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

$2\_6,3\_6,5\_6,11\_6,15\_6,21\_6,25\_6,31\_6,35\_6,45\_6,51\_6,$

$101\_6,105\_6,111\_6,115\_6,125\_6,ldots$

That is, for every prime number $p$ with $pne\; 2,3$, one has the modular arithmetic relations that either $pmod\; 6\; =\; 1$ or $pmod\; 6\; =\; 5$: the final digits is a 1 or a 5. Furthermore, all known perfect numbers besides 6 itself have 44 as the final two digits.

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 55_senary (35_decimal) 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, 34_senary is represented. This is equivalent to 3 × 6 + 4 which is 22_decimal.

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 the same senary number.

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 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

Natural languages

The Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6×2 = 12, nif means 36, and nif thef means 36×2 = 72.

References

Closely related number systems

• Hexatridecimal (base 36)
• Duodecimal (base 12)

See also

• Morse code
• Diceware has a way of encoding base 6 values into pronounceable words, using a standardized list of 7,776 unique words

External links

• Senary Base Conversion, includes fractional part, from Math Is Fun