A senary ()
numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
(also known as base-6, heximal, or seximal) has
six
6 is a number, numeral, and glyph.
6 or six may also refer to:
* AD 6, the sixth year of the AD era
* 6 BC, the sixth year before the AD era
* The month of June
Science
* Carbon, the element with atomic number 6
* 6 Hebe, an asteroid
People ...
as its
base. It has been adopted independently by a small number of cultures. Like
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
, 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 nu ...
, 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
In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to ''some'' positive power of itself than any other number. It is a stronger restriction than that of a highly composite ...
, many of the arguments made in favor of the
duodecimal
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wri ...
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. He ...
's and
Stephen Cole Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
'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
, with a
linear order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
. Let
be the
Kleene closure
In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics,
it is more commonly known as the free monoid ...
of
, where
is the operation of
string concatenation
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...
for
. The senary number system for
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
is the
quotient set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
equipped with a
shortlex order, where the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
is
. As
has a shortlex order, it is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the natural numbers
.
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 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 because the only ways ...
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
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
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 test
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radi ...
s for many numbers.
Furthermore, all even
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
s 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
, where
is prime.
Senary is also the largest number base ''r'' that has no
totative
In number theory, a totative of a given positive integer is an integer such that and is coprime to . Euler's totient function φ(''n'') counts the number of totatives of ''n''. The totatives under multiplication modulo ''n'' form the mu ...
s 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
Casting out nines is any of three arithmetical procedures:
*Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller th ...
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
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of the first two
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
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 55
senary (35
decimal) 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, 34
senary is represented. This is equivalent to 3 × 6 + 4, which is 22
decimal.
Additionally, this method is the least abstract way to count using two hands that reflects the concept of
positional notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
, 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
Chinese number gestures are a method to signify the natural numbers one through ten using one hand. This method may have been developed to bridge the many varieties of Chinese—for example, the numbers 4 () and 10 () are hard to distinguish ...
. 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
In United States colleges, top-tier basketball is governed by collegiate athletic bodies including National Collegiate Athletic Association (NCAA), the National Association of Intercollegiate Athletics (NAIA), the United States Collegiate Athle ...
, 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
Finger-counting, also known as dactylonomy, is the act of counting using one's fingers. There are multiple different systems used across time and between cultures, though many of these have seen a decline in use because of the spread of Arabic n ...
systems, such as
chisanbop
Chisanbop or chisenbop (from Korean ''chi (ji)'' finger + ''sanpŏp (sanbeop)'' calculation 지산법/指算法), sometimes called Fingermath, is an abacus-like finger counting method used to perform basic mathematical operations. According to ' ...
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: ...
, 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
Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English monk at the monastery of St Peter and its companion monastery of St Paul in the Kingdom o ...
, 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 (base 6). A problem from the 2007 International Linguistics Olympiad
The International Linguistics Olympiad (IOL) is one of ...
of
Indonesian New Guinea
Western New Guinea, also known as Papua, Indonesian New Guinea, or Indonesian Papua, is the western half of the Melanesian island of New Guinea which is administered by Indonesia. Since the island is alternatively named as Papua, the region ...
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
The Yam languages, also known as the Morehead River languages, are a family of Papuan languages. They include many of the languages south and west of the Fly River in Papua New Guinea and Indonesian West Papua.
Name
The name ''Morehead and Upp ...
. 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 6
6 for some of the languages. One example is
Komnzo with the following numerals: ''nibo'' (6
1), ''fta'' (6
2 6, ''taruba'' (6
3 16, ''damno'' (6
4 296
__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 frequent ...
, ''wärämäkä'' (6
5 776, ''wi'' (6
6 6656.
Some
Niger-Congo languages have been reported to use a senary number system, usually in addition to another, such as
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
or
vigesimal
vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'.
Places
In ...
.
[
]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 different ...
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
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
is convenient in that the digits can be represented using the Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...
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 Italy ...
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
Diceware is a method for creating passphrases, passwords, and other cryptographic variables using ordinary dice as a hardware random number generator. For each word in the passphrase, five rolls of a six-sided die are required. The numbers from ...
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
In cryptography, the ADFGVX cipher was a manually applied field cipher used by the Imperial German Army during World War I. It was used to transmit messages secretly using wireless telegraphy. ADFGVX was in fact an extension of an earlier cipher ca ...
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