List of numbers

This is a list of notable numbers and articles about notable numbers. The list does not contain all numbers in existence as most of the number sets are infinite. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers have qualities which could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property. This is known as the interesting number paradox.

The definition of what is classed as a number is rather diffuse and based on historical distinctions. For example, the pair of numbers (3,4) is commonly regarded as a number when it is in the form of a complex number (3+4i), but not when it is in the form of a vector (3,4). This list will also be categorised with the standard convention of types of numbers.

This list focuses on numbers as mathematical objects and is not a list of numerals, which are linguistic devices: nouns, adjectives, or adverbs that designate numbers. The distinction is drawn between the number five (an abstract object equal to 2+3), and the numeral five (the noun referring to the number).

Natural numbers
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. Often referred to as "the naturals", the natural numbers are usually symbolised by a boldface $N$ (or blackboard bold $$\mathbb{\N}$$, Unicode ).

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.

Mathematical significance
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.

3, 22-1, the first Mersenne prime. 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 in base 10.

12, the first sublime 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 33.

28, the second perfect number.

30, the smallest sphenic number.

32, the smallest nontrivial fifth power.

36, the smallest number which is a perfect power but not a prime power.

72, the smallest Achilles number.

255, 28 − 1, 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.

5040, the largest factorial (7! = 5040) that is also a highly composite number.

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".

7, the number of days in a week, and considered a "lucky" number in Western cultures.

8, considered a "lucky" number in Chinese culture due to its aural similarity to the term for prosperity.

12, a common grouping known as a dozen and the number of months in a year, of constellations of the Zodiac and astrological signs and of Apostles of Jesus.

13, considered an "unlucky" number in Western superstition. Also known as a "Baker's Dozen".

17, considered ill-fated in Italy and other countries of Greek and Latin origins.

18, considered a "lucky" number due to it being the value for life in Jewish numerology.

40, considered a significant number in Tengrism and Turkish folklore. Multiple customs, such as those relating to how many days one must visit someone after a death in the family, include the number forty.

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.

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.

12, the number base for measuring time in many civilizations.

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

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.

360, the number of sexagesimal degrees in a full circle.

365, the number of days in the common year, while there are 366 days in a leap year of the solar Gregorian calendar.

List of integers notable in computing

8, the number of bits in an octet.

256, The number of possible combinations within 8 bits, or an octet.

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.

Classes of natural numbers
Subsets of the natural numbers, such as the prime numbers, may be grouped into sets, for instance based on the divisibility of their members. Infinitely many such sets are possible. A list of notable classes of natural numbers may be found at classes of natural numbers.

Prime numbers
A prime number is a positive integer which has exactly two divisors: 1 and itself.

The first 100 prime numbers are:

Highly composite numbers
A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are often used in geometry, grouping and time measurement.

The first 20 highly composite numbers are:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560

Perfect numbers
A perfect number is an integer that is the sum of its positive proper divisors (all divisors except itself).

The first 10 perfect numbers:

1. 6

2. 28

3. 496

4. 8128

5. 33 550 336

6. 8 589 869 056

7. 137 438 691 328

8. 2 305 843 008 139 952 128

9. 2 658 455 991 569 831 744 654 692 615 953 842 176

10. 191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216

Integers
The integers are a set of numbers commonly encountered in arithmetic and number theory. There are many subsets of the integers, including the natural numbers, prime numbers, perfect numbers, etc. Many integers are notable for their mathematical properties. Integers are usually symbolised by a boldface $Z$ (or blackboard bold $$\mathbb{\Z}$$, Unicode ); this became the symbol for the integers based on the German word for "numbers" (Zahlen).

Notable integers include −1, the additive inverse of unity, and 0, the additive identity.

As with the natural numbers, the integers may also have cultural or practical significance. For instance, −40 is the equal point in the Fahrenheit and Celsius scales.

SI prefixes
One important use of integers is in orders of magnitude. A power of 10 is a number 10k, where k is an integer. For instance, with k = 0, 1, 2, 3, ..., the appropriate powers of ten are 1, 10, 100, 1000, ... Powers of ten can also be fractional: for instance, k = -3 gives 1/1000, or 0.001. This is used in scientific notation, real numbers are written in the form m × 10n. The number 394,000 is written in this form as 3.94 × 105.

Integers are used as prefixes in the SI system. A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre.

Rational numbers
A rational number is any number that can be expressed as the quotient or fraction $p/q$ of two integers, a numerator $p$ and a non-zero denominator $q$. Since $q$ may be equal to 1, every integer is trivially a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface $Q$ (or blackboard bold $$\mathbb{Q}$$, Unicode ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".

Rational numbers such as 0.12 can be represented in infinitely many ways, e.g. zero-point-one-two (0.12), three twenty-fifths ($3⁄25$), nine seventy-fifths ($9⁄75$), etc. This can be mitigated by representing rational numbers in a canonical form as an irreducible fraction.

A list of rational numbers is shown below. The names of fractions can be found at numeral (linguistics).

Irrational numbers
The irrational numbers are a set of numbers that includes all real numbers that are not rational numbers. The irrational numbers are categorised as algebraic numbers (which are the root of a polynomial with rational coefficients) or transcendental numbers, which are not.

Irrational but not known to be transcendental
Some numbers are known to be irrational numbers, but have not been proven to be transcendental. This differs from the algebraic numbers, which are known not to be transcendental.

Real numbers
The real numbers are a superset containing the algebraic and the transcendental numbers. The real numbers, sometimes referred to as "the reals", are usually symbolised by a boldface $R$ (or blackboard bold $$\mathbb{\R}$$, Unicode ). For some numbers, it is not known whether they are algebraic or transcendental. The following list includes real numbers that have not been proved to be irrational, nor transcendental.

Numbers not known with high precision
Some real numbers, including transcendental numbers, are not known with high precision.


 * The constant in the Berry–Esseen Theorem: 0.4097 < C < 0.4748
 * De Bruijn–Newman constant: 0 ≤ Λ ≤ 0.2
 * Chaitin's constants Ω, which are transcendental and provably impossible to compute.
 * Bloch's constant (also 2nd Landau's constant): 0.4332 < B < 0.4719
 * 1st Landau's constant: 0.5 < L < 0.5433
 * 3rd Landau's constant: 0.5 < A ≤ 0.7853
 * Grothendieck constant: 1.67 < k < 1.79
 * Romanov's constant in Romanov's theorem: 0.107648 < d < 0.49094093, Romanov conjectured that it is 0.434

Hypercomplex numbers
Hypercomplex number is a term for an element of a unital algebra over the field of real numbers. The complex numbers are often symbolised by a boldface $C$ (or blackboard bold $$\mathbb{\Complex}$$, Unicode ), while the set of quaternions is denoted by a boldface $H$ (or blackboard bold $$\mathbb{H}$$, Unicode ).

Algebraic complex numbers

 * Imaginary unit: $i=\sqrt{-1}$
 * nth roots of unity: $\xi_{n}^{k}=\cos\bigl(2\pi\frac{k}{n}\bigr)+i\sin\bigl(2\pi\frac{k}{n}\bigr)$, while $0 \leq k \leq n-10$ , GCD(k, n) = 1

Other hypercomplex numbers

 * The quaternions
 * The octonions
 * The sedenions
 * The dual numbers (with an infinitesimal)

Transfinite numbers
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.
 * Aleph-null: א$1⁄1$: the smallest infinite cardinal, and the cardinality of $$\mathbb{N}$$, the set of natural numbers
 * Aleph-one: א$− 1⁄12$: the cardinality of ω1, the set of all countable ordinal numbers
 * Beth-one: ב$1⁄2$ the cardinality of the continuum 2$1⁄2$
 * ℭ or $$\mathfrak c$$: the cardinality of the continuum 2$22⁄7$
 * Omega: ω, the smallest infinite ordinal

Numbers representing physical quantities
Physical quantities that appear in the universe are often described using physical constants.
 * Avogadro constant:
 * Electron mass:
 * Fine-structure constant:
 * Gravitational constant:
 * Molar mass constant:
 * Planck constant:
 * Rydberg constant:
 * Speed of light in vacuum:
 * Vacuum electric permittivity:

Numbers representing geographical and astronomical distances

 * $1⁄6$, the average equatorial radius of Earth in kilometers (following GRS 80 and WGS 84 standards).
 * $0.618$, the length of the Equator in kilometers (following GRS 80 and WGS 84 standards).
 * $1.059$, the semi-major axis of the orbit of the Moon, in kilometers, roughly the distance between the center of Earth and that of the Moon.
 * $1.26$, the average distance between the Earth and the Sun or Astronomical Unit (AU), in meters.
 * $1.304$, one light-year, the distance travelled by light in one Julian year, in meters.
 * $1.325$, the distance of one parsec, another astronomical unit, in whole meters.

Numbers without specific values
Many languages have words expressing indefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, as placeholder names, or when precision is unnecessary or undesirable. One technical term for such words is "non-numerical vague quantifier". Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".

Named numbers

 * Eddington number, ~1080
 * Googol, 10100
 * Googolplex, 10(10 100)
 * Graham's number
 * Hardy–Ramanujan number, 1729
 * Kaprekar's constant, 6174
 * Moser's number
 * Rayo's number
 * Shannon number
 * Skewes's number
 * TREE(3)