The numeral system used by most of the modern world today is called the decimal system, involving ten digits ("Base 10"). Sometimes, if a writer wants to portray a society as being significantly alien to our own, they will include a mention of an alternative numeral system for this society, with the "base" being a number other than ten.
This may be used to indicate the collective intelligence of the society that produced it, if it is portrayed as more sophisticated or more primitive than our system. There may also be an inferred correlation between the ten digits in our numeral system and the ten digits on the average pair of human hands. Therefore, a race of aliens with Four-Fingered Hands may use a base eight numeral system. Finally, it is very common for robots or other computer-based intelligences to count in base two.
This sort of thing is generally used as an insignificant throwaway joke, with "ordinary" numbers applying before and after, as it may be a difficult concept for some viewers to grasp.
Examples:
- Getting the Shinigami Eyes in Death Note gives you the ability to see a person's name and remaining lifespan when you look at their face. Sadly the lifespan only appears in a number system recognisable to Shinigami that just looks like random numbers to a human.
- In ElfQuest, the elves use base 8 because of their Four-Fingered Hands — that page has more details.
- The same goes for Albedo: Erma Felna EDF, albeit by Translation Convention all the numbers are translated to base-10 for the readers' sake. This became later into a plot point when the characters find an human ship who uses base-10 for obvious reasons.
- Aeon Natum Engel: The narration from the Migou POV and their dialogue goes to great lengths to convey their alien thought processes, including a base-36 numbering system.
- In Hoofstuck, Twilight Sparkle mentions that Equestria uses base eight rather than ten, since there's nothing on a pony you can use to count to ten (whereas four hooves times two make more sense). They used to run with base four originally, symbolising the four races (earth ponies, pegasi, unicorns, alicorns) but the earth ponies took offense at being considered the "zeroes".
- Averted in the Triptych Continuum. Equestria uses Base Ten. Neither Celestia nor Luna have any idea why.
- In The Elements of Friendship, the Equestrians use base-12, complete with neologisms for the new names for numbers of years. The author even wrote a blog post explaining the system.
- Discord notably doesn't do this, and at one point offhandedly derides the ponies for not using the decimal system.
- In Friendship is Sufficiently Advanced, an MLP: FIM/Eldraeverse crossover, ponies use base eight, counting the joints in equine limbs, while eldrae use base 12 as mentioned below. In chapter VII Twilight and the eldrae main character compare numerals when the former notes that the latter's ship designation (1129) is written in only three digits (seven-ten-one, or two-one-five-one in equine numbers).
- Behind-the-scenes material on Avatar claims that the Na'vi also use base 8 for the same reason as the ElfQuest elves.
- Vogons in The Hitchhiker's Guide to the Galaxy have a unary system, meaning that writing the number 1,000 means writing 1 a thousand times.
- In Star Wars Legends, the Hutts use a base 8 system due to only having four fingers on each hand. Being Hutts, they don't always tell this to their business partners, most of whom use base 10.
- In the Discworld books, trolls apparently have a "base Many" system (actually base four). As in, "one, two, three, many, many-one, many-two..." This is revealed in Men at Arms, when it turns out that Detritus is not incompetent with numbers, he is in fact very good at counting in powers of two.
- Even Discworld's trolls are better with numbers than the tree frogs of the Nomes Trilogy, who can only grasp "one" as a number. Faced with the quandary of two different bromeliads, a frog genius makes a breakthrough when it comprehends them as "one, and one more one".
- A subversion in The Restaurant at the End of the Universe: When Douglas Adams revealed that the question which produced the Ultimate Answer (42) was What do you get if you multiply six by nine?, somebody pointed out that the math actually did add up, using base 13. Adams responded, "I may be a sad individual, but I don't make jokes in base 13."
- In The Lord of the Rings:
- According to the appendices, elves habitually count in base 12.
- The Wild Men seem to use a base-20 system. When Ghan-buri-Ghan counts the number of soldiers in the Rohan army, he expresses it as "A score of scores counted ten times and five" (6,000).
- The gukuy in Eric Flint's Mother of Demons count using an eight-base system.
- It's never mentioned in the books themselves, or the series, but the "Gallifreyan numerals" used on the spines and chapter headings of the Doctor Who New Series Adventures (9th and 10th Doctors) are in base 7.
- In Leo Frankowski's The Cross Time Engineer series, the new civilization Conrad Stargard starts up in Medieval Europe uses base 12 mathematics, because Stargard believes it's more useful than decimal.
- In Learning the World by Ken MacLeod, the aliens are four-fingered, and count in base 8. When they learn that humans use base 10, their reaction is that having a base that isn't a power of two must be awfully inconvenient.
- In A Fire Upon the Deep by Vernor Vinge, the doglike Tines have two different numeral systems: one where they count "by legs" (in base 4) and one where they count "by fore-claws" (in base 10). Confusion between these two systems leads to the accidental meeting of two of the major characters. Amdiranifani is housed in room 33, Jefri is supposed to be imprisoned in room 15 (33 in base 4), and the guard who's taking him there uses the wrong numbering system.
- On Gor, the alien species the Kurii use base-12, presumably because they have 12 digits on their "hands."
- Little Fuzzy, a series begun by H. Beam Piper, uses a modified form of base 5. 1, 2, 3, 4, one hand. At 125, they've reached a hand of hands. It then goes to many, and many many. The fuzzies soon adopt the humans' base 10 system.
- Larry Niven and Jerry Pournelle's The Mote in God's Eye. The Moties have a total of 12 digits on their right hands and use base 12.
- The Kzinti of Larry Niven's Known Space count in base eight.
- The Culture uses a nonary (base nine, that is) system for their writing, although it's actually binary in a 3x3 square. Binary is mentioned as being used by some civilisations, and powers of two are the closest thing to a universally-regarded 'round number'.
- In Stranger in a Strange Land, the Martians apparently have a numerical system based on three and powers of three. "Three fulfilled," they call it...
- According to The Klingon Dictionary, the Klingons used to count in a ternary (base-three) system, but have since switched over to decimal.
- In Footfall, the Fithp use base 8.
- In Contact, a message is encoded in base 11, by someone or something capable of messing with the values of mathematical constants.
- The aliens in the Rama books appear to use base 3.
- In The Iron Standard by Henry Kuttner, the six-fingered Venusians use base-12.
- Out of the Dark gives us the Shongairi, who count in base-twelve. Possibly.
- Zamonien in Walter Moers' novels uses base 8. The author even created new symbols for the numbers.
- In the Fighting Fantasy book Rebel Planet, the two-fingered Arcadians count in binary. Converting binary to decimal is necessary to solve a couple of puzzles, though the reader is fortunately provided with a handy grid.
- Centaurians in The Pentagon War have 4 tentacle-fingers on each of 4 hands. When a Centaurian named Torra Zorra reads that a cable's diameter is 3 x 10^{-5} meters, we get this parenthetical aside:
(Curse the human penchant for powers of ten! Torra always had to mentally convert their numbers to base sixteen, just to get a handle on them.)
- The Megarites of the Star Trek universe use base 8, according to Star Trek: Ex Machina.
- In Battlefield Earth the Psychlos use a base-11 system (they have an extra finger on one hand). While this causes the human characters endless frustration when trying to work with Psychlo mathematics, it gets nonsensical when other alien races share these complaints, lauding the values of a sensible base-10 system and concluding that the Psychlos were just being perverse when designing their math.
It doesn't help that Psychlo mathematics requires that the user have a basic knowledge of the Psychlo homeworld's geography. Yet another bit of evidence that they're just trolling the universe. - Gully dwarves from Dragonlance, being astonishingly stupid, can only count up to two. Any higher quantity is described as "One and one and no more than two."
- In Childhood's End by Arthur C. Clarke, the Overlords are mentioned to count in base 14 (their hands have five fingers and two thumbs).
- Greg Egan's Orthogonal trilogy is presumably fed to the reader through a thick soup of Translation Convention, and thus all numbers are in decimal. Despite this, it's easy to infer that the alien race uses a duodecimal (base-12) numeral system. For one thing, all of their units (length, mass, etc.) increment in powers of 12.
- The rabbits of Watership Down can count up to four, presumably because that's how many paws they have. The number "hrair" in their language of Lapine means "any quantity greater than four" and/or "too many to count"; depending on context, it may also be translated into English as "five" or "a thousand."
- At one point in The Fangs of K'aath, the Funny Animal civilization's use of base 8 is explained (due, again, to their fingers) by one character to another; they write 8 as "10". They are apparently fond of mathematics, too, which is fitting, as this particular Fantasy Counterpart Culture is based on the medieval Islamic lands.
- In the Eldraeverse, the Eldrae use base 12, and had a brief flirtation with base 16. Not because of their number of digits (baseline Eldrae have as many as humans), but because their language was standardized by mathematical philosophers.
- In a throwaway joke in Agatha H. and the Clockwork Princess, Prince Aaronev Sturmvoraus sends a message to one of his subjects stating that if he's caught making change in base 8 one more time, he'll be paying his taxes in base 12.
- Currency in the Harry Potter universe is a mixture of base 17 (17 sickles to the galleon) and base 29 (29 knuts to the sickle). This is possibly a joke based on the currency units of RL Britain until the 70s (though at least they didn't use prime numbers).
- Empire from the Ashes: During the scenes set on the Achuultani ships, The characters mention "Twelves, Higher Twelves and Greater Twelves".
- One of Ian Stewart's popular maths books features an alien race who count in base seven. They're keen on cricket, but they go mad with excitement just before a player scores a half-century, because to them, he's just scored a century.
- In some of the Chakona Space stories, Caitians are mentioned to be using base 8 math and the less mathematically minded ones struggle with everyone else's usage of base 10 math.
- Stargate SG-1:
- A throwaway comment in the episode "The Fifth Race" implies that the Ancients counted in base eight.
- Also invoked when they nearly set off a Tobin mine by entering in the wrong code due to Daniel failing to factor in a zero into his translation. He argues the Phoenicians they were descended from never used a number zero, but Carter points out in order to program something as complex as a mine, the Tobin's would have had to have added a zero component.
Carter: Trust me; it's a math thing.
- There's a fan theory that the constellations on the gate actually correspond to digits in a base-38 numeral system. Which means that a gate address is not six arbitrary points in space with the destination at the intersection,^{note } but three two-digit base-38 coordinates.
- In The Frantics' sketch "Roman Numerals" a Roman citizen is baffled by the new decimal system.
Customer: How much is "forty-four" in real numbers?
Shopkeeper: XLIV ^{note } .
Customer: Well why don't you just say XLIV? Who can remember "forty-four?"- When Arabic positional decimal notation (i.e. today's numbers) were first introduced in Europe, some regions passed laws against the use of "deceptive ciphers" and mandated continuing the use of "real quantities" (i.e. Roman numerals).
- In Star Trek: The Next Generation, the Bynars use base 2 (i.e. binary).
- Babylon 5:
- The Minbari use base eleven (a byproduct of using fingers, toes, and the head as "digits" for counting).
- According to tie-in RPG materials, the Dilgar used base 25.
- In Traveller, the various alien species use different base mathematics. The Aslan use Base 8, the Hivers use Base 16, and the Droyne use Base 6. Most of the various Human Aliens, as well as the Vargr, use Base 10.
- The Warhammer 40,000, the Tau use base 8 owing to their Four-Fingered Hands, leading to a delicious little Stealth Pun: their battlesuits' designations are based on their size, such as XV25 for the smaller Stealth suits, XV8 for the main Crisis suit, or XV88 for the heavy Broadside variant. Which means that the new XV9 Hazard Close Support Armour units are taking things Up to Eleven.
- According to the Dragon article "The Ecology of the Spellweaver", these Dungeons & Dragons monsters count in base six, although they also have a decimal notation. The number six is really significant to spellweavers, who not only have six arms, but can regenerate on death six times, following which they produce six offspring.
- In the Myst games:
- The D'ni have a base 25 system, in keeping with the games' general tendency to use powers of five as Arc Numbers. It has a base 5 subsystem: the symbol for 5 is a 1 rotated, 10 is a rotated 2, and so on, up to a unique figure for 25.
- In Myst III: Exile, Saavedro's journal pages are numbered in a base 5 system similar to tallies, but with a leaf motif in keeping with the rest of his culture..
- Obduction, Myst's Spiritual Successor, features a base-four system associated with one of the setting's alien races. They also use a particular notation based on drawing lines among a diagonal grid of dots: lines attached to the central dot are ones; the dot above represents fours, the one on the right represents sixteens, and so on in an expanding spiral.
- The aliens in Iji use a ternary numeral system.
- During Portal's boss fight, after GLaDOS takes a missile hit she'll announce "Two plus two equals... ten! In base four I'm fine!"
- There is a popular theory among Half-Life fans that the Combine use a base-17 system, based on how prominent 17 seems to be. If nothing, it reinforces the utterly alien nature of the Combine.
- The Kilrathi from Wing Commander use Base 8 for their numbering system, given that they have a total of eight fingers. For the most part this isn't really mentioned much, but it's prominent in the dates for history of the Kilrathi war from their viewpoint as done in the manual for Armada, "Voices of War".
- Halo fans speculate that the Forerunners might have counted using a Base-7 counting system.
- A significant portion of the game Rama, based on Arthur C. Clarke's novel series of the same name, involves solving mathematical puzzles based on the native numeral systems of the Avians (base 16) and Octospiders (base 8). (Which is at odds with the Ramans, who use base 3. Neither the Avians nor the Octospiders are Ramans. They're just samples of other space-faring species that the Ramans had gathered.)
- One puzzle in Star Trek: 25th Anniversary involves figuring out the alien code that is more difficult because the aliens in question used a base 3 system.
- Nine Hours, Nine Persons, Nine Doors:
- One of the puzzles in the Kitchen requires you to use base 16 in order to figure out the code to open a locked safe.
- The Nonary Game itself, per its name, indirectly involves base 9 via digital roots. The last door, represented by the letter q, is locked behind at least base 27.
- Mass Effect mentions math systems not based on ten, such as a superweapon that relies on specialized base-12 mathematics and a krogan NPC muttering about the stupidity of base-10 math and the consequences of having extra fingers.
- One puzzle in the freeware interactive fiction game The Muldoon Legacy can only be solved when you realize that you're dealing with an example of this, rather than Base-10.
- In Girl Genius' world of mad science there's bound to be examples showing up: "Tell the landlord that he is to stop trying to make change in base eight, or he'll pay his taxes in base twelve."
- Cowbirds in Love points out that every base is base 10 (from the perspective of anyone using it). For bonus points, the alien in the comic uses base 4 and appears to be four-fingered.
- In Kevin & Kell when Dorothy took over accounting for Aby's business she converted all the numbers to "rabbit numbers" to keep the mostly carnivorous employees from getting ideas. By which she meant base six, for their average litter size.
- Schlock Mercenary: Implied. An alien with an odd number of fingers (and an odd number of bodies) says a long, complicated number will be easy to remember since it's a round number. It's certainly not round in Base 10. Most of the galaxy uses Base 10, likely because of human cultural supremacy.
- Web Video/Numberphile: Discussed in videos:
- There's a video about base 12 (a.k.a. duodecimal, dozenal)
- one about base 16 (a.k.a. hexadecimal)
- one by Tom Scott that explores the surprisingly many different ways numbers are expressed in different cultures, including a group in Papua New Guinea that uses base-27. And naturally, any video relating to computer science will at least mention binary or hexadecimal.
- Related to the above examples, "A better way to count" by jan Misali discusses base six, otherwise known as Seximal.
- No Evil uses a variation of Mayan numerals, as seen in this video.
- In Futurama, robots sometimes use base 2.
- The Schoolhouse Rock! music video for "Little Twelvetoes" briefly touches on the idea of what counting with a base-twelve system would be like, and demonstrates with the titular twelve-fingered alien character.
- In The Simpsons Principal Skinner wishes he had enough funding for his school to buy, amongst other things, "math books that don't have that base 6 crap in them". This is almost certainly a reference to New Math, a way of teaching mathematics that was briefly popular in The '60s but very rapidly fell out of favour.
- Done subtly in Animaniacs. Snow White imprisons Dot for being too cute, and Dot counts the time in hashmarks that only have four strokes each... presumably because she's a cartoon character.
- Computers work in Base 2 because the only input signals they can distinguish between are "on" and "off". Each one is called a "bit". The de facto standard of a byte establishes it as 8 bits, prompting people familiar with computer science to use the hexadecimal system (base 16) to represent a byte of information in two digits.
- Bases that are powers of 2 are fairly popular in computer science, including the above-mentioned hexadecimal (base 16) system and octal (base 8) which is most commonly used with the UNIX chmod command for setting file permissions.
- There have been computers working in base 3 (called ternary or trinary computers) with three possible values: -1, 0, +1. The popularity of binary computers have turned them into a historical footnote, but their efficiency may yet bring them back, especially as we get into quantum computing.
- And then there's Base64, which is mainly used as an encoding scheme to increase the information density when storing or transmitting data, especially in a text-based format.^{note }
- Roman numerals could be considered a combination of base 5 and base 10, with unique symbols for 1 (I), 5 (V), 10 (X), 50 (L), 100 (C), 500 (D), and 1,000 (M), all other numbers being an additive or subtractive combination of the unique signs.
- DNA is encoded (at the lowest level) in base 4. This is thought to provide an optimal compromise between error correction/detection and information density.
- "New math" had learning to do operations in bases other than 10 as part of its syllabus. It was also one of the things it was mocked for.
- The Mayan languages use a base-20 system. Interestingly so do the Nahuatl-speaking ("Aztec") cultures, which no other members of that language-group do (Hopi, O'odham, and Shoshone are base-10), so it might be something the Aztecs borrowed from the Maya (or, just possibly, that both borrowed from the Olmec, who also probably had base-20).
- Ancient Babylonians counted in base 60. This is reflected in the modern measurement of time (hours, minutes and seconds), as well as angular measure (degrees, minutes, seconds), and indirectly, the basis for scoring in tennis (Love, 15, 30, 40, and Game, which used to be 0, 15, 30, 45, and 60). It came about because of two tribes. One used base 10, counting each finger, and a neighboring tribe used base 12, on one hand they would use the thumb to count the remaining twelve segments of the remaining four fingers. Since they had different systems, they would convert to a larger base. Sixty is the smallest common multiple of 10 and 12.
- Vestiges of base 20 remain in English ("four score and seven years ago") and in French ("quatre-vingts"^{note } , also in the name of the Parisian hospital Quinze-Vingts^{note } which was originally founded to house 300 patients). In addition, numbers after sixty and eighty go up to sixty-nineteen (soixante-dix-neuf, instead of seventy-nine) and eighty-nineteen (quatre-vingts-dix-neuf, instead of ninety-nine). It's thought to be either a Celtic or a Basque influence, or an influence of the "Vasconic language-group" (of which Basque is the only surviving member) on Celtic.
- Yan Tan Tethera is a system for counting sheep from the same Celtic origins, once widespread in England but now rare & found mainly in the North. It goes up to "jiggit" for 20, then the shepherd makes a mark, places a stone etc. and begins again.
- Many languages count in little endian up to a certain point and switch to big endian for larger numbers. For example, in English "16" is sixteen but "36" is thirty six.
- Some languages are weirder: Arabic stays little-endian until 100. So "16" is sitta-`ashar (six-ten) and "36" is sitta-wa-thalaathuun (six-and-thirty), but "136" is mi'ah wa sitta wa thalaathuun (a-hundred-and-six-and-thirty). This, however, is a relatively modern construction. Like the language itself, the numbers were read from right to left, e.g. "sitta wa thalaathuun wa mi'ah" (six-and-thirty-and-hundred).
- In older English, it's also not uncommon to see constructions like "six and thirty", which is still how German, parts of Danish and old-fashioned Norwegian do it. This persists in nursery rhymes: (Four and twenty blackbirds...) and in rugby songs (four and twenty virgins...)
- Due to the influence of German, both "thirty-six" and "six and thirty" are gramatically correct in Czech, although the big-endian version is more common.
- Spanish is another language that zig-zags. Your little endians go all the way to fifteen, switch to big-endian to twenty (veinte), switch again to little-endian, and then end again at thirty (treinta), so you have your singulars (uno, seis 1, 6) your teens (doce, catorce, 12, 14), your big-endian teens (diecisiete, "ten and seven"), and then your twenties (veintiuno, veintiocho, 21, 28) before it goes entirely to big endian from there (treinta y nueve, "thirty-and-nine"). The cycle repeats once you make it to triple-digits, as they are referred to a lot closer to English but with the aforementioned endian cycle (dos cientos cuarenta y tres, "Two hundred forty-and-three").
- The Dozenal Society of America is the premier American organization for the promotion of base 12.
- Up until the 1970s, British currency was a mix of base 12 (12 pence to the shilling) and base 20 (20 shillings to the pound). They were the last First World nation to switch to a decimal based currency.
- The origin of base 10 is very likely just that we (usually) have a total of 10 fingers in our hands, like other terrestrial vertebrates in their forward claws or whatever^{note } , as a legacy of the ancestor of them. Had that creature had a different number of them, it's very likely we'd be counting in base (# of fingers in our two hands) and base 10 would be something reserved for mathematicians.