Senary

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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 peopleTemplate: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

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

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.Template:Overline 0.2 3 1/3
1/4 2 0.25 0.13 2 1/4
1/5 5 0.2 0.1111... = 0.Template:Overline 5 1/5
1/6 2, 3 0.1Template:Overline 0.1 2, 3 1/10
1/7 7 0.Template:Overline 0.Template:Overline 11 1/11
1/8 2 0.125 0.043 2 1/12
1/9 3 0.Template:Overline 0.04 3 1/13
1/10 2, 5 0.1 0.0Template:Overline 2, 5 1/14
1/11 11 0.Template:Overline 0.Template:Overline 15 1/15
1/12 2, 3 0.08Template:Overline 0.03 2, 3 1/20
1/13 13 0.Template:Overline 0.Template:Overline 21 1/21
1/14 2, 7 0.0Template:Overline 0.0Template:Overline 2, 11 1/22
1/15 3, 5 0.0Template:Overline 0.0Template:Overline 3, 5 1/23
1/16 2 0.0625 0.0213 2 1/24
1/17 17 0.Template:Overline 0.Template:Overline 25 1/25
1/18 2, 3 0.0Template:Overline 0.02 2, 3 1/30
1/19 19 0.Template:Overline 0.Template:Overline 31 1/31
1/20 2, 5 0.05 0.01Template:Overline 2, 5 1/32
1/21 3, 7 0.Template:Overline 0.0Template:Overline 3, 11 1/33
1/22 2, 11 0.0Template:Overline 0.0Template:Overline 2, 15 1/34
1/23 23 0.Template:Overline 0.0Template:Overline 35 1/35
1/24 2, 3 0.041Template:Overline 0.013 2, 3 1/40
1/25 5 0.04 0.Template:Overline 5 1/41
1/26 2, 13 0.0Template:Overline 0.0Template:Overline 2, 21 1/42
1/27 3 0.Template:Overline 0.012 3 1/43
1/28 2, 7 0.03Template:Overline 0.01Template:Overline 2, 11 1/44
1/29 29 0.Template:Overline 0.Template:Overline 45 1/45
1/30 2, 3, 5 0.0Template:Overline 0.0Template:Overline 2, 3, 5 1/50
1/31 31 0.Template:Overline 0.Template:Overline 51 1/51
1/32 2 0.03125 0.01043 2 1/52
1/33 3, 11 0.Template:Overline 0.0Template:Overline 3, 15 1/53
1/34 2, 17 0.0Template:Overline 0.0Template:Overline 2, 25 1/54
1/35 5, 7 0.0Template:Overline 0.Template:Overline 5, 11 1/55
1/36 2, 3 0.02Template:Overline 0.01 2, 3 1/100

Finger counting

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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

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

See also

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

Related number systems

References

  1. Template:Cite web
  2. Template:Cite web
  3. {{#invoke:citation/CS1|citation |CitationClass=citation }}

External links

de:Senär#Senäres Zahlensystem