

Alternative Number System

The number 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 number 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 number system and the ten digits on the average pair of human hands. Therefore, a race of aliens with FourFingered Hands may use a base eight number system. Finally, it is very common for robots or other computerbased 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:
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Comicbooks
Fan Works
 Aeon Natum Engel: The narration from the Migou POV and their dialogue goes to great lengths to convey their alien thought processes, including a base36 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".
Film
Literature
LiveAction TV
Tabletop Games
 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 FourFingered 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.
Video Games
 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.
 The aliens in Iji use a ternary number 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 HalfLife fans that the Combine use a base17 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 Base7 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 number systems of the Avians (base 16) and Octospiders (base 8). (Which is at odds with the Ramans, who use base 3.)
 Of course, both the Avians and Octospiders aren't Ramans. They're just samples of other spacefaring 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.
 One of the puzzles in the Kitchen in Nine Hours, Nine Persons, Nine Doors requires you to use base 16 in order to figure out the code to open a locked safe.
Webcomics
 In Housepets!, a mouse named Spo came from a very large family. How large? The sibling born immediately after him was named Spp...
 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 fourfingered.
Web Originals
 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.
Western Animation
 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 basetwelve system would be like, and demonstrates with the titular twelvefingered 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 Sixties but very rapidly fell out of favour.
Real Life
 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 abovementioned 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 popularit 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.
 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 base20 system. Interestingly so do the Nahuatlspeaking ("Aztec") cultures, which no other members of that languagegroup do (Hopi, O'odham, and Shoshone are base10), so it might be something the Aztecs borrowed from the Maya (or, just possibly, that both borrowed from the Olmec, who also probably had base20).
 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). 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 least common denominator between 10 and 12.
 Vestiges of base 20 remain in English ("four score years and ten ago") and in French ("quatrevingts"^{note }4*20=80, also in the name of the Parisian hospital QuinzeVingts^{note }15*20 which was originally founded to house 300 patients). In addition, numbers after sixty and eighty go up to sixtynineteen (soixantedixneuf, instead of seventynine) and eightynineteen (quatrevingtsdixneuf, instead of ninetynine). It's thought to be either a Celtic or a Basque influence, or an influence of the "Vasconic languagegroup" (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 littleendian until 100. So "16" is sitta`ashar (sixten) and "36" is sittawathalaathuun (sixandthirty), but "136" is mi'ah wa sitta wa thalaathuun (ahundredandsixandthirty). This, however, is a relatively modern construction. Like the language itself, the numbers were read from right to left; "sitta wa thalaathuun wa mi'ah" (sixandthirtyandhundred)
 In older English, it's also not uncommon to see constructions like "six and thirty", which is still how the other Germanic languages usually do it.
 The Dozenal Society of America is the premier American organization for the promotion of base 12.

