The natural numbers are a subset of the integers and are of historical and pedagogical value as they can be used for counting and often have ethno-cultural significance (see below). Beyond this, natural numbers are widely used as a building block for other number systems including the integers, rational numbers and real numbers. Natural numbers are those used for counting (as in "there are six (6) coins on the table") and ordering (as in "this is the third (3rd) largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers". Defined by the Peano axioms, the natural numbers form an infinitely large set.
The inclusion of 0 in the set of natural numbers is ambiguous and subject to individual definitions. In set theory and computer science, 0 is typically considered a natural number. In number theory, it usually is not. The ambiguity can be solved with the terms "non-negative integers", which includes 0, and "positive integers", which does not.
Natural numbers may be used as cardinal numbers, which may go by various names. Natural numbers may also be used as ordinal numbers.
Natural numbers may have properties specific to the individual number or may be part of a set (such as prime numbers) of numbers with a particular property.
List of mathematically significant natural numbers
- 1, the multiplicative identity. Also the only natural number (not including 0) that isn't prime or composite.
- 2, the base of the binary number system, used in almost all modern computers and information systems. Also notable as the only even prime number.
- 3, 22-1, the first Mersenne number. It is the first odd prime, and it is also the 2 bit integer maximum value.
- 4, the first composite number
- 6, the first of the series of perfect numbers, whose proper factors sum to the number itself.
- 9, the first odd number that is composite
- 11, the fifth prime and first palindromic multi-digit number
- 17, the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.
- 24, all Dirichlet characters mod n are real if and only if n is a divisor of 24.
- 25, the first centered square number besides 1 that is also a square number
- 27, the cube of 3, the value of , where is Knuth's up-arrow notation.
- 28, the second perfect number.
- 30, the smallest sphenic number.
- 32, the smallest nontrivial fifth power.
- 36, the smallest number which is perfect power but not prime power.
- 72, the smallest Achilles number.
- 255, 28 − 1, a Mersenne number and the smallest perfect totient number that is neither a power of three nor thrice a prime; it is also the largest number that can be represented using an 8-bit unsigned integer
- 341, the smallest base 2 Fermat pseudoprime.
- 496, the third perfect number.
- 1729, the Hardy–Ramanujan number, also known as the second taxicab number; that is, the smallest positive integer that can be written as the sum of two positive cubes in two different ways.
- 8128, the fourth perfect number.
- 142857, the smallest base 10 cyclic number.
- 9814072356, the largest perfect power that contains no repeated digits in base ten.
Cultural or practical significance
Along with their mathematical properties, many integers have cultural significance or are also notable for their use in computing and measurement. As mathematical properties (such as divisibility) can confer practical utility, there may be interplay and connections between the cultural or practical significance of an integer and its mathematical properties.
List of integers notable for their cultural meanings.
- 3, significant in Christianity as the Trinity. Also considered significant in Hinduism (Trimurti, Tridevi). Holds significance in a number of ancient mythologies.
- 4, considered an "unlucky number" in modern China, Japan and Korea due to its audible similarity to the word "Death."
- 5, number of fingers or toes for almost all amphibians, reptiles and mammals
- 7, considered a "lucky" number in Western cultures.
- 8, considered a "lucky" number in Chinese culture.
- 9, a significant number in Norse Mythology.
- 12, the number base for some ancient counting systems and the basis for some modern measuring systems. Known as a dozen.
- 13, considered an "unlucky" number in Western superstition. Also known as a "Baker's Dozen".
- 15, the number of players on a rugby union team. It is also the first point received in tennis.
- 16, the base of the hexadecimal number system which is utilized within many programming languages.
- 18, age of majority in most countries in the world.
- 19, the length of one side of a Go board.
- 21, the legal drinking age in the United States.
- 22, the namesake of catch-22, a paradoxical condition in which there is no escape due to mutually conflicting or dependent conditions.
- 23, number of chromosomes in a human haploid. Other human cells have 23 pairs of chromosomes.
- 26, the number of English letters,
bijective base 26 is used in the columns of Microsoft Excel.
- 42, the "answer to the ultimate question of life, the universe, and everything" in the popular 1979 science fiction work The Hitchhiker's Guide to the Galaxy.
- 69, used as slang to refer to a sexual act.
- 86, a slang term that is used in the American popular culture as a transitive verb to mean throw out or get rid of.
- 108, considered sacred by the Dharmic Religions. Approximately equal to the ratio of the distance from Earth to Sun and diameter of the Sun.
- 420, a code-term that refers to the consumption of cannabis.
- 616, the true value of "Number of the Beast".
- 666, the Number of the Beast from the Book of Revelation.
- 786, regarded as sacred in the Muslim Abjad numerology.
- 5040, mentioned by Plato in the Laws as one of the most important numbers for the city.
List of integers notable for their use in units, measurements and scales.
- 10, the number of digits in the decimal number system.
- 14, the number of days in a fortnight.
- 16, the number of digits in the hexadecimal number system.
- 24, number of hours in a day
- 25, the number of cents in a quarter.
- 31, the number of days most months of the year have.
- 60, the number base for some ancient counting systems, such as the Babylonians', and the basis for many modern measuring systems.
- 365, the number of days in the common year.
List of integers notable in computing
- 8, the number of bits in a byte
- 256, The number of possible combinations within 8 bits, or a byte.
- 1024, the number of bytes in a kibibyte. It's also the number of bits in a kibibit.
- 65535, 216 − 1, the maximum value of a 16-bit unsigned integer.
- 65536, 216, the number of possible 16-bit combinations.
- 65537, 216 + 1, the most popular RSA public key prime exponent in most SSL/TLS certificates on the Web/Internet.
- 16777216, 224, or 166; the hexadecimal "million" (0x1000000), and the total number of possible color combinations in 24/32-bit True Color computer graphics.
- 2147483647, 231 − 1, the maximum value of a 32-bit signed integer using two's complement representation.
- 9223372036854775807, 263 − 1, the maximum value of a 64-bit signed integer using two's complement representation.