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.

Comparison to other radices
A quinary multiplication table
×1234101112131420
11234101112131420
2241113202224313340
3311142230334144102110
441322314044103112121130
1010203040100110120130140200
1111223344110121132143204220
12122441103120132144211223240
13133144112130143211224242310
141433102121140204223242311330
202040110130200220240310330400
Numbers zero to twenty-five in standard quinary
Quinary 012341011121314202122
Binary 01101110010111011110001001101010111100
Decimal 0123456789101112
Quinary 232430313233344041424344100
Binary 11011110111110000100011001010011101001010110110101111100011001
Decimal 13141516171819202122232425
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 = 0.010
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[4] use quinary number systems, including Gumatj, Nunggubuyu,[5] Kuurn Kopan Noot,[6] Luiseño[7] and Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:[5]

NumberBase 5Numeral
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. Seven is VII and seventy is LXX.

Many versions of the abacus, such as the 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.

Quadquinary

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

Calculators and programming languages

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.

Python's int() function supports conversion of numeral systems from any base to base 10. Thus, the quinary number 101 is evaluated using int('101',5) as 26.[8]

See also

References
  1. "Archived copy" (PDF). Archived (PDF) from the original on 2017-07-12. Retrieved 2017-06-05.CS1 maint: archived copy as title (link)
  2. "Archived copy" (PDF). Archived (PDF) from the original on 2016-02-22. Retrieved 2017-06-05.CS1 maint: archived copy as title (link)
  3. "Archived copy" (PDF). Archived (PDF) from the original on 2017-07-12. Retrieved 2017-06-05.CS1 maint: archived copy as title (link)
  4. Harald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are omnipresent." doi:10.1515/9783110220933.11
  5. Harris, John (1982), Hargrave, Susanne (ed.), "Facts and fallacies of aboriginal number systems" (PDF), Work Papers of SIL-AAB Series B, 8: 153–181, archived from the original (PDF) on 2007-08-31
  6. Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
  7. Closs, Michael P. Native American Mathematics. ISBN 0-292-75531-7.
  8. "Convert base-2 binary number string to int". Stack Overflow. Archived from the original on 24 November 2017. Retrieved 5 May 2018.
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