"I became a comedian because I was the only Asian in my high school... that failed math."
— Dat Phan, Vietnamese stand-up comedian.
Pretty much any South or East Asian character in western media who isn't a martial artist or a gangster will be highly intelligent or unusually geeky... if not both. If done clearly and more on the intelligent side than geeky, this will be an exaggeration of the Brainy Brunette.
The polar opposite of this, becoming more common in recent years, is the Asian Airhead — an almost Always Female character archetype centering on an Asian girl who is gorgeous, popular, thoroughly Americanized and dumb as a brick.
The reasoning behind this stereotype is that most Asian cultures regard education and smarts as very important things. Parents passed down these beliefs to their kids over generations, and eventually it became this trope.
See also Bollywood Nerd and Jewish and Nerdy. The Other Wiki has an article on the "model minority" stereotype from which this trope is derived. Contrast All Asians Know Martial Arts, as most Asian and Nerdy characters lack the athletic ability for martial arts. For related tropes, see Nerd and Black and Nerdy.
Examples:
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Anime and Manga
As anime and manga are Asian-dominated mediums, their Asian characters are likely to be less stereotypical than those found in West-dominated media. They'll still have nerdy characters, but they will be mixed in with more non-nerdy Asian characters.
Bruce (Masi Oka's character) in the 2008 Get Smart movie. Of course, he was flanked by two geeky white men: Lloyd and Max himself. Also, in the direct-to-video spin off Bruce and Lloyd: Out of Control, Bruce is apparently more successful with the ladies than Lloyd.
The 1981 film The Cannonball Run featured a pair of racers of indeterminate Asian origin (one of whom was played by Jackie Chan!). Their car is a 1980-81 Subaru hatchback modified to a James Bond-esque gizmo-mobile which even turns into a submersible at one point. The humor is pretty gentle for 1981, although their high-speed, unsubtitled dialogue and exaggerated facial expressions are definitely unfortunate. Something of a Reverse Funny Aneurysm considering Subaru's later rallying success. The only thing missing is that the movie car is black instead of blue-and-yellow.
Sixteen Candles. Long Duk Dong - though it's subverted by the time the film is over, as he's gone into Full Wacky Pseudo-Stoner Mode.
Also, Lau in The Dark Knight was hired by the mobs to launder their money because he was "good at calculations". This, with a few other examples above, grates on the nerves of Shirt Guy Dom. In particular, he was an accountant who lived in a country where he couldn't really be touched by the US. Still fits this trope, but it's justified.
The "Asian Nerds" in Mean Girls. Averted by the "Cool Asians."
Subverted in 21 and Over. At first, we're led to believe that Jeff Chang is a straight-A student who tutors people in science. We later find out that he's on the verge of failing out of college and is being tutored in science.
Literature
Aversion: Claudia Kishi of The Baby Sitters Club: artsy but terrible in all her other subjects. Her sister definitely qualifies, though.
Matt Scudder's occasional hacker-for-hire Jimmy Hong.
This was double subverted in the weirdest way possible in Degrassi Junior High, where Yick Yu, a dorky bespectacled Asian kid, had to work extra hard in class to break the stereotype (that apparently exists in the Degrassi world) of Asians as dimwitted jocks. Uh, what?
Originally to be played straight, but the actor complained.
Arguably, Grant Imahara of Mythbusters plays up to this stereotype. He builds robots for fun and in the jailhouse rope episode had pi on his "prisoner" outfit as a "prisoner number". At least he's not a stereotypical bespectacled dork; he's very handsome (see picture above). When they tested lie detectors, Grant was asked if he had ever thought of building a robot girlfriend. He said "no", and the machine flagged it as a lie.
If you need any additional evidence, in this picture, not only is he dressed as Spock in that picture, but when he was inducted into the 501st Legion (think Civil War reenactors, only with Stormtroopers) he attended his ceremony dressed as the Tenth Doctor.
The 2009 live-action Scooby-DooContinuity Reboot has cast a Japanese-American actress as Velma. However there was no effort to change the character into an Asian, and furthermore, when she's in the Velma costume, you wouldn't know she's Asian.◊
While she's thoroughly Americanized and acts like an Asian Airhead, Zoe from the short lived Knight Rider 2008 series speaks 9 languages and is a genius at computers.
Evelyn Kwong from Ned's Declassified School Survival Guide is the smartest person at James K. Polk besides Cookie, leading to a lot of competition. She subverts a lot of the accompanying tropes that go with this though, she's competitive, rude, loud, mean and a sometimes Stalker with a Crush.
Vince Masuka of Dexter. However, he subverts some of the stereotypes by showing interest in and, occasionally, success with women.
And by not being very Asian, except in appearance.
The Amazing Race prefers to cast this type of Asian team. They generally speak Chinese, too.
Lt. Tao of The Closer is probably second only to Imahara in how incredibly this trope he is on contemporary TV.
Averted in the Disney Channel movie Wendy Wu: Homecoming Warrior. Wendy herself averts this trope, but because of the film's title falls into the stereotype of martial artist. Her brother, however, averts both stereotypes as a popular and somewhat dim jock.
In Community, the school's Math Club appears to be comprised solely of this trope, as lampshaded when Jeff and Chang are pinned down by them during a paintball tournament:
Chang: Relax. I have a plan. [louder] Hey, Math Club! I'm Asian! Are you guys Asian? Math Club Member: ... That's pretty racist, dude. Chang: That wasn't a no! I'm coming over!
Played with in regards to Trini in Mighty Morphin' Power Rangers as she is shown to be a good student though the biggest nerd in the group is Billy - she was his Translator Buddy early in the series. She's more like a combination of Billy and Valley Girl Kimberly.
Glenn from both iterations of the The Walking Dead is a subversion, possibly a slight Deconstruction of most of the older Asian stereotypes. Before the outbreak, he was very far from the innocent, quietly nerdy Asian archetype. In fact, he never spoke to his parents, and was so far in debt from college and other purposes, he had to resort to car theft with his friends, and various low-paying jobs such as pizza delivery to make ends meet. After the outbreak, he quickly becomes the Guile Hero of his survivor group, as a comparatively rational thinker with a preference for sneaking around the walkers.
Dr Chi Park, third-generation team member on House, is an over-achieving meganekko. In a twist of events, she's asked resident Chick Magnet Robert Chase out on a date, and he's said yes.
In the short-lived Some of My Best Friends, Camp Gay Vern attempts to fix a broken TV. Another character dubiously asks if he can really fix it, and he responds, "Hello! I'm Asian!" Turns out he has no clue.
Kevin Tran in Supernatural. In his first episode, he's introduced playing the cello in his bedroom (which is filled with honors and awards). When his phone starts ringing, he doesn't pick up until a timed alarm on his computer tells him practice time is over, at which point he starts talking to his nearly-as-nerdy girlfriend. Then he gets hit by a celestial lightning bolt and appointed translator of the Word of God. Of all the things to happen on exam day...
Suki from Tower Prep. Justified in that the series takes place at a school for the gifted.
New Media
I Love Bees has nerdy medical student Hiroyuki, roommate of the Middle Eastern and Nerdy protagonist Kamal.
Newspaper Comics
Asok the Intern in Dilbert. Scott Adams has said that if he keeps getting complaints about stereotyping, he'll make Asok a drug dealer.
Chinese stand-up comedian Joe Wong, with his slight frame, wire-rimmed glasses and strong accent reinforcing his background as a chemical engineer who graduated from two universities in China before furthering his studies at Rice University in Texas and is never afraid to portray himself as a Funny Foreigner in his routines. Some of his material deals with the perception of all Asians as this trope.
Vietnamese-American comedian Dat Phan, as evidenced by the page quote.
Video Games
Jenny from My Sims has an Asian appearance. Her Interests are Geeky and Studious. More specifically, she "is president of five science fiction clubs, seven video game clans, and a prolific fan fiction author."
Mei Ling from the Metal Gear series is a Chinese-American data analyst going to MIT at the time Metal Gear Solid game takes place.
The entire point of the song 'Nobody's Asian in the movies' by Maurissa Tancharoen from the Dr. Horrible's Sing-Along Blog musical commentary, Commentary! The Musical. The song ends with "I'm gonna go play with my violin and... math!"
Possibly a jokey jab at the lead actress of Dr Horrible, Felicia Day, who is not Asian but was a violin performance/math double major.
One article from The Onion interviewed a pair of Asian students who, due to lazy and stupid behavior, were praised for breaking stereotypes.
Generator (Jade Sinclair) of the Whateley Universe is the Crazy Awesome variant of this. She once invented real, working shoulder angels. The campus nearly didn't survive.
Toshi, one of Steve's friends in American Dad, is a highly intelligent Asian. Who talks in Japanese all the time and wants to kill, humiliate or mutilate Steve, usually.
Averted by Jerk Jock Vince Chung, who is "so cool that you forget he's Asian."
Averted in (of all things) Danny Phantom, where the most visible Asian student is a friendly, but dimwitted jock. Same in The Spectacular Spider Man, with Kenny "King" Kong.
It should be noted that this is a Race Lift. In the Ultimate Universe (he doesn't exist in the main Marvel verse) Kong is very much Caucasian.
Averted in W.I.T.C.H. with Hay Lin, who is an artist and a bit of a scatterbrain. But also shown with nerd of the team being Taranee who is half-Asian.
Averted in 6teen with the character of Nikki Wong, who, while certainly not stupid, is somewhat lazy, and also happens to be a Snark Knight.
Futurama: "A Big Piece Of Garbage". Professor Wernstrum requests a group of interns, and that most of them be Chinese. Likely because it makes him look smarter (or so it has been suggested by tie-in comics) Subverted by Amy Wong and family, who are either dense or clumsy, or both.
Though Amy seems to be more of a Genius Ditz than a straight-up Asian Airhead, now that she's finally got her degree. Her research even managed to doom and save the Earth!
There's a minor character in South Park named Kevin Stoley. He's a Chinese-American and is quite a Star Wars nerd. Barely more than a Living Prop, we don't know his intellect though.
There's also the rather weird fact that other than explicitly saying he's Chinese-American, nothing else implies this, including surname or appearance. It's not like South Park is usually subtle with this sort of thing...
Frequently an Enforced Trope (albeit unintentionally), as 80% of the scholarships for Asians are for those wishing to become doctors, lawyers, or scientists. The 20% of scholarships for ANY other field are taken so quickly that some people think all Asian scholarships are for those three fields.
Because of this stereotype, discrimination against Asians is very prevalent in higher education as they are accused of taking up enrollment slots since they are too smart for others to compete. Infamously, statistics have shown that Ivy League schools in the US have implicitly capped the percentage of Asian enrollment and Maclean's magazine in Canada had their "Too Asian?" article which received major backlash across the country for reporting that Asians take away university culture from white students because they are overly focused on their studies. This is part of a wider Double Standard in which a white person's success will be attributed to a strong work ethic while similar successes coming from an Asian person will be attributed to them being overworked in some way (e.g. the Education Mama, an oppressive state training regime, an "overachiever" personality).
South Koreans, Japanese and more recently Chinese and Taiwanese are often assumed to be at least one step higher on the competitive gaming ladder than any Western gamer. The common explanation is that they apply their traditional determination, dedication and competitive nature to videogames as well as (or in lieu of) Real Life. As with other generalizations, however, the truth of this belief varies from person to person.
On the flip side, the general social trends of Eastern cultures which emphasize the group over the individual tends to result in typically more successful groups. In the space of modern genres like the MMO, this attitude can prove fairly dominating.
Dr. David Oh, flight director of NASA's Curiosity Mars rover, has to stay on Mars time (24h40min days) for the job. So his entire family went on Mars time and his son is keeping a blog about it.
One of the roots of East Asia's Education Mama culture was the Chinese Imperial Examination, an extremely difficult test to obtain a prestigious and well-paid government position. Only five percent of test-takers passed.
Japan tends to be extremely fond of the Game of Nerds. Individual people vary, of course, but as a whole the nation is surprisingly into the Western sport.