# Quinary

Quinary (base-5 or pental[1][2][3]) is a numeral system with five as the base. A possible origination of a quinary system is that there are five fingers on either hand.

In the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100 and sixty is written as 220.

As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods.

Today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a sub-base system, is sexagesimal, base 60, which used 10 as a sub-base.

Each quinary digit has log25 (approx. 2.32) bits of information.[4]

Few calculators support calculations in the quinary system, except for some Sharp models (including some of the EL-500W and EL-500X series, where it is named the pental system[1][2][3]) since about 2005, as well as the open-source scientific calculator WP 34S. The Python programming language supports conversion of a string to quinary using the int function. For example, if s='101' then the function print(int('101',5)) would return 26.[5]

 × 1 2 3 4 10 11 12 13 14 20 1 1 2 3 4 10 11 12 13 14 20 2 2 4 11 13 20 22 24 31 33 40 3 3 11 14 22 30 33 41 44 102 110 4 4 13 22 31 40 44 103 112 121 130 10 10 20 30 40 100 110 120 130 140 200 11 11 22 33 44 110 121 132 143 204 220 12 12 24 41 103 120 132 144 211 223 240 13 13 31 44 112 130 143 211 224 242 310 14 14 33 102 121 140 204 223 242 311 330 20 20 40 110 130 200 220 240 310 330 400
Quinary Binary Decimal 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 10 11 12 13 14 20 21 22 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 23 24 30 31 32 33 34 40 41 42 43 44 100 1101 1110 1111 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001
 Decimal (periodic part) Quinary (periodic part) Binary (periodic part) 1/2 = 0.5 1/2 = 0.2 1/10 = 0.1 1/3 = 0.3 1/3 = 0.13 1/11 = 0.01 1/4 = 0.25 1/4 = 0.1 1/100 = 0.01 1/5 = 0.2 1/10 = 0.1 1/101 = 0.0011 1/6 = 0.16 1/11 = 0.04 1/110 = 0.010 1/7 = 0.142857 1/12 = 0.032412 1/111 = 0.001 1/8 = 0.125 1/13 = 0.03 1/1000 = 0.001 1/9 = 0.1 1/14 = 0.023421 1/1001 = 0.000111 1/10 = 0.1 1/20 = 0.02 1/1010 = 0.00011 1/11 = 0.09 1/21 = 0.02114 1/1011 = 0.0001011101 1/12 = 0.083 1/22 = 0.02 1/1100 = 0.0001 1/13 = 0.076923 1/23 = 0.0143 1/1101 = 0.000100111011 1/14 = 0.0714285 1/24 = 0.013431 1/1110 = 0.0001 1/15 = 0.06 1/30 = 0.013 1/1111 = 0.0001 1/16 = 0.0625 1/31 = 0.0124 1/10000 = 0.0001 1/17 = 0.0588235294117647 1/32 = 0.0121340243231042 1/10001 = 0.00001111 1/18 = 0.05 1/33 = 0.011433 1/10010 = 0.0000111 1/19 = 0.052631578947368421 1/34 = 0.011242141 1/10011 = 0.000011010111100101 1/20 = 0.05 1/40 = 0.01 1/10100 = 0.000011 1/21 = 0.047619 1/41 = 0.010434 1/10101 = 0.000011 1/22 = 0.045 1/42 = 0.01032 1/10110 = 0.00001011101 1/23 = 0.0434782608695652173913 1/43 = 0.0102041332143424031123 1/10111 = 0.00001011001 1/24 = 0.0416 1/44 = 0.01 1/11000 = 0.00001 1/25 = 0.04 1/100 = 0.01 1/11001 = 0.00001010001111010111

## Usage

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.

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

## Quinary systems all over the world languages [1]

#### Asia

• Sumerian: gesh (1), min (2), esh (3), limmou (4), ia (5), ash (6), i-min (7 = 5+2), i-ssou (8 = 5+3), i-llimou (9 = 5+4)
• Palmyrene
• Khmer: moy (1), pee (2), bay (3), buan (4), pram (5), pram moy (6 = 5+1), pram pii (7), pram buan (8), pram buan (9)
• Bantawa: ʉk (1), hʉk (2), sum- (3), reʈ?(4), ʉk-chuk ("one hand"), ʉkchuk ʉk (6 = 1 hand + 1), ʉkchuk sum- (7=5+2), ʉkchuk sum- (8=5+3), ʉkchuk reʈ (9=5+4), hʉa-chuk ("2 hands")
• Abui: nuku (1), ayoku (2), sua (3), buti (4), yeting (5), talaama (6), yeting-ayoku (7), yeting-sua (8), yeting-buti (9)
• Sawila: sundana / dana (1), yaku (2), tuo (3), araasiku (4), yootine (5), yooting-suno (6=5+1), yooting-yaku (7=5+2), yootingtuo (8=5+3), yootaraasiku (9=5+4)
• Inupiaq: atausiq (1), malġuk (2), piŋasut (3), sisamat (4), tallimat (5), itchaksrat (6), tallimat malġuk (7 = 5+2), tallimat piŋasut (8), quliŋuġutaiḷaq (9)
• Yupik: atauciq (1), malruk (2), pingayun (3), cetaman (4), talliman (5), arvin-glegen (6), malrun-glegen (7 = 2+5), pingayun-legen (8=3+5), qulngunritaraan (9) en alutiik 29: allringuq, mal’uk, pingayun, staama, talliman (5), arwilgen (6), mallruungin (7=2+5), Inglulgen (=3+5), qulnguyan (9)
• Alamblak 30: rpat (1), hosf (2), hos-fi-rpat (3=2+1), hos-fi-hosf (4=2+2), tir yohtt (5), tir yohtti rpat (6=5+1), tir yohtti hosf (7=5+2), tir yohtti hosfirpat (8=5+3), tir yohtti hosfihosf (9=5+4)
• Alyutor (Chukotko-Kamchatka): ǝnnan (1), ŋitaq (2), ŋǝruqqǝ (3), ŋǝraqqǝ (4), mǝllǝŋin (5), ǝnnan-mǝllǝŋ(in) (6=1+5), ŋitaqmǝllǝŋ(in) (7=2+5), ŋǝruqmǝllǝŋ(in) (8=3+5), ŋǝraqmǝllǝŋ(in) (9=4+5)
• Chukchi ( Chukotko-Kamchatka): ynnėn (1), ṇirėq (2), ṇyroqq (3), ṇyraqq (4), mėtḷyṇėn (5), ynnan-mytḷyṇėn (1+5), ṇėr’amytḷyṇėn (2+5), ṇyr’omytḷyṇėn (3+5), ṇyr’amytḷyṇėn (9)
• Itelmen ( Chukotko-Kamchatka): qniŋ (1), касҳ (2), č’oq (3), č’aq (4), kuvumnuk (5), kelvuk (6), etuk-tunuk (7), čo’oq-tunuk (8=3+5), ča’aq-tanak (9=4+5)
• Proto-aleoutian : *attaqa-n (1), *aallax (2), *qaanku (3), *sitsin (4), *tsang (5), *atsung (6), *allung (7), qamtsing (8), *sitsing (9)
• Proto-eskimo: *ataʁuciʁ (1), *malʁugh (2), *pingayut (3), *citama (4), *tadlimat (5), *aʁvinelegh (6), *malʁughneng aʁvinelegh (7=2+5), *ningayuneng aʁvinelegh (8=3+5), *qulengnguʁutengit (9=4+5) en naukanski yuit: ataasiq (1), maalghut (2), pingayut (3), sitamat (4), taɫimat (5), aghvinelek (6), maalghugneng aghvinelek (7=2+5), pingayuneng aghvinelek (8=3+5), qulngughutngilnguq (9)

#### Africa

• Proto-berber: yn (1), sn (2), krad (3), okkoz (4), fuss (5), fuss d yn (6=5+1), fuss d sn (7=5+2), fuss d krad (8=5+3), addad meraw (10-1)
• Soso: kérén (1), ̀fírín (2), sàxán (3), náání (4), súlí (5), sénní, sóló-fèré (7=5+2), sólómá-sàxán (8=3+5), sólómá-náání (9=4+5)
• Herero: oimwe (1), ombari (2), ondatu (3), oine (4), ondano (5), ohamb-oumwe (6=5+1), ohamb-ombari (7=5+2), ohamb-ondatu (8=5+3), omuvyu (9)
• Makua: mosa (1), pili (2), tharu (3), sheshe (4), thanu (5), thanu na mosa (6 = 5+1), thanu na pili (7), thanu na tharu (8), thanu na sheshe (9) en fulani / peul47 (langue nigéro-congolaise): goo (1), zizi (2), tati (3), nayi (4), joyi (5), jee-gom (6 = 5+1), jee-zizi (7), je-tati (8), je-nayi (9)
• Chichewa / chiwa: chimodzi (1), ziwiri (2), zitatu (3), zinayi (4), zisanu (5), zisanu n’chimodzi (6 = 5+1) zisanu n’ziwiri (7), zisanu n'zitatu (8), zisanu n'zinayi (9)
• Supyire: nìŋkìn (1), shùùnì (2), tàànrè (3), sìcyἑἑrè (4), kaŋkuro (5), baa-nì (6 = 5+1), baa-shùùnì (7 = 5+2), baa-tàànrè (8 = 5+3), baa-rìcyἑἑrè (9 = 5+4)
• Vai: dóndo(1), féra (2), ságba (3), náni (4), sōru (5), sūn-dóndo (6=5+1), sum-féra (7=7+2), sūn-ságba (8=5+3), sūn-náni (9=5+4).
• Nu gbaya: kpɔk (1), yiitoo (2), taar(3), naar (4), mɔɔrɔ (5), mɔɔrɔ ɗoŋ kpɔk (6=5+1), mɔɔrɔ ɗoŋ yiito (7=5+2), mɔɔrɔ ɗoŋ taar (8=5+3), mɔɔrɔ ɗoŋ naar (9=5+4)
• Pygmée de la shanga:
• Sherbro: buul (1), ting (2), ra (3), yol (4), men (5), men-buul (6=5+1), men-ting (7=5+2), men-ra (8=5+3), men-yol (9=5+4) en laalaa: wíinó (1), kanak (2), kaahaay (3), niikiis (4), yítus (5), yítus na wíinó (6=5+1), yítus na kanak (7=5+2), yítus na kaahaay (8=5+3), yítus na niikis (9=5+4) 50 en bambara (famille mandé):

#### Oceania

• Lote: éllé (1), nai (2), mol (3), hénél (4), limé (5), limé pe ratele éllé (6), limé pe ratele nai (7), limé pe ratele mol (8), limé pe ratele hénél (9)
• Ronga : dha (1), zhua (2), telu (3), wutu (4), lima (5), lima esa (6=5+1), lima-zhua (7), zhuam-butu (8), taraesa (9)
• Wandamen 57: siri (1), muandu (2), toru (3), at (4), rim (5), rim e siri (6 = 5+1), rim e muandu (7 = 5+2), rim e toru (8 = 5+3), rim e at (9 = 5+4)
• Nyelâyu 58: pwa-laic (1), pwa-du (2), pwa-jen (3), pwa-lavaac (4), pwa-nem (5), pwanem nua pwalaix (6=5+1), pwanem nua pwadu (7=5+2), pwanem nua pwajen (8=5+3), pwanem nua pwalavaac (9=5+4)
• Sobei: tes (1), daidu (2), tou (3), fau (4), dim (5), dim-tes (6=5+1), dim-daidu (7=5+2), dim-tou (8 = 5+3), dim-fau (9=5+4) En sio 60: taitu (1), rua (2), ŋato (3), ŋapa (4), lima (5), lima kanaŋo taitu (6=5+1), lima kanaŋo rua (7=5+2), lima kanaŋo ŋato (8=5+3), lima kanaŋo ŋapa (9=5+4)
• Efaté du sud (Vanuatu): i-skei (1), i-nru (2), i-tul (3), i-fat (4), i-lim (5), i-ɪa-tes (6=5+1), i-ɪa-ru (7), i-ɪa-tul (8), i-ɪ-fot (9)
• Maskelynes: e-sua (1), e-ru (2), i-tor (3), i-vat (4), e-rim (5), (e-)malve-tes (6), (e-)məlev-ru (7), (e-)məlev-tor (8), (e-)məlev-pat (9)
• Lenakel (Vanuatu): karena (1), kiu (2), kesil (3), kuver (4), katilum (5), katilum karena (6=5+1), katilum kiu (7=5+2), katilum kesil (8=5+3), katilum kuver (9=5+4)
• Paama: tai (1), elu (2), etel (3), ehat (4), elim (5), lahi-tai (6=5+1), lau-lu (7=5+2), lau-tel (8=5+3), lau-hat (9=5+4)
• Nume: tuwal (1) l-ru (2), i-tol (3), i-ßet (4), tißi-lim (5) leße-te (6=5+1), leße-ru (7=5+2), leße-tol (8=5+3), leße-ßet (9=5+4)

#### America

• Tupinambá: oîepé (1), mokõî (2), mosapyr (3), irundyk (4), po (5), pó-oîepé (6=5+1), pó-mokõi (7=5+2), pó-mosapyr (8=5+3), pó-irundyk (9=5+4)
• Old Tupi: poteī (6), pokōi (7), poapy (8), porundy (9), kuā (10), kuateī (11), kuakōi (12), kuapy (13), kuarundy (14), kuapo (15), kuapoteī (16), mokoikuā (20), mokoikuā poapy (28), porundyma porundysā porundykuā porundy (9.999)
• Nheengatu: iepé (1), mukũi (2), mukũi (3), irundi (4), iepe-pu (5), pu-iepé (6=5+1), pu-mukũi (7=5+2), pu-musapíri (8=5+3), pu-irundi (9=5+4)
• Pemon: töukin (1), sakü'ne (2), seutawö’ne (3), saköröro’ne (4), mia töukin (5, "hand one"), mia töukin töukin (6 = 5+1), mia töukin sakü’ne (7=5+2), mia töukin seutawö’ne (8=5+3), mia töukin saköröro’ne (9 = 5+4)
• Paresi: hatita (1), hinama (2), hanama (3), zalakakoa (4), hakake (5, "one hand"), 6 (5+1), 7 (5+2)
• Arawak: aba (1), bian (2), kabyn (3), bithi (4), (a)badakhabo (5), (a)ba-thian (1+5), bian-thian (7=2+5), kabyn-thian (8=3+5), bithi-thian (9=4+5)
• Zuni: dobinde (1), kwili (2), ha'i (3), a:wide (4), apde (5), doba-lekk'ya (6 = 1+5), kwilelekkya (7=2+5), ha'elekk'ya (8=3+5), dena-lekk'ya
• Choctaw: achvffa (1), tuklo (2), tuchena (3), ushta (4), talhapi (5), hvnnali (6), on-tuklo (7=5+2), on-tuchena (8=5+3), chakkali (9)
• Mikasuki: łáàmen (1), toklan (2), tocheenan (3), shéetaaken (4), chahkeepan (5), ee-paaken (6), kola-paaken (7=2+5), toshna-paaken (8=3+5), oshta-paaken (9)
• Garifuna: 70 = 3x20+10 (ürüwa wein disi), 90 = 4x20+10 (gádürü wein disi)
• Nahuatl: ce (1), ome (2), ëyi (3), nawi (4), macuilli (5), chicu-ace (6 = 5+1), chic-ome (7), chicu-ëyi (8), chico-nawi (9)
• Sumo (Ulwa): aslah (1), bû (2), bâs (3), arungka (4), singka (5), ting as kau as (6 = 5+1), ting as hau bû (7), ting as kau bâs (8), ting as kau arungka (9)
• Kiliwa: msig (1), juwak (2), jmi’k (3), mnak (4), salchipam (5), msigl paayp (6=1+5), juwakl paayp (7=5+2), jmi’kl paayp (8=3+5), msigl tmat (9)
• Quetzaltepec 76: tu’uk, määjtsk, tëkëëk, taxk (4), mëkooxk (5), të-tuujk (6=1+5), wx-tuujk (7=2+5), tuk-tuujk (8=3+5), tax-tuujk (9=4+5)
• Kalina 77: òwin (1), oko (2), oruwa (3), okupàen (4), ainatone (5), òwin-tòima (6), oko-tòima (7), oruwa-tòima (8), okupàen-tòima (9) en ojibwé 78: bezhig (1), niizh (2), niswi (3), naanan (5), [n]ingodwaaswi (6), niizh-waaswi (7 = 2+5), [n]ish-waaswi (8=3+5), zhaangaswi (9)
• Montagnais (innu-aimum) 79: peik (1), nish (1), nisht (3), neu (4), patetat (5), kutu-asht (6), nishu-asht (7=2+5), nishu-aush (8=3+5), peikushteu (9)
• Mohican: nuqut (1), nis (2), shwi (3), yáw (4), nupáw (5), qut-ôsk (6=1+5), nis-ôsk (7=2+5), shw-ôsk (8=3+5), pásukokun (9)
• Cheyenne: na’êstse (1), neše (2), na’he (3), neve (4), noho (5), naesoh-to (6=1+5), nésoh-to (7=2+5), na’noh-to (8=3+5), sóoh-to (9)
• Atikamekw 82: peikw (1), nicw (2), nicto (3), new (4), niaran (5), nikot-waso (6), nico-waso (7=2+5), nic-waso (8=3+5), cakitato (9)
• Tlingit: (tléil) (1), tléix' (2), déix̱ (3), nás'k (4), daax'oon (5), keijín (6), tlei-dooshú (7=2+5), dax̱a-dooshú (8=3+5), nas'ga-dooshú (9=4+5)
• Xa islakala: n̓áukʷ (1), má՚ elaukʷ (2), yúdukʷ (3), mukʷ (4), sek̕áukʷ (5), q̓etl̕áukʷ (6), malh-áus (2+5), yudexʷ-áus (8=3+5), muw-áus (9=4+5)
• Groendlandais: ataaseq (1), marluk (2) pingasut (3), sisamat (4), tallimat (5), arfinillit (6), arfineq marluk (7=5+2), arfineq pingasut (8=5+3), qulingiluat (9=5+4)