for college age (ie legal to have sex) students, E=mc^{2} is a reasonable equation for the coursework. Sometimes the person just gets lazy and simply writes "1+2=3" on the chalkboard which really gets your Fridge Logic going. Of course you're not supposed to be paying attention to the chalkboard...
E=mc^{2} abuse is played for laughs in Young Einstein, a fictional biography that is deliberately wrong in just about every way. Whenever a character reads it aloud, they pronounce it "emk".
In one scene, Albert walks in on a class in progress, where the teacher is writing "E=" and an insanely complex mess of scribbles covering the rest of the board. Albert helpfully wipes out a clean space and writes a big "mc^{2}". He's physically thrown out of the room in response.
In Kids in the Hall Brain Candy, Dr. Cooper is first seen inventing the anti-depression drug by finishing a gigantic equation on the chalkboard with "=HAPPY".
Subverted in Sherlock Holmes: A Game of Shadows. While Moriarty's chalkboard was covered with mathematical symbols and formulas, the writers consulted with actual mathematicians in order to make sure they were scientifically accurate for the time period. According to one of the scholars involved, the original design for the chalkboard would have invoked this trope as explained here.
There's also a nod to the Holmes canon - Moriarty is said to be known for his work on the binomial theorem, and the blackboard includes Pascal's triangle, which is one way of finding binomial coefficients.
In The Absent-Minded Professor (the original), the eponymous professor has a giant equation on the blackboard behind him. This example also contains a rare sign error: he realizes one of his minuses should be a plus. This eventually leads him to the discovery of flubber.
Literature
In the Sword Of The Galaxy book series, a simple algebraic expression can be used by the Trakkorians to enter hyperspace. When the author received complaints about this piece of Fridge Logic, he recited the MST3K Mantra.
Played for laughs in The Maelstrom by Roger E. Moore (from Cloakmaster Cycle). Two Words: Tinker Gnomes.
His eyes were flashing on and off when they pulled him out, so presumably the cold overcame him just before he could write the answer.
The Maelstrom by Roger E. Moore (Cloakmaster Cycle) has a good parody of the "E=mc^{2} says it all" version.
Live-Action TV
The trope's name comes from a pastiche of E=mc^{2} that appeared in a scene of Farscape: The Peacekeeper Wars where Harvey (who was named after Not-So-Imaginary Friend of the movie Harvey) writes "E = MC Hammer" on a blackboard of other nonsense. Especially funny in that Crichton, as a physicist and an astronaut, would know exactly what E=mc^{2} actually means, and Harvey is likely just dicking around.
The same exact equation was said aloud by Dina on the Nickelodeon sitcom Salute Your Shorts, during a scene intended to make fun of the "growing and learning" activities going on during her camp experience.
In the NCIS episode "Red Cell", Abby and McGee get into an argument about whether or not a set of equations is homology or cohomology. But the operator symbols are different in these two concepts, and what are subscripts in homology are superscripts in cohomology. No one with sufficient education (as Abby and McGee have) can mistake one for the other.
In the Covert Affairs episode "Walter's Walk", Auggie refers to a sequence of numbers as being a "complex cylic permutation". But a permutation is a rearrangement of things, not a sequence. Moreover cyclic permutations are the simplest of all permutations, and talking about them being complex makes no sense.
In an episode of Torchwood, a integral equation was used to open the rift that led Jack and Tosh into the past. It didn't seem very realistic, though at least it had integration variables.
In "The Three Doctors," after coming through the black hole, the physicist Dr. Tyler writes the famous equation in the sand. Justified, too—he's trying to come to grips with the fact that he's just traveled at the speed of light.
One of the forms of Gallifreyan (the Time Lords language) appears to be this, specifically Old High Gallifreyan.
In the pilot episode of Sliders, the main character leaves his blackboard covered in equations, not knowing what to write after the equals sign. When he comes back, his double has solved it, and the expression he has written as an infinity symbol on the denominator. That's right. He just needed to divide by infinity.
One of the clips in the Star Trek: Enterprise opener shows a black and white image of a scientist writing complex equations across a blackboard. It looks very fancy until the glaring mathematical error.
In his original pitch for Star Trek, Gene Roddenberry wanted to use the Drake equation to demonstrate how likely it was we'd encounter aliens. Unfortunately, he couldn't remember the Drake equation, so he just made something up (including two variables being raised to the first power). In the Star Trek: Voyager episode "Future's End" Rain Robinson's work station has both equations on the wall.
The original series episode "Court-Martial" refers to amplifying the sound of a heartbeat by a factor of "one to the fourth power"... which is one, meaning no amplification (or attenuation) at all.
In LOST Seasons 4 & 5 you get both Daniel Faraday (4) & his mother (5) just constantly writing jibberish on the chalkboard while conducting experiments.
Terrifyingly, the DVD shows that a real professor of physics wrote that gibberish up for them.
A sketch in series 1 episode 3 of A Bit of Fry and Laurie involves a "hilarious blooper" from a 1970s Open University programme, in which Hugh Laurie's presenter talks us through a blackboard full of nonsensical integral-like equations. What's more, the flub involves the equation giving a "resultant modular quantity" of 0.567395, rather than 0.567359 as Laurie's character states, when the integral in question has no limits and wouldn't have a numerical solution, not to mention that the programme appears to be a physics lecture about wave theory, which wouldn't involve numbers that precise as solutions anyway. ... Hmm, come to think of it, that's a hilarious blooper in itself! I don't believe it! Ha ha ha ha!
In an Imagine Spot, Everybody Hates Chris has a Sadist Teacher demand to know, giving no context whatsoever, "What is a^{2} plus b^{2}?!?". The correct answer is apparently "c^{2}". In other words, the protagonist has internalized the most common expression of the Pythagorean theorem, but with no clue of what it actually means.
NUMB3RS gets a lot of credit for having both correct (syntactically) and relevant (in the context of the episode) math equations. Generally, it does — when the equations are displayed in the foreground. Whenever we see one of the fancy animations while Charlie explains something, though, expect at least half of the math in the background to be random and irrelevant. It gets to the point where even people who know nothing about math should be able to figure out that the equations are meaningless, because they appear even when the explanation Charlie is giving is not mathematical in nature (like the infamous IRC explanation, which doubles as being wholly inaccurate itself).
Spoofed in an episode of Mythbusters. Grant sketched out a complex but valid mathematical formula on the side of a test vehicle. After Grant finished the calculations and announced the result, this exchange occurred:
Rimmer does this to himself in Red Dwarf when he covers his arms and legs in astrophysics equations that are beyond his capacity to understand to cheat on a test he is clearly not qualified to pass.
Rimmer: Right, they're bound to ask the right thigh, which is 10 per cent. They must ask the left thigh, which is 20 per cent. They've got to ask one of the forearms. Which means I've passed already; anything on the left shin's a bonus! Right. *Looks at one arm* `CUTIE'. Current under tension is ... what's this? Current under tension is equal? Current under tension is expendable? Current under tension is expensive? What does this mean? *Begins to panic* What does any of it mean? I've covered my body in complete and utter and total absolute nonsense gibberish!
In That '70s Show, Eric's parents see fit to check that he's not neglecting school, so Red asks him sternly, "What is X?". Eric just stares at him, then very politely explains that it's a variable that can represent a given number. The parents think it sounds right, but neither of them actually know what answer they were expecting.
Music
Mariah Carey went so far as to name her most recent album E=MC^{2}. She claims it stands for "(E) Emancipation (=) equals (MC) Mariah Carey to the second power." Yeah right.
"Weird Al" Yankovic's video "White and Nerdy". Just to show how smart Alfred is. Ironically, it contains an error.
Alas, in the song "Pancreas" he gets Newton's equation for the gravitational force between two masses wrong.
Pinball
To track what ingredients are needed in the Gilligan's Island pinball to make Volcano Seltzer, the Professor's formula is shown iconically on the playfield, with nonsense like "...[seashell] +/- (mix well and add) [rope] - ( [shrunken head] < [turtle eggs] ) / [banana]..."
Video Games
The first few levels of Super Paper Mario feature "joke" equations in the background, made up of random numbers and mathematical symbols combined with famous Mario icons such as the Fire Flower and mushrooms.
In EarthBound, Doctor Andonuts has a big chalkboard in his lab with nothing written on it but a big "E=mc^{2}".
Half-Life has a number of blackboards adorned with Newton's equation for gravity. That's like Ernest Hemingway trying to come to grips with "Hop On Pop".
This is lampshaded and blasted to bits in Freeman's Mind.
In Portal 2, there are posters around that list paradoxes in the event of a rogue AI. One of the listed paradoxes is "Does a set of all sets contain itself?" ^{note }This is clearly based on Russell's Paradox: "Does a set of all sets that do not contain themselves contain itself?" The way it's written, though, it's a nonstarter. The "set" of Russell's Paradox is a subset of this "set", and thus, the "set" described isn't a set.^{note }Although it does actually function as a version of the Liar Paradox. The Obvious Rule Patch for Russell's Paradox is that a set cannot contain itself. Thus the set of all sets does not contain itself, and is not the set of all sets.
Knowing the dev team's sense of humor, it might be deliberate.
While chalkboards in Team Fortress 2 do sport real equations (including the ridiculously complicated equation they use for rendering rim lights), it also boasts "equations" like "FIRE = QQ".
Thisxkcd strip includes an integral without a variable of integration, as part of a clearly erroneous equation — but in this case the error is integral to the punchline.
The variable of integration is obvious, since there's only one variable; the bigger issue is somehow getting a constant out of an indefinite integral (this would only happen if you nonsensically integrated zero, and even then you wouldn't get a specific constant because the derivative of any constant is zero). But this is also part of the joke, and the Alt Text takes it further: "It's pi plus C, of course."
A more elaborate variant appears in this cartoon by Tom Tomorrow. Instead of meaningless numerical symbols, it's an invalid proof that involves dividing by zero, perhaps to complement the scientist's fallacious reasoning.
The invalidity of the proof is probably intentional, but it still appears to be copied from an online source by someone who doesn't understand mathematical notation. It writes "a^2" instead of "a^{2}" (the former notation is a substitute that is only ever used when technical constrains make it impossible to write the latter, which is not the case for blackboards).
In Two Lumps, Eben is occasionally shown contemplating equations right before Snooch interrupts with his latest insanity. In at least one instance readers have submitted corrections for the math displayed.
In Lovecraft Is Missing, the characters fill in blackboards translating to mathematical concepts both music composition and the works of H.P. Lovecraft - apparently old Howard hid some spells in tales. While translating, there's something of an example of something that could be Misplaced a Decimal Point or Carry the One: "Aw, now that don't make no kinda sense... Wait a minute. Yes it does. I was flat out wrong. I thought I made a misake when I hadn't."
Web Original
The title card for The Angry Video Game Nerd video Chronologically Confused features the Nerd in front of a board filled with nonsensical equations and formulae, including at points a Triforce and a drawing of Mario.
Episode 1 of Echo Chamber has Tom and Dana (and possibly Zack) trying to figure out how to "show a Show Within a Show" and covering an entire blackboard with ideas, including nonsensical chemical and mathematical formulae that have little to do with...anything they are trying to discuss.
The blog Blackboards in Porn is devoted to pointing out examples of this trope in pornography.
Manatee Girl The Movie has "science" on the board behind Hunky Marine Biologist Boyfriend. The "science" in question involves a picture of a crudely drawn fish being added to numbers and EC equals MC squared randomly on the side.
Fillmore! has an episode set around a maths test. When the teacher gives her class an assignment, it reads: 2+x=5, x-y= 16, -2x-y=-14, 2x+4y=12, -2x+ 5y=5. She asks them to solve both x and y, and tells them to spend the rest of the lesson doing so. This amount of time would make sense, as it's actually impossible—the value of x and y changes in those equations from one example to the next.
And what is worse, the moral of the story is that "the value of x remains constant". Arghh!
The intro to Pinky and the Brain has a scene where Brain is writing his "theory of everything" on a chalkboard, which is basically a bunch of pseudo-mathematical mumbo-jumbo, including "THX=1138".
Spoofed in another episode which has Brain reveal that his latest plan to take over the world was hinged on an equation he had just uncovered. Pinky askes him if it is something complicated like E=mc^{2}, and Brain replies that it is in fact even simpler, just E.
Monkey Dust has a chalkboard full of geometry in a class teaching cottaging! (For non-UK readers, that's anonymous gay sex in public bathrooms). True to the title of this trope, the board also contains "run=dmc".
The Simpsons is fond of this one. When Homer has a bright idea, the camera would occasionally do a close-up of his head, revealing two chimpanzees in graduation gowns and mortarboards writing E=mc^{2} on a chalkboard. Otherwise, the chimps would be grooming each other and eating the lice.
Parodied when Homer is an inventor. During a montage, he's shown writing equations on a blackboard. After he's done, the camera moves to shot of the house—where there's a massive explosion. Cut back to Homer: who examines his equation and crosses out the offending section, a drawing of a stick of dynamite, which he then replaces with something else. This results in another, bigger explosion.
Also done in Phineas and Ferb. The boys finally solved it by replacing the bomb with a smiley face.
And subverted in another episode: Homer puts on some Nerd Glasses he finds in the power plant's toilets and immediately recites what he seems to think is the Pythagorean Theorem (he's actually quoting The Wizard of Oz movie):
Homer: The sum of the square roots of any two sides of an isoceles triangle is equal to the square root of the remaining side! Co-worker in a stall: That's a right triangle, y'idiot! Homer: D'oh!
Note that the co-worker's correction is nowhere near complete, becase the formula refers to the squares, nor the square roots, and "the remaining side" has to be the hypotenuse. But that wouldn't be as funny.
Variation, in the episode in which the Simpsons discover the third dimension, "P=NP" can be read in the 3d-CGI world.
Also parodied in "Moneybart":
Nelson: Lisa's a genius! She can do the kind of math that has letters! Watch: what does X equal, Lisa? Lisa: Well, that depends... Nelson: She could do it yesterday, I swear!
The board in Arthur's class would often be filled with formulas like the Pythagorean theorem or simple derivatives, even though the characters are respectively three and six years too young for them. Apparently, this is part of Mr. Ratburn's Sadist Teacher reputation.
In Rocko's Modern Life in an episode where Rocko keeps being late for work. While he gets berated for it by his boss, there's a flipchart in the background that contains, amongst other things, a clipart of a clock followed by a plus-minus sign and a square root containing "MC" and a clipart of a hammer.
In Code Lyoko, the same board (y=ax+b) is used for every science course. Including one about parthenogenesis.
In Robot Chicken, they parody this trope in a sketch. Where two scientists are laughing their heads off to a bizarre problem on the board, only for one of them to stop, erase a symbol, add another, and then both burst out laughing again.
One episode of The New Adventures of Winnie the Pooh centered on Gopher's "equation for the ultimate tunnel." How and why the layout for a tunnel took the form of a chalkboard full of mathematical-looking scrawlings was never explained.
Real Life
Even children's math workbooks have this problem. Some of them that go from K-12 will have the same picture with pi, percents, and algebra symbols on every book regardless of if it is a 11th grade (age 16-17) or 3rd grade (age 8-9) book.
This extends even to the university level. The cover of late-undergraduate/early-graduate level "Introduction to Lie Algebras" from the Springer series is adorned with many generic calculus and linear algebra equations not particularly related to the topic at hand. Subverted in that the equations, while complicated and arguably "pretty", should not be mysterious to anyone capable of understanding the contents of the book.
Speaking about Truth in Television - imagine what goes on when TV people come and ask for "some formulas" as a background for an interview with someone related to maths/physics education (and their request is granted just to please them).
Exceptions
General
Clausius-Clapeyron and the derivation via the Maxwell relations is extremely common, usually as Mad Science. It's easy to copy verbatim, and looks scary because it's an important equation in Thermodynamics and requires high level chemistry, physics and mathematics to understand.
Anime & Manga
The opening credits of The Melancholy Of Haruhi Suzumiya feature a number of diagrams and equations, including the obligatory E=mc^{2}, the Drake Equation and Shannon's source entropy formula, appropriately enough (as well as plenty most people have never heard of). In "Snow Mountain Syndrome" of the later novels, though, solving an instance of Euler's planar graph formula becomes a matter of great importance, so this is to be expected. That and the guy that writes the Light Novel likes math.
In a chapter of Pluto by Naoki Urasawa, we see Atom to write on a wall a huge quantity of equations. While they are probably correct (the notation is correct, for instance and any single expression makes sense), it is not probable that they are "the formula of the anti-proton bomb": some are mathematical definitions, or Fourier tranforms, other appears to be basic equations of quantum physics, but surely not a project of bomb.
Subverted in a Crowning Moment Of Awesome in Tobaku Datenroku Zero. Near the end of a psychotic quiz game in which a super-sharp pendulum is lowered when the answer is wrong, Ukai Zero is answering a trivia question about the period of a pendulum. He's shown to be under pressure, thinking up a bunch of random equations which have nothing to do with the relevant speeds of rotation. And he gets the question wrong. However, it turns out that all those equations were him using the measurements of his body parts to ascertain that, with his next incorrect answer, the anchor would crash into the block his head was resting on, effectively winning the game. The MC was not happy.
Not actually an equation, but the blackboard in computer science class in Serial Experiments Lain shows an entirely plausible and workable (partial) code for a Conway's Game of Life implementation. It's a bit too straightforward and not very efficient, but then, it is a teaching example. What's interesting here is that the code is written in Common Lisp — a somewhat obscure programming language, but one with a long history of an intellectual superiority. Which is entirely in tune with the show's Viewers Are Geniuses approach.
Comics
One strip of the newspaper comic Foxtrot has Insufferable Genius Jason Fox, a ten-year-old, solving a simple division problem on his class chalkboard. He rewrites the equation in about a dozen different ways, many of which would require knowledge of trigonometry or calculus to understand. Given that the cartoonist Bill Amend majored in physics in college, all of them are correct.
Another strip has Jason serving as Paige's math tutor and the latter asking him what is the cosine of 60 degrees. Jason then starts rattling off a really long sum and only stops when Paige reminds him she's not paying him by the hour. Said sum is the actual Taylor series expansion of the cosine meaning of course he could continue going forever. The answer she was looking for is 0.5.
Yet another strip has Jason presenting Paige with an alphanumeric cipher with a twist: the key is comprised of 26 math problems, one for each letter of the alphabet. One of the clues involves integral calculus (but of course it's for Q, which doesn't get used much). The answer is "PAIGE FOX IS BAD AT MATH".
And another strip shows Jason doing a problem in class where he has to calculate the area of a farm enclosed by a fence of some length and width. Naturally, he draws out a coordinate plane for the farm and does an integration to find the area under the curve. That's definitely the method you all would choose, right?
Then there's the strip showing Peter holding several sheets of paper covered in equations, and a diagram of a catapult. The purpose? Allow Jason and Marcus to fly into Paige's room with squirt guns.
Peter: From such smarts, such stupidity.
Films — Live-Action
A very common mockery thrown at the movie A Beautiful Mind is that in promo posters for the movie, Russell Crowe as John Nash is sitting making a thinking face behind a glass wall covered in equations. Right on his forehead is the statement "0 < pi < 1". Geeks had a lot of fun mocking the apparent total lack of understanding of math in Hollywood and coming up for ways this statement might be justified (often speculating on the shape of Russell Crowe's penis). In reality, there was a perfectly good explanation, in fact: The math consultant for the movie had been asked to help prepare the shoot for this poster by covering the glass wall with impressive-looking equations, and for "authenticity's" sake had done so using real equations from the real John Nash's papers. John Nash was a game theorist, and he had written a paper involving an imaginary game with 24 players. Since there are 24 letters in the Greek alphabet, he playfully decided to name the imaginary players after Greek letters. The "0 < pi < 1" statement comes from a portion of the paper describing an imaginary situation involving placement of turn order within the game. Notice that Nash was specifically doing this in order to undermine mathematicians insistence on using symbols to mean just one thing.
Also note that symbols such as pi have by now been used in pretty much every subject of math and physics. Often in quantum field theory textbooks, for example, there are equations with pi taken to mean momentum density expressed as a formula that includes pi (the number). Surprisingly, people are never confused. The context is sufficient to eliminate any doubt — which is exactly why viewers of A Beautiful Mind were confused: the statement was presented without a context.
Averted in the romantic comedy It's My Turn, which begins with a proof of the Snake Lemma. In fact Weibel's An Introduction to Homological Algebra refuses to print a proof, referring the readers instead to the first scene of this move.
Literature
Robert A. Heinlein was meticulous about everything he ever wrote about. While much of his work has been victim to Science Marches On, he would often spend whole chapters explaining why something does or doesn't work in his future technologies. One of his most mindboggling and in depth explanations (for a layman) was the short story And He Built a Crooked House, where he actively explains how to create a tesseract house in four dimensions. He mentions in his non-fiction writing that when he was doing his earliest juvenile novels - mostly wham-bang space opera involving teenage boys fighting Nazis on the Moon and such - he and his wife Virginia still made complete orbital calculations for the spacecraft, working longhand on rolls of butcher's paper. He did make some errors in the end: In Starman Jones, he mentions that one planet had a moon, thereby making it possible to measure the planet's mass. He forgot that the spacecraft itself would serve (as the equivalent of the moon) to measure the mass. Furthermore, measuring the mass of any planet would be an essential part of traveling through the star system.
Live-Action TV
Considering that the show NUMB3RS has, well, numbers as its unique gimmick, it would be pretty insulting if this were the case. They keep a mathematician on staff who writes all of the equations seen.
For the physics class scenes in 3rd Rock From The Sun, much of Dick's dialogue was written by Elegant Universe author Dr. Brian Greene.
Seriously averted with The Big Bang Theory, who have a theoretical physics professor on-staff at all times. Not only are all of the equations shown in the programme entirely correct (and used in the correct contexts to boot), but some of the ones that are shown (especially displayed in the background on Sheldon and Leonard's whiteboards in their apartment) are actual assignments that the university professor sets to his physics students the week before. They can actually watch the show and learn at the same time!
An episode of Stargate SG-1 had Major Carter guest-lecturing Air Force cadets on a multiple-universe theory, in front of the "chalk board full of formulae" version of this trope. According to the episode's commentary, the mathematics on the board were real. Carter made an original mistake as well — she accidentally inverted one of the ratios.
Only time in the entire series she screws up math.
A cadet actually walks up and stares at the board before pointing out the mistake, only to be chastised by the professor for correcting a scholar. Carter then re-examines the problem and also sees the mistake, noting that it completely changes the problem. She is then very interested in the cadet for being smart enough to spot it.
Averted in Kyle XY. There's a real math problem there, which is typically supposed to take months to solve (looked it up, but can't remember).
Not so certain about his solution, though.
University of Oklahoma chemistry professor Donna Nelson serves as the science consultant for Breaking Bad, so the chemistry Walter teaches his high school class is accurate. They intentionally change some of the steps for making methamphetamine so that viewers don't learn how to make meth from the show, and change the color of the meth in order to make it more visually interesting.
On a similar note, in Portal 2's cooperative mode, some whiteboards have Fourier transforms written on them. This would be quite useful for a lab that screws with physics as much as Aperture does.
And early in the single-player game, a wall in one of Ratman's Rooms Full of Crazy shows a cat jumping (or perhaps tunneling? {probably tunneling in a reference to the phenomenon of quantum tunneling}) out of a box, along with various Quantum Physics equations, a reference to the Schrödinger's Cat thought experiment that was played with in the tie-in Lab Rat comic.
Averted in Black Mesa, the long-developed Source-based Fan Remake of Half-Life, where whiteboards in the offices and labs have actual believable notes, diagrams, calculus equations, chemical formulas, etc. Hard to say how accurate they are, but the point is, they look real.
One splash screen in the "bodychecking hoverboarders" game Pararena has the quite relevant "F = ma" hidden upside-down in the corner.
The opening animation and title screen of Game Theory features a complicated equation that somehow involves Mario jumping over a Goomba.
Western Animation
Futurama has quite the opposite. Several of the writers are math people, so there are math jokes in the background that laymen won't even notice, like the number 1729 (yes, that's a joke).
They also parody and subvert on occasion too. For example, in the first season episode "Mars University" Fry signs up for the Professor's course, entitled "The Mathematics of Quantum Neutrino Fields" which Fry mishears as "Wonton Burrito Meals". When Fry finally turns up to the class he finds the Professor teaching complete nonsense, using an illustration named Witten's Dog to explain why electrons taste like grapeade (a parody of Schrödinger's cat). This is especially amusing when you consider that earlier in the episode the Professor himself refers to genetic engineering as "preposterous science-fiction mumbo-jumbo".
The Professor also claimed that the name itself was an E=MC Hammer; he just made up some imposing-sounding nonsense so that no one would take his class, because he's a professor and doesn't know how to teach, and indeed Fry's motivation for taking the course is apparently because Professor Farnsworth is teaching it.
Bender has a box of P and a box of nP in his closet.
When Bender first meets Flexo, they find it greatly amusing that both of them have serial numbers expressible as a sum of two cubes (like 1729 above).
In one episode, Bender is terrified when the number 666 mysteriously appears on the wall... in binary.
But since it's written backwards, not until he sees it in a mirror.
Bender also freaks out over a dream where he thought he saw the number 2. Fortunately, Fry is right there to reassure Bender there's no such thing as 2.
Loew's Aleph-naught-Plex. That is all.
One of the most awesome(ly geeky) parts of the entire series is a half hour extra on the first movie's dvd, where the writing crew bring in a mathematician to explain some of the math jokes in the series to the viewer.
In the new episode where Bender replicates into grey goo, the professor puts up an equation for the total mass of the Benders as the successive generations replicate, and all the employees (except Fry, of course) understand it and gasp because it doesn't converge. Being Futurama, this equation actually IS a mathematical representation of the infinite series at hand, which grows indefinitely rather than converges.
In the new (sixth) season, in the body-swapping episode, when Bubblegum Tate has worked out a solution to their body-swapping problem, there is actually a correct and reasonably rigorous proof on their floating holographic blackboard.
The entire proof can be found on the wiki. The writer who wrote the Theorem has a Ph.D in mathematics.
The PJs actually used the trope title in-context. When Thurgood is attempting to prepare to go back to school, a neighbor helps him catch up on his science. The neighbor writes "E=MC___" on the board, and Thurgood subsequently writes in "Hammer". After a second attempt, he writes in "Nuggets" (geddit?)
In the Teen Titans episode "Stranded", when Cyborg is explaining to Beast Boy how to fix the crashed T-ship a string of mathematical symbols emerges from his mouth — the point being that Beast Boy does not understand science. But the symbols themselves are in fact a mathematically accurate series expansion.
The blackboard◊ in Cheerilee's classroom features an equation that, while meaningless, resembles those seen in an undergraduate-level course on electromagnetics. Odd, considering that everything else points to the students being more like elementary school kids.
In "It's About Time", Twilight writes the formulae for time dilation on her blackboard (well, mostly; she crashes and burns computing the integral, but at least they're valid mathematical expressions). Keep in mind that at that point, Twilight still hadn't figured out how it worked yet.
On Animaniacs, Einstein is inspired when Yakko, Wakko and Dot sing the ACME Song and Wakko write it backwards, with the "a" resembling a "2", resulting in Emc2. Al merely added an "=" between the "E" and "m".
In the Regular Show episode Video 101, Eileen is seen solving a complex-looking problem in advanced calculus. The problem is a triple integral that converts from cartesian coordinates to cylindrical coordinates, exactly the kind of problem you'd see in a Calculus III class.
Real Life
Nobel Prize winning physicist Leon Lederman speculated in The God Particle that physicists in trouble could write the number 137 on a sign and expect other physicists to come to their assistance. The reason: 137 is (very close to) the reciprocal of alpha, the fine structure constant, and one of the most arbitrary-seeming constants in physics.^{note }This is because alpha is "dimensionless" which means it comes out the same no matter which units of measure you use. A simple example is the ratio between the electrical charge of an electron and that of a down-quark, which is exactly 3:1 no matter which units you use for measuring charge. Similarly, alpha is the square of the ratio between the charge of an electron and the "Planck charge" (which is a property of free space). However, whereas it is easy to assume that there is some simple mathematical reason for the ratio 3:1 — even without knowing what that reason is — it is appreciably harder to imagine how the pure mathematical structure of the Theory-Of-Everything could force 1/alpha to be exactly 137-and-a-bit. Many have lost their way trying to prove it was exactly 137 before it was measured more accurately...