Senary Did You Mean Senary?

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Numeral systems

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Unary (1)
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Ternary (3)
Quinary (5)
Senary (6)
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Duodecimal (12)
Hexadecimal (16)
Vigesimal (20)
Quadrovigesimal (24)
Hexavigesimal (26)
Septemvigesimal (27)
Hexatridecimal (36)
Sexagesimal (60)

In mathematics, a senary numeral system is a base-six numeral system. The name heximal is also valid for such a numeral system, but is deprecated to avoid confusion with the more often used hexadecimal number base, colloquially known as 'hex'.

Senary may be considered useful in the study of prime numbers since all primes, when expressed in base-six, other than 2 and 3 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

26,3 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. Furthermore, all known perfect numbers besides 6 itself have 44 as the final two digits.

Base six can be counted on hands by using each hand to represent a digit, 0 to 5. The digit zero is represented by a fist (no fingers extended). Using this method, it is possible for one person to represent values from zero to 55_senary (= 35_decimal), rather than the usual ten obtained in standard finger-counting.

Interestingly, the number 3333_senary is equal to 777_decimal.

Fractions

Due to the fact that 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   Senary
  1/2     1/2   =  0.3
  1/3     1/3   =  0.2
  1/4     1/4   =  0.13
  1/5     1/5   =  0.1111 recurring
  1/6     1/10  =  0.1
  1/7     1/11  =  0.05050505 recurring
  1/8     1/12  =  0.043
  1/9     1/13  =  0.04
  1/10    1/14  =  0.03333 recurring
  1/12    1/20  =  0.03
  1/14    1/22  =  0.023232323 recurring
  1/15    1/23  =  0.022222222 recurring
  1/16    1/24  =  0.0213
  1/18    1/30  =  0.02
  1/20    1/32  =  0.014444444 recurring

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