Overview

The integer six is a natural counting number that follows five and precedes seven. As an element of the positive integers, it is even and composite: its positive divisors are 1, 2 and 3, which multiply to produce 6 and also sum to the number itself.

Basic mathematical properties

Six holds several simple but notable mathematical identities and roles. It is:

  • an even number, because it is divisible by 2;
  • composite (2 × 3); the product of the first two prime numbers;
  • a triangular number (1 + 2 + 3 = 6) and also a pronic number (2 × 3 = 6);
  • 3! (three factorial), that is, 3 × 2 × 1 = 6;
  • the first perfect number, since the sum of its proper divisors 1 + 2 + 3 equals 6.

Representations and notation

Six can be written in many numeral systems and notations. In common Western usage it appears as the Arabic digit "6"; in Roman numerals it is written VI. In binary it is 110, in octal it is 6, and in hexadecimal it is 6 as well. The next known perfect number after six is 28, showing how rare perfect numbers are.

Algebra, geometry and combinatorics

Geometrically, figures with six sides (hexagons) and sixfold symmetry are widespread in natural and human-made structures. Regular hexagons tile the plane and are efficient packing shapes, as seen in honeycombs. In algebra, groups and combinatorial objects built from six elements often exhibit distinctive behavior; for example, the symmetric group on six elements is notable in group theory for exceptional properties among symmetric groups.

Uses, examples and cultural notes

Six appears frequently in everyday contexts: dice faces go from one to six on a standard cube, and many systems use six-part groupings (e.g., six-pack, hexagonal arrangements). It is an index of symmetry and balance in design, and because it is the first perfect number it has attracted attention since antiquity in number theory and philosophical discussions about harmony. For further reading on related concepts such as the natural numbers and perfect numbers, see links to basic references on natural numbers and perfect numbers.

Notable distinctions

Beyond its elementary properties, six is interesting because it simultaneously satisfies several simple classifications: it is factorial, triangular, pronic and perfect. These overlapping identities make it a small-number example that is often used to illustrate broader definitions in arithmetic and elementary number theory.

For concise reference and definitions related to this number consult introductory resources on counting numbers, divisibility and polygonal figures; additional links appear here for quick navigation: six overview, natural number, five, seven, perfect number, 28, Roman numerals, even number.