Senary

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The senary numeral system (also known as base-6 or heximal) has six as its base. It has been adopted independently by a couple of cultures. Like decimal, it is a semiprime, though being the product of the only two consecutive numbers that are both prime (2,3) it has a high degree of mathematical properties for its size. As six is a superior highly composite number, many of the arguments made in favor of the dozenal system also apply to this base.

Mathematical properties[edit]

Senary multiplication table
× 1 2 3 4 5 10
1 1 2 3 4 5 10
2 2 4 10 12 14 20
3 3 10 13 20 23 30
4 4 12 20 24 32 40
5 5 14 23 32 41 50
10 10 20 30 40 50 100

Senary may be considered useful in the study of prime numbers since all primes other than 2 and 3, when expressed in base-six, have 1 or 5 as the final digit. In base-six 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, ...

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 digits 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 base 6, which is proven by the fact that every even perfect number is of the form 2p−1(2p−1) where 2p−1 is prime.

Senary is also the largest number base to lack non-unitary, non-neighbor totatives, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. Similarly, it maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do.

Fractions[edit]

Because six is the product of the first two prime numbers and is adjacent to the next two prime numbers, many senary fractions have simple representations:

Decimal base
Prime factors of the base: 2, 5
Prime factors of one below the base: 3
Prime factors of one above the base: 11
Other Prime factors: 7 13 17 19 23 29 31
Senary base
Prime factors of the base: 2, 3
Prime factors of one below the base: 5
Prime factors of one above the base: 11
Other Prime factors: 15 21 25 31 35 45 51
Fraction Prime factors
of the denominator
Positional representation Positional representation Prime factors
of the denominator
Fraction
1/2 2 0.5 0.3 2 1/2
1/3 3 0.3333... = 0.3 0.2 3 1/3
1/4 2 0.25 0.13 2 1/4
1/5 5 0.2 0.1111... = 0.1 5 1/5
1/6 2, 3 0.16 0.1 2, 3 1/10
1/7 7 0.142857 0.05 11 1/11
1/8 2 0.125 0.043 2 1/12
1/9 3 0.1 0.04 3 1/13
1/10 2, 5 0.1 0.03 2, 5 1/14
1/11 11 0.09 0.0313452421 15 1/15
1/12 2, 3 0.083 0.03 2, 3 1/20
1/13 13 0.076923 0.024340531215 21 1/21
1/14 2, 7 0.0714285 0.023 2, 11 1/22
1/15 3, 5 0.06 0.02 3, 5 1/23
1/16 2 0.0625 0.0213 2 1/24
1/17 17 0.0588235294117647 0.0204122453514331 25 1/25
1/18 2, 3 0.05 0.02 2, 3 1/30
1/19 19 0.052631578947368421 0.015211325015211325 31 1/31
1/20 2, 5 0.05 0.014 2, 5 1/32
1/21 3, 7 0.047619 0.014 3, 11 1/33
1/22 2, 11 0.0045 0.01345242103 2, 15 1/34
1/23 23 0.0434782608695652173913 0.001322030441 35 1/35
1/24 2, 3 0.0416 0.013 2, 3 1/40
1/25 5 0.04 0.01235 5 1/41
1/26 2, 13 0.0384615 0.0121502434053 2, 21 1/42
1/27 3 0.037 0.012 3 1/43
1/28 2, 7 0.03571428 0.0114 2, 11 1/44
1/29 29 0.0344827586206896551724137931 0.01124045443151 45 1/45
1/30 2, 3, 5 0.03 0.01 2, 3, 5 1/50
1/31 31 0.032258064516129 0.010545 51 1/51
1/32 2 0.03125 0.01043 2 1/52
1/33 3, 11 0.03 0.01031345242 3, 15 1/53
1/34 2, 17 0.02941176470588235 0.01020412245351433 2, 25 1/54
1/35 5, 7 0.0285714 0.01 5, 11 1/55
1/36 2, 3 0.027 0.01 2, 3 1/100

Finger counting[edit]

Main article: 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.

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.

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, 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 notion to young students.

More abstract finger counting systems, such as chisanbop or finger binary, 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, in the first chapter of De temporum ratione, (725), titled "Tractatus de computo, vel loquela per gestum digitorum,"[1][2] allowed counting up to 9,999 on two hands.

Natural languages[edit]

Despite the rarity of cultures which 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"[3]), with 1-6 often being pure forms and numerals thereafter being constructed or borrowed.[4]

The Ndom language of Papua New Guinea is reported to have senary numerals.[5] 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 are the Morehead-Maro 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 Kómnzo with the following numerals: nimbo (61), féta (62), tarumba (63), ntamno (64), wärämäkä (65), wi (66).

Some Niger-Congo languages have been reported to use a senary number system, usually in addition to another such as ten or twenty.[4]

Proto-Uralic has also been suspected to have had senary numerals, with a numeral for 7 being borrowed later, though evidence for constructing larger numerals (8 & 9) subtractively from ten suggests this may not be so.[4]

See also[edit]

  • Diceware method to encode base 6 values into pronounceable passwords.
  • Base 36 encoding scheme

Related number systems[edit]

References[edit]

  1. ^ "Dactylonomy". Laputan Logic. 16 November 2006. Retrieved May 12, 2012. 
  2. ^ Bloom, Jonathan M. (2001). "Hand sums: The ancient art of counting with your fingers". Yale University Press. Retrieved May 12, 2012. 
  3. ^ http://www.jstor.org/discover/10.1086/430579
  4. ^ a b c http://ling.uni-konstanz.de/pages/home/plank/for_download/publications/151_Plank_SenerySummary_2009.pdf
  5. ^ Owens, Kay (2001), "The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania", Mathematics Education Research Journal 13 (1): 47–71, doi:10.1007/BF03217098 

External links[edit]