Senary

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Senary, or heximal, refers to a base-6 numeral system. It is based on a semiprime of the first two prime numbers. It has been adopted independently by a couple of cultures. Some people[who?] advocate for its use due to many common fractions terminating, its simple to learn addition and multiplication tables, and its economy relative to higher bases.

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

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.

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]

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.

Other 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]

The Ndom language of Papua New Guinea is reported to have senary numerals.[3] Mer means 6, mer an thef means 6×2 = 12, nif means 36, and nif thef means 36×2 = 72. Proto-Uralic language is also suspected to have used senary numerals.[citation needed]

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. ^ 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]