Numeral Systems Part 11
Beyond Ten
PART 01 PART 1010
decimal
Quick refresher on the decimal system.
11
undecimal
Where previously we lost access to digits, now we need to start adding new ones, as there are now more single-digit numbers before reaching “10”. The most common way to do this is to add letters, starting with A.
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(11) 1 2 3 4 5 6 7 8 9 A 10 11 12 13 14 15 16 17 18 19 1A 20 21...
^ ^(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(11) 22 23 24 25 26 27 28 29 2A 30 31 32 33 34 35 36 37 38 39 3A...
^ ^2,048 in undecimal => 1,5A2 1 x 1331 = 1331
5 x 121 = 605
A(10) x 11 = 110
+ 2 x 1 = 2
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2048
12
duodecimal, AKA dozenal
Languages in use: Chepang (Nepal), Dhivehi (Maldives), Gwandare (Northern Nigeria), Kainji languages — Janji, Gbiri-Niragu, and Bishi (West-Central Nigeria)
Not dewey decimal. Duodecimal is used in time-keeping (12 hours/clock face, 12 months/year), measurement (12 inches/foot), and various zodiacs.
Base-12 used to give decimal a run for its money where it came to counting, and still used in some cases, like donuts. Dozen →12, gross →12² = 144, great gross →12³ = 1,728. Other wierd groupings were also in more common use: long hundred (AKA small gross, AKA twelfty)→120, long thousand → 1,200.
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(12) 1 2 3 4 5 6 7 8 9 A B 10 11 12 13 14 15 16 17 18 19 1A 1B...2,048 in duodecimal => 1,228 1 x 1728 = 1728
2 x 144 = 288
2 x 12 = 24
+ 8 x 1 = 8
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2048
13
tridecimal
Spooky.
14
tetradecimal
Also fairly meh. A fortnight. Any time you hear a British person say something like “nine stone seven pounds” you can (somewhat) easily convert that to 133 pounds, when you know a stone is fourteen pounds.
15
pentadecimal
Languages in use: Huli language
Used in some telecom magic.
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(15) 1 2 3 4 5 6 7 8 9 A B C D E 10 11 12 13 14 15 16 17 18...2,048 in pentadecimal => 918 9 x 225 = 2025
1 x 15 = 15
+ 8 x 1 = 8
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2048
16
hexadecimal
Used in computing, measurement (16 ounces/pound).
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(16) 1 2 3 4 5 6 7 8 9 A B C D E F 10 11 12 13 14 15 16 17...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(16) 18 19 1A 1B 1C 1D 1E 1F 20 21 22 23 24 25 26 27 28 29 2A 2B...(10) 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108...
(16) 5C 5D 5E 5F 60 61 62 63 64 65 66 67 68 69 6A 6B 6C...(10) 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166...
(16) 98 99 9A 9B 9C 9D 9E 9F A0 A1 A2 A3 A4 A5 A6...(10) 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262...
(16) F8 F9 FA FB FC FD FE FF 100 101 102 103 104 105 106...
I like hexadecimal a lot.
2,048 in hexadecimal => 800 8 x 256 = 2048
0 x 16 = 0
+ 0 x 1 = 0
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2048
17, 18, 19
Meh. Boring.
20
vigesimal
Languages in use: Ainu (Northern Japan), Atong (Northeast India and Bangladesh), Aztec (central Mexico — extinct), Basque millers (Spainish/French Pyrenees), Burushaski (Pakistan and India), Chukchi (Northeastern Siberia), Dzongkha (Bhutan), Kaktovik Inupiaq numerals in the Inuit language (Alaska), Maya (Mesoamerica — extinct), Santali (India and Bangladesh), Tlingit (Pacific Northwest)
You have probably heard the quote —
Four score and seven years ago our fathers brought forth on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal.
It sounds better than “eighty-seven years ago”, doesn’t it? Like both quinary (5) and decimal (10), vigesimal enjoys usage all over the world, likely due to humans having a combined 20 fingers and toes. Vigesimal is also used in a folk rhyming/counting system used by shepherds in northern England called Yan Tan Tethera.
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(20) 1 2 3 4 5 6 7 8 9 A B C D E F G H I J 10 11 12 13...2,048 in vigesimal => 528 5 x 400 = 2000
2 x 20 = 40
+ 8 x 1 = 2
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2048
21, 22, 23
More meh.
24
tetravigesimal
Languages in use: Kaugel (Papua New Guinea)
Used in timekeeping (24 hours/day).
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(24) 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(24) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29...2,048 in tetravigesimal => 3D8 3 x 576 = 1728
D (13) x 24 = 312
+ 8 x 1 = 8
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2048
25
pentavigesimal
More compact quinary (5).
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(25) 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(25) O 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H 1I...2,048 in pentavigesimal => 36N 3 x 625 = 1875
6 x 25 = 150
+ N (23) x 1 = 23
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2048
26
hexavigesimal
cell #ing?, alphanumerics…
Uh, a compact notation for tridecimal (13).
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(26) 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(26) O P 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H...2,048 in hexavigesimal => 30G 3 x 676 = 2028
0 x 26 = 0
+ G (16) x 1 = 16
-----------------------
2048
An alternate way to represent hexavigesimal is called bijective numeration. Bijective means essentially that every integer has only one possible way it can be represented by a set of digits. The decimal system is not bijective, since you can lead numbers with any number of zeroes (or trailing, after a decimal point) without changing the value of the number. So in practical terms you can take bijective to mean a numeral system with no symbol for zero. The most common scenario where you actually see bijective base-26 is the numbering of columns in spreadsheet software.
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(26) A B C D E F G H I J K L M N O P Q R S T U V W...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(26) X Y Z AA AB AC AD AE AF AG AH AI AJ AK AL AM AN AO AP AQ...2,048 in bijective hexavigesimal => BZT C (3) x 676 = 2028
0 x 26 = 0 WRONG! no zero in bijective
+ P (16) x 1 = 16
-----------------------
2048
--> B (2) x 676 = 1352
Z (26) x 26 = 676
+ T (20) x 1 = 20
-----------------------
2048
27
septavigesimal
Languages in use: Oksapmin (Papua New Guinea) and Telefol (Papua New Guinea)
These counting systems are interesting in that they operate in a similar way to finger counting, just with a lot more body parts! Including arms, elbows, wrists, neck, ears, eyes and nose in addition to all the fingers.
Septavigesimal can be used as a compact representation of ternary (3) and nonary (9).
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(27) 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(27) O P Q 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26...2,048 in septavigesimal => 2LN 2 x 729 = 1458
L (21) x 27 = 567
+ N (23) x 1 = 23
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2048
32
duotrigesimal
Languages in use: Ngiti (Congo)
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(32) 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(32) O P Q R S T U V 10 11 12 13 14 15 16 17 18 19 20 21...2,048 in duotrigesimal => 200 2 x 1024 = 2048
0 x 32 = 0
+ 0 x 1 = 0
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2048
Doubling the base from hexadecimal makes it even more compact. However, it does become harder to keep track of what values all of the letters map to.
2 4 8 16 32 10
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1 0000 0000 ~ 1 0000 ~ 400 ~ 100 ~ 80 256
0 1111 1111 ~ 0 3333 ~ 777 ~ 0FF ~ 7V 255
0 1000 0000 ~ 0 2000 ~ 200 ~ 080 ~ 40 128
0 0111 1111 ~ 0 1333 ~ 177 ~ 07F ~ 3V 127
0 0100 0000 ~ 0 1000 ~ 100 ~ 040 ~ 20 64
0 0010 0000 ~ 0 0200 ~ 040 ~ 020 ~ 10 32
0 0001 0000 ~ 0 0100 ~ 020 ~ 010 ~ 0G 16
0 0000 1111 ~ 0 0033 ~ 017 ~ 00F ~ 0F 15
0 0000 1000 ~ 0 0020 ~ 010 ~ 008 ~ 08 8
0 0000 0100 ~ 0 0010 ~ 004 ~ 004 ~ 04 4
0 0000 0010 ~ 0 0002 ~ 002 ~ 002 ~ 02 2
0 0000 0001 ~ 0 0001 ~ 001 ~ 001 ~ 01 136
hexatrigesimal
Uses all the letters and digits. A compact notation for senary (6).
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(36) 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N...(10) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43...
(36) O P Q R S T U V W X Y Z 10 11 12 13 14 15 16 17...2,048 in hexatrigesimal => 1KW 1 x 1296 = 1296
K (20) x 36 = 720
+ W (32) x 1 = 32
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2048
60
sexagesimal
Used in Sumerian and then Babylonian mathematics, not actually 60 unique digit symbols but used 10 as a sub-base. We still use this sexagesimal today to measure time (60 seconds/minute, 60 minutes/hour) and angles (60 seconds/minute, 60 minutes/degree). Super convenient for representing fractions as 60 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.
(10) 1 2 3 4 5 6 7 8 9 10 11...
(60) I II III IIII IIIII IIIIII IIIIIII IIIIIIII IIIIIIIII X XI...(10) 12 13 14 15 16 17 18 19...
(60) XII XIII XIIII XIIIII XIIIIII XIIIIIII XIIIIIIII XIIIIIIIII...(10) 20 21 22 23 24 25 26 27
(60) XX XXI XXII XXIII XXIIII XXIIIII XXIIIIII XXIIIIIII...2,048 in sexigesimal => XXIIII IIIIIIII XXXIIII (34) x 60 = 2040
+ IIIIIIII (8) x 1 = 8
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2048
easy peasy
64
tetrasexigesimal
2,048 in tetrasexigesimal => w0 w (32) x 64 = 2048
+ 0 x 1 = 0
----------------------
2048
Used in the I Ching hexagrams. Conveniently encoded by all the characters in the Latin alphabet, numbers, and two special characters.
That’s all folks.
