# Vigesimal Numbers

The vigesimal numbers system or base 20 system is a numbering system based on the number twenty (in the same way that the decimal system is based on the number ten).

#### Notation system

The vigesimal notation system needs twenty digits or symbols, ten more than the decimal system. A modern system, similar to that used in computing by the hexadecimal system , is to use the ten digits “0-9” and the first ten letters of the alphabet “AJ” (or the letters “AK” without using the letter “I “, as it could be confused with the number “1”). In this way, for example, the number eleven is written as A 20 (the 20 means base 20), or the number twenty is written as 10 20.

##### According to this notation:
20 20 means forty in decimal {= (2 × 20¹ + (0 × 20 0 )}
DA 20 means two hundred seventy in decimal {= (13 × 20¹) + (10 × 20 0 }
100 20 means four hundred in decimal {= (1 × 20²) + (0 × 20¹) + (0 × 20 0 )}.

In the rest of this article numbers without the suffix 20 express quantities in decimal notation unless otherwise specified. For example, 10 means ten, 21 means twenty-one.

#### The usage of Vigesimal Numbers

Base 20 is at the root of many numbering systems (since 20 are the fingers and toes used for counting). This has conditioned the linguistic structure in the names of certain numbers. Remnants of this are still found in many modern European languages, such as French, in the words for certain numbers (80 for example), and even more so in Basque, which calls 40, 60 and 80 as to multiples of 20.

The Maya numerals are an example of a base-20 numeral system or Vigesimal Numbers. Original: Neuromancer2K4 Vector: Bryan Derksen, CC BY-SA 3.0, via Wikimedia Commons

#### The multiplication table of Vigesimal

Vigesimal multiplication table
1 2 3 4 5 6 7 8 9 A B C D E F G H I J 10
2 4 6 8 A C E G I 10 12 14 16 18 1A 1C 1E 1G 1I 20
3 6 9 C F I 11 14 17 1A 1D 1G 1J 22 25 28 2B 2E 2H 30
4 8 C G 10 14 18 1C 1G 20 24 28 2C 2G 30 34 38 3C 3G 40
5 A F 10 15 1A 1F 20 25 2A 2F 30 35 3A 3F 40 45 4A 4F 50
6 C I 14 1A 1G 22 28 2E 30 36 3C 3I 44 4A 4G 52 58 5E 60
7 E 11 18 1F 22 29 2G 33 3A 3H 44 4B 4I 55 5C 5J 66 6D 70
8 G 14 1C 20 28 2G 34 3C 40 48 4G 54 5C 60 68 6G 74 7C 80
9 I 17 1G 25 2E 33 3C 41 4A 4J 58 5H 66 6F 74 7D 82 8B 90
A 10 1A 20 2A 30 3A 40 4A 50 5A 60 6A 70 7A 80 8A 90 9A A0
B 12 1D 24 2F 36 3H 48 4J 5A 61 6C 73 7E 85 8G 97 9I A9 B0
C 14 1G 28 30 3C 44 4G 58 60 6C 74 7G 88 90 9C A4 AG B8 C0
D 16 1J 2C 35 3I 4B 54 5H 6A 73 7G 89 92 9F A8 B1 BE C7 D0
E 18 22 2G 3A 44 4I 5C 66 70 7E 88 92 9G AA B4 BI CC D6 E0
F 1A 25 30 3F 4A 55 60 6F 7A 85 90 9F AA B5 C0 CF DA E5 F0
G 1C 28 34 40 4G 5C 68 74 80 8G 9C A8 B4 C0 CG DC E8 F4 G0
H 1E 2B 38 45 52 5J 6G 7D 8A 97 A4 B1 BI CF DC E9 F6 G3 H0
I 1G 2E 3C 4A 58 66 74 82 90 9I AG BE CC DA E8 F6 G4 H2 I0
J 1I 2H 3G 4F 5E 6D 7C 8B 9A A9 B8 C7 D6 E5 F4 G3 H2 I1 J0
10 20 30 40 50 60 70 80 90 A0 B0 C0 D0 E0 F0 G0 H0 I0 J0 100

## Examples of the mathematical use of Vigesimal Numbers

In base 20, twenty digits (or graphic signs) are used, which is ten more than in the usual decimal system. As with any base number greater than 10, the symbols to represent digits beyond nine are obtained using the letters of the alphabet, starting from A for ten, and up to J for nineteen. Another method of notation skips the letter I (capital i equals eighteen) to avoid confusion with the number 1. Thus the number eighteen is written J and 19 is written K for advance of the notation used.

##### The number is followed by the subscript 20 to indicate the base used. In summary:
Desimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Duodecimal or twelfth 0 1 2 3 4 5 6 7 8 9 A B 10 11 12 13 14 15 16 17 18
Vigesimal 0 1 2 3 4 5 6 7 8 9 A B C D E F G H I J 10
other method 0 1 2 3 4 5 6 7 8 9 A B C D E F G H J K 10
##### As in any number system, the number equal to the value of the base is written 10, in this case twenty is written 1020.
Notation rating breakdown:
• 2020 = 3412 = 4010 (in fact, 2×20 = 40)
• 6F20 = B312 = 13510 (in fact, 6×20 + 15 = 135)
• DA20 = 1A612 = 27010 (in fact, 13×20 + 10 = 270)
• 10020 = 29412 = 40010 (in fact, 1×202 = 400)
• 4J920 = 119912 = 198910 (in fact, 4×202 + 19×201 + 9 = 1989)
• 51420 = 120812 = 202410 (in fact, 5×202 + 1×201 + 4 = 2024)
• 8CG20 = 200012 = 345610 (in fact, 8×202 + 12×201 + 16 = 3456)
• 100020 = 476812 = 800010 (in fact, 1×203 = 8000)
• 234020 = A00012 = 1728010 (in fact, 2×203 + 3×202 + 4×201 = 17280).
• 7FA820 = 3000012 = 6220810 (in fact, 7×203 + 15×202 + 10×201 + 8 = 62208).
• 1000020 = 7871412 = 16000010 (in fact, 1×204 = 160000)
• 10EE820 = 8000012 = 16588810 (in fact, 1×204 + 0×203 + 14×202 + 14×201 + 8 = 165888).
• 1B21C20 = 10000012 = 24883210 (in fact, 1×204 + 11×203 + 2×202 + 1×201 + 12 = 248832).
##### Examples of arithmetic operations
Examples of arithmetic operations
Decimal Vigesimal Duodecimal
270 + 45 = 315 DA + 25 = FF 1A6 + 39 = 223
1989 – 135 = 1854 4J9 – 6F = 4CE 1199 – B3 = 10A6
184 × 11 = 2024 94 × B = 514 134 × B = 1208
17280 ÷ 9 = 1920 2340 ÷ 9 = 4G0 A000 ÷ 9 = 1140
##### Calculation example
• Hexadecimal : 100 ÷ 5 = 20
• Vigesimal: CG ÷ 5 = 2B.4
• Duodecimal 1/13 indivisible
• Duodecimal: 2000 ÷ 13 = 172.4972…
• Vigesimal: 8CG ÷ F = BA.8

### Exponentiation (power)

• Senary: 10 = 2×3 (A senary system is a base six number system)
• Decimal: 10 = 2×5
• Duodecimal: 10 = 4×3
• Vigesimal: 10 = 4×5
##### Power of twenty by vicesimal notation
Power of twenty by vicesimal notation
Exposant Vigesimal Duodecimal equivalent Decimal equivalent Senary equivalent
1 10 18 20 32
2 100 182 = 294 202 = 400 322 = 1 504
3 1 000 183 = 4 768 203 = 8 000 323 = 101 012
4 10 000 184 = 78 714 204 = 160 000 324 = 3 232 424
5 100 000 185 = 1 0A3 A28 205 = 3 200 000 325 = 152 330 452
6 1 000 000 186 = 19 525 054 206 = 64 000 000 3210 = 10 203 424 144
7 10 000 000 187 = 2B8 804 8A8 207 = 1 280 000 000 3211 = 331 002 501 532
8 100 000 000 188 = 4 B65 47A 994 208 = 25 600 000 000 3212 = 15 432 132 502 304
9 1 000 000 000 189 = 83 28B 920 368 209 = 512 000 000 000 3213 = 1 031 113 132 522 212
A 10 000 000 000 18A = 1 194 6B7 345 B14 2010 = 10 240 000 000 000 3214 = 33 440 105 133 555 224
B 100 000 000 000 18B = 1A B77 741 75A 628 2011 = 204 800 000 000 000 3215 = 2 003 323 453 211 540 052
C 1 000 000 000 000 1810 = 323 488 2A8 596 454 2012 = 4 096 000 000 000 000 3220 = 104 155 244 231 222 522 544
-1 0.1 1/18 0.05 1/32
-2 0.01 1/294 0.0025 1/1504
-3 0.001 1/4768 0.000125 1/101012

### Fractions and divisions

Vigesimal fraction is the same as decimal, their prime factors are 2 and 5. The biggest difference between decimal system and vicesimal system is divisibility by 4. Decimal 10 is indivisible by 4 and its quotient is 2.5. But, vicesimal 10 is divisible by 4, its quotient is 5.

In addition, divisibility by 4 and an odd number is the common property of duodecimal and vicesimal: duodecimal 10÷3 = 4, vicesimal 10÷5 = 4.

##### The divisor is 1020 or less:
• 1 / 2 = 0,A
• 1 / 3 = 0,6D6D6D6D répétition
• 1 / 4 = 0,5
• 1 / 5 = 0,4
• 1 / 6 = 0,36D6D6D6D répétition
• 1 / 7 = 0.2H2H2H2H répétition
• 1 / 8 = 0,2A
• 1 / 9 = 0,248HFB248HFB répétition
• (1/1010) 1 / A = 0,2
• (1/1210) 1 / C = 0,1D6D6D6D6 répétition
• (1/1510) 1 / F = 0,16D6D6D6D répétition
• (1/1610) 1 / G = 0,15
• (1/1810) 1 / I = 0,1248HFB248HFB répétition
• (1/2010) 1 / 10 = 0,1
##### Divisor is 1120 or greater:
• (1/2510) 1 / 15 = 0,0G
• (1/4010) 1 / 20 = 0,0A
• (1/5010) 1 / 2A = 0,08
##### Main fraction:
• 1 / 2 = 0,A
• 1 / 3 = 0,6D6D…
• 2 / 3 = 0,D6D6…
• 1 / 4 = 0,5
• 3 / 4 = 0,F
• 1 / 5 = 0,4
• 2 / 5 = 0,8
• 3 / 5 = 0,C
• 4 / 5 = 0,G

Sources: PinterPandai, Britannica, Convertworld

Photo credit: Mdsats at English Wikipedia (Public Domain) via Wikimedia Commons

Photo description: the equation “5 + 8 equal 13” written with Maya numerals.