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Formulaic Magic

"Using the power of Mathemagic, you can multiply power, subtract hope, and cuberoot the dreams of man himself."
Modern science loves explaining complex physical phenomena through comparatively simplelooking mathematical abstractions. Motion of an object? F = ma. Energy content of a mass? E = mc^{2}. State of a gas? pV = nRT. Exceptions to these rules are themselves often explainable through math — more intimidating calculations, of course, but still math. These equations are only useful, however, because they accurately describe physical interactions in nature; without this connection, they're just meaningless jumbles of symbols.
Authors often neglect to establish this link between math and reality, instead treating equations as if they possess a power all on their own. This often manifests in two ways:
Such "magical math" (or "mathemagic" or " Mathamancy") doesn't have to be connected to any aspect of reality, so it will often be rather nonsensical as a result. In this way, such math is somewhat related to the Numerological Motif, in which numbers get intrinsic properties.
On the flip side, writers will occasionally make systems of magic describable through math, much like their physical science counterparts. This use of math is more readily justified, though usually still not explained in sufficient detail to make mathematical sense.
This may be Functional Magic under rule magic. Arguably Truth in Television if you consider science to be a form of magic, and that its formulas are being applied to create technology, thus hacking the universe and having magiclike powers over things.
Examples
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Anime and Manga
 Witches in Soul Eater use "calculation magic" to make their spells more effective.
 It's revealed in the supplementary manga of Lyrical Nanoha that constructing the Instant Runes needed for magic in that setting requires a very good grasp of math. In fact, one of the reasons why Nanoha and Fate are such powerful mages is because they're both math geniuses.
 The reality warping abilities of human interfaces in Haruhi Suzumiya seem to work like this.
 Grimoires and parchments in A Certain Magical Index seem to contain complex formulas which need to be deciphered in order for it to work — kind of like math books, except those don't attempt to destroy their user.
 Part 7 of Jojos Bizarre Adventure takes geometry very seriously. A short 2 page lesson on the golden ratio gives the protagonist the power to create miniature wormholes.
Fan Fics
 Magic in the version of the Nanoha multiverse the fanfic The New Math is set in works via complex math equations. The most powerful mages are also really, really good at math.
Comic Books
 Multiple times in The DCU:
 The AntiLife Equation is usable as a method of Mass Hypnosis. It runs Loneliness + Alienation + Fear + Despair + Selfworth ÷ Mockery ÷ Condemnation ÷ Misunderstanding x Guilt x Shame x Failure x Judgment, n=y where y=Hope and n=Folly, Love=Lies, Life=Death, Self=DARKSEID.
 The Pre Crisis DCU Round Robin maxiseries DC Challenge had a series of numbers in the first issue (written by Mark Evanier) that were somehow important — but none of the other writers could ever really figure out how. Several of them later used the numbers in formulas for this, that, and the other. The secret? Add them up on a calculator and turn the calculator upside down; it spelled out the name "ELI ELLIS."
 Golden Age superhero Johnny Quick accessed his superspeed powers by reciting the mathematical formula "3X2(9YZ)4A". More recent materials have retconned this as a mantra that allows him to tap into the mystical Speed Force.
 One member of the Green Lantern Corps is a sentient mathematical equation, and so is his ring.
 In the Marvel Universe, Amadeus Cho (the seventhsmartest person on the planet) sees the world as mathematical formulas. The Incredible Hulk does as well, according to Cho (albeit unconsciously), which explains how he somehow doesn't hurt any innocent bystanders while Hulksmashing.
 Small time Marvel villain Mathemanic has an array of math psychopowers. For example, making people aware of galacticscale distances so that they can't aim properly.
 This is a severe understatement. On his first appearance, he made Firestar believe she was the equivalent distance between the earth and the sun away from EVERYTHING else. And I mean the smallest speck of matter. She promptly went catatonic.
 There was a Fantastic Four arc where Reed whipped up a mathematical equation that became sentient. It was a lowgrade Reality Warper that dealt with reality in mathematical terms and demanded that Reed create a formula that equaled himself.
Film
 π: Max starts the film with a dogmatic belief that "mathematics is the language of nature," which straddles the line between "math is capable of explaining everything in existence" and "math determines our existence." He later encounters a 216digit number that seems to have a catastrophic effect on anything it touches: the stock market, his mentor's health, and eventually his (already tenuous) sanity.
 The John Carpenter horror movie Prince of Darkness, much like H.P. Lovecraft's stories, has ancient, complex mathematical equations of extraterrestrial origin with seemingly magical powers. They predict the existence of a God and AntiGod and, once understood, they open up whole range of bizarre phenomena, including leaping headlong through mirrors into an abyssal darkness on the other side.
Literature
 Magic in HP Lovecraft's mythos is often related to mathematics, and it makes sense considering his preference for Alien Geometries. In Dreams in the Witch House, for instance, the protagonist is a mathematician who discovers an equation that would allow him to travel outside angled space (basically, to create wormholes). If you can understand the true nature of the universe, you can use that knowledge to do things that seem physically impossible to us. Provided you don't first Go Mad from the Revelation or attract the attention of some Eldritch Abomination, of course.
 Connected to the above, The Laundry Series by Charles Stross also shows magic as mathematics, to the point where computers solving certain equations can warp reality as per magical spells. Becomes a bit of a problem when the walls around reality start weakening, to the worstcase scenario of somebody solving equations in their head running the risk of accidentally summoning an Eldritch Abomination.
 Jack L. Chalker's Well of Souls series: The Great Equation. A couple of supercomputers are capable of warping reality retroactively (that is, those who didn't see the change actually happen are incapable of realizing that anything actually changed) by "altering" the Equation, which basically is reality. By moving a few numbers in the equation, the result — that is, our reality — changes to suit.
 The Harry Potter series has an Arithmancy class, whose name would imply divination through arithmetic. (Ironic, because Hermione actively loathes Divination itself.) The reader doesn't get to hear much about the subject, though, beyond the fact that Hermione's studying it (and apparently enjoys it).
 In Fanon, Arithmancy is a common choice for Intelligent!Harry stories. In these stories, its usually presented as discussing (and learning) the rules that govern magic itself. For instance, Arithmancy masters can cast wandlessly more easily, because they understand the way the magic flows and can create shortcuts, similar to simplifying a complicated equation. Its usually also implied that math in the magic world is dreadfully behind that of Muggles, with Trig being 'the last lesson before Mastery'. Cue Harry (and Hermione, more often than not), learning Calc.
 The Compleat Enchanter books by L. Sprague de Camp are heavily based on mathematics causing magic.
 In the Young Wizards series, all magic is based in math and science, and the kids have quite highlevel discussions of these things, because part of magic is being able to completely describe what you want to change.
 In "Career Day", one of the stories in the Chicks in Chainmail anthology, the protagonist comes from a world where magic is done with mathematical formulas. She brings her daughter's class there on a field trip to observe her in her career as a barbarian swordswoman. When her opponent in a duel cheats by hiring a wizard to help him, the other chaperon on the trip, her daughter's math teacher, counteracts the magic with his knowledge of calculus. It Makes Sense in Context. Later expanded to a full novel, called Mathemagics with this entire premise for the magical system.
 The Hitchhiker's Guide to the Galaxy, in the later books, introduces the field of Bistromathics, math that can be used to warp spacetime for purposes of Fasterthanlight travel by taking advantages of the unique properties numbers take on when written down within the confines of an Italian restaurant.
 The Aons (runes) neccessary to make the magic system from Elantris work are very much like a combination of mathematical symbols and a very complex alphabet. Learning magic is incredibly difficult for this reason, and it can take weeks for even an experienced practitioner to write out the more complex spell "equations".
 The premise behind Simon Bloom
 In Monday Begins on Saturday, a book called Equations of Mathmagic is mentioned.
Live Action TV
 The Doctor Who story Logopolis features a planet where they've developed a branch of mathematics in which the act of performing the calculations changes the fabric of reality. The entire population takes shifts in calculating a neverending formula that holds the end of the universe at bay.
 Bonus points to the Master for destroying an entire quarter of the Universe by killing a couple of the guys doing the math.
 A more abstract example is seen in the episode "The Shakespeare Code"; the Carrionites use words as magic, like witches, and the Doctor explains this as completely nonmagical by comparing it to mathematics on earth; "With the right string of numbers you can split the atom!" Except, of course, that saying the number out loud will not cause an atom to spontaneously divide in two.
 Unless you're Logopolitan, of course.
 While not stated on LOST itself,the Alternate Reality Game The Lost Experience states that the numbers (4, 8, 15, 16, 23, 42) are parameters in the Valenzetti equation used to derive the time remaining before humanity's extinction.
Music
 Rhapsody of Fire song "The Mystic Prophecy of the Demonknight " has the narrator mention the heroes using "black geometry" to create a bridge over a deep chasm. When being pursued on the way back, they don't have enough time to work it out and are forced to make a Leap of Faith.
Newspaper Comics
 Bloom County: Oliver Wendell Jones writes a formula to explain the universe and accidentally wipes out Opus when he realises that, under this formula, flightless water fowl could not exist. He then realises he had forgotten to carry a one, causing an unimpressed Opus to pop back into existence when he makes the correction.
Tabletop Games
 In Complete Arcane, the 3.5 Edition of Dungeons & Dragons rulebook, there is a Prestige Class called the Geometer. They learn to describe magic as using abstract geometric designs. This is an advantage, if nothing else, because they can save a ton on spellbooks: they need only a single page to depict any spell, regardless of complexity, whereas with the traditional method one would need more pages with higherlevel spells. They can also use scribe glyphs of these designs to cast a spell silently.
 In Vampire: The Masquerade, one branch of Tremere Thaumaturgy concerns itself with Kabbala, Jewish mysticism heavily based in numerology derived from the Torah. In the same way other magi would seek out a person or thing's True Name, Kabbalists use calculations to derive someone or something's True Number. Blood magic as derived through math. It's worth noting that, like basically all magic in the Old World of Darkness  blood, hedge, or mortal magick  this method only serves as a focus for the user's will. The equations need to have conscious intent in order to reshape reality. Hence why numbers scrawled on primary school blackboards don't end up melting the instructor's face.
Video Games
 In Dungeons of Dredmor, one of the skill sets you can choose at start is Mathemagics. It gives you access to fun spells such as Curse of the Golden Ratio and Beklam's Diminishing Calculus!
 The World Ends with You: Minamimoto summons a nuke by initializing the Level i Flare, which consists of reciting pi to 150 digits. Of course, there are a lot of people capable of reciting pi to 150 digits, but they aren't Minamimoto, and they don't have a graffiti covered dead end alleyway and eldritch tattoos.
 Fun bit of math/Final Fantasy info here. "Level x Flare" is a type of spell that casts Flare on everything with a level divisible by whatever number x is. i is a mathematical term referring to the square root of 1, the consummate imaginary number. In addition to that, as the root of 1, that also makes it a possible factor in every number, real or imaginary. Nothing escapes Level i Flare.
 That said, 1 holds that same property as well, because as the multiplicative identity, it is a factor of every number ever.
 The beauty of Level i Flare is that it's a factor of every number, not just real numbers, meaning it would affect everything, across all possible realities. Useful when your target can travel through dimensions.
 The Final Fantasy Tactics series has a Calculator class that uses various magical effects. How their abilities work from an inuniverse perspective is not specified.
 According to the remake for the PSP's tutorial, Calculators are attuned enough to the flow of mist that they can manipulate it in unique and bizarre ways which other classes just can't pull off. In other words: SE knew it made absolutely no sense and handwaved it.
 Final Fantasy VII has an interesting example: When Safer Sephiroth casts Super Nova, four equations fly by the screen. They are, roughly, the potential attractive force between the sun and the planet, the Earth's potential attractive force, and... the area of a circle. Still, 75% good scientific formulae being used in a magic spell is better than average.
 Fridge Logic (or Epileptic Trees, depending on who you talk to): the area of a circle can be used calculate the distribution of impact force over a target area, so could theoretically useful. Its also possible to use a derived equation, √(A/π)=R to calculate a radius based on a given cross sectional area for example, exactly how big the nova's blast would have to be to stretch to reach the battlefield and stop exactly as it does.
 This is echoed in Dissidia: When doing Supernova as his EX Burst, Sephiroth will begin chanting the equations, which briefly appear onscreen.
 Rita's nonspell special attacks in Tales of Vesperia take the form of mathematical equations.
 Reinhart Manx, the playable mage character in Dungeon Siege III employs both this and Magitek, using the power of math rather than fireballs.
Visual Novel
 In Umineko no Naku Koro ni, the Power Levels for Leviathan and Kyrie are determined in how many hours they've experienced intense envy. Ingame, Leviathan attempts the multiplication for the latter's multiplication formula, along with revealing her own, giving players the idea; in the TIPS, Kyrie is shown actually chanting her formula as she runs to give her husband (flirting with a younger girl) what for.
 Chaos;Head: fun^{10} × int^{40} = Ir2, the equation Takumi discovered in elementary school and the basis of the Noah Project.
Web Comics
Web Original
 Fine Structure uses science, not magic. In later stories, scientists fluent in Eka can unlock the "The simple knowledge of an equation is powerful in itself" kind of formulaic magic. It's also revealed that the universe, or more accurately the imprisoning god, doesn't like anyone messing around with Eka.
 Ra, by the same author, is about actual magic, which is a poorlyunderstood branch of physics. Magic is accessed by reciting specific incantations while performing mental exercises stated to be not dissimilar to solving differential equations. More generally, spells also require a lot of math to design, and resemble computer programs (right down to concepts like modularity, recursion, and writing interpreters to parse gestures as incantations).
 SCP033 of the SCP Foundation is an equation which adds up to a "missing number", an integer which nonparanormal mathematics has somehow missed. If it's recorded in any sort of computing device or written on machinemade paper, whatever it's recorded on starts to physically degrade.
Western Animation
 Looney Tunes has Egghead Junior, who can accomplish anything, even to the point of bending reality, so long as he has a few seconds to write down an appropriate equation for doing so.
 In an episode of Camp Lazlo, being flipped upside down gradually changes the Too Dumb to Live characters Chip and Skip into geniuses (it increases the blood flow to their heads). At the height of their intellect, they use their brains to flip themselves rightside up again: this involves rattling off a lengthy Newtonian formula, and then simply glowing with a pink light that levitates them into position. Unfortunately, this reduces the blood flow to their brains, and they soon turn right back into idiots.
 There's a Donald Duck cartoon called "Donald Duck's Adventures In Mathemagic Land," which is Donald exploring (with the Narrator) the history of mathematics, and math's contribution to things such as music, games, and the natural world.
Real Life
 From the times of ancient Egypt mathematics were associated with mystical properties and powers, and various theorems and systems passed in and out of cult fashion. The most famous of these were the Pythagoreans, who maintained a hermetic hierarchy back in the days when you could get away with executing your members for revealing their secrets.
 And don't even ask what they did to people who failed to box their answers.
 Legend has it that the Pythagoreans were so obsessed with all numbers being rational that, when someone worked out that the square roots of nonsquare integers were irrational (square roots being very important when dealing with, y'know, the Pythagorean theorem), they grabbed the guy and hurled him off a cliff.
 Another telling of the story mentions how he was so pleased with the discovery that he sacrificed every sheep in town.
 During the 19th century, 4th dimensional mathematics was seen by occultists as the key to understanding ghosts and the spirit world: if people could just teach themselves how to think and move in four dimensions rather than the normal three, they could become like ghosts themselves, teleporting and becoming intangible at will. While that's technically true (if oversimplified), today it's believed that, if higher spatial dimensions such as hyperspace exist, they're curled up far too small for human beings to move through, and/or necessitate immense amounts of energy in order to access.
 The notion to get as close as possible to Quantum Physics is quite strong in modern Chaos Magick, of which a prominent example is the Octavo: A Roundworld Edition, a spellbook consisting of eight formulaespells, by Peter Carroll.

