# User Contributed Dictionary

### Adjective

- Of sixth rank or order.
- 2001, Manish K. Gupta, David G. Glynn, and T. Aaron Gulliver,
On Senary Simplex Codes, in Applied Algebra, Algebraic Algorithms
and Error-Correcting Codes: proceedings of 14th International
Symposium, Serdar Boztaş, Igor E. Shparlinski (eds.), page 112
- In particular, one can construct mixed binary/ternary codes via senary codes by applying the Chinese Gray map (see Example 1).

- 2001, Manish K. Gupta, David G. Glynn, and T. Aaron Gulliver,
On Senary Simplex Codes, in Applied Algebra, Algebraic Algorithms
and Error-Correcting Codes: proceedings of 14th International
Symposium, Serdar Boztaş, Igor E. Shparlinski (eds.), page 112

#### See also

# Extensive Definition

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

## 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 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 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

senary in German: Hexalsystem

senary in French: Système sénaire

senary in Haitian: Sistèm senè

senary in Japanese: 六進法

senary in Finnish: Senaarijärjestelmä

senary in Thai: เลขฐานหก