Twenty (number) — properties, history and uses

Mathematical properties

In elementary arithmetic, 20 is composite and even. Its full prime factorization is 2² × 5. The positive divisors of 20 are listed below; each entry links to a short reference:

• 1
• 2
• 4
• 5
• 10
• 20

The sum of the proper divisors 1 + 2 + 4 + 5 + 10 = 22, which is greater than 20; therefore 20 is classified as an abundant number. Additional elementary facts include φ(20) = 8 (Euler's totient function), and 20 appears as a combinatorial value: the binomial coefficient C(6,3) = 20. It is also the fourth tetrahedral number, meaning it can be represented as a regular pyramid of points with four layers.

Numeral systems and notation

Some counting systems use 20 as a base; such systems are described as vigesimal. Anthropologists note vestiges of vigesimal counting in several languages and traditional measures. English preserves the word "score" to mean twenty, while French forms 80 as "quatre-vingts" (literally "four twenties"). The ancient Maya used a largely base-20 numerical system for arithmetic and calendrics. The use of 20 as a unit influenced older systems of money and weight; for example, in pre-decimal British currency there were twenty shillings in a pound. In Roman numerals, 20 is written as XX. For discussion of base-20 notation and its history, see a general entry on vigesimal systems.

Geometry and polyhedra

The number 20 is prominent in three-dimensional geometry as the face count of a well-known regular solid. A convex regular polyhedron with twenty triangular faces is the icosahedron, one of the five classical Platonic solids. The regular icosahedron has twenty faces, twelve vertices and thirty edges and is the dual of the dodecahedron. In broader terms, any solid described as having twenty faces may be referred to as a polyhedron with twenty faces, and such structures occur in natural and designed systems from crystals to molecular cages.

Cultural, biological and scientific occurrences

Twenty carries cultural and biological significance in many contexts. In Japan, the traditional coming-of-age celebration seijin no hi honors those who reach twenty years old, a milestone with long social importance. Human children typically develop twenty primary (milk) teeth, which are later replaced by permanent teeth. In chemistry, the element calcium is assigned atomic number 20, placing it among the alkaline earth metals and making it chemically and biologically important for bones and cellular processes.

Uses, examples and further notes

Because 20 factors into small primes and equals two tens, it is often used for packaging, scoring and grouping: examples include counts of items packaged in twenties and traditional phrases that count by scores. In elementary number theory and combinatorics the number serves as a useful small example: it is simultaneously composite, abundant and tetrahedral, and it appears as a binomial coefficient and in simple multiplicative contexts. For concise introductions to the concepts used above, consult linked entries on divisors and abundancy, numeral bases and the classical solids: divisor 1, divisor 2, divisor 4, divisor 5, divisor 10, abundant numbers, vigesimal, polyhedron, icosahedron, Platonic solids and Roman numerals.