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Asenary (/ˈsiːnəri,ˈsɛnəri/)numeral system (also known asbase-6,heximal, orseximal) hassix as itsbase. It has been adopted independently by a small number of cultures. Like thedecimal base 10, the base is asemiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is asuperior highly composite number, many of the arguments made in favor of theduodecimal system also apply to the senary system.
The standardset of digits in the senary system is, with thelinear order. Let be theKleene closure of, where is the operation ofstring concatenation for. The senary number system fornatural numbers is thequotient set equipped with ashortlex order, where theequivalence class is. As has a shortlex order, it isisomorphic to the natural numbers.
| × | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 10 |
| 2 | 2 | 4 | 10 | 12 | 14 | 20 |
| 3 | 3 | 10 | 13 | 20 | 23 | 30 |
| 4 | 4 | 12 | 20 | 24 | 32 | 40 |
| 5 | 5 | 14 | 23 | 32 | 41 | 50 |
| 10 | 10 | 20 | 30 | 40 | 50 | 100 |
When expressed in senary, allprime 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 numberp greater than 3, one has themodular arithmetic relations that eitherp ≡ 1 or 5 (mod 6) (that is, 6 divides eitherp − 1 orp − 5); the final digit is a 1 or a 5. This is proved by contradiction.
For any integern:
Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simpledivisibility tests for many numbers.
Furthermore, all evenperfect numbers besides 6 have 44 as the final two digits when expressed in senary, which is proven by the fact that every even perfect number is of the form2p – 1(2p – 1), where2p − 1 is prime.
Senary is also the largest number baser that has nototatives other than 1 andr − 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.
If a number is divisible by 2, then the final digit of that number, when expressed in senary, is 0, 2, or 4.If a number is divisible by 3, then the final digit of that number in senary is 0 or 3.A number is divisible by 4 if its penultimate digit is odd and its final digit is 2, or its penultimate digit is even and its final digit is 0 or 4.A number is divisible by 5 if the sum of its senary digits is divisible by 5 (the equivalent ofcasting out nines in decimal).If a number is divisible by 6, then the final digit of that number is 0.To determine whether a number is divisible by 7, one can sum its alternate digits and subtract those sums; if the result is divisible by 7, the number is divisible by 7, similar to the "11" divisibility test in decimal.
Because six is theproduct of the first twoprime numbers and is adjacent to the next two prime numbers, many senary fractions have simple representations. The table below shows fractional and positional representations in both decimal and senary, and has cells colored to indicate a certain level of complexity;Light grey cells indicate a rational number with a repeating cycle of length one or two,dark grey cells indicate a repeating cycle longer than two digits, and uncolored, white cells indicate no repeating cycle at all.
| Decimal base Prime factors of the base:2,5 Prime factors of one below the base:3 Prime factors of one above the base:11 | Senary base Prime factors of the base:2,3 Prime factors of one below the base:5 Prime factors of one above the base:7 (=116) | ||||
| 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 |
| 1/19 | 19 | 0.052631578947368421 | 0.015211325 | 31 | 1/31 |
| 1/20 | 2,5 | 0.05 | 0.014 | 2,5 | 1/32 |
| 1/21 | 3,7 | 0.047619 | 0.014 | 3,11 | 1/33 |
| 1/22 | 2,11 | 0.045 | 0.01345242103 | 2,15 | 1/34 |
| 1/23 | 23 | 0.0434782608695652173913 | 0.01322030441 | 35 | 1/35 |
| 1/24 | 2,3 | 0.0416 | 0.013 | 2,3 | 1/40 |
| 1/25 | 5 | 0.04 | 0.01235 | 5 | 1/41 |
| 1/26 | 2,13 | 0.0384615 | 0.0121502434053 | 2,21 | 1/42 |
| 1/27 | 3 | 0.037 | 0.012 | 3 | 1/43 |
| 1/28 | 2,7 | 0.03571428 | 0.0114 | 2,11 | 1/44 |
| 1/29 | 29 | 0.0344827586206896551724137931 | 0.01124045443151 | 45 | 1/45 |
| 1/30 | 2,3,5 | 0.03 | 0.01 | 2,3,5 | 1/50 |
| 1/31 | 31 | 0.032258064516129 | 0.010545 | 51 | 1/51 |
| 1/32 | 2 | 0.03125 | 0.01043 | 2 | 1/52 |
| 1/33 | 3,11 | 0.03 | 0.01031345242 | 3,15 | 1/53 |
| 1/34 | 2,17 | 0.02941176470588235 | 0.01020412245351433 | 2,25 | 1/54 |
| 1/35 | 5,7 | 0.0285714 | 0.01 | 5,11 | 1/55 |
| 1/36 | 2,3 | 0.027 | 0.01 | 2,3 | 1/100 |
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.
If the right hand is used to represent a unit (0 to 5), and the left to represent the multiples of 6, then 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 to3 × 6 +4, which is 22decimal.
Additionally, this method is the least abstract way to count using two hands that reflects the concept ofpositional notation, as the movement from one position to the next is done by switching from one hand to another. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as withChinese number gestures. As senary finger counting also deviates only beyond 5, this counting method rivals the simplicity of traditional counting methods, a fact which may have implications for the teaching of positional notation to young students.
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. Flipping the 'sixes' hand around to its backside may help to further disambiguate which hand represents the 'sixes' and which represents the units. 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 aunary system only requires the other party to count the number of extended fingers.
InNCAA 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 abstractfinger counting systems, such aschisanbop orfinger binary, allow counting to 99, 1023, or even higher depending on the method (though not necessarily senary in nature). The English monk and historianBede, described in the first chapter of his workDe 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]
TheNdom language ofWestern New Guinea,Indonesia, is reported to have senary numerals.[6][7]Mer means 6,mer an thef means 6 × 2 = 12,nif means 36, andnif thef means 36 × 2 = 72.
Another example fromPapua New Guinea are theYam languages. In these languages, counting is connected to ritualized yam-counting. These languages count from a base six, employing words for the powers of six; running up to 66 for some of the languages. One example isKomnzo with the following numerals:nibo (61),fta (62 [36]),taruba (63 [216]),damno (64 [1296]),wärämäkä (65 [7776]),wi (66 [46656]).
SomeNiger–Congo languages have been reported to use a senary number system, usually in addition to another, such asdecimal orvigesimal.[5]
Proto-Uralic has also been suspected to have had senary numerals, with a numeral for 7 being borrowed later, though evidence for constructing larger numerals (8 and 9) subtractively from ten suggests that this may not be so.[5]
For some purposes, senary might be too small a base for convenience. This can be worked around by using its square, base 36 (hexatrigesimal), as then conversion is facilitated by simply making the following replacements:
| Decimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Base 6 | 0 | 1 | 2 | 3 | 4 | 5 | 10 | 11 | 12 | 13 | 14 | 15 | 20 | 21 | 22 | 23 | 24 | 25 |
| Base 36 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F | G | H |
| Decimal | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 |
| Base 6 | 30 | 31 | 32 | 33 | 34 | 35 | 40 | 41 | 42 | 43 | 44 | 45 | 50 | 51 | 52 | 53 | 54 | 55 |
| Base 36 | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
Thus, the base-36 number WIKI36 is equal to the senary number 523032306. In decimal, it is 1,517,058.
The choice of 36 as aradix is convenient in that the digits can be represented using theArabic numerals 0–9 and theLatin letters A–Z; this choice is the basis of thebase36 encoding scheme. The compression effect of 36 being the square of 6 causes a lot of patterns and representations to be shorter in base 36: