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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 digits 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.

Comparison to other radices

[edit]
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

[edit]

Many languages[4] use quinary number systems, including Gumatj, Nunggubuyu,[5] Kuurn Kopan Noot,[6] Luiseño,[7] 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:[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 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

[edit]
In this section, the numerals are in decimal. For example, "5" means five, and "10" means ten.
Chinese Abacus or suanpan

A decimal system with two and five as a sub-bases is called biquinary and is found in Wolof and Khmer. Roman numerals are an early 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. 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 (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 and the IBM 650 to represent decimal numbers.

Calculators and programming languages

[edit]

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.

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Quinary

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The quinary numeral system, also known as base-5, is a method that uses five digits—0, 1, 2, 3, and 4—to represent numbers, where each position signifies a power of 5. For example, the number 5 in is written as 10 in quinary, 25 as 100, and 30 as 110. Historically, quinary systems trace their origins to early practices, particularly the use of a single hand's five fingers, making it one of the simplest and most intuitive bases for primitive s. Pure quinary scales are rare and often evolve into hybrid forms, such as quinary-vigesimal (base-20, treating two hands as a unit), as seen in various indigenous cultures where terms for numbers frequently reference body parts like "one hand" for 5 or "two hands" for 10. Examples include the Betoya language of , where 5 is "teente" (one hand) and 10 is "cayaente" (two hands), and the Tamanac system, with 5 as "amnaitone" (one hand). In , quinary bases appear in numerous non-European languages, such as Gumatj and Nunggubuyu in , and persist in regions like and among some Native American groups, though many transition to for larger quantities. Beyond traditional counting, quinary principles influence modern applications through biquinary variants, which combine base-5 with base-2 or base-10 for efficient representation. These appear in designs like the Chinese suanpan and Japanese , where rods facilitate calculations via quinary groupings, and in early hardware. Notably, bi-quinary coded was employed in World War II-era machines like the Colossus codebreaker and the magnetic drum calculator from the , allowing reliable processing in binary electronics by encoding each digit with a quinary structure. Today, quinary systems hold niche value in mathematics and for their information density—each digit carries approximately 2.32 bits—and in studying numeral across cultures.

Fundamentals

Definition and Positional Notation

The quinary numeral system, also known as base-5, is a that employs five as its and utilizes the digits 0, 1, 2, 3, and 4 exclusively to represent values. In this system, each digit's value depends on its position relative to the others, with the rightmost digit representing the units place and subsequent positions to the left indicating higher powers of 5. This structure allows for the compact representation of integers and other numerical quantities using only these five symbols. The place values in quinary follow the powers of 5, starting from the right: the units place is 50=15^0 = 1, the next position to the left is 51=55^1 = 5 (fives), followed by 52=255^2 = 25 (twenty-fives), 53=1255^3 = 125 (one hundred twenty-fives), and so on. For instance, the quinary number 10 represents 1×51+0×50=51 \times 5^1 + 0 \times 5^0 = 5 in , while 100 denotes 1×52+0×51+0×50=251 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 = 25 in , and 220 corresponds to 2×52+2×51+0×50=602 \times 5^2 + 2 \times 5^1 + 0 \times 5^0 = 60 in . To convert a number to quinary, repeatedly divide the number by 5, recording the at each step (which will be between and 4), and read the remainders from bottom to top to form the quinary digits. For example, converting 60 to quinary involves: 60 ÷ 5 = 12 ; 12 ÷ 5 = 2 2; 2 ÷ 5 = 0 2; yielding in quinary. Conversely, to convert a quinary number to , multiply each digit by the corresponding power of 5 based on its position and sum the results; for in quinary, this is 2×25+2×5+[0](/page/0)×1=602 \times 25 + 2 \times 5 + [0](/page/0) \times 1 = 60. A possible origin of the quinary system lies in the human hand, which has five digits, facilitating natural practices.

Arithmetic in Quinary

Arithmetic in the quinary system, or base-5, follows procedures analogous to those in base-10 but adapted to the digit set {0, 1, 2, 3, 4}, where the base value is 5. For , digits in each place value are summed column by column from right to left; if the sum exceeds 4, a carry of 1 is propagated to the next higher place, and the remainder ( 5) is written in the current place. For example, adding the single digits 3 and 2 yields 10 in quinary, as 3 + 2 = 5 (in ), which is equivalent to 1 × 5¹ + 0 × 5⁰. A multi-digit example is 342₅ + 132₅: units place 2 + 2 = 4; fives place 4 + 3 = 7 (decimal) = 12₅ (write 2, carry 1); twenty-fives place 3 + 1 + 1 (carry) = 5 (decimal) = 10₅ (write 0, carry 1); resulting in 1024₅. Subtraction in quinary proceeds column by column from right to left, borrowing 1 from the next higher place (equivalent to adding 5 to the current place) if the top digit is smaller than the bottom. For instance, subtracting 3 from 10₅ (which is 5 in ) gives 2₅, as no borrow is needed. In cases requiring borrowing, such as 3000₅ - 2342₅, the process involves finding complementary digits that sum to the base value per column, often using an table to determine borrows and carries; the result is 1033₅. Multiplication uses the standard algorithm of partial products and , but all intermediate results are expressed in base-5 with carries when exceeding 4. The single-digit , requiring only 25 entries due to the limited digit set, is as follows:
×01234
000000
101234
20241113
303111422
404132231
For example, 3 × 4 = 12 (decimal) = 22₅. A multi-digit case, such as 1422₅ × 21₅, involves computing partial products (e.g., 1 × 1422₅ = 1422₅; 2 × 1422₅ = 3344₅ with carries) and then shifting the second partial by one position to 33440₅ before adding them to yield 40412₅. Division in quinary employs , determining how many times the fits into partial dividends, with s less than the base (i.e., 0–4); the process mirrors base-10 but uses base-5 multiples. For example, 100₅ ÷ 10₅ = 10₅ with remainder 0, as 100₅ (25 decimal) is exactly 5 × 10₅ (5 decimal). In a more involved case like 243₅ ÷ 3₅, using manipulatives or algorithmic steps: 243₅ represents 2 × 25 + 4 × 5 + 3 = 73 decimal; dividing yields quotient 44₅ (4 × 5 + 4 = 24 decimal) with remainder 1₅, verified by 3₅ × 44₅ + 1₅ = 243₅. The quinary system's arithmetic is straightforward because the small digit set limits the number of basic facts to memorize—such as only 10 combinations and 25 products—facilitating mental once the carry and borrow rules are learned.

Historical and Cultural Context

Origins from Counting Practices

The quinary , with base 5, is hypothesized to have originated from early human practices of counting using the five fingers of one hand, naturally leading to groupings in fives as a fundamental unit. This method, known as dactylonomy, provided a simple and intuitive way to track quantities, where completing one hand's digits signified the number 5, often linguistically expressed as "one hand" in various cultures. Such practices likely emerged as one of the earliest cognitive tools for numeration, reflecting the physical constraints and affordances of the human body in prehistoric societies. Evidence for base-5 patterns in early counting appears in prehistoric artifacts, such as notched bones used as tally sticks, which suggest systematic groupings potentially aligned with finger-based increments. For instance, the , dating back approximately 42,000 years from the on the border of and , features 29 notches that represent early tallying practices linked to bodily counting methods. These artifacts indicate that quinary-like organization predated written numerals, serving as precursors to more complex systems in ancient societies across and beyond, though no pure quinary dominance is evident in major early civilizations like , where mixed bases incorporating factors of 5 appeared sporadically. In ancient developments, quinary elements served as building blocks in emerging numeral systems, influencing transitions to hybrid bases that addressed practical needs for larger counts. A notable evolution occurred through the extension of hand-based counting to include both hands or hands and feet, doubling the quinary unit to form vigesimal (base-20) systems, as seen in the Mayan numeral framework where 5 played a central role via symbols like the horizontal bar for five fingers. This shift from pure quinary to mixed scales, such as quinary-vigesimal, allowed for more expansive representation while retaining the foundational grouping of fives. The limited widespread adoption of pure quinary systems stems from their inherent inefficiency in handling , as the smaller base requires significantly more digits compared to (base-10), complicating arithmetic and record-keeping for growing societies. Consequently, quinary often became subordinate within larger bases like or , merging into hybrid forms rather than persisting independently, which favored scalability in trade, astronomy, and administration.

Usage in Languages and Societies

Quinary numeral systems appear in several indigenous languages, particularly those associated with practices that emphasize groupings of five. In , the Gumatj dialect of Yolngu Matha employs a pure quinary system extending to 25, where numbers are structured around the base term rulu for five, and higher values like 25 are denoted as 100 in quinary notation (now endangered). Similarly, Nunggubuyu, another language (now endangered), utilizes a quinary framework for counting up to 25, with terms compounding multiples of five, such as wulawa for 10 (equivalent to 2 × 5). Among North American indigenous languages, the Kwak'wala (also known as Kwakiutl, spoken on ) incorporates base-5 counting, often extending into quinary-vigesimal patterns for larger quantities (now endangered with about 150 speakers as of ). The Luiseño language of Uto-Aztecan speakers features a quinary-decimal system, where numerals from 6 to 9 are formed by adding units to five (pam for five, e.g., pama for six; now extinct). In , the Saraveca language, an Arawakan tongue once spoken in , stands out as one of the few documented pure quinary systems, with all numerals derived strictly from multiples of five without decimal admixture (extinct since around 2000, with speakers having shifted to Chiquitano by 1962). Quinary elements also integrate into mixed systems in other regions for practical everyday counting. The of employs a quinary-decimal structure, where numbers 6 through 9 are expressed as five plus the remainder (e.g., juroom for five, juroom been for six), facilitating oral tallying in markets and daily exchanges. Likewise, Khmer in uses a biquinary approach for spoken numerals up to 20, basing 6–9 on pram (five) plus units, which simplifies informal counting of goods and quantities in rural and urban settings. In these societies, quinary supports broader social functions beyond mere enumeration, aiding memory in through bundled groupings of five items, structuring classifications via fivefold categories of relations, and enhancing rituals like finger-based where sequences of five gestures encode narratives or songs. This integration stems from the intuitive use of fingers as a foundational counting tool.

Mathematical Properties

Comparisons with Other Bases

The quinary numeral system, or base-5, differs from more common systems like binary (base-2) and decimal (base-10) primarily in its radix and digit set, leading to variations in representation length and suitability for different applications. While binary excels in electronic implementation due to its simplicity with two states, decimal aligns with human anatomy via ten fingers, and quinary offers a middle ground with five digits (0-4), often tied to single-hand counting. These differences affect how numbers are encoded, with quinary providing shorter representations than binary for the same value but longer than decimal. To illustrate integer representations, the following table shows the decimal values 1 through 10 expressed in binary, quinary, and notations. This highlights quinary's intermediate digit count: for instance, 10 requires four binary digits but only two quinary digits.
DecimalBinaryQuinary
1111
21022
31133
410044
5101105
6110116
7111127
81000138
91001149
1010102010
Fractional representations also vary by base, as the terminating or repeating nature depends on the denominator's factors relative to the base. For example, 1/21/2 equals 0.120.1_2 in binary, a terminating since 2 divides the base, but 0.222…50.222\dots_5 (repeating) in quinary, as 2 does not fully factor into powers of 5. This repeating pattern in quinary contrasts with decimal's 0.5100.5_{10}, which terminates. In terms of efficiency, quinary strikes a balance for representing large integers: it requires fewer digits than binary (e.g., the number 220≈1,048,5762^{20} \approx 1,048,576 needs 21 binary digits but only about 9 quinary digits, since log⁡5(220)≈8.3\log_5(2^{20}) \approx 8.3) but more than decimal, making it less compact for extensive calculations yet simpler than unary's one-to-one tally marks. For human use, quinary's advantages include its alignment with one hand's five fingers, facilitating intuitive counting in cultures without tools, though decimal's familiarity from two hands and binary's electronic reliability render quinary less practical for widespread adoption. Unlike fully positional systems such as binary or , where digit value depends strictly on position, exhibit partial quinary influence through a bi-quinary structure (base-5 for units and base-2 for multiples), using symbols like (5) and X (10=2×5) in an additive-subtractive framework without true place value or . This hybrid approach, while efficient for small counts, lacks the scalability of pure positional quinary. Quinary's relative rarity stems from its suboptimal balance between simplicity and scalability: binary dominates computing due to reliable two-state hardware, decimal prevails in daily life for ergonomic reasons, and quinary, though finger-friendly, offers no unique divisibility advantages (unlike base-12) and complicates larger arithmetic without decimal's cultural entrenchment.

Fraction Representation and Periods

In base-5 (quinary) representation, fractions terminate if and only if the denominator in lowest terms is a power of 5, analogous to how denominators that are powers of 10 terminate in but adapted to the prime base 5. For example, 15=0.15\frac{1}{5} = 0.1_5 and 125=0.015\frac{1}{25} = 0.01_5, as these align directly with the place values of the base. For fractions with denominators coprime to 5, the quinary expansion is purely periodic with a repeating block whose length is the multiplicative order of 5 the denominator (or a prime factor thereof), defined as the smallest positive kk such that 5k≡1(modp)5^k \equiv 1 \pmod{p} for prime p≠5p \neq 5. This period length divides p−1p-1 and determines the repetend's cycle. Due to 5 being prime, expansions for other denominators recur; for instance, 12=0.2‾5\frac{1}{2} = 0.\overline{2}_5 with period 1 (order of 5 2 is 1), and 13=0.13‾5\frac{1}{3} = 0.\overline{13}_5 with period 2 (order of 5 3 is 2). Similarly, 17=0.032412‾5\frac{1}{7} = 0.\overline{032412}_5 with period 6 (order of 5 7 is 6). The period lengths in quinary tend to be relatively short for small primes, as 5 lies between the highly composite bases 4 and 6, facilitating quicker returns to 1 modulo pp compared to larger bases like 10; for primes up to 52−1=245^2 - 1 = 24, the longest such period is 22 (for p=23p = 23, where the order of 5 modulo 23 is 22). This positioning enhances fraction simplicity in quinary relative to decimal, where periods can grow longer (e.g., 1/7 has period 6 in both, but 1/19 has period 9 in quinary versus 18 in decimal), though binary fractions often terminate more readily for even denominators.

Variants and Extensions

Biquinary System

The biquinary system, also known as bi-quinary, is a numeral encoding scheme that represents digits using a mixed of base 2 and base 5, where each digit n is expressed as a pair xy such that n = 5x + y, with x in {0, 1} and y in {0, 1, 2, 3, 4}. This approach combines a binary indicator for multiples of 5 with a quinary component for the , enabling efficient representation in devices limited to simple binary or unary operations. Historically, the Roman numeral system exemplifies biquinary principles through its additive structure, where values are built from units of 1 (I) and 5 (V), scaled by powers of 10; for instance, VII denotes 5 (V) + 2 (II). Similarly, the of employs a biquinary counting method, grouping numbers in fives with additive compounds; numbers 6 through 9 are formed as juróóm (5) plus 1 through 4 (e.g., juróóm benn for 6, meaning "five one"). In traditional computing aids, the Chinese suanpan implements biquinary notation across its rods, with an upper deck of two beads valued at 5 each (for 0 or 5) and a lower deck of five beads valued at 1 each (for 0–4), allowing each column to represent digits 0–9 by positioning beads toward the central beam. This design facilitates rapid arithmetic by separating the 5-multiple from the , mirroring the system's core encoding logic. Early electronic computers adopted biquinary for due to its compatibility with relay-based hardware, which favored unary-like quinary increments and binary flags. The , introduced in , used a seven-bit biquinary per digit—two bits for the binary 0/5 indicator and five for the quinary 0–4 value—with error-checking to ensure exactly one bit from each set was active, enhancing reliability in its magnetic . To encode a digit in biquinary, decompose it into its 5-multiple and remainder; for example, 7 becomes 5 + 2, represented as binary flag 1 (for 5) paired with quinary 2, often visualized as two marks or active positions in a five-slot . This method, while less storage-efficient than pure (requiring more bits per digit), supported straightforward addition circuits in vacuum-tube and eras by handling carries at the 5-boundary.

Balanced Quinary

Balanced quinary is a signed-digit variant of the quinary numeral system, employing base 5 with place values that are powers of 5, but utilizing the digit set {-2, -1, 0, 1, 2} instead of the standard {0, 1, 2, 3, 4}. These digits are often symbolized as 2ˉ\bar{2}, 1ˉ\bar{1}, 0, 1, 2 to denote the negative values, allowing for a symmetric representation around zero that covers the range from −5n−12-\frac{5^n - 1}{2} to 5n−12\frac{5^n - 1}{2} for nn digits. This system generalizes the principles of balanced numeral systems to an odd base of 5, where each digit corresponds to the minimal residues modulo 5. In balanced quinary, every has a unique representation without leading zeros, as the digit set spans exactly one full set of residues 5. For example, the number 3 is represented as 12ˉ1\bar{2}, since 1×51+(−2)×50=5−2=31 \times 5^1 + (-2) \times 5^0 = 5 - 2 = 3. Similarly, 4 is 101ˉ10\bar{1}, equivalent to 1×51+(−1)×50=5−1=41 \times 5^1 + (-1) \times 5^0 = 5 - 1 = 4. This uniqueness arises because the digits are chosen to avoid redundancy, ensuring no two different digit sequences yield the same value. Arithmetic operations in balanced quinary offer advantages similar to those in , particularly in and , where digit-wise operations often avoid the need for carry or borrow in certain cases due to the symmetric digit range. The sum of two digits ranges from -4 to 4, which can be resolved by simple adjustment rules (e.g., converting sums outside -2 to 2 by adding or subtracting 5 and carrying over), facilitating efficient computation in theoretical models. To convert a number from standard quinary to balanced quinary, first obtain the standard representation using digits 0-4, then apply an adjustment rule: replace any 3 with 12ˉ1\bar{2} (adding 1 to the next higher digit), and any 4 with 101ˉ10\bar{1} (also adding 1 to the next higher digit), propagating carries as needed. For instance, the standard quinary 3 (which is just 3) becomes 12ˉ1\bar{2} after adjustment. Historically, balanced quinary remains a theoretical extension of balanced numeral systems, primarily discussed in studies of signed-digit representations for mathematical efficiency, with no evidence of major cultural or practical adoption in languages or societies. It serves as an analogous system to , extending the concept to higher odd bases for exploratory purposes in numeral .

Modern Applications

In Calculators and Devices

Historical devices such as the Chinese suanpan incorporate quinary elements through its structure of two upper beads representing multiples of five and five lower beads representing units, facilitating bi-quinary calculations that align with base-5 sub-operations within a framework. This design, dating back centuries, allowed efficient manual arithmetic by leveraging the quinary bead arrangement for rapid counting and addition in trade and commerce. Early mechanical calculators, while predominantly -based, occasionally experimented with base-5 mechanisms to mimic abacus efficiency, though such implementations remained rare and non-standardized compared to dominant base-10 gear systems. In modern calculators, Sharp's EL-W516 series, introduced around 2011, includes quinary mode for base conversion, enabling users to perform operations between base-5 numbers and other bases like and binary. Similarly, the WP 34S, an open-source for HP-30b hardware released in 2011, supports quinary , and arithmetic through its flexible BASE mode, which allows selection of base 5 among various bases (2-16) for precise numerical handling. These features cater to educational and specialized applications requiring non-decimal computations. The advantages of quinary support in such devices include a reduced digit set (0-4), which simplifies electronic implementations by minimizing the states needed per digit in basic circuits, potentially lowering complexity in early digital designs. However, remains limited due to the prevalence of binary and standards, restricting quinary to niche modes rather than primary operations. For example, on a Sharp EL-W516 calculator in , selecting Pental (base 5) via the N-base function allows entry of "10" (quinary), which the device converts and displays as 5 in , demonstrating straightforward base conversion for verification. On the WP 34S, invoking BASE 5 mode followed by entering 10 yields the same equivalent of 5, supporting arithmetic like addition in quinary directly.

In Computing and Programming

In early computing, biquinary coded decimal—detailed in the article's variants section—was used in machines like the Colossus and to facilitate decimal arithmetic on binary hardware. Support for quinary operations in modern programming languages is typically provided through general-purpose base conversion utilities rather than native data types, reflecting the dominance of binary architectures. In Python, the built-in int() function handles quinary strings directly by specifying base 5, enabling seamless conversion to integers. For example, the following code converts a quinary number to decimal:

python

def quinary_to_decimal(n): return int(n, 5) # Example usage result = quinary_to_decimal('123') # Converts 1*25 + 2*5 + 3 = 38 in decimal

def quinary_to_decimal(n): return int(n, 5) # Example usage result = quinary_to_decimal('123') # Converts 1*25 + 2*5 + 3 = 38 in decimal

This approach leverages Python's standard library for arbitrary base parsing (2-36). Libraries like SymPy extend this for symbolic computations, allowing quinary representations in algebraic manipulations via custom base handling. In specialized environments, such as HP's Reverse Polish Lisp (RPL) used in programmable calculators like the WP 34S (firmware released around 2011), quinary conversions can be implemented programmatically using RPL's stack-based operations and conditional logic, though direct built-in support is limited to common bases. Contemporary applications of quinary in remain niche, primarily in for demonstrating numeral systems and in experimental as of 2023. Tools like interactive simulators (e.g., those in curricula) use quinary to teach and conversions, highlighting its historical roots without requiring hardware changes. In , quinary polynomials appear in lattice-based schemes, such as variants of , where coefficients limited to {-1, 0, 1, 2, 3} (quinary range) reduce key sizes while maintaining security against hybrid attacks. However, quinary encoding for prime-based systems, as explored in some post-quantum prototypes, is rare outside research due to the need for custom implementations. Despite these uses, quinary faces significant challenges in binary-dominated computing environments, including conversion overhead that increases processing time and memory usage compared to native binary or hexadecimal operations. On binary hardware, quinary data requires repeated modulo-5 divisions for arithmetic, leading to inefficiencies that have confined it to legacy emulations or specialized simulations rather than mainstream systems.

References

  1. https://mathresearch.utsa.edu/wiki/index.php?title=Base_10,_Base_2_%26_Base_5
  2. https://etc.usf.edu/lit2go/pdf/passage/5612/the-number-concept-007-chapter-6.pdf
  3. https://pmc.ncbi.nlm.nih.gov/articles/PMC12536273/
  4. http://www.rr.cs.cmu.edu/HLF/HLFYT.ppsx
  5. https://faculty.etsu.edu/gardnerr/3040/Notes-Eves6/Eves6-1-2.pdf
  6. https://www.mathsisfun.com/numbers/bases.html
  7. https://www.cs.umb.edu/~haoyu/tests/practice1.pdf
  8. https://mathresearch.utsa.edu/wiki/index.php?title=Base_10%2C_Base_2_%26_Base_5
  9. http://mathcentral.uregina.ca/QQ/database/QQ.09.16/h/babawale1.html
  10. https://www.youtube.com/watch?v=jyCsjd3N0jY
  11. http://mathcentral.uregina.ca/QQ/database/QQ.09.14/h/nakita1.html
  12. http://mathcentral.uregina.ca/QQ/database/QQ.09.13/h/alise1.html
  13. https://www.concord.edu/wp-content/uploads/DIVISION.pdf
  14. http://mathcentral.uregina.ca/QQ/database/QQ.09.13/h/samuel1.html
  15. https://pmc.ncbi.nlm.nih.gov/articles/PMC8678391/
  16. https://www.laphamsquarterly.org/roundtable/early-history-counting
  17. https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/
  18. https://aiatsis.gov.au/sites/default/files/e_access/serial/m0029743_v_a.pdf
  19. https://www.researchgate.net/publication/391206601_Numbers_in_Australia
  20. https://kuscholarworks.ku.edu/entities/publication/399db531-42e1-4ee6-a67e-ead88aabea6a
  21. https://www.omniglot.com/language/numbers/kwakwala.htm
  22. https://www.jstor.org/stable/2972526
  23. https://www.researchgate.net/publication/374268705_Systems_of_cardinal_numerals_in_languages_around_the_world
  24. https://cejsh.icm.edu.pl/cejsh/element/bwmeta1.element.desklight-77d1bd17-2a73-4217-94c8-aa9f8a8871c5/c/7_African_Hum_57_new.pdf
  25. https://openresearch-repository.anu.edu.au/bitstream/1885/145401/1/PL-B101.pdf
  26. https://jscholarship.library.jhu.edu/server/api/core/bitstreams/234bcf73-5120-429e-94ec-568933c8312b/content
  27. https://files.eric.ed.gov/fulltext/ED184846.pdf
  28. https://openwa.pressbooks.pub/math131/chapter/0-3-exploring-other-base-systems/
  29. https://www.cut-the-knot.org/blue/frac_conv.shtml
  30. https://www.stat.berkeley.edu/~nolan/stat133/Spr04/chapters/representations.pdf
  31. https://www.themathdoctors.org/how-roman-numerals-work/
  32. https://people.csail.mit.edu/kuat/courses/decimal-expansions.pdf
  33. https://kconrad.math.uconn.edu/blurbs/ugradnumthy/ordersmodm.pdf
  34. https://mathworld.wolfram.com/RepeatingDecimal.html
  35. https://www.merriam-webster.com/dictionary/biquinary
  36. https://research.swtch.com/biquinary
  37. https://jangawolof.org/2007/11/23/wolof-numbers-counting/
  38. https://sliderulemuseum.com/Abaci.shtml
  39. https://www.ias.ac.in/article/fulltext/reso/023/12/1395-1410
  40. https://oeis.org/wiki/Balanced_quinary_numeral_system
  41. https://oeis.org/wiki/Balanced_numeral_systems
  42. http://infolab.stanford.edu/pub/voy/museum/pictures/display/2-5-Mechanical.html
  43. https://sharp.ca/uploads/product_downloads/ELW516XGBSL_OperationsManual_ENG-en.pdf
  44. https://literature.hpcalc.org/community/wp34s-om-en.pdf
  45. https://onlinelibrary.wiley.com/doi/10.1002/adts.202100304
  46. https://docs.sympy.org/latest/modules/core.html
  47. https://ntruprime.cr.yp.to/nist/ntruprime-20171130.pdf
  48. https://retrocomputing.stackexchange.com/questions/9678/why-did-the-ibm-650-use-bi-quinary
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