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]

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 = 0.010
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
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