# Senary Numbers

In science and computer science, the system of the senary numbers are a number system in which numbers are represented using the digits from 0 to 5 (0-5). It can be used as a checking tool, along with the octal system and the sexagesimal system. The senary system or sexagesimal system is based on a number (x).

### Exponentiation

Senary numbers use only six digits, digits increase faster than other bases. But, the exponents of two and three are equal, this can be expressed as: 10(6)n = 2(6)n × 3(6)n.

In particular, six and ten have the same structure whose prime factors have the same exponents. Also, six to the 4n-th power (104n) is close to ten to the 3n-th power (143n). For instance:

• 1.0000(6) = 1.296(10) (equivalent to kilogram)
• 1.0000.0000(6) = 1.679.616(10) (equivalent to mega)
• 1.0000.0000.0000(6) = 2.176.782.336(10) (equivalent to giga)
• 1.0000.0000.0000.0000(6) = 2.821.109.907.456(10) (equivalent to tera)
##### Power of six by senary notation
The power of six by senary notation
ExponentSenaryDecimal equivalentDuodecimal equivalentVigesimal equivalent
110666
210062 = 3662 = 3062 = 1G
31 00063 = 21663 = 16063 = AG
410 00064 = 1 29664 = 90064 = 34G
5100 00065 = 7 77665 = 4 60065 = J8G
101 000 00066 = 46 65666 = 23 00066 = 5 GCG
1110 000 00067 = 279 93667 = 116 00067 = 1E JGG
12100 000 00068 = 1 679 61668 = 690 00068 = A9 J0G
131 000 000 00069 = 10 077 69669 = 3 460 00069 = 32J E4G
1410 000 000 000610 = 60 466 1766A = 18 300 0006A = IHI 58G
15100 000 000 000611 = 362 797 0566B = A1 600 0006B = 5 D79 CCG
201 000 000 000 000612 = 2 176 782 336610 = 509 000 0006C = 1E 04H FGG
2110 000 000 000 000613 = 13 060 694 016611 = 2 646 000 0006D = A4 196 F0G
22100 000 000 000 000614 = 78 364 164 096612 = 13 230 000 0006E = 314 8G0 A4G
231 000 000 000 000 000615 = 470 184 984 576613 = 77 160 000 0006F = I76 CG3 18G
2410 000 000 000 000 000616 = 2 821 109 907 456614 = 396 900 000 0006G = 5 A3J GGI 8CG
25100 000 000 000 000 000617 = 16 926 659 444 736615 = 1 A94 600 000 0006H = 1D 13J 11A BGG
301 000 000 000 000 000 000618 = 101 559 956 668 416616 = B 483 000 000 0006I = 9I 73E 693 B0G

### Integer notation (whole number that can be positive, negative, or zero)

#### From senary to decimal

Here are the first numbers from 1 to 40 and from 91 to 110 expressed in senary and then decimal positional notation.

 Senary Decimal Senary Decimal Senary Decimal 1 2 3 4 5 10 11 12 13 14 15 20 21 22 23 24 25 30 31 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 33 34 35 40 41 42 43 44 45 50 51 52 53 54 55 100 101 102 103 104 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 231 232 233 234 235 240 241 242 243 244 245 250 251 252 253 254 255 300 301 302 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Senary numbers are expresses six as “10”, nine (9) as “13” i.e. “six plus three”, ten (decimal 10) as “14” i.e. “six plus four”, twelve (decimal 12) as “20” i.e. “two sixes”, sixteen (decimal 16) as “24” i.e. “two sixes and four“.

Digits for multiples of three end in 3 or 0, e.g. decimal 18 (eighteen) is expressed as “30” (three sixes), decimal 15 (ten and five) is expressed as “23” (two sixes and three), decimal 27 (two ten and seven) is expressed as “43” (four sixes and three).

Numbers over 100 (decimal 36), e.g. decimal 81 is expressed as “213” to say “two of six squares, six and three”, decimal 100 is expressed as “244” to say “two of six squares, four six and four“.

##### Notation rating breakdown:
• 810 = 126 = 1×6 + 2
• 1010 = 146 = 1×6 + 4
• 1210 = 206 = 2×6
• 2710 = 436 = 4×6 + 3
• 3010 = 506 = 5×6
• 3610 = 1006 = 1×62
• 4910 = 1216 = 1×62 + 2×61 + 1
• 5610 = 1326 = 1×62 + 3×61 + 2
• 6410 = 1446 = 1×62 + 4×61 + 4
• 8110 = 2136 = 2×62 + 1×61 + 3
• 10010 = 2446 = 2×62 + 4×61 + 4
• 10810 = 3006 = 3×62
• 12510 = 3256 = 3×62 + 2×61 + 5
• 14410 = 4006 = 4×62
• 17510 = 4516 = 4×62 + 5×61 + 1
• 18010 = 5006 = 5×62
• 21610 = 10006 = 1×63
• 25610 = 11046 = 1×63 + 1×62 + 0×61 + 4
• 56910 = 23456 = 2×63 + 3×62 + 4×61 + 5
• 72910 = 32136 = 3×63 + 2×62 + 1×61 + 3
• 100010 = 43446 = 4×63 + 3×62 + 4×61 + 4
• 102410 = 44246 = 4×63 + 4×62 + 2×61 + 4
• 108010 = 50006 = 5×63
• 129610 = 100006 = 1×64
• 194410 = 130006 = 1×64 + 3×63
• 200010 = 131326 = 1×64 + 3×63 + 1×62 + 3×61 + 2
• 500010 = 350526 = 3×64 + 5×63 + 0×62 + 5×61 + 2
• 656110 = 502136 = 5×64 + 0×63 + 2×62 + 1×61 + 3
##### Examples of arithmetic operations
Examples of arithmetic operations
DecimalSenary
1944 + 56 = 200013000 + 132 = 13132
100 – 64 = 36244 – 144 = 100
16 × 81 = 129624 × 213 = 10000
1080 ÷ 27 = 405000 ÷ 43 = 104
64 / 144 = 4 / 9144 / 400 = 4 / 13
38 = 6561312 = 50213
24 = 23×340 = 23×3
##### Date and hour
Date and hour
EventsDecimalSenary
The death of Alfred Nobel10 / 12 / 189614 / 20 / 12440
Atomic bombing of Hiroshima6 / 8 / 194510 / 12 / 13001
September 11, 2001 attacks11 / 9 / 200115 / 13 / 13133
##### From decimal to senary

Here are some benchmarks.

 Decimal Senary Decimal Senary 1 2 3 4 5 6 7 8 9 12 15 18 24 27 30 36 1 2 3 4 5 10 11 12 13 20 23 30 40 43 50 100 42 54 72 108 144 162 180 216 324 432 648 972 1080 1296 1944 2592 110 130 200 300 400 430 500 1000 1300 2000 3000 4300 5000 10000 13000 20000

As will be described in detail in the section on fractions, the upper-digit shift of senary numbers has a relationship of “four to nine” (4×13 = 100).

Therefore, four 130s will be 1000, nine 400s will be 10000, three quarters of 1000 will be 430, two ninths of 100 will be 12.

#### Doubles or duplicate detection

• All numbers ending in senary with a digit representing a multiple of 2 — i.e. 2, 4, 0 — are divisible by 2.
• All numbers ending in a digit representing a multiple of 3 — that is, 3 and 0 — divisible by 3.
• If the last two digits are a multiple of 4 {04, 12, 20, 24, 32, 40, 44, 52, 00} — that’s a multiple of 4. Nine (13 = 3²) types in all.
• If the sum of the digits is a multiple of 5 — it is a multiple of 5.
• If the last two digits are a multiple of 13 {13, 30, 43, 00} — it’s a multiple of 13 (nine). 4 (= 2²) types in all.

(as in decimal, all numbers ending in a digit representing a multiple of 2 — i.e. 2, 4, 6, 8, 0 are divisible by 2; and all numbers ending in a multiple of 5 — i.e. 5 and 0 — divisible by 5).

#### Prime number

A prime number other than 2 or 3 can therefore only end in senary with 1 or 5. (in decimal a prime number other than 2 or 5 can only end with 1, 3, 7 or 9).

• Prime numbers from 1 to 100 (to decimal 36)
• 2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51
• Prime numbers from 101 to 1000 (decimal 37 to 216)
• 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255
• 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551
• Compound numbers that are neither divisible by 2 nor by 3, from 1 to 1000 (to decimal 216)
• 41, 55, 121, 131, 145, 205, 221, 231, 235, 311, 315, 321, 325, 341, 355, 401, 415, 425, 441, 451, 505, 511, 535, 541, 545, 555

## Fractions and divisions

Six is the product of two prime numbers, namely 2 and 3. As a result, some properties of senary positional notation are reminiscent of those of decimal positional notation.

All fractions whose denominator knows no other prime factor than 2 and 3 are expressed in senary with a finite number of decimal places. (Compare with the role of 2 and 5 in decimal.) Six and ten are only even numbers, a quarter is expressed as two decimal places. Thus, senary and decimal system, the position of 3 and 5 is reversed. For example, “0.2” is 1/5 (i.e. two tenths) in decimal, but 1/3 (i.e. two sixths) in senary.

In senary notation, reciprocals of powers of 2 are powers of 3, reciprocals of powers of 3 are powers of 2, dividing by powers of 2 and 3 becomes easier than any notation. Thus, the powers of 3 become dominant, the powers of 5 become weak.

The senary fraction have the characteristic of “short repetition” as being the same as the decimal fraction. Decimal fraction have 3-3 requires 3 digit repeats, 3-4 requires 9 (=3-2) digit repeats. Like this, the senary fraction have 5-2 requires 5-digit repetitions. The number whose repetitions reach about twenty-seven is 3-5 in decimal (33, twenty-seven digits), 5-3 in senary (5², twenty-five digits).

### Unit fraction

FactorisationDécimalSénaire
21/2 = 0,51/2 = 0,3
31/3 = 0,33 repetition1/3 = 0,2
221/4 = 0,251/4 = 0,13
51/5 = 0,21/5 = 0,11 repetition
2×31/6 = 0,166 répétition1/10 = 0,1
111/7 = 0,142857142857 repetition1/11 = 0,0505 repetition
231/8 = 0,1251/12 = 0,043
321/9 = 0,11 repetition1/13 = 0,04
2×51/10 = 0,11/14 = 0,033 repetition
151/11 = 0,0909 repetition1/15 = 0,03134524210313452421 repetition
22×31/12 = 0,08333 repetition1/20 = 0,03
211/13 = 0,076923076923 repetition1/21 = 0,024340531215024340531215 repetition
2×111/14 = 0,0714285714285 repetition1/22 = 0,02323 repetition
3×51/15 = 0,066 repetition1/23 = 0,022 repetition
241/16 = 0,06251/24 = 0,0213
2×321/18 = 0,055 repetition1/30 = 0,02
22×51/20 = 0,051/32 = 0,0144 repetition
23×31/24 = 0,04166 repetition1/40 = 0,013
521/25 = 0,041/41 = 0.0123501235 repetition
331/27 = 0,037037 repetition1/43 = 0,012
251/32 = 0,031251/52 = 0,01043
22×321/36 = 0,0277 repetition1/100 = 0,01
23×51/40 = 0,0251/104 = 0,00522 repetition
24×31/48 = 0,020833 repetition1/120 = 0,0043
2×521/50 = 0,021/122 = 0,00415304153 repetition
2×331/54 = 0,0185185 repetition1/130 = 0,004
2101/64 = 0,0156251/144 = 0,003213
23×321/72 = 0,01388 répétition1/200 = 0,003
24×51/80 = 0,01251/212 = 0,002411 repetition
341/81 = 0,012345679012345679 répétition1/213 = 0,0024
25×31/96 = 0,0104166 repetition1/240 = 0,00213
22×521/100 = 0,011/244 = 0,002054320543 repetition
22×331/108 = 0,00925925 repetition1/300 = 0,002
531/125 = 0,0081/325 = 0,0014211153224043351545031
0014211153224043351545031 repetition
2111/128 = 0,00781251/332 = 0,0014043
24×321/144 = 0,006944 repetition1/400 = 0,0013
25×51/160 = 0,006251/424 = 0,0012033 repetition
2×341/162 = 0,0061728395061728395 repetition1/430 = 0,0012
210×31/192 = 0,00520833 repetition1/520 = 0,001043
23×521/200 = 0,0051/532 = 0,0010251402514 repetition
23×331/216 = 0,004629629 repetition1/1000 = 0,001

### Main fraction

Enter the decimal equivalent in parentheses.

Up to ninths (except sevenths and eighths)

 1/2 = 0,31/3 = 0,22/3 = 0,41/4 = 0,13 (9/36)3/4 = 0,43 (27/36)1/5 = 0.1111…2/5 = 0.2222…3/5 = 0.3333…4/5 = 0.4444… (1/6)dix = 1/10 = 0,1(5/6)dix = 5/10 = 0,5(1/9)dix = 1/13 = 0,04 (4/36)(2/9)dix = 2/13 = 0,12 (8/36)(4/9)dix = 4/13 = 0,24 (16/36)(5/9)dix = 5/13 = 0,32 (20/36)(7/9)dix = 11/13 = 0,44 (28/36)(8/9)dix = 12/13 = 0,52 (32/36) (approximation of decimal 0.9)
##### Tenths
• (1/10)ten = 1/14 = 0,0333…
• (3/10)ten = 3/14 = 0,1444…
• (7/10)ten = 11/14 = 0,4111…
• (9/10)ten = 13/14 = 0,5222…
##### Twelfths (1 / 22×3)
• (1/12)ten = 1/20 = 0,03 (3/36)
• (5/12)ten = 5/20 = 0,23 (15/36)
• (7/12)ten = 11/20 = 0,33 (21/36)
• (11/12)ten = 15/20 = 0,53 (33/36)
##### Eighteenth (1 / 2×32)
• (1/18)ten = 1/30 = 0,02 (3/36)
• (5/18)ten = 5/30 = 0,14 (10/36)
• (7/18)ten = 11/30 = 0,22 (14/36)
• (11/18)ten = 15/30 = 0,34 (22/36)
• (13/18)ten = 21/30 = 0,42 (26/36)
• (17/18)ten = 25/30 = 0,54 (34/36)
##### Eighths (2-3)
• (1/8)ten = 1/12 = 0,043 (27/216)
• (3/8)ten = 3/12 = 0,213 (81/216)
• (5/8)ten = 5/12 = 0,343 (135/216)
• (7/8)ten = 11/12 = 0,513 (189/216)
##### Twenty-sevenths  (3-3)
 (1/27)ten = 1/43 = 0,012 (8/216)(2/27)ten = 2/43 = 0,024 (16/216)(4/27)ten = 4/43 = 0,052 (32/216)(5/27)ten = 5/43 = 0,104 (40/216)(7/27)ten = 11/43 = 0,132 (56/216)(8/27)ten = 12/43 = 0,144 (64/216) (decimal approximation 0,3)(10/27)ten = 14/43 = 0,212 (80/216)(11/27)ten = 15/43 = 0,224 (88/216) (decimal approximation 0,4)(13/27)ten = 21/43 = 0,252 (104/216)(14/27)ten = 22/43 = 0,304 (112/216) (16/27)ten = 24/43 = 0,332 (128/216) (decimal approximation 0,6)(17/27)ten = 25/43 = 0,344 (136/216)(19/27)ten = 31/43 = 0,412 (152/216) (decimal approximation 0,7)(20/27)ten = 32/43 = 0,424 (160/216)(22/27)ten = 34/43 = 0,452 (176/216)(23/27)ten = 35/43 = 0,504 (184/216)(25/27)ten = 41/43 = 0,532 (200/216)(26/27)ten = 42/43 = 0,544 (208/216)

### Calculation examples

##### 1/2, 1/4
• Decimal: 49 ÷ 2 = 24,5
• Senary: 121 ÷ 2 = 40,3
• Decimal: 49 ÷ 4 = 12,25
• Senary: 121 ÷ 4 = 20,13
##### 1 / 23 (1/8 in decimal)
• Decimal: 27 ÷ 8 = 3,375
• Senary: 43 ÷ 12 = 3,213
• Decimal: 100 ÷ 8 = 12,5
• Senary: 244 ÷ 12 = 20,3
##### 1/3
• Octal: 100 ÷ 3 = 25,2525…
• Senary: 144 ÷ 3 = 33,2
• Decimal: 100 ÷ 3 = 33,3333…
• Senary: 244 ÷ 3 = 53,2
• Hexadecimal: 100 ÷ 3 = 55,5555…
• Senary: 1104 ÷ 3 = 221,2
##### 1/9, 1/100 in senary (1/36 in decimal)
• Octal: 100 ÷ 11 = 7,0707…
• Senary: 144 ÷ 13 = 11,04
• Decimal: 1000 ÷ 9 = 111,1111…
• Senary: 4344 ÷ 13 = 303,04
• Hexadecimal: 100 ÷ 9 = 1C,71C71C…
• Senary: 1104 ÷ 13 = 44,24
• Decimal: 19 ÷ 36 = 0,52777…
• Senary: 31 ÷ 100 = 0,31
##### 1 / 33 (1/27 in decimal), 1/1000 in senary (1/216 in decimal)
• Decimal: 8 ÷ 27 = 0,296296…
• Senary: 12 ÷ 43 = 0,144
• Decimal: 100 ÷ 27 = 3,703703…
• Senary: 244 ÷ 43 = 3,412
• Hexadecimal: 100 ÷ 1B = 9.7B425ED097B425ED09…
• Decimal: 256 ÷ 27 = 9,481481…
• Senary: 1104 ÷ 43 = 13,252
• Decimal: 125 ÷ 216 = 0,578703703…
• Senary: 325 ÷ 1000 = 0,325
##### 1/5
• Octal : 100 ÷ 5 = 14.63146314…
• Decimal: 64 ÷ 5 = 12,8
• Senary: 144 ÷ 5 = 20,4444…
• Hexadecimal: 100 ÷ 5 = 33,3333…
• Decimal: 256 ÷ 5 = 51,2
• Senary: 1104 ÷ 5 = 123,1111…
##### 1 / 52 (1/25 in decimal), 1/100 in decimal
• Decimal: 8 ÷ 25 = 0,32
• Senary: 12 ÷ 41 = 0,1530415304…
• Hexadecimal: 100 ÷ 19 = A.3D70A3D70A…
• Decimal: 256 ÷ 25 = 10,24
• Senary: 1104 ÷ 41 = 14,1235012350…
• Decimal: 53 ÷ 100 = 0,53
• Senary: 125 ÷ 244 = 0,310251402514…
##### 1 / 24 (1/16 in decimal)
• Decimal: 11 ÷ 16 = 0,6875
• Senary: 15 ÷ 24 = 0,4043
• Decimal: 2023 ÷ 16 = 126,4375
• Senary: 13211 ÷ 24 = 330,2343
• Decimal: 6561 ÷ 16 = 410,0625
• Senary: 50213 ÷ 24 = 1522,0213
##### 1 / 34 (1/81 in decimal)
• Decimal: 32 ÷ 81 = 0,395061728395061728…
• Senary: 52 ÷ 213 = 0,2212
• Decimal: 256 ÷ 81 = 3,160493827160493827…
• Senary: 1104 ÷ 213 = 3,0544
• Decimal: 625 ÷ 81 = 7,716049382716049382…
• Senary: 2521 ÷ 213 = 11,4144

Sources: PinterPandai,