Senary

In mathematics, a senary numeral system is a base-6 numeral system.

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. Writing out the prime numbers in base-six (and using the subscript 6 to denote that these are senary numbers), the first few primes are

$2_6,3_6,5_6,11_6,15_6,21_6,25_6,31_6,35_6,45_6,51_6,$
$101_6,105_6,111_6,115_6,125_6,\ldots$

That is, for every prime number p with $p\ne 2,3$, one has the modular arithmetic relations that either $p\mod 6 = 1$ or $p\mod 6 = 5$: the final digits is a 1 or a 5. This is proved by contradiction. For integer n:

• If n mod 6 = 0, 6|n
• If n mod 6 = 2, 2|n
• If n mod 6 = 3, 3|n
• If n mod 6 = 4, 2|n

Furthermore, all known perfect numbers besides 6 have 44 as the final two digits when expressed in base 6.

Finger counting

 * 1 2 3 4 5 10 11 12 1 1 2 3 4 5 10 11 12 2 2 4 10 12 14 20 22 24 3 3 10 13 20 23 30 33 40 4 4 12 20 24 32 40 44 52 5 5 14 23 32 41 50 55 104 10 10 20 30 40 50 100 110 120 11 11 22 33 44 55 110 121 132 12 12 24 40 52 104 120 132 144

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), entitled "Tractatus de computo, vel loquela per gestum digitorum",[1][2] which allowed counting up to 9,999 on two hands.

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 Senary base Prime factors of the base: 2, 3 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

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 is also suspected to have used senary numerals.[citation needed]