# 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
123456789ABCDEFGHIJ10
2468ACEGI10121416181A1C1E1G1I20
369CFI1114171A1D1G1J2225282B2E2H30
48CG1014181C1G2024282C2G3034383C3G40
5AF10151A1F20252A2F30353A3F40454A4F50
6CI141A1G22282E30363C3I444A4G52585E60
7E11181F22292G333A3H444B4I555C5J666D70
8G141C20282G343C40484G545C60686G747C80
9I171G252E333C414A4J585H666F747D828B90
A101A202A303A404A505A606A707A808A909AA0
B121D242F363H484J5A616C737E858G979IA9B0
C141G28303C444G58606C747G88909CA4AGB8C0
D161J2C353I4B545H6A737G89929FA8B1BEC7D0
E18222G3A444I5C66707E88929GAAB4BICCD6E0
F1A25303F4A55606F7A85909FAAB5C0CFDAE5F0
G1C2834404G5C6874808G9CA8B4C0CGDCE8F4G0
H1E2B3845525J6G7D8A97A4B1BICFDCE9F6G3H0
I1G2E3C4A58667482909IAGBECCDAE8F6G4H2I0
J1I2H3G4F5E6D7C8B9AA9B8C7D6E5F4G3H2I1J0
102030405060708090A0B0C0D0E0F0G0H0I0J0100

## 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:
Desimal01234567891011121314151617181920
Duodecimal or twelfth0123456789AB101112131415161718
Vigesimal0123456789ABCDEFGHIJ10
other method0123456789ABCDEFGHJK10
##### 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
DecimalVigesimalDuodecimal
270 + 45 = 315DA + 25 = FF1A6 + 39 = 223
1989 – 135 = 18544J9 – 6F = 4CE1199 – B3 = 10A6
184 × 11 = 202494 × B = 514134 × B = 1208
17280 ÷ 9 = 19202340 ÷ 9 = 4G0A000 ÷ 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
ExposantVigesimalDuodecimal equivalentDecimal equivalentSenary equivalent
110182032
2100182 = 294202 = 400322 = 1 504
31 000183 = 4 768203 = 8 000323 = 101 012
410 000184 = 78 714204 = 160 000324 = 3 232 424
5100 000185 = 1 0A3 A28205 = 3 200 000325 = 152 330 452
61 000 000186 = 19 525 054206 = 64 000 0003210 = 10 203 424 144
710 000 000187 = 2B8 804 8A8207 = 1 280 000 0003211 = 331 002 501 532
8100 000 000188 = 4 B65 47A 994208 = 25 600 000 0003212 = 15 432 132 502 304
91 000 000 000189 = 83 28B 920 368209 = 512 000 000 0003213 = 1 031 113 132 522 212
A10 000 000 00018A = 1 194 6B7 345 B142010 = 10 240 000 000 0003214 = 33 440 105 133 555 224
B100 000 000 00018B = 1A B77 741 75A 6282011 = 204 800 000 000 0003215 = 2 003 323 453 211 540 052
C1 000 000 000 0001810 = 323 488 2A8 596 4542012 = 4 096 000 000 000 0003220 = 104 155 244 231 222 522 544
-10.11/180.051/32
-20.011/2940.00251/1504
-30.0011/47680.0001251/101012

### 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.