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A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like decimal, it is a
semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...
, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to senary. In turn, the senary logic refers to an extension of
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. ...
's and Stephen Cole Kleene's ternary logic systems adjusted to explain the logic of statistical tests and missing data patterns in sciences using empirical methods.


Formal definition

The standard
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of digits in senary is given by \mathcal_6 = \lbrace 0, 1, 2, 3, 4, 5\rbrace, with a linear order 0 < 1 < 2 < 3 < 4 < 5. Let \mathcal_6^* be the Kleene closure of \mathcal_6, where ab is the operation of string concatenation for a, b \in \mathcal^*. The senary number system for natural numbers \mathcal_6 is the quotient set \mathcal_6^* / \sim equipped with a shortlex order, where the equivalence class \sim is \lbrace n \in \mathcal_6^*, n \sim 0n \rbrace. As \mathcal_6 has a shortlex order, it is isomorphic to the natural numbers \mathbb.


Mathematical properties

When expressed in senary, all
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s other than 2 and 3 have 1 or 5 as the final digit. In senary, the prime numbers are written :2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, ... That is, for every prime number ''p'' greater than 3, one has the modular arithmetic relations that either ''p'' ≡ 1 or 5 (mod 6) (that is, 6 divides either ''p'' − 1 or ''p'' − 5); the final digit is a 1 or a 5. This is proved by contradiction. For any integer ''n'': * If ''n'' ≡ 0 (mod 6), 6 , ''n'' * If ''n'' ≡ 2 (mod 6), 2 , ''n'' * If ''n'' ≡ 3 (mod 6), 3 , ''n'' * If ''n'' ≡ 4 (mod 6), 2 , ''n'' Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simple divisibility tests for many numbers. Furthermore, all even perfect numbers besides 6 have 44 as the final two digits when expressed in senary, which is proven by the fact that every even perfect number is of the form 2^(2^-1), where 2^-1 is prime. Senary is also the largest number base ''r'' that has no totatives other than 1 and ''r'' − 1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do. If a number is divisible by 2, then the final digit of that number, when expressed in senary, is 0, 2, or 4. If a number is divisible by 3, then the final digit of that number in senary is 0 or 3. A number is divisible by 4 if its penultimate digit is odd and its final digit is 2, or its penultimate digit is even and its final digit is 0 or 4. A number is divisible by 5 if the sum of its senary digits is divisible by 5 (the equivalent of casting out nines in decimal). If a number is divisible by 6, then the final digit of that number is 0. To determine whether a number is divisible by 7, one can sum its alternate digits and subtract those sums; if the result is divisible by 7, the number is divisible by 7


Fractions

Because six is the product of the first two
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s and is adjacent to the next two prime numbers, many senary fractions have simple representations:


Finger counting

Each regular human hand may be said to have six unambiguous positions; a fist, one finger (or thumb) extended, two, three, four, and then all five extended. If the right hand is used to represent a unit, and the left to represent the "sixes", it becomes possible for one person to represent the values from zero to 55senary (35decimal) with their fingers, rather than the usual ten obtained in standard finger counting. e.g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 × 6 + 4, which is 22decimal. Additionally, this method is the least abstract way to count using two hands that reflects the concept of positional notation, as the movement from one position to the next is done by switching from one hand to another. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. As senary finger counting also deviates only beyond 5, this counting method rivals the simplicity of traditional counting methods, a fact which may have implications for the teaching of positional notation to young students. Which hand is used for the 'sixes' and which the units is down to preference on the part of the counter, however when viewed from the counter's perspective, using the left hand as the most significant digit correlates with the written representation of the same senary number. Flipping the 'sixes' hand around to its backside may help to further disambiguate which hand represents the 'sixes' and which represents the units. The downside to senary counting, however, is that without prior agreement two parties would be unable to utilize this system, being unsure which hand represents sixes and which hand represents ones, whereas decimal-based counting (with numbers beyond 5 being expressed by an open palm and additional fingers) being essentially a unary system only requires the other party to count the number of extended fingers. In NCAA basketball, the players' uniform numbers are restricted to be senary numbers of at most two digits, so that the referees can signal which player committed an infraction by using this finger-counting system. More abstract finger counting systems, such as chisanbop or
finger binary Finger binary is a system for counting and displaying binary numbers on the fingers of either or both hands. Each finger represents one binary digit or bit. This allows counting from zero to 31 using the fingers of one hand, or 1023 using both: t ...
, allow counting to 99, 1,023, or even higher depending on the method (though not necessarily senary in nature). The English monk and historian Bede, described in the first chapter of his work ''De temporum ratione'', (725), titled "''Tractatus de computo, vel loquela per gestum digitorum''," a system which allowed counting up to 9,999 on two hands.


Natural languages

Despite the rarity of cultures that group large quantities by 6, a review of the development of numeral systems suggests a threshold of numerosity at 6 (possibly being conceptualized as "whole", "fist", or "beyond five fingers"), with 1–6 often being pure forms, and numerals thereafter being constructed or borrowed. The
Ndom language Ndom is a language spoken on Yos Sudarso Island in Papua province, Indonesia. It is reported to have numbers in senary A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independ ...
of Indonesian New Guinea is reported to have senary numerals. ''Mer'' means 6, ''mer an thef'' means 6 × 2 = 12, ''nif'' means 36, and ''nif thef'' means 36 × 2 = 72. Another example from
Papua New Guinea Papua New Guinea (abbreviated PNG; , ; tpi, Papua Niugini; ho, Papua Niu Gini), officially the Independent State of Papua New Guinea ( tpi, Independen Stet bilong Papua Niugini; ho, Independen Stet bilong Papua Niu Gini), is a country i ...
are the Yam languages. In these languages, counting is connected to ritualized yam-counting. These languages count from a base six, employing words for the powers of six; running up to 66 for some of the languages. One example is Komnzo with the following numerals: ''nibo'' (61), ''fta'' (62 6, ''taruba'' (63 16, ''damno'' (64
296 __NOTOC__ Year 296 ( CCXCVI) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Diocletian and Constantius (or, less freque ...
, ''wärämäkä'' (65
776 __NOTOC__ Year 776 ( DCCLXXVI) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 776 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
, ''wi'' (66 6656. Some Niger-Congo languages have been reported to use a senary number system, usually in addition to another, such as decimal or vigesimal.
Proto-Uralic Proto-Uralic is the unattested reconstructed language ancestral to the modern Uralic language family. The hypothetical language is believed to have been originally spoken in a small area in about 7000–2000 BCE, and expanded to give differenti ...
has also been suspected to have had senary numerals, with a numeral for 7 being borrowed later, though evidence for constructing larger numerals (8 and 9) subtractively from ten suggests that this may not be so.


Base 36 as senary compression

For some purposes, senary might be too small a base for convenience. This can be worked around by using its square, base 36 (hexatrigesimal), as then conversion is facilitated by simply making the following replacements: Thus, the base-36 number WIKIPEDIA36 is equal to the senary number 5230323041222130146. In decimal, it is 91,730,738,691,298. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the
Latin letters The Latin script, also known as Roman script, is an alphabetic writing system based on the letters of the classical Latin alphabet, derived from a form of the Greek alphabet which was in use in the ancient Greek city of Cumae, in southern ...
A–Z: this choice is the basis of the
base36 Base36 is a binary-to-text encoding scheme that represents binary data in an ASCII string format by translating it into a radix-36 representation. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0� ...
encoding scheme. The compression effect of 36 being the square of 6 causes a lot of patterns and representations to be shorter in base 36: 1/910 = 0.046 = 0.436 1/1610 = 0.02136 = 0.2936 1/510 = 0.6 = 0.36 1/710 = 0.6 = 0.{{overline, 536


See also

* Diceware method to encode base-6 values into pronounceable passwords. *
Base36 Base36 is a binary-to-text encoding scheme that represents binary data in an ASCII string format by translating it into a radix-36 representation. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0� ...
encoding scheme * ADFGVX cipher to encrypt text into a series of effectively senary digits


References


External links


Shack's Base Six Dialectic

Senary base conversion

Website about Seximal
Positional numeral systems Finger-counting de:Senär#Senäres Zahlensystem