Wersja z 2019-12-25

Grzegorz Jagodziński

Pozycyjne systemy liczbowe

System liczbowy można rozumieć jako system liczebników, czyli system nazywania liczb. O różnych systemach liczebników mowa w innym artykule.

Zasadniczo jednak system liczbowy to system zapisywania liczb przy pomocy skończonej liczby znaków, zwanych cyframi. Istnieją systemy liczbowe pozycyjne i addytywne. Przykładem addytywnego systemu liczbowego jest rzymski system zapisu liczb. Systemami takimi nie będziemy się tu zajmować.

W pozycyjnym systemie liczbowym istnieje liczba zwana bazą, której naturalne potęgi zapisuje się przy pomocy cyfry 1 oraz tylu zer, ile wynosi potęga. System pozycyjny wymaga tylu różnych cyfr, ile wynosi baza. Wyjaśnimy to poniżej.

System dziesiętny

Na co dzień liczymy, posługując się systemem o bazie „10”, zwanym systemem dziesiętnym, a określanym też jako decymalny. Oznacza to, że liczbę „dziesięć” zapisujemy „10”. Piszemy cyfrę „1” oraz jedno zero, ponieważ „dziesięć” to baza w pierwszej potędze. Liczbę „sto” zapisujemy przy pomocy jedynki i dwóch zer, ponieważ „sto” jest to „dziesięć” podniesione do drugiej potęgi, `100 = 10^2`. Z kolei „tysiąc” to jedynka i trzy zera, ponieważ `1000 = 10^3`.

Zaważmy, że ilość cyfr danej liczby w zapisie dziesiętnym jest większa o jeden od najwyższej użytej potęgi dziesiątki, ponieważ ostatnia pozycja ma numer zero. Stąd liczba opisująca trzecią potęgę bazy ma cztery cyfry, odpowiadające pozycjom 3, 2, 1 i 0. Mówimy też, że liczba `1000` ma cyfrę `1` na pozycji tysięcy (tj. na pozycji `10^3` lub na pozycji 3), cyfrę `0` na pozycji setek (tj. na pozycji `10^2` lub na pozycji 2), cyfrę `0` na pozycji dziesiątek (tj. na pozycji `10^1` lub na pozycji 1) i wreszcie cyfrę `0` na pozycji jedności (tj. na pozycji `10^0` lub na pozycji 0). Do problemu wrócimy, analizując inne systemy liczbowe.

W systemie dziesiętnym istnieje dziesięć cyfr, czyli tyle, ile wynosi baza. Są to kolejno `0, 1, 2, 3, 4, 5, 6, 7, 8, 9`.

System dwójkowy

System o bazie „2”, czyli system dwójkowy, zwany też binarnym, używa tylko dwóch cyfr, `0` i `1`. System taki używany jest w urządzeniach elektronicznych, ponieważ bazuje na dwóch tylko stanach: włączony (`1`) i wyłączony czyli zgaszony (`0`), co jest łatwe do odczytu i zapisu. Ponieważ bazą tego systemu jest „dwa”, liczba ta w zapisie dwójkowym ma postać `10`. Jeśli z kontekstu nie wynika, w jakim systemie zapisano liczbę, używamy po nim zapisu bazy (w systemie dziesiętnym), podanego w dolnej linii pisma. Możemy więc zapisać `2_10 = 10_2`, co oznacza, że liczbie `2` zapisanej w systemie dziesiętnym odpowiada zapis `10` (czytany „jeden zero”, a nie „dziesięć”!) w systemie dwójkowym.

Aby zapisać daną liczbę w dowolnym systemie, trzeba pamiętać wartości kolejnych potęg bazy. Chcąc zatem posługiwać się np. systemem dwójkowym, można zapamiętać wartości kolejnych potęg dwójki: `2^0 = 1`, `2^1 = 2 = 10_2`, `2^2 = 4 = 100_2`, `2^3 = 8 = 1000_2`, `2^4 = 16 = 10000_2`, `2^5 = 32 = 100000_2`, `2^6 = 64 = 1000000_2` itd. W systemie dwójkowym zamiast pozycji jedności, dziesiątek, setek itd. mamy więc pozycję jedności, dwójek, czwórek, ósemek, szesnastek itd.

Liczby niebędące potęgami dwójki zapisujemy jako sumy odpowiednich potęg. Jeśli w sumie występuje dana potęga, w zapisie dwójkowym występuje jedynka, jeśli jej nie ma, występuje zero. Np. dziesiętne `10` możemy zapisać w postaci sumy potęg dwójki `8 + 2`, co daje `2^3 + 2^1`. Zatem zapis dwójkowy będzie miał 4 pozycje (odpowiadające potęgom dwójki od 3 do 0), przy czym na pozycjach 3 i 1 będą jedynki, a na pozycjach 2 i 0 będą zera: `1010`. Cały rachunek można zapisać tak:

`10 = 8 + 2 = 2^3 + 2^1 = 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 = 1010_2`

Możemy też liczyć, dodając potęgi bazy zapisane w systemie dwójkowym:

`10 = 8 + 2 = 2^3 + 2^1 = 1000_2 + 10_2 = 1010_2`

Istnieje też druga, wygodniejsza metoda zamiany, niewymagająca pamiętania wartości potęg danej liczby. W metodzie tej dzielimy daną liczbę przez bazę bez znajdowania cyfr po przecinku i zapisujemy resztę. Powtarzamy procedurę, dzieląc całkowitoliczbową część uprzednio otrzymanego wyniku. Postępujemy w ten sposób tak długo, aż otrzymamy w wyniku zero. Kolejne reszty zapisujemy od prawej do lewej.

Poddajmy tej procedurze liczbę `10`, zamieniając ją na zapis dwójkowy. Mamy zatem `10 : 2 = 5` bez reszty, z czego wnioskujemy, że ostatnią cyfrą w zapisie dwójkowym jest `0`. Teraz poddajemy operacji wynik `5` i wykonujemy dzielenie: `5 : 2 = 2` i reszta `1`, którą dopisujemy z lewej strony, otrzymując `10`. W trzecim kroku dzielimy wynik `2` (bez uwzględnienia reszty), otrzymując `2 : 2 = 1` z resztą `0`, które dopisujemy z lewej strony: `010`. Musimy jeszcze podzielić `1 : 2 = 0` z resztą `1`, którą znów dopisujemy z lewej, dostając `1010`. A ponieważ całkowitoliczbowym wynikiem dzielenia jest zero, kończymy procedurę, co pozwala nam zapisać `10_10 = 1010_2`.

Spróbujmy w podobny sposób zapisać w systemie dwójkowym liczbę `1000`. Pierwsza metoda prowadzi do następującego rozkładu:

`1000 = 512 + 488 = 512 + 256 + 232 = 512 + 256 + 128 + 104 = 512 + 256 + 128 + 64 + 40 = 512 + 256 + 128 + 64 + 32 + 8`

`1000 = 1 * 2^9 + 1 * 2^8 + 1 * 2^7 + 1 * 2^6 + 1 * 2^5 + 1 * 2^3`

`1000_10 = 1111101000_2`

Druga metoda prowadzi do następujących dzieleń:

`1000 : 2 = 500` r `0`
`500 : 2 = 250` r `0`
`250 : 2 = 125` r `0`
`125 : 2 = 62` r `1`
`62 : 2 = 31` r `0`
`31 : 2 = 15` r `1`
`15 : 2 = 7` r `1`
`7 : 2 = 3` r `1`
`3 : 2 = 1` r `1`
`1 : 2 = 0` r `1`

z czego po odczytaniu reszt od dołu ku górze wynika, że `1000_10 = 1111101000_2`.

Odczytanie liczby zapisanej w systemie dwójkowym (krótko: liczby dwójkowej lub liczby binarnej), to znaczy jej zamiana na doskonale nam znany system dziesiętny, jest dość proste, ale wymaga znajomości potęg bazy, czyli dwójki. Np. liczba dwójkowa `1010_2` ma jedynkę na pozycji ósemek, zero na pozycji czwórek, jedynkę na pozycji dwójek, i zero na pozycji jedności, przedstawia więc liczbę dziesiętną `1 * 8 + 0 * 4 + 2 * 2 + 0 * 1 = 8 + 2 = 10`.

Jedenastocyfrowa liczba binarna `10110111001_2` oznacza z kolei liczbę dziesiętną:

`1 * 2^10 + 1 * 2^8 + 1 * 2^7 + 1 * 2^5 + 1 * 2^4 + 1 * 2^3 + 1 * 2^0 = 1024 + 256 + 128 + 32 + 16 + 8 + 1 = 1465`

(w rachunku pomijamy te potęgi, na pozycjach których występują zera).

Liczby binarne oznacza się odpowiednim indeksem dolnym, ale w informatyce stosuje się także inne sposoby, np. dopisanie prefiksu % lub 0b, albo sufiksu b. Zatem np. dziesiętne `4143` można przestawić jako `1000000101111_2`, `"%"1000000101111`, `0"b"1000000101111` lub `1000000101111"b"`.

Z uwagi na stosowanie w informatyce systemu dwójkowego przyjmuje się, że wielokrotności jednostki ilości informacji lub pojemności pamięci, bajta, nie są potęgami tysiąca, czyli `10^3`, ale `1024`, czyli `2^10`. Stąd kilobajt ma `1024` bajty, a nie `1000` bajtów, megabajt to `2^20 = 1 048 576` bajtów, gigabajt to `2^30 = 1 073 741 824` bajty, a terabajt to `2^40 = 1 099 511 627 776` bajtów.

Producenci twardych dysków ze względów marketingowych (tzn. chcąc oszukać klientów) nie stosują się do tej konwencji, i zamiast przedrostków binarnych, stosują zwykłe, dziesiętne. Dysk o pojemności `1` gigabajta ma zatem w rzeczywistości zaledwie `953`,`67` megabajtów pojemności. Podobnie podając szybkość transferu danych (w bodach czyli bitach na sekundę) używa się przedrostków dziesiętnych.

Aby uporządkować ten niewątpliwy bałagan, przyjmuje się zapis `1` KB dla kilobajta binarnego (liczącego `1024` bajty), a `1` kB dla kilobajta dziesiętnego (liczącego `1000` bajtów). Jednak megabajtów, gigabajtów czy terabajtów w ten sposób odróżnić się nie da. Dlatego zaproponowano, by kilobajty binarne oznaczać skrótem KiB, podobnie megabajty binarne skrótem MiB, gigabajty binarne skrótem GiB itd. Propozycja ta z wolna znajduje uznanie. Znacznie oporniej przyjmują się propozycje nazw jednostek binarnych: kibibajt, mebibajt, gibibajt, tebibajt.

A oto jak przeliczać jednostki binarne i decymalne:

kilobajt 1 kB103 B1000 B0,977 KiB  
  1,024 kB210 B1024 B1 KiB kibibajt
megabajt 1 MB106 B1 000 000 B0,954 MiB  
  1,049 MB220 B1 048 576 B1 MiB mebibajt
gigabajt 1 GB109 B1 000 000 000 B0,931 GiB  
  1,074 GB230 B1 073 741 824 B1 GiB gibibajt
terabajt 1 TB1012 B1 000 000 000 000 B0,909 TiB  
  1,100 TB240 B1 099 511 627 776 B1 TiB tebibajt
petabajt 1 PB1015 B1 000 000 000 000 000 B0,888 PiB  
  1,126 PB250 B1 125 899 906 842 624 B1 PiB pebibajt

System ósemkowy

Liczby zapisane w systemie dwójkowym są długie i dlatego niepraktyczne w stosowaniu. Dość szybko pojawiły się więc w praktyce programistycznej inne systemy, które dawały się łatwo przekształcić do zapisu dwójkowego, ale dawały krótsze zapisy.

Języki europejskie stosują na ogół tysięczno-dziesiętny system liczebników, co oznacza, że czwarta potęga bazy bywa nazywana złożonym terminem „dziesięć tysięcy”. W ślad za tą właściwością języka matematyka europejska stosuje zapisy w rodzaju `10` `000`, dzieląc liczby na trzycyfrowe grupy, począwszy od prawej strony, i oddzielając te grupy przerwami. Podobną procedurę zaczęto stosować dla systemu dwójkowego, pisząc `1` `111` `101` `000` zamiast `1111101000` (co oznacza `1000` w zapisie dziesiętnym). Dla wygody i oszczędzenia miejsca postanowiono pójść jeszcze dalej, i zamiast trzech cyfr binarnych stanowiących grupę zaczęto pisać jedną cyfrę, teoretycznie dziesiętny odpowiednik danej trzycyfrowej liczby dwójkowej. Stąd zamiast `1` `111` `101` `000` otrzymalibyśmy `1750` (gdzie `1` to binarne `1`, `7` to binarne `111` czyli `4 + 2 + 1`, dalej `5` to binarne `101`, wreszcie `0` odpowiada grupie `000`.

Zauważmy, że pisząc w ten sposób, będziemy wykorzystywać jedynie cyfry od `0` do `7`, a liczbę (dziesiętną) `8` zapiszemy jako `10`. Rzeczywiście, `8_10 = 1000_2`, co po oddzieleniu trzech zer i zastąpieniu ich jednym da `10`. Zastosowany przez nas system będzie więc w istocie ósemkowy, czyli oktalny. W systemie tym `8_10 = 10_8`, a na przykład `10_10 = 12_8`. Rzeczywiście tak jest, skoro `10_10 = 1010_2`, a ponieważ dwójkowe `010` odpowiada dziesiętnej cyfrze `2`, to `1010_2 = 12_8`.

Przeliczanie odwrotne, z systemu ósemkowego na dwójkowy, nie wymaga żadnych obliczeń, a jedynie zastępowania jedna po drugiej cyfr ósemkowych ich odpowiednikami dwójkowymi. Na przykład `144_8 = 1` `100` `100_2` (chodzi oczywiście o `100_10`).

Natomiast algorytm przejścia z systemu dziesiętnego na ósemkowy jest analogiczny do tego używanego dla otrzymania zapisu binarnego. Pokażemy to, posługując się metodą drugą (dzielenia z resztą).

Znajdźmy zapis liczby (dziesiętnej) `100` w systemie ósemkowym. W tym celu dzielimy `100 : 8 = 12` r `4`, i zapisujemy tę resztę na ostatniej pozycji (najbardziej prawej) szukanej liczby oktalnej. Dzielimy teraz wynik: `12 : 8 = 1` r `4`. Zapisujemy więc kolejną cyfrę `4`, na lewo od poprzednio zapisanej. Na końcu dzielimy `1 : 8 = 0` r `1`, i tym sposobem otrzymujemy ostatecznie, że `100_10 = 144_8`.

Wreszcie „odczytanie” liczby oktalnej, tj. jej zapis w systemie dziesiętnym, wymaga znajomości potęg ósemki (albo ich każdorazowego obliczania). Np. `220_8 = 2 * 8^2 + 2 * 8^1 + 0 * 8^0 = 2 * 64 + 2 * 8 = 128 + 16 = 144`.

Liczby ósemkowe oznacza się odpowiednim indeksem dolnym, ale w informatyce stosuje się także inne sposoby, np. dopisanie prefiksu 0, o, q, 0o, \, & lub sufiksu o. Zatem np. dziesiętne `4143` można przestawić jako `10057_8`, `010057`, `"o"10057`, `"q"10057`, `0"o"10057`, `"\"10057`, `&10057` lub `10057"o"`.

System szesnastkowy

System liczebników tysięczno-dziesiętny panuje w cywilizacji zachodniej, ale nie w Chinach. Tam bowiem liczba „dziesięć tysięcy” ma swoją własną, prostą nazwę. Można by ją przetłumaczyć jako „miriada”. „Sto tysięcy” to w konsekwencji „dziesięć miriad”.

Liczby w kręgu cywilizacji chińskiej zapisuje się stosownie do ich nazw, a zatem oddzielając przerwami cztery, a nie trzy kolejne cyfry. Dlatego „milion” (`10^6`) zapiszemy po chińsku nie jako `1` `000` `000`, ale jako `100` `0000`, co odzwierciedla nazwę tej liczby: „sto miriad”.

Oddzielanie od siebie cyfr binarnych co cztery zamiast co trzy pozycje, zbieżne z chińskim zapisem wielkich liczb, doprowadziło do przyjęcia systemu szesnastkowego, zwanego też heksadecymalnym. Obecnie system ten wyparł niemal zupełnie ósemkowy i panuje niepodzielnie w informatyce. System ten podobnie łatwo jak ósemkowy daje się konwertować na binarny, a zapis liczb jest jeszcze bardziej zwięzły. Wymaga jednak aż 16 cyfr. Innymi słowy, takie liczby jak 10, 13 czy 15 są w nim zapisywane pojedynczymi symbolami. W tym celu wykorzystuje się kolejne liczby alfabetu (małe lub duże), używane w funkcji cyfr szesnastkowych: `"a"_16 = 10_10`, `"b"_16 = 11_10`, `"c"_16 = 12_10`, `"d"_16 = 13_10`, `"e"_16 = 14_10`, `"f"_16 = 15_10`. Oczywiście „szesnaście”, będące bazą systemu, zapisywane jest jako `10`.

Liczby szesnastkowe oznacza się odpowiednim indeksem dolnym, ale w informatyce stosuje się także inne sposoby, m.in. dopisanie prefiksów 0x, \x, $ lub sufiksu h. Zatem np. dziesiętne `4143` można przestawić jako `102"f"_16`, `0"x"102"f"`, `"\x"102"f"`, `$102"f"` lub `102"fh"`.

Duże liczby w systemie szesnastkowym zapisuje się, oddzielając cyfry po cztery, licząc od prawej strony, jak liczby decymalne w zapisie chińskim. Np. `131 328_10 = 2 0100_16`, lub `3 805 184_10 = 3"a" 1000_16`.

:

Liczebniki występują w większości języków świata. Za wyjątek uchodzi język pirahã z dżungli amazońskiej. W języku tym występują tylko wyrazy hoi – mało i baagiso – dużo. Nie występuje też liczba mnoga rzeczowników i zaimków[4].

W językach indoeuropejskich, semickich, ałtajskich i chińsko-tybetańskich system liczebnika jest oparty na układzie dziesiętnym. Rozpowszechnienie tej bazy wynika z faktu, że ludzie mają dziesięć palców u rąk.

Wiele plemion w Afryce i Oceanii używa systemu opartego na liczebnikach o bazie 5. W niektórych językach Nowej Gwinei system jest oparty na bazie 4 lub 8[5]. Prekolumbijska cywilizacja Majów używała systemu dwudziestkowego, a Sumerowie – sześćdziesiątkowego.

2: binary
Binary systems are base 2, often using zeros and ones. With only two symbols binary is useful for logical systems like computers.

 

3: ternary
Main article: Ternary numeral system § Practical usage
Base 3 counting has practical usage in some analog logic, in baseball scoring and in self–similar mathematical structures.

 

4: quaternary
Main article: Quaternary numeral system
Some Austronesian and Melanesian ethnic groups, some Sulawesi and some Papua New Guineans, count with the base number four, using the term asu and aso, the word for dog, as the ubiquitous village dog has four legs.[11] This is argued by anthropologists to be also based on early humans noting the human and animal shared body feature of two arms and two legs as well as its ease in simple arithmetic and counting. As an example of the system's ease a realistic scenario could include a farmer returning from the market with fifty asu heads of pig (200), less 30 asu (120) of pig bartered for 10 asu (40) of goats noting his new pig count total as twenty asu: 80 pigs remaining. The system has a correlation to the dozen counting system and is still in common use in these areas as a natural and easy method of simple arithmetic.[11][12]

Many or all of the Chumashan languages originally used a base 4 counting system, in which the names for numbers were structured according to multiples of 4 and 16 (not 10). There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819.[1]

The Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10.

Base-4 systems are attested on four continents:
North America: Some extinct Chumash languages (Chumashan, USA) show
original base-4 systems, running up to 32 (Mamet 2005:113-115) (Beeler
1967, 1963, Hughes 1974). Base-4-8 is also documented with the older
generation in the now extinct Yuki (Isolate, USA). For Yuki, Kroeber
(1925) describes how base-4 is related to hand-counting by considering
the spaces between the ?ngers (cf. Hinton 1994) 7 . The Chumashan lan-
guages and Yuki are both in California but quite distantly apart, with
Yuki in the north and Chumashan in the south, and other language
families intervening.

South America: The extinct Lule (Isolate, Argentina) of Clark (1937:102)
and Machoni de Cerdeña (1732:84-86) as well as the poorly attested ex-
tinct Charrúa (Charruan, Uruguay) reported in (Ibarra Grasso 1939b:202)
appear to have had base-4 up to 10, at which point the system turns
into a commonplace 5-10-20 system with hands and feet. It cannot be
inferred from the data hand that there was ever true base-4 system here,
beyond 10. A couple of descriptions of a Guaraní variety in Paraguay
(Tupí-Guaraní/Tupi, Paraguay) show base-4 up to 10, but the expres-
sions for numbers above 10 are not shown (Ibarra Grasso 1938:278)
(Ibarra Grasso 1939a:590). Other old and new descriptions of any vari-
eties of Guaraní (too many to list) do not show any traces of base-4. Iso-
lated vocabularies of Mocovi and Toba (Guaicuruan, Argentina) show
base-4 up to 8 and 10 respectively (Koch-Grünberg 1903:114-124), but
the vast majority of vocabularies for these languages (too many to list)
show no trace of this. The extinct Payaguá (Isolate 8 , Paraguay) has
one attestation with alternative base-4 forms up to 20 (Koch-Grünberg
1903:114-124). All these cases occur within a relatively small area of
South America, but there is otherwise little evidence for an areal con-
nection.
Indonesia/Papua New Guinea: An indeterminate number of languages
in the highlands have a variations of a base-4 system (Lean 1986a,c,
1992:13-86,15-59,Ch. 5), where at least one, Kakoli (Hagen/Trans New
Guinea, Papua New Guinea) is attested with as base 4-24 (Bowers and
Lepi 1975). Kewa (Engan/Trans New Guinea, Papua New Guinea) has
several parallel numeral system, one of them being base-4 (Franklin and
Franklin 1962) and goes at least up to 20, and beyond that it may be
combined with a body-tally system to form higher numbers in units of
four (Pumuge 1975). The word for '4' is 'hand', i.e., four ?ngers con-
stitute one hand and the thumb is separate. The traditional counting
system of Mbowamb (Hagen/Trans New Guinea, Papua New Guinea)
near Mt. Hagen has been been described with a fair amount of detail.
It is clearly a 2-4-8 system, for which Vicedom and Tischner (1948:268-
270) gives expressions up to 24, and says the system can be used up to
about 80. Another description seems to indicate that after 20, counting
can be done in units of 20 (Strauss 1962:315-318), cf. also Lancy and
Strathern (1981). As in Kewa, the base-4 is connected with counting
the ?ngers of one hand, the thumb counted separately. The origin of
the highland base-4 system(s) has not been systematically investigated,
but given the geographical proximity and the fact that the Engan and
Hagen languages are not closely related, an areal connection seems
likely even if this is not directly observable in the forms in question.
On the north coast, around the border between Indonesian Papua and
Papua New Guinea, base-4 is also present variously in the Sko lan-
guages (most of the data is collected in Lean 1986b), ? see Donohue
(2008) for a good attestation of 4-12-24 in Skou ? as well as 4-24 in To-
bati (Sarmi-Jayapura Bay/Austronesian) for which the best attestion
is Moolenburgh (1904). Given the proximity of the languages and the
fact that they are genetically unrelated, there is almost certainly an
areal connection between base-4 in Skou and the Sarmi-Jayapura Bay
Oceanic languages.
Africa: An indeterminate number of languages in the northeastern Demo-
cratic Republic of the Congo (DRC) have (traces of) a base-4 sys-
tem. The ?rst attestation appears to be a Nyali (Bantu, DRC) va-
riety for which Stuhlmann (1894:624) notes that 8=2*4, 9=2*4+1,
13=12+1, 14=12+2, 16=2*8, 17=2*8+1 but 20=2*10. Later reports
of related Bantu varieties show that there was/is a fully systematic 4-
24 or 4-32 underlying these forms (van Geluwe 1960, Kalunga Mwela-
Ubi 1999, Bokula and Ngandi 1985). Furthermore, thanks to Kutsch
Lojenga (1994:353-357), we have a full attestation of almost obsolete
Ngiti (Lendu/Central Sudanic, DRC) and Lendu (Lendu/Central Su-
danic, DRC) 4-32 systems (p.c. Constance Kutsch Lojenga 2007). Var-
ious wordlists attest traces of the same base-4 systems in decay or
amalgamation with base-10 and base-20 in closely related Bantu and
Central Sudanic languages (Johnston 1922b, Struck 1910, Johnston
1904, Bokula 1970, Harries 1959, Lojenga 1994, Schebesta 1966, 1934,
Asangama 1983, Czekanowski 1924, Stuhlmann 1917) and unpublished
SIL survey lists.
In addition, there are a number of languages which have been claimed to
be base-4 in the literature but which are not base-4 according to the de?nition
used in this paper. We will mention a few of the most important ones here.

The language called
?
Afúdu (Unassigned 9 , West Africa) by Koelle (1854) uses
some additions with 4 in the numbers below 10 but is decimal in the range
10-20. Bodo and Deuri (Bodo-Garo/Sino-Tibetan, India) have vestiges of
base-4 counting extending higher than 20 and Bai (Bai/Sino-Tibetan, China)
is documented with a base-4-16-80 system for shell money in medieval times
(Mazaudon 2007). Yiwom (West Chadic A/Afro-Asiatic, Nigeria) has 7-
9 as 4+3,4+4,4+5 but no other forms are based on 4 (Ibriszimow 1988).
de Castelnau (1851a:10-13) reports base-4 (actually base-2-4) in Apinayé
(Jê/Jê-Jabutí, Brazil) but no actual forms are given (de Castelnau 1851b:270-
274) and is likely to be spurious in the absense of corroborating data in
this rather well-documented language (too many references to list). Base-4
for counting special objects is widely attested in the Oceanic languages of
Melanesia (Kolia 1975, Friederici 1912, Parkinson 1907).
 

5: quinary
Main article: Quinary
Quinary systems are based on the number 5. It is almost certain the quinary system developed from counting by fingers (five fingers per hand).[13] An example are the Epi languages of Vanuatu, where 5 is luna 'hand', 10 lua-luna 'two hand', 15 tolu-luna 'three hand', etc. 11 is then lua-luna tai 'two-hand one', and 17 tolu-luna lua 'three-hand two'.

5 is a common auxiliary base, or sub-base, where 6 is 'five and one', 7 'five and two', etc. Aztec was a vigesimal (base-20) system with sub-base 5.

Many languages[6] use quinary number systems, including Gumatj, Nunggubuyu,[7] Kuurn Kopan Noot,[8] Luiseño[9] and Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:[7]

Number Base 5 Numeral
1 1 wanggany
2 2 marrma
3 3 lurrkun
4 4 dambumiriw
5 10 wanggany rulu
10 20 marrma rulu
15 30 lurrkun rulu
20 40 dambumiriw rulu
25 100 dambumirri rulu
50 200 marrma dambumirri rulu
75 300 lurrkun dambumirri rulu
100 400 dambumiriw dambumirri rulu
125 1000 dambumirri dambumirri rulu
625 10000 dambumirri dambumirri dambumirri rulu
In the video game Riven and subsequent games of the Myst franchise, the D'ni language uses a quinary numeral system.

Biquinary
A decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system. The numbers 1, 5, 10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX.

Most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary. Units of currencies are commonly partially or wholly biquinary.

Quadquinary
A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, Kaktovik Inupiaq numerals and the Maya numerals.

 

6: senary
Main article: Senary
The Morehead-Maro languages of Southern New Guinea are examples of the rare base 6 system with monomorphemic words running up to 66. Examples are Kanum and Kómnzo. The Sko languages on the North Coast of New Guinea follow a base-24 system with a sub-base of 6.

Despite the rarity of cultures that group large quantities by 6, a review of the development of numeral systems suggests a threshold of numerosity at 6 (possibly being conceptualized as "whole", "fist", or "beyond five fingers"[5]), with 1–6 often being pure forms, and numerals thereafter being constructed or borrowed.[6]

The Ndom language of Papua New Guinea is reported to have senary numerals.[7] Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36 × 2 = 72.

Another example from Papua New Guinea are the Yam languages. In these languages, counting is connected to ritualized yam-counting. These languages count from a base six, employing words for the powers of six; running up to 66 for some of the languages. One example is Komnzo with the following numerals: nibo (61), fta (62), taruba (63), damno (64), wärämäkä (65), wi (66).

Some Niger-Congo languages have been reported to use a senary number system, usually in addition to another, such as decimal or vigesimal.[6]

Proto-Uralic has also been suspected to have had senary numerals, with a numeral for 7 being borrowed later, though evidence for constructing larger numerals (8 and 9) subtractively from ten suggests that this may not be so.[6]

Base-6 systems are attested on Kolopom island (formerly Frederik-Hendrik-
Eiland) in southwest Indonesian Papua, as well as in the Kanum and Nambu
languages in southern New Guinea around the Indonesian-Papua New Guinea
border. Their origins have been discussed extensively (Donohue 2008, Evans
2009, Hammarström 2009, Plank 2009) and need not be repeated here.
In addition, there are a number of languages which have been claimed to
be base-6 in the literature but which are not base-6 according to the de?nition
used in this paper (cf. Plank 2009, Gamble 1980, Beeler 1961, Ibarra Grasso
1939b). A few require comment. One early attestation of Balanta (Northern
Atlantic/Atlantic-Congo, Senegal/Guinea Bissau) has additions of 6 for the
numbers 7-12 (Koelle 1854). But since we do not know the continuation
beyond 12, it is unsure whether the 6:s generalize (cf. Wilson 1961a). Also,
later attestations give di?erent, non-base-6, forms (Wilson 1961b, Quintina
1961, Fudeman 1999). Similarly, Less Traditional Tiwi (Isolate, Australia)
may have formed some numbers in the range 7-10 with 6 (Lee 1987:96-100),
but not further.

 

7: septenary
Septenary systems are very rare, as few natural objects consistently have seven distinctive features. Traditionally, it occurs in week-related timing. It has been suggested that the Palikur language has a base-seven system, but this is dubious.[14]

 

8: octal
Main article: Octal
Octal counting systems are based on the number 8. Examples can be found in the Yuki language of California and in the Pamean languages of Mexico, because the Yuki and Pame keep count by using the four spaces between their fingers rather than the fingers themselves.[15]

By Native Americans
The Yuki language in California and the Pamean languages[1] in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves.[2]

By Europeans
It has been suggested that the reconstructed Proto-Indo-European word for "nine" might be related to the PIE word for "new". Based on this, some have speculated that proto-Indo-Europeans used an octal number system, though the evidence supporting this is slim.[3]
In 1668 John Wilkins in An Essay towards a Real Character, and a Philosophical Language proposed use of base 8 instead of 10 "because the way of Dichotomy or Bipartition being the most natural and easie kind of Division, that Number is capable of this down to an Unite".[4]
In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number system based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult and instead proposed 8 as the base. In 1718 Swedenborg wrote (but did not publish) a manuscript: "En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10" ("A new arithmetic (or art of counting) which changes at the Number 8 instead of the usual at the Number 10"). The numbers 1-7 are there denoted by the consonants l, s, n, m, t, f, u (v) and zero by the vowel o. Thus 8 = "lo", 16 = "so", 24 = "no", 64 = "loo", 512 = "looo" etc. Numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule.[5]
Writing under the pseudonym "Hirossa Ap-Iccim" in The Gentleman's Magazine, (London) July 1745, Hugh Jones proposed an octal system for British coins, weights and measures. "Whereas reason and convenience indicate to us an uniform standard for all quantities; which I shall call the Georgian standard; and that is only to divide every integer in each species into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho' all nations count universally by tens (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision ten is uncapable...." In a later treatise on Octave computation (1753) Jones concluded: "Arithmetic by Octaves seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic."[6]
In 1801, James Anderson criticized the French for basing the metric system on decimal arithmetic. He suggested base 8, for which he coined the term octal. His work was intended as recreational mathematics, but he suggested a purely octal system of weights and measures and observed that the existing system of English units was already, to a remarkable extent, an octal system.[7]
In the mid 19th century, Alfred B. Taylor concluded that "Our octonary [base 8] radix is, therefore, beyond all comparison the "best possible one" for an arithmetical system." The proposal included a graphical notation for the digits and new names for the numbers, suggesting that we should count "un, du, the, fo, pa, se, ki, unty, unty-un, unty-du" and so on, with successive multiples of eight named "unty, duty, thety, foty, paty, sety, kity and under." So, for example, the number 65 (101 in octal) would be spoken in octonary as under-un.[8][9] Taylor also republished some of Swedenborg's work on octal as an appendix to the above-cited publications.
 

Northern Pame (Otopamean/Otomanguean, Mexico), the sole case of a base-
8 language (attested up to 32) which does not have 4 as a sub-base is pre-
sented and discussed in Avelino (2006), though 5-8 have etymologies which
involve 5.

 

9: nonary
It has been suggested that Nenets has a base-nine system.[14]

 

10: decimal
Main article: Decimal
A majority of traditional number systems are decimal. This dates back at least to the ancient Egyptians, who used a wholly decimal system. Anthropologists hypothesize this may be due to humans having five digits per hand, ten in total.[13][16] There are many regional variations including:

Western system: based on thousands, with variants (see English numerals)
Indian system: crore, lakh (see Indian numbering system. Indian numerals)
East Asian system: based on ten-thousands (see below)
 

12: duodecimal
Main article: Duodecimal
Duodecimal systems are based on 12.

These include:

Chepang language of Nepal,
Mahl language of Minicoy Island in India
Nigerian Middle Belt areas such as Janji, Kahugu and the Nimbia dialect of Gwandara.
Melanesia[citation needed]
reconstructed proto-Benue–Congo
Duodecimal numeric systems have some practical advantages over decimal. It is much easier to divide the base digit twelve (which is a highly composite number) by many important divisors in market and trade settings, such as the numbers 2, 3, 4 and 6.

Because of several measurements based on twelve,[17] many Western languages have words for base-twelve units such as dozen, gross and great gross, which allow for rudimentary duodecimal nomenclature, such as "two gross six dozen" for 360. Ancient Romans used a decimal system for integers, but switched to duodecimal for fractions, and correspondingly Latin developed a rich vocabulary for duodecimal-based fractions (see Roman numerals). A notable fictional duodecimal system was that of J. R. R. Tolkien's Elvish languages, which used duodecimal as well as decimal.

Languages in the Nigerian Middle Belt Janji, Gbiri-Niragu, Piti, and the Nimbia dialect of Gwandara; Chepang language of Nepal, and the Mahl dialect of Maldivian; dozen-gross-great gross counting; 12-hour clock and months timekeeping; years of Chinese zodiac; foot and inch; Roman fractions.

Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara;[2] the Chepang language of Nepal,[3] and the Maldivian language (Dhivehi) of the people of the Maldives and Minicoy Island in India are known to use duodecimal numerals.

Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they come from Proto-Germanic *ainlif and *twalif (meaning, respectively one left and two left), suggesting a decimal rather than duodecimal origin.[4][5]

Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day (although at some point this was changed to 24). Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 12 old British pence in a shilling, 24 (12×2) hours in a day, and many other items counted by the dozen, gross (144, square of 12) or great gross (1728, cube of 12). The Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. Pre-decimalisation, Ireland and the United Kingdom used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling or Irish pound), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.

Dhivehi (Indo-Aryan/Indo-European, Maldives) has an early attested (Gray
1878) but long extinct base-12 which is attested up to 96 thanks to the e?orts
of Fritz (2002:107-123). 10 Apart from that case, there are base-12 systems in
the Plateau area of northern Nigeria. The ?rst known attestations of such
systems 11 come from the famous Polyglotta Africana by Koelle (1854) which
includes numerals 1-20 in a number of West African languages and the ?rst
proclamation of duodecimality as a system appears to be Schubert (1888).
As shown in Table 3, we have tried to collect all independent attestations
that have been published, or, unpublished but available on the internet. 12
However, not all of them are necessarily independent as this information is
not always deducible from the text. It is likely that there are a few more
attestations in publications that we do not have access to. For many, if not
all, other sources on the same varieties attest base-10 rather than base-12,
which means that the base-12 systems are currently under pressure.

The base-12 systems occur only in languages in the area of Jos plateau
of Nigeria, but which belong to di?erent (sub-)families, namely Plateau
(Atlantic-Congo), East Kainji (Atlantic-Congo), West Chadic (Afro-Asiatic),
Adamawa (Atlantic-Congo) and Jarawan Bantu (Atlantic-Congo). A root
resembling #sok for 12, with plausible sound correspondences (Gerhardt re-
constructs *suak), is widespread in Plateau, wherefore it is very likely that
base-12 is old in Plateau. The same root occurs in Jarawan Bantu and Ron
of Da?o, both of which are isolated instance of this root, or indeed base-
12, in their respective families, so borrowing from (proto-southwest) Plateau
is highly likely (if not certain, as concluded by Maddieson and Williamson
1975:136 and Gerhardt 1997:140-141 for Jarawan Bantu). In East Kainji and
the Beromic subgroup of Plateau, a root #kuri occurs for 12, which makes
a borrowing in either direction likely. Furthermore, #piri is 12 in Gure and
Kahugu (East Kainji) and #zowa is 12 in Ake and Koro (Plateau) and yet
other roots for 12 appear in the remaining West Chadic cases. Since base-12
is so rare in the languages of the world, the variety of non-ancient roots sug-
gest that a base-12 system may be borrowed even without key morphemes.
The root for 12 in the alleged Mumuye variety with base-12 allegation is not
known.
There are no obvious clues as to the unusual choice of 12 as a base. A
few of the base-12 languages in Meek (1931) have hand gestures that often
are used accompanying the spoken expression. A combination of ?ngers and
eyes make up 12 in at least one of these cases, but no traces of words meaning
eye, hand or ?nger can be found in the corresponding spoken expressions. On
the other hand, although not a base, 12 bears a special position in several
modern European languages too, with a special word like 'dozen' and an
elevated frequency (Dehaene and Mehler 1992). The reason(s) for this is not
well-understood either.

15

There appears to be only one case of a language attested as base-15, at least
for a number of decades, namely Huli (East New Guinea Highlands/Trans
New Guinea, Papua New Guinea) of the southern highland fringes. It is
clearly an original body-tally system with a cycle of 29 ? midway/centerpoint
is thus 15 ? which under in?uence from a Tok Pisin base-system turned into
base-15 (Cheetham 1978, Lomas 1988).


 

16: hexadecimal
Main article: hexadecimal
Hexadecimal systems are based on 16.

The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hexadecimal calculations such as additions and subtractions.[18]

South Asian monetary systems were base-16. One rupee in Pakistan and India was divided into 16 annay. A single anna was subdivided into four paisa or twelve pies (thus there were 64 paise or 192 pies in a rupee). The anna was demonetised as a currency unit when India decimalised its currency in 1957, followed by Pakistan in 1961.

 

20: vigesimal
Main article: Vigesimal
Vigesimal numbers use the number 20 as the base number for counting. Anthropologists are convinced the system originated from digit counting, as did bases five and ten, twenty being the number of human fingers and toes combined.[13][19] The system is in widespread use across the world. Some include the classical Mesoamerican cultures, still in use today in the modern indigenous languages of their descendants, namely the Nahuatl and Mayan languages (see Maya numerals). A modern national language which uses a full vigesimal system is Dzongkha in Bhutan.

Partial vigesimal systems are found in some European languages: Basque, Celtic languages, French (from Celtic), Danish, and Georgian. In these languages the systems are vigesimal up to 99, then decimal from 100 up. That is, 140 is 'one hundred two score', not *seven score, and there is no numeral for 400.

The term score originates from tally sticks, and is perhaps a remnant of Celtic vigesimal counting. It was widely used to learn the pre-decimal British currency in this idiom: "a dozen pence and a score of bob", referring to the 20 shillings in a pound. For Americans the term is most known from the opening of the Gettysburg Address: "Four score and seven years ago our fathers...".

In many European languages, 20 is used as a base, at least with respect to the linguistic structure of the names of certain numbers (though a thoroughgoing consistent vigesimal system, based on the powers 20, 400, 8000 etc., is not generally used).

The Open Location Code, used for encoding geographic areas uses a base 20 encoding of coordinates.[1]
Africa
Vigesimal systems are common in Africa, for example in Yoruba.

Ogún, 20, is the basic numeric block. Ogójì, 40, (Ogún-meji) = 20 multiplied by 2 (èjì). Ogota, 60, (Ogún-mẹ̀ta) = 20 multiplied by 3 (ẹ̀ta). Ogorin, 80, (Ogún-mẹ̀rin) = 20 multiplied by 4 (ẹ̀rin). Ogorun, 100, (Ogún-màrún) = 20 multiplied by 5 (àrún).

16 (Ẹẹ́rìndílógún) = 4 less than 20.

17 (Etadinlogun) = 3 less than 20.

18 (Eejidinlogun) = 2 less than 20.

19 (Okandinlogun) = 1 less than 20.

21 (Okanlelogun) = 1 increment on 20.

22 (Eejilelogun) = 2 increment on 20.

23 (Etalelogun) = 3 increment on 20.

24 (Erinlelogun) = 4 increment on 20.

25 (Aarunlelogun) = 5 increment on 20.

Americas
Twenty was a base in the Maya and Aztec number systems. The Maya used the following names for the powers of twenty: kal (20), bak (202 = 400), pic (203 = 8,000), calab (204 = 160,000), kinchil (205 = 3,200,000) and alau (206 = 64,000,000). See also Maya numerals and Maya calendar, Mayan languages, Yucatec. The Aztec called them: cempoalli (1 × 20), centzontli (1 × 400), cenxiquipilli (1 × 8,000), cempoalxiquipilli (1 × 20 × 8,000 = 160,000), centzonxiquipilli (1 × 400 × 8,000 = 3,200,000) and cempoaltzonxiquipilli (1 × 20 × 400 × 8,000 = 64,000,000). Note that the ce(n/m) prefix at the beginning means "one" (as in "one hundred" and "one thousand") and is replaced with the corresponding number to get the names of other multiples of the power. For example, ome (2) × poalli (20) = ompoalli (40), ome (2) × tzontli (400) = ontzontli (800). The -li in poalli (and xiquipilli) and the -tli in tzontli are grammatical noun suffixes that are appended only at the end of the word; thus poalli, tzontli and xiquipilli compound together as poaltzonxiquipilli (instead of *poallitzontlixiquipilli). (See also Nahuatl language.)
The Tlingit people use base 20.

Inuit numerals
The Kaktovik Inupiaq numerals uses a base 20 system. In 1994, Students from Kaktovik, Alaska, came up with the Kaktovik Inupiaq numerals in 1994. Before the numerals had been developed, the Inuit names had been falling out of favor.[2]
Asia
Dzongkha, the national language of Bhutan, has a full vigesimal system, with numerals for the powers of twenty 20, 400, 8,000, and 160,000.
Atong, a language spoken in the South Garo Hills of Meghalaya state, Northeast India, and adjacent areas in Bangladesh, has a full vigesimal system that is nowadays considered archaic.[3]
In Santali, a Munda language of India, "fifty" is expressed by the phrase bār isī gäl, literally "two twenty ten."[4] Likewise, in Didei, another Munda language spoken in India, complex numerals are decimal to 19 and decimal-vigesimal to 399.[5]
The Burushaski number system is base 20. For example, 20 altar, 40 alto-altar (2 times 20), 60 iski-altar (3 times 20) etc.
In East Asia, the Ainu language also uses a counting system that is based around the number 20. “hotnep” is 20, “wanpe etu hotnep” (ten more until two twenties) is 30, “tu hotnep” (two twenties) is 40, “ashikne hotnep” (five twenties) is 100. Subtraction is also heavily used, e.g. “shinepesanpe” (one more until ten) is 9.[citation needed]
The Chukchi language has a vigesimal numeral system.[6]
Oceania
There is some evidence of base-20 usage in the Māori language of New Zealand as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tama-hokotahi, referring to a great warrior ("the one man equal to 20").

In Europe
Etymology
Vigesimal is derived from the Latin adjective vicesimus.

Examples
Twenty (vingt) is used as a base number in the French language names of numbers from 70 to 99, except in the French of Belgium, Switzerland, the Democratic Republic of the Congo, Rwanda, the Aosta Valley and the Channel Islands. For example, quatre-vingts, the French word for "80", literally means "four-twenties"; soixante-dix, the word for "70", is literally "sixty-ten"; soixante-quinze ("75") is literally "sixty-fifteen"; quatre-vingt-sept ("87") is literally "four-twenties-seven"; quatre-vingt-dix ("90") is literally "four-twenties-ten"; and quatre-vingt-seize ("96") is literally "four-twenties-sixteen". However, in the French of Belgium, Switzerland, the Democratic Republic of the Congo, Rwanda, the Aosta Valley, and the Channel Islands, the numbers 70 and 90 generally have the names septante and nonante. Therefore, the year 1996 is "mille neuf cent quatre-vingt-seize" in Parisian French, but it is "mille neuf cent nonante-six" in Belgian French. In Switzerland, "80" can be quatre-vingts (Geneva, Neuchâtel, Jura) or huitante (Vaud, Valais, Fribourg); octante is also in use in rural parts of Southern France.[citation needed]
Twenty (tyve) is used as a base number in the Danish language names of numbers from 50 to 99. For example, tres (short for tresindstyve) means 3 times 20, i.e. 60. However, Danish numerals are not vigesimal since it is only the names of some of the tens that are etymologically formed in a vigesimal way. In contrast with e.g. French quatre-vingt-seize, the units only go from zero to nine between each ten which is a defining trait of a decimal system. For details, see Danish numerals.
Twenty (ugent) is used as a base number in the Breton language names of numbers from 40 to 49 and from 60 to 99. For example, daou-ugent means 2 times 20, i.e. 40, and triwec'h ha pevar-ugent (literally "three-six and four-twenty") means 3×6 + 4×20, i.e. 98. However, 30 is tregont and not *dek ha ugent ("ten and twenty"), and 50 is hanter-kant ("half-hundred").
Twenty (ugain) is used as a base number in the Welsh language, although in the latter part of the 20th century[citation needed] a decimal counting system has come to be preferred. However, the vigesimal system exclusively is used for ordinal numbers. Deugain means 2 times 20 i.e. 40, trigain means 3 times 20 i.e. 60, etc. Dau ar bymtheg ar ddeugain means 57 (two upon fifteen upon twoscore). Prior to its withdrawal from circulation in 1970, papur chweugain (note of sixscore) was the nickname for the ten-shilling (= 120 pence) note.
Twenty (fichead) is traditionally used as a base number in Scottish Gaelic, with deich ar fhichead or fichead 's a deich being 30 (ten over twenty, or twenty and ten), dà fhichead 40 (two twenties), dà fhichead 's a deich 50 (two twenty and ten) / leth-cheud 50 (half a hundred), trì fichead 60 (three twenties) and so on up to naoidh fichead 180 (nine twenties). Nowadays a decimal system is taught in schools, but the vigesimal system is still used by many, particularly older speakers.
Twenty (njëzet) is used as a base number in the Albanian language. The word for 40 (dyzet) means two times 20 (some Gheg subdialects, however, use 'katërdhetë'). The Arbëreshë in Italy may use 'trizetë' for 60. Formerly, 'katërzetë' was also used for 80. Today Cham Albanians in Greece use all zet numbers. Basically 20 means 1 zet, 40 means 2 zet, 60 means 3 zet and 80 means 4 zet.
Twenty (otsi) is used as a base number in the Georgian language. For example, 31 (otsdatertmeti) literally means, twenty-and-eleven. 67 (samotsdashvidi) is said as, “three-twenty-and-seven”.
Twenty (tqa) is used as a base number in the Nakh languages.
Twenty (hogei) is used as a base number in the Basque language for numbers up to 100 (ehun). The words for 40 (berrogei), 60 (hirurogei) and 80 (laurogei) mean "two-score", "three-score" and "four-score", respectively. For example, the number 75 is called hirurogeita hamabost, lit. "three-score-and ten-five". The Basque nationalist Sabino Arana proposed a vigesimal digit system to match the spoken language,[7] and, as an alternative, a reform of the spoken language to make it decimal,[8] but both are mostly forgotten.[9]
Twenty (dwisti or dwujsti) is used as a base number in the Resian dialect of the Slovenian language in Italy's Resia Valley. 60 is expressed by trïkrat dwisti (3×20), 70 by trïkrat dwisti nu dësat (3×20 + 10), 80 by štirikrat dwisti (4×20) and 90 by štirikrat dwisti nu dësat (4×20 + 10).[10][11]
In the old British currency system (pre-1971), there were 20 shillings (worth 12 pence each) to the pound. Under the decimal system introduced in 1971 (1 pound equals 100 new pence instead of 240 pence in the old system), the shilling coins still in circulation were re-valued at 5 pence (no more were minted and the shilling coin was demonetised in 1990).
In the imperial weight system there are twenty hundredweight in a ton.
In English, counting by the score has been used historically, as in the famous opening of the Gettysburg Address "Four score and seven years ago…", meaning eighty-seven (87) years ago. In the Authorised Version of the Bible the term score is used over 130 times although only when prefixed by a number greater than one while a single "score" is always expressed as twenty. The use of the term score to signify multiples of twenty has fallen into disuse in modern English.
Other languages have terms similar to the old English score, for example Danish and Norwegian snes.
In regions where traces of the Brythonic Celtic languages have survived in dialect, sheep enumeration systems that are vigesimal are recalled to the present day. See Yan Tan Tethera.
Related observations
Among multiples of 10, 20 is described in a special way in some languages. For example, the Spanish words treinta (30) and cuarenta (40) consist of "tre(3)+inta (10 times)", "cuar(4)+enta (10 times)", but the word veinte (20) is not presently connected to any word meaning "two" (although historically it is[12]). Similarly, in Semitic languages such as Arabic and Hebrew, the numbers 30, 40 ... 90 are expressed by morphologically plural forms of the words for the numbers 3, 4 ... 9, but the number 20 is expressed by a morphologically plural form of the word for 10. The Japanese language has a special word (hatachi) for 20 years (of age), and for the 20th day of the month (hatsuka).
In some languages (e.g. English, Slavic languages and German), the names of the two-digit numbers from 11 to 19 consist of one word, but the names of the two-digit numbers from 21 on consist of two words. So for example, the English words eleven (11), twelve (12), thirteen (13) etc., as opposed to twenty-one (21), twenty-two (22), twenty-three (23), etc. In French, this is true up to 16. In a number of other languages (such as Hebrew), the names of the numbers from 11-19 contain two words, but one of these words is a special "teen" form, which is different from the ordinary form of the word for the number 10, and it may in fact be only found in these names of the numbers 11-19.
Cantonese[13] and Wu Chinese frequently use the single unit 廿 (Cantonese yàh, Shanghainese nyae or ne, Mandarin niàn) for twenty, in addition to the fully decimal 二十 (Cantonese yìh sàhp, Shanghainese el sah, Mandarin èr shí) which literally means "two ten". Equivalents exist for 30 and 40 (卅 and 卌 respectively: Mandarin sà and xì), but these are more seldom used. This is a historic remnant of a vigesimal system.[citation needed]
Although Khmer numerals have represented a decimal positional notation system since at least the 7th century, Old Khmer, or Angkorian Khmer, also possessed separate symbols for the numbers 10, 20, and 100. Each multiple of 20 or 100 would require an additional stroke over the character, so the number 47 was constructed using the 20 symbol with an additional upper stroke, followed by the symbol for number 7. This suggests that spoken Angkorian Khmer used a vigesimal system.
Thai uses the term ยี่สิบ (yi sip) for 20. Other multiples of ten consist of the base number, followed by the word for ten, e.g. สามสิบ (sam sip), lit. three ten, for thirty. The yi of yi sip is different from the number two in other positions, which is สอง (song). Nevertheless, yi sip is a loan word from Chinese.
Lao similarly forms multiples of ten by putting the base number in front of the word ten, so ສາມສິບ (sam sip), litt. three ten, for thirty. The exception is twenty, for which the word ຊາວ (xao) is used. (ซาว sao is also used in the North-Eastern and Northern dialects of Thai, but not in standard Thai.)
The Kharosthi numeral system behaves like a partial vigesimal system.


 

24: quadrovigesimal
The Sko languages have a base-24 system with a sub-base of 6.

 

32: duotrigesimal
Main article: Duotrigesimal
Ngiti has base 32.

 

60: sexagesimal
Main article: Sexagesimal
Ekari has a base-60 system. Sumeria had a base-60 system with a decimal sub-base (perhaps a conflation of the decimal and a duodecimal systems of its constituent peoples), which was the origin of the numbering of modern degrees, minutes, and seconds.

 

80: octogesimal
Supyire is said to have a base-80 system; it counts in twenties (with 5 and 10 as sub-bases) up to 80, then by eighties up to 400, and then by 400s (great scores).

kàmpwóò ŋ̀kwuu sicyɛɛré ná béé-tàànre ná kɛ́ ná báár-ìcyɛ̀ɛ̀rè
fourhundred eighty four and twenty-three and ten and five-four
799 [i.e. 400 + (4 x 80) + (3 x 20) + {10 + (5 + 4)}]’

Rare Second Bases
Some rarities in the next higher bases after 5, 10 or 20 are as follows:
10-40: Pech (Paya/Chibchan, Honduras) as of Conzemius (1928:264-265)
and Hawaiian (Oceanic/Austronesian, USA) until it restructured to 10-
100 under foreign pressure (von Chamisso 1837, Dwight 1848, Hughes
1982).
5-20-40: Southwestern Pomo (Pomoan, USA) in one attestation (Closs 1986:35-
41).
10-60: Attested (Drabbe 1952) in Ekagi (Paniai Lakes/Trans New Guinea,
Indonesia) and Ntomba (Bantu/Atlantic-Congo, DRC) until it restruc-
tured to 10-100 under foreign pressure (Gilliard 1928, 1924).
5-10-20-60: Famously known from the long extinct Sumerian (Isolate, Iraq),
see, e.g. Powell (1972).
(5-)10-20-(60/)80: Attested in Mande (Monteil 1905, Dombrowski and
Dombrowski 1991, Delafosse 1928, Hartner 1943), Dogon (Calame-
Griaule 1968), Gur (Carlson 1994, Welmers 1950:167-169) and Bangi
Me (Blench 2005) languages in a relatively small area in West Africa,
wherefore an areal connection is almost certain. In the Mande attes-
tations, the systems vary between 60 and 80 as per a certain root that
sometimes means 60 and sometimes 80.

5-25: Gumatj (Anindilyakwa, Australia) is described, with ample examples,
to be 5-25 (upto 625). However, one would not usually use exact num-
bers for counting this high in this language and there is a certain like-
lihood that the system was extended this high only at the time of
elicitation with one single speaker (Harris 1982, Sobek 2005), espe-
cially since an earlier attestation, if anything, shows a commonplace
vigesimal count (Tindale 1928:128-129). At least one speaker of Biwat
(Yuat River, Papua New Guinea) appears to have made the same 5-25
innovation (McElvenny 2006), as two other earlier attestations rather
show a commonplace 5-20 system (Haberland and Seyfarth 1974, Mead
1932) 13 . It is remarkable that there is no 14 incontestable attestation of
a 5-25 system that extends to a whole speech community. The con-
trast with 5-20 systems, which are ubiquitous, reveals much as to the
evolution of normed number expression within a community.

2 Binary Digital computing, imperial and customary volume (bushel-kenning-peck-gallon-pottle-quart-pint-cup-gill-jack-fluid ounce-tablespoon)
3 Ternary Cantor set (all points in [0,1] that can be represented in ternary with no 1s); counting Tasbih in Islam; hand-foot-yard and teaspoon-tablespoon-shot measurement systems; most economical integer base
4 Quaternary Data transmission, DNA bases and Hilbert curves; Chumashan languages, and Kharosthi numerals
5 Quinary Gumatj, Ateso, Nunggubuyu, Kuurn Kopan Noot, and Saraveca languages; common count grouping e.g. tally marks
6 Senary Diceware, Ndom, Kanum, and Proto-Uralic language (suspected)
7 Septenary Weeks timekeeping
8 Octal Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, compact notation for binary numbers, Xiantian (I Ching, China)
9 Nonary Base9 encoding; compact notation for ternary
10 Decimal / Denary(Computing) Most widely used by modern civilizations[8][9][10]
11 Undecimal Jokingly proposed during the French Revolution to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal; check digits in ISBN
12 Duodecimal Languages in the Nigerian Middle Belt Janji, Gbiri-Niragu, Piti, and the Nimbia dialect of Gwandara; Chepang language of Nepal, and the Mahl dialect of Maldivian; dozen-gross-great gross counting; 12-hour clock and months timekeeping; years of Chinese zodiac; foot and inch; Roman fractions
13 Tridecimal Base13 encoding; Conway base 13 function
14 Tetradecimal Programming for the HP 9100A/B calculator[11] and image processing applications;[12] pound and stone
15 Pentadecimal Telephony routing over IP, and the Huli language
16 Hexadecimal Base16 encoding; compact notation for binary data; tonal system; ounce and pound
17 Heptadecimal Base17 encoding
18 Octodecimal Base18 encoding
19 Enneadecimal Base19 encoding
20 Vigesimal Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
21 Unvigesimal Base21 encoding
22 Duovigesimal Base22 encoding
23 Trivigesimal Kalam language, Kobon language[citation needed]
24 Tetravigesimal 24-hour clock timekeeping; Kaugel language
25 Pentavigesimal Base25 encoding
26 Hexavigesimal Base26 encoding; sometimes used for encryption or ciphering,[13] using all letters
27 Heptavigesimal Septemvigesimal Telefol and Oksapmin languages. Mapping the nonzero digits to the alphabet and zero to the space is occasionally used to provide checksums for alphabetic data such as personal names,[14] to provide a concise encoding of alphabetic strings,[15] or as the basis for a form of gematria.[16] Compact notation for ternary.
28 Octovigesimal Base28 encoding; months timekeeping
29 Enneavigesimal Base29
30 Trigesimal The Natural Area Code, this is the smallest base such that all of 1/2 to 1/6 terminate, a number n is a regular number if and only if 1/n terminates in base 30
31 Untrigesimal Base31
32 Duotrigesimal Base32 encoding and the Ngiti language
33 Tritrigesimal Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong
34 Tetratrigesimal Using all numbers and all letters except I and O
35 Pentatrigesimal Using all numbers and all letters except O
36 Hexatrigesimal Base36 encoding; use of letters with digits
37 Heptatrigesimal Base37; using all numbers and all letters of the Spanish alphabet
38 Octotrigesimal Base38 encoding; use all duodecimal digits and all letters
40 Quadragesimal DEC Radix-50₈ encoding used to compactly represent file names and other symbols on Digital Equipment Corporation computers. The character set is a subset of ASCII consisting of space, upper case letters, the punctuation marks "$", ".", and "%", and the numerals.
42 Duoquadragesimal Base42 encoding
45 Pentaquadragesimal Base45 encoding
48 Octoquadragesimal Base48 encoding
49 Enneaquadragesimal Related to base 7
50 Quinquagesimal Base50 encoding
52 Duoquinquagesimal Base52 encoding, a variant of Base62 without vowels[17]
54 Tetraquinquagesimal Base54 encoding
56 Hexaquinquagesimal Base56 encoding, a variant of Base58[18]
57 Heptaquinquagesimal Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[19] or I, 1, l, 0, and O [20]
58 Octoquinquagesimal Base58 encoding
60 Sexagesimal Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[21] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages
62 Duosexagesimal Base62 encoding, using 0–9, A–Z, and a–z
64 Tetrasexagesimal Base64 encoding; I Ching in China.
This system is conveniently coded into ASCII by using the 26 letters of the Latin alphabet in both upper and lower case (52 total) plus 10 numerals (62 total) and then adding two special characters (for example, YouTube video codes use the hyphen and underscore characters, - and _ to total 64).[citation needed]
72 Duoseptagesimal Base72 encoding
80 Octogesimal Base80 encoding
81 Unoctogesimal Base81 encoding, using as 81=34 is related to ternary
85 Pentoctogesimal Ascii85 encoding. This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME-64 encoding, since 855 is only slightly bigger than 232. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.
90 Nonagesimal Related to Goormaghtigh conjecture for the generalized repunit numbers.
91 Unnonagesimal Base91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).
92 Duononagesimal Base92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[22]
93 Trinonagesimal Base93 encoding, using all of ASCII printable characters except for "," (0x27) and "-" (0x3D) as well as the Space character. "," is reserved for delimiter and "-" is reserved for negation.[23]
94 Tetranonagesimal Base94 encoding, using all of ASCII printable characters.[24]
95 Pentanonagesimal Base95 encoding, a variant of Base94 with the addition of the Space character.[25]
96 Hexanonagesimal Base96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits
100 Centesimal As 100=102, these are two decimal digits
120 Centevigesimal Base120 encoding
121 Centeunvigesimal Related to base 11
125 Centepentavigesimal Related to base 5
128 Centeoctovigesimal Using as 128=27
144 Centetetraquadragesimal Two duodecimal digits
256 Duocentehexaquinquagesimal Base256 encoding, as 256=28
360 Trecentosexagesimal Degrees for angle

:

    `n^2` `n^3` `n^4` `n^5`   `2^n` `3^n` `4^n` `5^n`   `n!`
4   `2^2`         `2^2`          
6                       `3!`
8     `2^3`       `2^3`          
9   `3^2`           `3^2`        
16   `4^2`   `2^4`     `2^4`   `4^2`      
24                       `4!`
25   `5^2`               `5^2`    
27     `3^3`         `3^3`        
32         `2^5`   `2^5`          
36   `6^2`                    
49   `7^2`                    
64   `8^2` `4^3`       `2^6`   `4^3`      
81   `9^2`   `3^4`       `3^4`        
100   `10^2`                    
120                       `5!`
121   `11^2`                    
125     `5^3`             `5^3`    
128             `2^7`          
144   `12^2`                    
169   `13^2`                    
196   `14^2`                    
216     `6^3`                  
225   `15^2`                    
243         `3^5`     `3^5`        
256   `16^2`   `4^4`     `2^8`   `4^4`      
289   `17^2`                    
324   `18^2`                    
343     `7^3`                  
361   `19^2`                    
400   `20^2`                    
441   `21^2`                    
484   `22^2`                    
512     `8^3`       `2^9`          
529   `23^2`                    
576   `24^2`                    
625   `25^2`   `5^4`           `5^4`    
676   `26^2`                    
720                       `6!`
729   `27^2` `9^3`         `3^6`        
784   `28^2`                    
841   `29^2`                    
900   `30^2`                    
961   `31^2`                    
1000     `10^3`                  
1024   `32^2`     `4^5`   `2^10`   `4^5`      

    `n^2` `n^3` `n^4` `n^5`   `2^n` `3^n`   `n!` `n!!`   `7n` `11n` `13n` `17n` `19n` `23n` `29n`
4   `2^2`         `2^2`                        
6                   `3!`                  
8     `2^3`       `2^3`       `4!!`                
9   `3^2`           `3^2`                      
15                     `5!!`                
16   `4^2`   `2^4`     `2^4`                        
21                         `7*3`            
24                   `4!`                  
25   `5^2`                                  
27     `3^3`         `3^3`                      
32         `2^5`   `2^5`                        
33                           `11*3`          
35                         `7*5`            
36   `6^2`                                  
39                             `13*3`        
48                     `6!!`                
49   `7^2`                     `7*7`            
51                               `17*3`      
55                           `11*5`          
57                                 `19*3`    
64   `8^2` `4^3`       `2^6`                        
65                             `13*5`        
69                                   `23*3`  
77                         `7*11` `11*7`          
81   `9^2`   `3^4`       `3^4`                      
85                               `17*5`      
87                                     `29*3`
91                         `7*13`   `13*7`        
95                                 `19*5`    
100   `10^2`                                  
105                     `7!!`                
115                                   `23*5`  
119                         `7*17`     `17*7`      
120                   `5!`                  
121   `11^2`                       `11*11`          
125     `5^3`                                
128             `2^7`                        
133                         `7*19`       `19*7`    
143                           `11*13` `13*11`        
144   `12^2`                                  
145                                     `29*5`
161                         `7*23`         `23*7`  
169   `13^2`                         `13*13`        
187                           `11*17`   `17*11`      
196   `14^2`                                  
203                         `7*29`           `29*7`
209                           `11*19`     `19*11`    
216     `6^3`                                
217                         `7*31`            
221                             `13*17` `17*13`      
225   `15^2`                                  
243         `3^5`     `3^5`                      
247                             `13*19`   `19*13`    
253                           `11*23`       `23*11`  
256   `16^2`   `4^4`     `2^8`                        
259                         `7*37`            
287                         `7*41`            
289   `17^2`                           `17*17`      
299                             `13*23`     `23*13`  
301                         `7*43`            
319                           `11*29`         `29*11`
323                               `17*19` `19*17`    
324   `18^2`                                  
329                         `7*47`            
341                           `11*31`          
343     `7^3`                   `7*7*7`            
361   `19^2`                             `19*19`    
371                         `7*53`            
377                             `13*29`       `29*13`
384                     `8!!`                
391                               `17*23`   `23*17`  
400   `20^2`                                  
403                             `13*31`        
407                           `11*37`          
413                         `7*59`            
427                         `7*61`            
437                                 `19*23` `23*19`  
441   `21^2`                                  
451                           `11*41`          
469                         `7*67`            
473                           `11*43`          
481                             `13*37`        
484   `22^2`                                  
493                               `17*29`     `29*17`
497                         `7*71`            
511                         `7*73`            
512     `8^3`       `2^9`                        
517                           `11*47`          
527                               `17*31`      
529   `23^2`                               `23*23`  
533                             `13*41`        
539                         `7*7*11` `11*7*7`          
551                                 `19*29`   `29*19`
553                         `7*79`            
559                             `13*43`        
576   `24^2`                                  
581                         `7*83`            
583                           `11*53`          
589                                 `19*31`    
611                             `13*47`        
623                         `7*89`            
625   `25^2`   `5^4`                              
629                               `17*37`      
637                         `7*7*13`   `13*7*7`        
649                           `11*59`          
667                                   `23*29` `29*23`
671                           `11*61`          
676   `26^2`                                  
679                         `7*97`            
689                             `13*53`        
697                               `17*41`      
703                                 `19*37`    
707                         `7*101`            
713                                   `23*31`  
720                   `6!`                  
721                         `7*103`            
729   `27^2` `9^3`         `3^6`                      
731                               `17*43`      
737                           `11*67`          
749                         `7*107`            
763                         `7*109`            
767                             `13*59`        
779                                 `19*41`    
781                           `11*71`          
784   `28^2`                                  
791                         `7*113`            
793                             `13*61`        
799                               `17*47`      
803                           `11*73`          
817                                 `19*43`    
841   `29^2`                                 `29*29`
847                         `7*11*11` `11*7*11`          
851                                   `23*37`  
869                           `11*79`          
871                             `13*67`        
889                         `7*127`            
893                                 `19*47`    
899                                     `29*31`
900   `30^2`                                  
901                               `17*53`      
913                           `11*83`          
917                         `7*131`            
923                             `13*71`        
931                         `7*7*19`            
943                                   `23*41`  
945                     `9!!`                
949                             `13*73`        
959                         `7*137`            
961   `31^2`                                  
973                         `7*139`            
979                           `11*89`          
989                                   `23*43`  
1000     `10^3`                                
1001                         `7*11*13` `11*7*13` `13*7*11`        
1003                               `17*59`      
1007                                 `19*53`    
1024   `32^2`     `4^5`   `2^10`