Editing Quinary

{{short description|Base five numeral system}}
{{Table Numeral Systems}}
'''Quinary''' ('''base 5''' or '''pental'''<ref name="Sharp_EL-W531"/><ref name="Sharp_EL-W506-W516-W546"/><ref name="Sharp_EL-W531X"/>) is a [[numeral system]] with [[5 (number)|five]] as the [[radix|base]]. A possible origination of a quinary system is that there are five [[finger|digits]] on either [[hand]].

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

==Comparison to other radices==
{| class="wikitable"; text-align:center"
|+ 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
|}

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

{|  class="wikitable"
|+ '''Fractions in quinary'''
|'''Decimal''' (<u>periodic part</u>) ||'''Quinary''' (<u>periodic part</u>)
|'''Binary''' (<u>periodic part</u>)
|-
|1/2 = 0.5 
|'''1/2''' = 0.<u>2</u>
|1/10 = 0.1
|-
|1/3 = 0.<u>3</u>
|'''1/3''' = 0.<u>13</u>
|1/11 = 0.<u>01</u>
|-
|1/4 = 0.25
|'''1/4''' = 0.<u>1</u>
|1/100 = 0.01
|-
|1/5 = 0.2
|'''1/10''' = 0.1
|1/101 = 0.<u>0011</u>
|-
|1/6 = 0.1<u>6</u>
|'''1/11''' = 0.<u>04</u>
|1/110 = 0.0<u>01</u>
|-
|1/7 = 0.<u>142857</u>
|'''1/12''' = 0.<u>032412</u>
|1/111 = 0.<u>001</u>
|-
|1/8 = 0.125
|'''1/13''' = 0.<u>03</u>
|1/1000 = 0.001
|-
|1/9 = 0.<u>1</u>
|'''1/14''' = 0.<u>023421</u>
|1/1001 = 0.<u>000111</u>
|-
|1/10 = 0.1
|'''1/20''' = 0.0<u>2</u>
|1/1010 = 0.0<u>0011</u>
|-
|1/11 = 0.<u>09</u>
|'''1/21''' = 0.<u>02114</u>
|1/1011 = 0.<u>0001011101</u>
|-
|1/12 = 0.08<u>3</u>
|'''1/22''' = 0.<u>02</u>
|1/1100 = 0.00<u>01</u>
|-
|1/13 = 0.<u>076923</u>
|'''1/23''' = 0.<u>0143</u>
|1/1101 = 0.<u>000100111011</u>
|-
|1/14 = 0.0<u>714285</u>
|'''1/24''' = 0.<u>013431</u>
|1/1110 = 0.0<u>001</u>
|-
|1/15 = 0.0<u>6</u>
|'''1/30''' = 0.0<u>13</u>
|1/1111 = 0.<u>0001</u>
|-
|1/16 = 0.0625
|'''1/31''' = 0.<u>0124</u>
|1/10000 = 0.0001
|-
|1/17 = 0.<u>0588235294117647</u>
|'''1/32''' = 0.<u>0121340243231042</u>
|1/10001 = 0.<u>00001111</u>
|-
|1/18 = 0.0<u>5</u>
|'''1/33''' = 0.<u>011433</u>
|1/10010 = 0.0<u>000111</u>
|-
|1/19 = 0.<u>052631578947368421</u>
|'''1/34''' = 0.<u>011242141</u>
|1/10011 = 0.<u>000011010111100101</u>
|-
|1/20 = 0.05
|'''1/40''' = 0.0<u>1</u>
|1/10100 = 0.00<u>0011</u>
|-
|1/21 = 0.<u>047619</u>
|'''1/41''' = 0.<u>010434</u>
|1/10101 = 0.<u>000011</u>
|-
|1/22 = 0.0<u>45</u>
|'''1/42''' = 0.<u>01032</u>
|1/10110 = 0.0<u>0001011101</u>
|-
|1/23 = 0.<u>0434782608695652173913</u>
|'''1/43''' = 0.<u>0102041332143424031123</u>
|1/10111 = 0.<u>00001011001</u>
|-
|1/24 = 0.041<u>6</u>
|'''1/44''' = 0.<u>01</u>
|1/11000 = 0.000<u>01</u>
|-
|1/25 = 0.04
|'''1/100''' = 0.01
|1/11001 = 0.<u>00001010001111010111</u>
|}

==Usage==
Many languages<ref name="rarities">{{Cite book |last=Hammarström |first=Harald |url=https://www.degruyter.com/document/doi/10.1515/9783110220933.11/html |title=Rethinking Universals |date=March 26, 2010 |publisher=De Gruyter Mouton |isbn=9783110220933 |volume=45 |pages=11–60 |chapter=Rarities in numeral systems |doi=10.1515/9783110220933.11 |access-date=May 14, 2023 |url-access=registration }}</ref> use quinary number systems, including [[Gumatj language|Gumatj]], [[Nunggubuyu language|Nunggubuyu]],<ref name="harris">{{Cite web |last=Harris |first=John W. |date=December 1982 |others=Work Papers of SIL-AAB |title=Facts and fallacies of Aboriginal number system |url=http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf |url-status=dead |archive-url=https://web.archive.org/web/20070831202737/http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf |archive-date=August 31, 2007 |access-date=May 14, 2023 |website=www1.aiatsis.gov.au |pages=153–181 }}</ref> [[Kuurn Kopan Noot language|Kuurn Kopan Noot]],<ref>{{Cite book |last=Dawson |first=James |url=https://archive.org/details/australianabori00dawsgoog |title=Australian aborigines : the languages and customs of several tribes of aborigines in the western district of Victoria, Australia |publisher=Canberra City, ACT, Australia : Australian Institute of Aboriginal Studies; Atlantic Highlands, NJ : Humanities Press [distributor] |others=University of Michigan |year=1981 |access-date=May 14, 2023}}</ref> [[Luiseño language|Luiseño]],<ref>{{Cite book |author-last=Closs |author-first=Michael P. |title=Native American Mathematics |year=1986 |isbn=0-292-75531-7}}</ref> and [[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:<ref name="harris"/>

{| class="wikitable" border="1" style="text-align:center"
! Number !! Base 5 !! Numeral
|-
! style="text-align:right" | 1 
| 1
| wanggany
|-
! style="text-align:right" | 2
| 2
| marrma
|-
! style="text-align:right" | 3
| 3
| lurrkun
|-
! style="text-align:right" | 4
| 4
| dambumiriw
|-
! style="text-align:right" | 5
| 10
| wanggany rulu
|-
! style="text-align:right" | 10
| 20
| marrma rulu
|-
! style="text-align:right" | 15
| 30
| lurrkun rulu
|-
! style="text-align:right" | 20
| 40
| dambumiriw rulu
|-
! style="text-align:right" | 25
| 100
| dambumirri rulu
|-
! style="text-align:right" | 50
| 200
| marrma dambumirri rulu
|-
! style="text-align:right" | 75
| 300
| lurrkun dambumirri rulu
|-
! style="text-align:right" | 100
| 400
| dambumiriw dambumirri rulu
|-
! style="text-align:right" | 125
| 1000
| dambumirri dambumirri rulu
|-
! style="text-align:right" | 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 [[Mundugumor language|Biwat language]] as a similar case (previously attested as 5-20, but with one speaker recorded as making an innovation to turn it 5-25).<ref name=rarities/>

==Biquinary==

: ''In this section, the numerals are in decimal. For example, "5" means [[5|five]], and "10" means [[10|ten]].''

[[File:Chinese-abacus.jpg|thumb|right|Chinese Abacus or suanpan]]
A [[decimal]] system with two and five as a sub-bases is called [[biquinary]] and is found in [[Wolof language|Wolof]] and [[Khmer language|Khmer]]. [[Roman numeral]]s are an early biquinary system. The numbers [[1]], [[5]], [[10]], and [[50 (number)|50]] are written as '''I''', '''V''', '''X''', and '''L''' respectively. Seven is '''VII''', and seventy is '''LXX'''. The full list of symbols is:
{| class="wikitable" style="text-align:center"
|'''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 (symbol)|vinculum]] (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]], such as the [[suanpan]] and [[soroban]], 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.

[[Bi-quinary coded decimal]] is a variant of biquinary that was used on a number of early computers including [[Colossus computer|Colossus]] and the [[IBM 650]] to represent decimal numbers.

==Calculators and programming languages==
Few [[calculator]]s support calculations in the quinary system, except for some [[Sharp Corporation|Sharp]] models (including some of the [[Sharp EL-500W series|EL-500W]] and [[Sharp EL-500X series|EL-500X]]<!-- EL-531XH(GR) --> series, where it is named the ''pental system''<ref name="Sharp_EL-W531"/><ref name="Sharp_EL-W506-W516-W546"/><ref name="Sharp_EL-W531X"/>) since about 2005, as well as the open-source scientific calculator [[WP 34S]].

==See also==
*{{annotated link|Pentadic numerals}}
*[[Bi-quinary coded decimal]]

== References ==
{{reflist|refs=
<ref name="Sharp_EL-W531">{{cite web |url=http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/OperationGuide_ELW531.pdf |title=SHARP |access-date=2017-06-05 |url-status=live |archive-url=https://web.archive.org/web/20170712182220/http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/OperationGuide_ELW531.pdf |archive-date=2017-07-12 }}</ref>
<ref name="Sharp_EL-W506-W516-W546">{{cite web |url=http://www.sharp.de/cps/rde/xbcr/documents/documents/om/30_cal/ELW506-W516-W546_OM_DE.pdf |title=Archived copy |access-date=2017-06-05 |url-status=live |archive-url=https://web.archive.org/web/20160222014019/http://www.sharp.de/cps/rde/xbcr/documents/documents/om/30_cal/ELW506-W516-W546_OM_DE.pdf |archive-date=2016-02-22 }}</ref>
<ref name="Sharp_EL-W531X">{{cite web |url=http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/scientific_calculator_operation_guide.pdf |title=SHARP |access-date=2017-06-05 |url-status=live |archive-url=https://web.archive.org/web/20170712124336/http://www.sharp-world.com/contents/calculator/support/guidebook/pdf/scientific_calculator_operation_guide.pdf |archive-date=2017-07-12 }}</ref>
}}

== External links ==
* [http://www.mathsisfun.com/numbers/convert-base.php?to=quinary Quinary Base Conversion], includes fractional part, from Math Is Fun
* {{commons category-inline|Quinary numeral system}}
* [http://www.florestica.com/hpotd/dni_calculator/index.html Quinary-pentavigesimal and decimal calculator], uses [[D'ni]] numerals from the [[Myst]] franchise, integers only, fan-made.

[[Category:Positional numeral systems]]
[[Category:5 (number)]]