# Quinary

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Base five numeral system

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**Quinary** (**base 5** or **pental**[1][2][3]) is a [numeral system](/source/Numeral_system) with [five](/source/5_(number)) as the [base](/source/Radix). A possible origination of a quinary system is that there are five [digits](/source/Finger) on either [hand](/source/Hand).

In the quinary place system, five numerals, from [0](/source/0_(number)) to [4](/source/4_(number)), are used to represent any [real number](/source/Real_number). According to this method, [five](/source/5_(number)) is written as 10, [twenty-five](/source/25_(number)) is written as 100, and [sixty](/source/60_(number)) 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](/source/Highly_composite_number) ([4](/source/4_(number)) and [6](/source/6_(number))) guarantees that many recurring fractions have relatively short periods.

## Comparison to other radices

A quinary multiplication table × 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

Numbers zero to twenty-five in standard quinary Quinary 0 1 2 3 4 10 11 12 13 14 20 21 22 Binary 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 Quinary 23 24 30 31 32 33 34 40 41 42 43 44 100 Binary 1101 1110 1111 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001 Decimal 13 14 15 16 17 18 19 20 21 22 23 24 25

Fractions in quinary 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 = 0.001 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[4] use quinary number systems, including [Gumatj](/source/Gumatj_language), [Nunggubuyu](/source/Nunggubuyu_language),[5] [Kuurn Kopan Noot](/source/Kuurn_Kopan_Noot_language),[6] [Luiseño](/source/Luise%C3%B1o_language),[7] and [Saraveca](/source/Saraveca). Gumatj has been reported to be a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:[5]

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

However, Harald Hammarström reports that "one would not usually use exact numbers for counting this high in this language and there is a certain likelihood that the system was extended this high only at the time of elicitation with one single speaker," pointing to the [Biwat language](/source/Mundugumor_language) as a similar case (previously attested as 5-20, but with one speaker recorded as making an innovation to turn it 5-25).[4]

## Biquinary

- *In this section, the numerals are in decimal. For example, "5" means [five](/source/5), and "10" means [ten](/source/10).*

Chinese Abacus or suanpan

A [decimal](/source/Decimal) system with two and five as a sub-bases is called [biquinary](/source/Biquinary) and is found in [Wolof](/source/Wolof_language) and [Khmer](/source/Khmer_language). [Roman numerals](/source/Roman_numeral) are an early biquinary system. The numbers [1](/source/1), [5](/source/5), [10](/source/10), and [50](/source/50_(number)) are written as **I**, **V**, **X**, and **L** respectively. Seven is **VII**, and seventy is **LXX**. The full list of symbols is:

Roman I V X L C D M Decimal 1 5 10 50 100 500 1000

Note that these are not positional number systems. In theory, a number such as 73 could be written as IIIXXL (without ambiguity) and as LXXIII. To extend Roman numerals to beyond thousands, a [vinculum](/source/Vinculum_(symbol)) (horizontal overline) was added, multiplying the letter value by a thousand, e.g. overlined **M̅** was one million. There is also no sign for zero. But with the introduction of inversions like IV and IX, it was necessary to keep the order from most to least significant.

Many versions of the [abacus](/source/Abacus), such as the [suanpan](/source/Suanpan) and [soroban](/source/Soroban), use a biquinary system to simulate a decimal system for ease of calculation. [Urnfield culture numerals](/source/Urnfield_culture_numerals) and some [tally mark](/source/Tally_mark) systems are also biquinary. Units of [currencies](/source/Currencies) are commonly partially or wholly biquinary.

[Bi-quinary coded decimal](/source/Bi-quinary_coded_decimal) is a variant of biquinary that was used on a number of early computers including [Colossus](/source/Colossus_computer) and the [IBM 650](/source/IBM_650) to represent decimal numbers.

## Calculators and programming languages

Few [calculators](/source/Calculator) support calculations in the quinary system, except for some [Sharp](/source/Sharp_Corporation) models (including some of the [EL-500W](/source/Sharp_EL-500W_series) and [EL-500X](https://en.wikipedia.org/w/index.php?title=Sharp_EL-500X_series&action=edit&redlink=1) series, where it is named the *pental system*[1][2][3]) since about 2005, as well as the open-source scientific calculator [WP 34S](/source/WP_34S).

## See also

- [Bi-quinary coded decimal](/source/Bi-quinary_coded_decimal)

- [Pentadic numerals](/source/Pentadic_numerals) – Scandinavian numeral system

## References

1. ^ [***a***](#cite_ref-Sharp_EL-W531_1-0) [***b***](#cite_ref-Sharp_EL-W531_1-1) ["SHARP"](http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/OperationGuide_ELW531.pdf) (PDF). [Archived](https://web.archive.org/web/20170712182220/http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/OperationGuide_ELW531.pdf) (PDF) from the original on 2017-07-12. Retrieved 2017-06-05.

1. ^ [***a***](#cite_ref-Sharp_EL-W506-W516-W546_2-0) [***b***](#cite_ref-Sharp_EL-W506-W516-W546_2-1) ["Archived copy"](http://www.sharp.de/cps/rde/xbcr/documents/documents/om/30_cal/ELW506-W516-W546_OM_DE.pdf) (PDF). [Archived](https://web.archive.org/web/20160222014019/http://www.sharp.de/cps/rde/xbcr/documents/documents/om/30_cal/ELW506-W516-W546_OM_DE.pdf) (PDF) from the original on 2016-02-22. Retrieved 2017-06-05.{{[cite web](https://en.wikipedia.org/wiki/Template:Cite_web)}}: CS1 maint: archived copy as title ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_archived_copy_as_title))

1. ^ [***a***](#cite_ref-Sharp_EL-W531X_3-0) [***b***](#cite_ref-Sharp_EL-W531X_3-1) ["SHARP"](http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/scientific_calculator_operation_guide.pdf) (PDF). [Archived](https://web.archive.org/web/20170712124336/http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/scientific_calculator_operation_guide.pdf) (PDF) from the original on 2017-07-12. Retrieved 2017-06-05.

1. ^ [***a***](#cite_ref-rarities_4-0) [***b***](#cite_ref-rarities_4-1) Hammarström, Harald (March 26, 2010). "Rarities in numeral systems". [*Rethinking Universals*](https://www.degruyter.com/document/doi/10.1515/9783110220933.11/html). Vol. 45. De Gruyter Mouton. pp. 11–60. [doi](/source/Doi_(identifier)):[10.1515/9783110220933.11](https://doi.org/10.1515%2F9783110220933.11). [ISBN](/source/ISBN_(identifier)) [9783110220933](https://en.wikipedia.org/wiki/Special:BookSources/9783110220933). Retrieved May 14, 2023.

1. ^ [***a***](#cite_ref-harris_5-0) [***b***](#cite_ref-harris_5-1) Harris, John W. (December 1982). ["Facts and fallacies of Aboriginal number system"](https://web.archive.org/web/20070831202737/http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf) (PDF). *www1.aiatsis.gov.au*. Work Papers of SIL-AAB. pp. 153–181. Archived from [the original](http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf) (PDF) on August 31, 2007. Retrieved May 14, 2023.

1. **[^](#cite_ref-6)** Dawson, James (1981). [*Australian aborigines : the languages and customs of several tribes of aborigines in the western district of Victoria, Australia*](https://archive.org/details/australianabori00dawsgoog). University of Michigan. Canberra City, ACT, Australia : Australian Institute of Aboriginal Studies; Atlantic Highlands, NJ : Humanities Press [distributor]. Retrieved May 14, 2023.

1. **[^](#cite_ref-7)** Closs, Michael P. (1986). *Native American Mathematics*. [ISBN](/source/ISBN_(identifier)) [0-292-75531-7](https://en.wikipedia.org/wiki/Special:BookSources/0-292-75531-7).

## External links

- [Quinary Base Conversion](http://www.mathsisfun.com/numbers/convert-base.php?to=quinary), includes fractional part, from Math Is Fun

- Media related to [Quinary numeral system](https://commons.wikimedia.org/wiki/Category:Quinary_numeral_system) at Wikimedia Commons

- [Quinary-pentavigesimal and decimal calculator](http://www.florestica.com/hpotd/dni_calculator/index.html), uses [D'ni](/source/D'ni) numerals from the [Myst](/source/Myst) franchise, integers only, fan-made.

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Adapted from the Wikipedia article [Quinary](https://en.wikipedia.org/wiki/Quinary) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Quinary?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.