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

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:

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

Calculation example

• Hexadecimal 1/5 indivisible
  • 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

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.