List of numbers

{{main|Natural number}}

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 {{Math|N}} (or blackboard bold $\backslash mathbb\{\backslash N\}$, Unicode {{Unichar|2115|DOUBLE-STRUCK CAPITAL N}}).

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.{{Collapsible list

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| bullets = on|1, the multiplicative identity. Also the only natural number (not including 0) that is not prime or composite.|2, the base of the binary number system, used in almost all modern computers and information systems. Also the only natural even number to also be prime.|3, 2²-1, the first Mersenne prime and first Fermat number. It is the first odd prime, and it is also the 2 bit integer maximum value.|4, the first composite number.|5, the sum of the first two primes and only prime which is the sum of 2 consecutive primes. The ratio of the length from the side to a diagonal of a regular pentagon is the golden ratio.|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.|70, the smallest weird number.|72, the smallest Achilles number.|108, the second 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.{{cite web

|url=http://mathworld.wolfram.com/Hardy-RamanujanNumber.html

|title=Hardy–Ramanujan Number

|last=Weisstein

|first=Eric W.

|archive-url=https://web.archive.org/web/20040408221409/http://mathworld.wolfram.com/Hardy-RamanujanNumber.html

|archive-date=2004-04-08

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| 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.

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Along with their mathematical properties, many integers have cultural significance{{Cite journal|last1=Ayonrinde|first1=Oyedeji A.|last2=Stefatos|first2=Anthi|last3=Miller|first3=Shadé|last4=Richer|first4=Amanda|last5=Nadkarni|first5=Pallavi|last6=She|first6=Jennifer|last7=Alghofaily|first7=Ahmad|last8=Mngoma|first8=Nomusa|date=2020-06-12|title=The salience and symbolism of numbers across cultural beliefs and practice|journal=International Review of Psychiatry|volume=33|issue=1–2|pages=179–188|doi=10.1080/09540261.2020.1769289|issn=0954-0261|pmid=32527165|s2cid=219605482}} 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.

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|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" in their respective languages.

|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 Chinese 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".{{Cite web |title=Demystified {{!}} Why a baker's dozen is thirteen |url=https://www.britannica.com/video/213933/Demystified-why-is-bakers-dozen-thirteen |access-date=2024-06-05 |website=www.britannica.com |language=en}}

|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 the Hebrew word 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, a slang term for reciprocal oral sex.

|86, a slang term that is used in the American popular culture as a transitive verb to mean throw out or get rid of.{{cite web | url = http://www.merriam-webster.com/dictionary/86 | title = Eighty-six – Definition of eighty-six | work = Merriam-Webster |archive-url = https://web.archive.org/web/20130408004615/http://www.merriam-webster.com/dictionary/86 |archive-date = 2013-04-08 | url-status = live }}

|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.

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| bullets = on|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.|1000, the scale factor of most metric prefixes.

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|2, radix of binary numbers used in most digital electronics.

|4, the number of bits in a nibble.

|8, the number of bits in an octet and usually in a byte.

|16, used as a radix in hexadecimal notation. Also frequently used memory bus width (in bits) in older systems.

|32 and 64, typical memory bus widths (in bits) in contemporary systems.

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

|1024, the number of bytes in a kibibyte, and 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.

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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.

{{Main|Prime number|List of prime numbers}}

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

The first 100 prime numbers are:

{{main|Highly composite number}}

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

{{main|Perfect number}}

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

The first 10 perfect numbers:

{{ordered list|type=decimal

|  6

|  28

|  496

|  8128

|  33 550 336

|  8 589 869 056

|  137 438 691 328

|  2 305 843 008 139 952 128

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

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

}}

{{main|Integer}}

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 {{Math|Z}} (or blackboard bold $\backslash mathbb\{\backslash Z\}$, Unicode {{Unichar|2124|DOUBLE-STRUCK CAPITAL Z}}); 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.

One important use of integers is in orders of magnitude. A power of 10 is a number 10^k, 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.

{{main|Rational number}}

A rational number is any number that can be expressed as the quotient or fraction {{math|p/q}} of two integers, a numerator {{math|p}} and a non-zero denominator {{math|q}}.{{cite book|title=Discrete Mathematics and its Applications|last=Rosen|first=Kenneth|publisher=McGraw-Hill|year=2007|isbn=978-0-07-288008-3|edition=6th|location=New York, NY|pages=105, 158–160}} Since {{math|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 {{math|Q}} (or blackboard bold $\backslash mathbb\{Q\}$, Unicode {{unichar|211A|DOUBLE-STRUCK CAPITAL Q}});{{cite web|url=http://searchdatacenter.techtarget.com/definition/Mathematical-Symbols|title=Mathematical Symbols|last1=Rouse|first1=Margaret|access-date=1 April 2015}} 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 ({{sfrac|3|25}}), nine seventy-fifths ({{sfrac|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).

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Real numbers are least upper bounds of sets of rational numbers that are bounded above, or greatest lower bounds of sets of rational numbers that are bounded below, or limits of convergent sequences of rational numbers. Real numbers that are not rational numbers are called irrational numbers. The real numbers are categorised as algebraic numbers (which are the root of a polynomial with rational coefficients) or transcendental numbers, which are not; all rational numbers are algebraic.

{{main|Algebraic number}}

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{{main|Transcendental number}}

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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.

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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.

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{{See also|Normal number|Uncomputable number}}

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

{{main|Hypercomplex number}}

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 {{Math|C}} (or blackboard bold $\backslash mathbb\{\backslash Complex\}$, Unicode {{Unichar|2102|DOUBLE-STRUCK CAPITAL C}}), while the set of quaternions is denoted by a boldface {{Math|H}} (or blackboard bold $\backslash mathbb\{H\}$, Unicode {{Unichar|210D|DOUBLE-STRUCK CAPITAL H}}).

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

• The quaternions
• The octonions
• The sedenions
• The trigintaduonions
• The dual numbers (with an infinitesimal)

{{main|Transfinite number}}

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: ℵ₀, the smallest infinite cardinal, and the cardinality of $\backslash mathbb\{N\}$, the set of natural numbers
• Aleph-one: ℵ₁, the cardinality of ω₁, the set of all countable ordinal numbers
• Beth-one: $\backslash beth\_1$ or $\backslash mathfrak\; c$, the cardinality of the continuum 2{{sup|ℵ₀}}
• Omega: ω, the smallest infinite ordinal

{{main|Physical constant|List of physical constants}}

Physical quantities that appear in the universe are often described using physical constants.

• Avogadro constant: {{physconst|NA|symbol=yes}}
• Electron mass: {{physconst|me|symbol=yes}}
• Fine-structure constant: {{physconst|alpha|symbol=yes}}
• Gravitational constant: {{physconst|G|symbol=yes}}
• Molar mass constant: {{physconst|Mu|symbol=yes}}
• Planck constant: {{physconst|h|symbol=yes}}
• Rydberg constant: {{physconst|Rinf|symbol=yes}}
• Speed of light in vacuum: {{physconst|c|symbol=yes}}
• Vacuum electric permittivity: {{physconst|eps0|symbol=yes}}

• Equator#Exact length, the average equatorial radius of Earth in kilometers (following GRS 80 and WGS 84 standards).
• Equator#Exact length, the length of the Equator in kilometers (following GRS 80 and WGS 84 standards).
• Lunar distance (astronomy), 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.
• Astronomical Unit, the average distance between the Earth and the Sun or Astronomical Unit (AU), in meters.
• Light-year, one light-year, the distance travelled by light in one Julian year, in meters.
• Parsec, the distance of one parsec, another astronomical unit, in whole meters.

{{Main|Indefinite and fictitious numbers}}

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".[http://versita.metapress.com/content/t98071387u726916/?p=1ad6a085630c432c94528c5548f5c2c4&pi=1 "Bags of Talent, a Touch of Panic, and a Bit of Luck: The Case of Non-Numerical Vague Quantifiers" from Linguista Pragensia, Nov. 2, 2010] {{webarchive|url=https://archive.today/20120731092211/http://versita.metapress.com/content/t98071387u726916/?p=1ad6a085630c432c94528c5548f5c2c4&pi=1 |date=2012-07-31 }} Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".[https://www.bostonglobe.com/ideas/2016/07/13/the-surprising-history-indefinite-hyperbolic-numerals/qYTKpkP9lyWVfItLXuTHdM/story.html Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"]

• Hardy–Ramanujan number, 1729
• Kaprekar's constant, 6174
• Eddington number, ~10⁸⁰
• Googol, 10¹⁰⁰
• Shannon number
• Centillion, 10³⁰³
• Skewes's number
• Googolplex, 10^(10100)
• Mega/Circle(2)
• Moser's number
• Graham's number
• TREE(3)
• SSCG(3)
• Rayo's number

{{Portal|Mathematics}}

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• Absolute infinite
• English numerals
• Floating-point arithmetic
• Fraction
• Integer sequence
• Interesting number paradox
• Large numbers
• List of mathematical constants
• List of prime numbers
• List of types of numbers
• Mathematical constant
• Metric prefix
• Names of large numbers
• Names of small numbers
• Negative number
• Numeral (linguistics)
• Numeral prefix
• Order of magnitude
• Orders of magnitude (numbers)
• Ordinal number
• The Penguin Dictionary of Curious and Interesting Numbers
• Perfect numbers
• Power of two
• Power of 10
• Surreal number
• Table of prime factors

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{{reflist}}

• {{citation|last=Finch|first=Steven R.|title=Mathematical Constants (Encyclopedia of Mathematics and its Applications, Series Number 94)|url=https://isbnsearch.org/isbn/0521818052|pages=[https://archive.org/details/mathematicalcons0000finc/page/130 130–133]|year=2003|contribution=Anmol Kumar Singh|publisher=Cambridge University Press|isbn=0521818052}}
• {{Citation

| first = Roger

| last = Apéry

| title = Irrationalité de $\backslash zeta(2)$ et $\backslash zeta(3)$

| year = 1979

| journal = Astérisque

| volume = 61

| pages = 11–13

}}.

• Kingdom of Infinite Number: A Field Guide by Bryan Bunch, W.H. Freeman & Company, 2001. {{isbn|0-7167-4447-3}}

• [http://www.archimedes-lab.org/numbers/Num1_69.html What's Special About This Number? A Zoology of Numbers: from 0 to 500]
• [http://www.isthe.com/chongo/tech/math/number/number.html Name of a Number]
• [http://www.mathcats.com/explore/reallybignumbers.html See how to write big numbers]
• {{webarchive |url=https://web.archive.org/web/20101127194324/http://pages.prodigy.net/jhonig/bignum/indx.html |title=About big numbers |date=27 November 2010}}
• [http://www.mrob.com/pub/math/largenum.html Robert P. Munafo's Large Numbers page]
• [http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm Different notations for big numbers – by Susan Stepney]
• [http://www.ibiblio.org/units/large.html Names for Large Numbers], in How Many? A Dictionary of Units of Measurement by Russ Rowlett
• [https://erich-friedman.github.io/numbers.html What's Special About This Number?] (from 0 to 9999)

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Category:Mathematical tables