A senary (/ˈsiːnəri, ˈsɛnəri/) numeral system (also known as base-6, heximal, or seximal) has six as its base.
When expressed in senary, all prime numbers other than 2 and 3 have 1 or 5 as the final digit.
In senary, the prime numbers are written: 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 digit is a 1 or a 5.
For any integer n: Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simple divisibility tests for many numbers.
Senary is also the largest number base r that has no totatives other than 1 and r − 1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table.
This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do.
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: Each regular human hand may be said to have six unambiguous positions; a fist, one finger extended, two, three, four, and then all five fingers extended.
Additionally, this method is the least abstract way to count using two hands that reflects the concept of positional notation, as the movement from one position to the next is done by switching from one hand to another.
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.
The downside to senary counting, however, is that without prior agreement two parties would be unable to utilize this system, being unsure which hand represents sixes and which hand represents ones, whereas decimal-based counting (with numbers beyond 5 being expressed by an open palm and additional fingers) being essentially a unary system only requires the other party to count the number of extended fingers.
In NCAA basketball, the players' uniform numbers are restricted to be senary numbers of at most two digits, so that the referees can signal which player committed an infraction by using this finger-counting system.
[1] More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99, 1023, or even higher depending on the method (though not necessarily senary in nature).
The English monk and historian Bede, described in the first chapter of his work De temporum ratione, (725), titled "Tractatus de computo, vel loquela per gestum digitorum," a system which allowed counting up to 9,999 on two hands.
[2][3] Despite the rarity of cultures that group large quantities by 6, a review of the development of numeral systems suggests a threshold of numerosity at 6 (possibly being conceptualized as "whole", "fist", or "beyond five fingers"[4]), with 1–6 often being pure forms, and numerals thereafter being constructed or borrowed.
[5] The Ndom language of Western New Guinea, Indonesia, is reported to have senary numerals.
One example is Komnzo with the following numerals: nibo (61), fta (62 [36]), taruba (63 [216]), damno (64 [1296]), wärämäkä (65 [7776]), wi (66 [46656]).
Some Niger–Congo languages have been reported to use a senary number system, usually in addition to another, such as decimal or vigesimal.
The compression effect of 36 being the square of 6 causes a lot of patterns and representations to be shorter in base 36: