Quinary


Quinary 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 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 bits of information.

Comparison to other radices

Quinary012341011121314202122
Binary01101110010111011110001001101010111100
Decimal0123456789101112
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Quinary232430313233344041424344100
Binary11011110111110000100011001010011101001010110110101111100011001
Decimal13141516171819202122232425

Decimal Quinary Binary
1/2 = 0.51/2 = 0.21/10 = 0.1
1/3 = 0.31/3 = 0.131/11 = 0.01
1/4 = 0.251/4 = 0.11/100 = 0.01
1/5 = 0.21/10 = 0.11/101 = 0.0011
1/6 = 0.161/11 = 0.041/110 = 0.010
1/7 = 0.1428571/12 = 0.0324121/111 = 0.001
1/8 = 0.1251/13 = 0.031/1000 = 0.001
1/9 = 0.11/14 = 0.0234211/1001 = 0.000111
1/10 = 0.11/20 = 0.021/1010 = 0.00011
1/11 = 0.091/21 = 0.021141/1011 = 0.0001011101
1/12 = 0.0831/22 = 0.021/1100 = 0.0001
1/13 = 0.0769231/23 = 0.01431/1101 = 0.000100111011
1/14 = 0.07142851/24 = 0.0134311/1110 = 0.0001
1/15 = 0.061/30 = 0.0131/1111 = 0.0001
1/16 = 0.06251/31 = 0.01241/10000 = 0.0001
1/17 = 0.05882352941176471/32 = 0.01213402432310421/10001 = 0.00001111
1/18 = 0.051/33 = 0.0114331/10010 = 0.0000111
1/19 = 0.0526315789473684211/34 = 0.0112421411/10011 = 0.000011010111100101
1/20 = 0.051/40 = 0.011/10100 = 0.000011
1/21 = 0.0476191/41 = 0.0104341/10101 = 0.000011
1/22 = 0.0451/42 = 0.010321/10110 = 0.00001011101
1/23 = 0.04347826086956521739131/43 = 0.01020413321434240311231/10111 = 0.00001011001
1/24 = 0.04161/44 = 0.011/11000 = 0.00001
1/25 = 0.041/100 = 0.011/11001 = 0.00001010001111010111

Usage

Many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:
NumberBase 5Numeral
11wanggany
22marrma
33lurrkun
44dambumiriw
510wanggany rulu
1020marrma rulu
1530lurrkun rulu
2040dambumiriw rulu
25100dambumirri rulu
50200marrma dambumirri rulu
75300lurrkun dambumirri rulu
100400dambumiriw dambumirri rulu
1251000dambumirri dambumirri rulu
62510000dambumirri 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 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 as 26.