View source for List of numbers

{{Short description|Notable numbers}}
{{pp|small=yes}}
{{Dynamic list}}
This is a list of notable [[number]]s and articles about notable numbers. The list does not contain all numbers in existence as most of the [[Set (mathematics)|number sets]] are infinite. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers have qualities that 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 (mathematics and physics)|vector]] (3,4). This list will also be categorized with the standard convention of [[list of types of numbers|types of numbers]].

This list focuses on numbers as [[mathematical objects]] and is ''not'' a list of [[Numeral (linguistics)|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 ==
{{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 number#Formal construction|rational numbers]] and [[Construction of the real numbers|real numbers]]. Natural numbers are those used for [[counting]] (as in "there are ''six'' (6) coins on the table") and [[total order|ordering]] (as in "this is the ''third'' (3rd) largest city in the country").  In common language, words used for counting are "[[cardinal number (linguistics)|cardinal numbers]]" and words used for ordering are "[[ordinal number (linguistics)|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]] <math>\mathbb{\N}</math>, Unicode {{Unichar|2115|DOUBLE-STRUCK CAPITAL N}})''.''

The inclusion of [[zero|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 [[numeral (linguistics)|various names]]. Natural numbers may also be used as [[ordinal numbers]].

<!-- Per consensus established so long ago I can't find it, this section only includes ''articles'', not ''redirects'', and is intended to include articles at least pointing to all the articles on "small" (less than 10^10) numbers.  If someone has a different idea of consensus for what should be in this section, please discuss. -->
{| class="wikitable sortable mw-collapsible" style="text-align:center;"
|+ class="nowrap" |Table of small natural numbers
|-
|[[0]]
|[[1]]
|[[2]]
|[[3]]
|[[4]]
|[[5]]
|[[6]]
|[[7]]
|[[8]]
|[[9]]
|-
|[[10]]
|[[11 (number)|11]]
|[[12 (number)|12]]
|[[13 (number)|13]]
|[[14 (number)|14]]
|[[15 (number)|15]]
|[[16 (number)|16]]
|[[17 (number)|17]]
|[[18 (number)|18]]
|[[19 (number)|19]]
|-
|[[20 (number)|20]]
|[[21 (number)|21]]
|[[22 (number)|22]]
|[[23 (number)|23]]
|[[24 (number)|24]]
|[[25 (number)|25]]
|[[26 (number)|26]]
|[[27 (number)|27]]
|[[28 (number)|28]]
|[[29 (number)|29]]
|-
|[[30 (number)|30]]
|[[31 (number)|31]]
|[[32 (number)|32]]
|[[33 (number)|33]]
|[[34 (number)|34]]
|[[35 (number)|35]]
|[[36 (number)|36]]
|[[37 (number)|37]]
|[[38 (number)|38]]
|[[39 (number)|39]]
|-
|[[40 (number)|40]]
|[[41 (number)|41]]
|[[42 (number)|42]]
|[[43 (number)|43]]
|[[44 (number)|44]]
|[[45 (number)|45]]
|[[46 (number)|46]]
|[[47 (number)|47]]
|[[48 (number)|48]]
|[[49 (number)|49]]
|-
|[[50 (number)|50]]
|[[51 (number)|51]]
|[[52 (number)|52]]
|[[53 (number)|53]]
|[[54 (number)|54]]
|[[55 (number)|55]]
|[[56 (number)|56]]
|[[57 (number)|57]]
|[[58 (number)|58]]
|[[59 (number)|59]]
|-
|[[60 (number)|60]]
|[[61 (number)|61]]
|[[62 (number)|62]]
|[[63 (number)|63]]
|[[64 (number)|64]]
|[[65 (number)|65]]
|[[66 (number)|66]]
|[[67 (number)|67]]
|[[68 (number)|68]]
|[[69 (number)|69]]
|-
|[[70 (number)|70]]
|[[71 (number)|71]]
|[[72 (number)|72]]
|[[73 (number)|73]]
|[[74 (number)|74]]
|[[75 (number)|75]]
|[[76 (number)|76]]
|[[77 (number)|77]]
|[[78 (number)|78]]
|[[79 (number)|79]]
|-
|[[80 (number)|80]]
|[[81 (number)|81]]
|[[82 (number)|82]]
|[[83 (number)|83]]
|[[84 (number)|84]]
|[[85 (number)|85]]
|[[86 (number)|86]]
|[[87 (number)|87]]
|[[88 (number)|88]]
|[[89 (number)|89]]
|-
|[[90 (number)|90]]
|[[91 (number)|91]]
|[[92 (number)|92]]
|[[93 (number)|93]]
|[[94 (number)|94]]
|[[95 (number)|95]]
|[[96 (number)|96]]
|[[97 (number)|97]]
|[[98 (number)|98]]
|[[99 (number)|99]]
|-
|[[100]]
|[[101 (number)|101]]
|[[102 (number)|102]]
|[[103 (number)|103]]
|[[104 (number)|104]]
|[[105 (number)|105]]
|[[106 (number)|106]]
|[[107 (number)|107]]
|[[108 (number)|108]]
|[[109 (number)|109]]
|-
|[[110 (number)|110]]
|[[111 (number)|111]]
|[[112 (number)|112]]
|[[113 (number)|113]]
|[[114 (number)|114]]
|[[115 (number)|115]]
|[[116 (number)|116]]
|[[117 (number)|117]]
|[[118 (number)|118]]
|[[119 (number)|119]]
|-
|[[120 (number)|120]]
|[[121 (number)|121]]
|[[122 (number)|122]]
|[[123 (number)|123]]
|[[124 (number)|124]]
|[[125 (number)|125]]
|[[126 (number)|126]]
|[[127 (number)|127]]
|[[128 (number)|128]]
|[[129 (number)|129]]
|-
|[[130 (number)|130]]
|[[131 (number)|131]]
|[[132 (number)|132]]
|[[133 (number)|133]]
|[[134 (number)|134]]
|[[135 (number)|135]]
|[[136 (number)|136]]
|[[137 (number)|137]]
|[[138 (number)|138]]
|[[139 (number)|139]]
|-
|[[140 (number)|140]]
|[[141 (number)|141]]
|[[142 (number)|142]]
|[[143 (number)|143]]
|[[144 (number)|144]]
|[[145 (number)|145]]
|[[146 (number)|146]]
|[[147 (number)|147]]
|[[148 (number)|148]]
|[[149 (number)|149]]
|-
|[[150 (number)|150]]
|[[151 (number)|151]]
|[[152 (number)|152]]
|[[153 (number)|153]]
|[[154 (number)|154]]
|[[155 (number)|155]]
|[[156 (number)|156]]
|[[157 (number)|157]]
|[[158 (number)|158]]
|[[159 (number)|159]]
|-
|[[160 (number)|160]]
|[[161 (number)|161]]
|[[162 (number)|162]]
|[[163 (number)|163]]
|[[164 (number)|164]]
|[[165 (number)|165]]
|[[166 (number)|166]]
|[[167 (number)|167]]
|[[168 (number)|168]]
|[[169 (number)|169]]
|-
|[[170 (number)|170]]
|[[171 (number)|171]]
|[[172 (number)|172]]
|[[173 (number)|173]]
|[[174 (number)|174]]
|[[175 (number)|175]]
|[[176 (number)|176]]
|[[177 (number)|177]]
|[[178 (number)|178]]
|[[179 (number)|179]]
|-
|[[180 (number)|180]]
|[[181 (number)|181]]
|[[182 (number)|182]]
|[[183 (number)|183]]
|[[184 (number)|184]]
|[[185 (number)|185]]
|[[186 (number)|186]]
|[[187 (number)|187]]
|[[188 (number)|188]]
|[[189 (number)|189]]
|-
|[[190 (number)|190]]
|[[191 (number)|191]]
|[[192 (number)|192]]
|[[193 (number)|193]]
|[[194 (number)|194]]
|[[195 (number)|195]]
|[[196 (number)|196]]
|[[197 (number)|197]]
|[[198 (number)|198]]
|[[199 (number)|199]]
|-
|[[200 (number)|200]]
|[[201 (number)|201]]
|[[202 (number)|202]]
|[[203 (number)|203]]
|[[204 (number)|204]]
|[[205 (number)|205]]
|[[206 (number)|206]]
|[[207 (number)|207]]
|[[208 (number)|208]]
|[[209 (number)|209]]
|-
|[[210 (number)|210]]
|[[211 (number)|211]]
|[[212 (number)|212]]
|[[213 (number)|213]]
|[[214 (number)|214]]
|[[215 (number)|215]]
|[[216 (number)|216]]
|[[217 (number)|217]]
|[[218 (number)|218]]
|[[219 (number)|219]]
|-
|[[220 (number)|220]]
|[[221 (number)|221]]
|[[222 (number)|222]]
|[[223 (number)|223]]
|[[224 (number)|224]]
|[[225 (number)|225]]
|[[226 (number)|226]]
|[[227 (number)|227]]
|[[228 (number)|228]]
|[[229 (number)|229]]
|-
|[[230 (number)|230]]
|[[231 (number)|231]]
|[[232 (number)|232]]
|[[233 (number)|233]]
|[[234 (number)|234]]
|[[235 (number)|235]]
|[[236 (number)|236]]
|[[237 (number)|237]]
|[[238 (number)|238]]
|[[239 (number)|239]]
|-
|[[240 (number)|240]]
|[[241 (number)|241]]
|[[242 (number)|242]]
|[[243 (number)|243]]
|[[244 (number)|244]]
|[[245 (number)|245]]
|[[246 (number)|246]]
|[[247 (number)|247]]
|[[248 (number)|248]]
|[[249 (number)|249]]
|-
|[[250 (number)|250]]
|[[251 (number)|251]]
|[[252 (number)|252]]
|[[253 (number)|253]]
|[[254 (number)|254]]
|[[255 (number)|255]]
|[[256 (number)|256]]
|[[257 (number)|257]]
|[[258 (number)|258]]
|[[259 (number)|259]]
|-
|[[260 (number)|260]]
|[[261 (number)|261]]
|[[262 (number)|262]]
|[[263 (number)|263]]
|[[264 (number)|264]]
|[[265 (number)|265]]
|[[266 (number)|266]]
|[[267 (number)|267]]
|[[268 (number)|268]]
|[[269 (number)|269]]
|-
|[[270 (number)|270]]
|[[271 (number)|271]]
|[[272 (number)|272]]
|[[273 (number)|273]]
|[[274 (number)|274]]
|[[275 (number)|275]]
|[[276 (number)|276]]
|[[277 (number)|277]]
|[[278 (number)|278]]
|[[279 (number)|279]]
|-
|[[280 (number)|280]]
|[[281 (number)|281]]
|[[282 (number)|282]]
|[[283 (number)|283]]
|[[284 (number)|284]]
|[[285 (number)|285]]
|[[286 (number)|286]]
|[[287 (number)|287]]
|[[288 (number)|288]]
|[[289 (number)|289]]
|-
|[[290 (number)|290]]
|[[291 (number)|291]]
|[[292 (number)|292]]
|[[293 (number)|293]]
|[[294 (number)|294]]
|[[295 (number)|295]]
|[[296 (number)|296]]
|[[297 (number)|297]]
|[[298 (number)|298]]
|[[299 (number)|299]]
|-
|[[300 (number)|300]]
|[[301 (number)|301]]
|[[302 (number)|302]]
|[[303 (number)|303]]
|[[304 (number)|304]]
|[[305 (number)|305]]
|[[306 (number)|306]]
|[[307 (number)|307]]
|[[308 (number)|308]]
|[[309 (number)|309]]
|-
|[[310 (number)|310]]
|[[311 (number)|311]]
|[[312 (number)|312]]
|[[313 (number)|313]]
|[[314 (number)|314]]
|<!-- Please do not add numbers without articles; see Talk: -->
|
|
|[[318 (number)|318]]
|
|-
|
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|
|[[400 (number)|400]]
|[[500 (number)|500]]
|[[600 (number)|600]]
|[[700 (number)|700]]
|[[800 (number)|800]]
|[[900 (number)|900]]
|-
|
|[[1000 (number)|1000]]
|[[2000 (number)|2000]]
|[[3000 (number)|3000]]
|[[4000 (number)|4000]]
|[[5000 (number)|5000]]
|[[6000 (number)|6000]]
|[[7000 (number)|7000]]
|[[8000 (number)|8000]]
|[[9000 (number)|9000]]
|-
|
|[[10,000]]
|[[20,000]]
|[[30,000]]
|[[40,000]]
|[[50,000]]
|[[60,000]]
|[[70,000]]
|[[80,000]]
|[[90,000]]
|-
||[[100,000|10<sup>5</sup>]]
|[[1,000,000|10<sup>6</sup>]]
|[[10,000,000|10<sup>7</sup>]]
|[[100,000,000|10<sup>8</sup>]]
|[[1,000,000,000|10<sup>9</sup>]]
| colspan="5"| [[Trillion|10<sup>12</sup>]]
|-
| colspan="10" |[[Order of magnitude|larger numbers]], including [[Googol|10<sup>100</sup>]] and [[Googolplex|10<sup>10<sup>100</sup></sup>]]
|}

=== 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.{{Collapsible list
| expand = y
| framestyle = 
| titlestyle = 
| title = List of mathematically significant natural numbers
| liststyle = 
| hlist = true
| bullets = on|[[1 (number)|1]], the multiplicative identity. Also the only natural number (not including 0) that is not prime or composite.|[[2 (number)|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 (number)|3]], 2<sup>2</sup>-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 (number)|4]], the first [[composite number]].|[[5 (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 (number)|6]], the first of the series of [[perfect number]]s, whose proper factors sum to the number itself.|[[9 (number)|9]], the first [[Parity (mathematics)|odd]] number that is [[Composite number|composite]].|[[11 (number)|11]], the fifth prime and first palindromic multi-digit number in base 10.|[[12 (number)|12]], the first [[sublime number]].|[[17 (number)|17]], the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.|[[24 (number)|24]], all [[Dirichlet character]]s [[Modular arithmetic|mod]] ''n'' are [[real number|real]] [[if and only if]] ''n'' is a divisor of 24.|[[25 (number)|25]], the first [[centered square number]] besides 1 that is also a square number.|[[27 (number)|27]], the [[Cube (algebra)|cube]] of 3, the value of 3<sup>3</sup>.|[[28 (number)|28]], the second [[perfect number]].|[[30 (number)|30]], the smallest [[sphenic number]].|[[32 (number)|32]], the smallest nontrivial [[fifth power (algebra)|fifth power]].|[[36 (number)|36]], the smallest number which is a [[perfect power]] but not a [[prime power]].|[[70 (number)|70]], the smallest [[weird number]].|[[72 (number)|72]], the smallest [[Achilles number]].|[[108 (number)|108]], the second [[Achilles number]].|[[255 (number)|255]], 2<sup>8</sup> − 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 (computer science)|integer]].|[[341 (number)|341]], the smallest base 2 [[Fermat pseudoprime]].|[[496 (number)|496]], the third [[perfect number]].|[[1729 (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.<ref>{{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
|url-status=live
}}</ref>
| [[5040 (number)|5040]], the largest [[factorial]] (7! = 5040) that is also a [[highly composite number]].|[[8128 (number)|8128]], the fourth perfect number.|[[142857 (number)|142857]], the smallest [[base 10]] [[cyclic number]].|[[9814072356 (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 [[culture|cultural]] significance<ref>{{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}}</ref> 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.

{{Collapsible list
| expand = y
| framestyle = 
| titlestyle = 
| title = List of integers notable for their cultural meanings
| liststyle = 
| hlist = 
| bullets = on
|[[3 (number)|3]], significant in [[Christianity]] as the [[Trinity]]. Also considered significant in [[Hinduism]] ([[Trimurti]], [[Tridevi]]). Holds significance in a number of ancient mythologies.
|[[4 (number)|4]], considered an [[Tetraphobia|"unlucky" number]] in modern China, Japan and Korea due to its audible similarity to the word "death" in their respective languages.
|[[7 (number)|7]], the number of days in a week, and considered a "lucky" number in Western cultures.
|[[8 (number)|8]], considered a [[Chinese numerology#Eight|"lucky" number in Chinese culture]] due to its aural similarity to the Chinese term for prosperity.
|[[12 (number)|12]], a common grouping known as a [[dozen]] and the number of months in a year, of constellations of the [[Zodiac]] and [[astrological sign]]s and of [[Apostles in the New Testament|Apostle]]s of [[Jesus]].
|[[13 (number)|13]], considered an [[Triskaidekaphobia|"unlucky" number]] in Western superstition. Also known as a "[[Baker's dozen]]".<ref>{{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}}</ref>
|[[17 (number)|17]], considered [[Heptadecaphobia|ill-fated]] in Italy and other countries of Greek and Latin origins.
|[[18 (number)|18]], considered a "lucky" number due to it being the value for the [[Chai (symbol)|Hebrew word for life]] in [[gematria|Jewish numerology]].
|[[40 (number)|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 (number)|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 (number)|69]], a slang term for reciprocal [[oral sex]].
|[[86 (number)|86]], a slang term that is used in the American popular culture as a transitive verb to mean throw out or get rid of.<ref name="mw">{{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 }}</ref>
|[[108 (number)|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 (number)|420]], a code-term that refers to the consumption of [[420 (cannabis culture)|cannabis]].
|[[666 (number)|666]], the [[number of the beast]] from the [[Book of Revelation]].
|[[786 (number)|786]], regarded as sacred in the Muslim [[Abjad numerals|Abjad numerology]].
|[[5040 (number)|5040]], mentioned by [[Plato]] in the ''[[Laws (dialogue)|Laws]]'' as one of the most important numbers for the city.
}}

{{Collapsible list
| expand = y
| framestyle = 
| titlestyle = 
| title = List of integers notable for their use in units, measurements and scales
| liststyle = 
| hlist = 
| bullets = on
|[[10]], the number of digits in the [[decimal]] number system.
|[[12 (number)|12]], the [[duodecimal|number base]] for measuring time in many civilizations.
|[[14 (number)|14]], the number of days in a [[fortnight]].
|[[16 (number)|16]], the number of digits in the [[hexadecimal]] number system.
|[[24 (number)|24]], number of [[hour]]s in a [[day]].
|[[31 (number)|31]], the number of days most months of the year have.
|[[60 (number)|60]], the [[sexagesimal|number base]] for some ancient counting systems, such as the [[Babylonian numerals|Babylonians']], and the basis for many modern measuring systems.
|[[360 (number)|360]], the number of [[Degree (angle)|sexagesimal degree]]s in a full [[circle]].
|[[365 (number)|365]], the number of days in the common year, while there are 366 days in a [[leap year]] of the solar [[Gregorian calendar]].
}}

{{Collapsible list
| expand = y
| framestyle = 
| titlestyle = 
| title = List of integers notable in computing
| liststyle =
| hlist =
| bullets = on
|[[4]], the number of [[bit]]s in a [[nibble]].
|[[8]], the number of bits in an [[Octet (computing)|octet]] and usually in a [[byte]].
|[[256 (number)|256]], The number of possible combinations within [[8-bit|8 bits]], or an octet.
|[[1024 (number)|1024]], the number of bytes in a [[kibibyte]], and bits in a [[kibibit]].
|[[65535 (number)|65535]], 2<sup>16</sup> − 1, the maximum value of a [[16-bit]] unsigned integer.
|[[65536 (number)|65536]], 2<sup>16</sup>, the number of possible [[16-bit]] combinations.
|[[65537 (number)|65537]], 2<sup>16</sup> + 1, the most popular RSA public key prime exponent in most SSL/TLS certificates on the Web/Internet.
|[[16777216 (number)|16777216]], 2<sup>24</sup>, or 16<sup>6</sup>; the hexadecimal "million" (0x1000000), and the total number of possible color combinations in 24/32-bit [[24-bit color|True Color]] computer graphics.
|[[2147483647]], 2<sup>31</sup> − 1, the maximum value of a [[32-bit]] [[Integer (computer science)|signed integer]] using [[two's complement]] representation.
|[[9223372036854775807]], 2<sup>63</sup> − 1, the maximum value of a [[64-bit]] [[Integer (computer science)|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 [[Template:Classes of natural numbers|classes of natural numbers]].

=== Prime numbers ===
{{Main|Prime number|List of prime numbers}}
A prime number is a positive integer which has exactly two [[divisor]]s: 1 and itself.

The first 100 prime numbers are:

{| class="wikitable sortable mw-collapsible" style="text-align:center;"
|+ class="nowrap" |Table of first 100 prime numbers
|-
 |&nbsp;&nbsp;[[2 (number)|2]]||&nbsp;&nbsp;[[3 (number)|3]]||&nbsp;&nbsp;[[5 (number)|5]]||&nbsp;&nbsp;[[7 (number)|7]]||&nbsp;[[11 (number)|11]]||&nbsp;[[13 (number)|13]]||&nbsp;[[17 (number)|17]]||&nbsp;[[19 (number)|19]]||&nbsp;[[23 (number)|23]]||&nbsp;[[29 (number)|29]]
 |-
 |&nbsp;[[31 (number)|31]]||&nbsp;[[37 (number)|37]]||&nbsp;[[41 (number)|41]]||&nbsp;[[43 (number)|43]]||&nbsp;[[47 (number)|47]]||&nbsp;[[53 (number)|53]]||&nbsp;[[59 (number)|59]]||&nbsp;[[61 (number)|61]]||&nbsp;[[67 (number)|67]]||&nbsp;[[71 (number)|71]]
 |-
 |&nbsp;[[73 (number)|73]]||&nbsp;[[79 (number)|79]]||&nbsp;[[83 (number)|83]]||&nbsp;[[89 (number)|89]]||&nbsp;[[97 (number)|97]]||[[101 (number)|101]]||[[103 (number)|103]]||[[107 (number)|107]]||[[109 (number)|109]]||[[113 (number)|113]]
 |-
 |[[127 (number)|127]]||[[131 (number)|131]]||[[137 (number)|137]]||[[139 (number)|139]]||[[149 (number)|149]]||[[151 (number)|151]]||[[157 (number)|157]]||[[163 (number)|163]]||[[167 (number)|167]]||[[173 (number)|173]]
 |-
 |[[179 (number)|179]]||[[181 (number)|181]]||[[191 (number)|191]]||[[193 (number)|193]]||[[197 (number)|197]]||[[199 (number)|199]]||[[211 (number)|211]]||[[223 (number)|223]]||[[227 (number)|227]]||[[229 (number)|229]]
 |-
 |[[233 (number)|233]]||[[239 (number)|239]]||[[241 (number)|241]]||[[251 (number)|251]]||[[257 (number)|257]]||[[263 (number)|263]]||[[269 (number)|269]]||[[271 (number)|271]]||[[277 (number)|277]]||[[281 (number)|281]]
 |-
 |[[283 (number)|283]]||[[293 (number)|293]]||[[307 (number)|307]]||[[311 (number)|311]]||[[313 (number)|313]]||[[317 (number)|317]]||[[331 (number)|331]]||[[337 (number)|337]]||[[347 (number)|347]]||[[349 (number)|349]]
 |-
 |[[353 (number)|353]]||[[359 (number)|359]]||[[367 (number)|367]]||[[373 (number)|373]]||[[379 (number)|379]]||[[383 (number)|383]]||[[389 (number)|389]]||[[397 (number)|397]]||[[401 (number)|401]]||[[409 (number)|409]]
 |-
 |[[419 (number)|419]]||[[421 (number)|421]]||[[431 (number)|431]]||[[433 (number)|433]]||[[439 (number)|439]]||[[443 (number)|443]]||[[449 (number)|449]]||[[457 (number)|457]]||[[461 (number)|461]]||[[463 (number)|463]]
 |-
 |[[467 (number)|467]]||[[479 (number)|479]]||[[487 (number)|487]]||[[491 (number)|491]]||[[499 (number)|499]]||[[503 (number)|503]]||[[509 (number)|509]]||[[521 (number)|521]]||[[523 (number)|523]]||[[541 (number)|541]]
 |}

=== Highly composite numbers ===
{{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 (number)|1]], [[2 (number)|2]], [[4 (number)|4]], [[6 (number)|6]], [[12 (number)|12]], [[24 (number)|24]], [[36 (number)|36]], [[48 (number)|48]], [[60 (number)|60]], [[120 (number)|120]], [[180 (number)|180]], [[240 (number)|240]], [[360 (number)|360]], [[720 (number)|720]], [[840 (number)|840]], [[1260 (number)|1260]], [[1680 (number)|1680]], [[2520 (number)|2520]], [[5040 (number)|5040]], [[7560 (number)|7560]]

=== Perfect numbers ===
{{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
|&nbsp; [[6 (number)|6]]
|&nbsp; [[28 (number)|28]]
|&nbsp; [[496 (number)|496]]
|&nbsp; [[8128 (number)|8128]]
|&nbsp; 33 550 336
|&nbsp; 8 589 869 056
|&nbsp; 137 438 691 328
|&nbsp; 2 305 843 008 139 952 128
|&nbsp; 2 658 455 991 569 831 744 654 692 615 953 842 176
|&nbsp; 191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216
}}

== Integers ==
{{main|Integer}}

The integers are a [[set (mathematics)|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]] <math>\mathbb{\Z}</math>, Unicode {{Unichar|2124|DOUBLE-STRUCK CAPITAL Z}}); this became the symbol for the integers based on the German word for "numbers" (''[[wiktionary:Zahlen|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 (number)|−40]] is the equal point in the [[Fahrenheit]] and [[Celsius]] scales.

=== SI prefixes ===
One important use of integers is in [[orders of magnitude (numbers)|orders of magnitude]]. A [[power of 10]] is a number 10<sup>''k''</sup>, where ''k'' is an integer. For instance, with ''k''&nbsp;=&nbsp;0, 1, 2, 3, ..., the appropriate powers of ten are 1, 10, 100, 1000, ... Powers of ten can also be fractional: for instance, ''k''&nbsp;=&nbsp;-3 gives 1/1000, or 0.001. This is used in [[scientific notation]], real numbers are written in the form ''m''&nbsp;×&nbsp;10<sup>''n''</sup>. The number 394,000 is written in this form as 3.94&nbsp;×&nbsp;10<sup>5</sup>.

Integers are used as [[Metric prefix|prefixes]] in the [[SI system]]. A '''metric prefix''' is a [[unit prefix]] that precedes a basic unit of measure to indicate a [[multiple (mathematics)|multiple]] or [[fraction (mathematics)|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.

{|class="wikitable sortable"
|-
! Value
! 1000<sup>''m''</sup>
! Name
!Symbol
|-
| style="text-align:right;"|{{gaps|1|000}}|| 1000<sup>1</sup>||[[Kilo-|Kilo]]
|k
|-
| style="text-align:right;"|{{gaps|1|000|000}}|| 1000<sup>2</sup>||[[Mega-|Mega]]
|M
|-
| style="text-align:right;"|{{gaps|1|000|000|000}}|| 1000<sup>3</sup>||[[Giga-|Giga]]
|G
|-
| style="text-align:right;"|{{gaps|1|000|000|000|000}}|| 1000<sup>4</sup>||[[Tera-|Tera]]
|T
|-
| style="text-align:right;"|{{gaps|1|000|000|000|000|000}}|| 1000<sup>5</sup>||[[Peta-|Peta]]
|P
|-
| style="text-align:right;"|{{gaps|1|000|000|000|000|000|000}}|| 1000<sup>6</sup>||[[Exa-|Exa]]
|E
|-
| style="text-align:right;"|{{gaps|1|000|000|000|000|000|000|000}}|| 1000<sup>7</sup>||[[Zetta-|Zetta]]
|Z
|-
| style="text-align:right;"|{{gaps|1|000|000|000|000|000|000|000|000}}|| 1000<sup>8</sup>||[[Yotta-|Yotta]]
|Y
|-
| style="text-align:right;"|{{gaps|1|000|000|000|000|000|000|000|000|000}}|| 1000<sup>9</sup>||[[Ronna-|Ronna]]
|R
|-
| style="text-align:right;"|{{gaps|1|000|000|000|000|000|000|000|000|000|000}}|| 1000<sup>10</sup>||[[Quetta-|Quetta]]
|Q
|}

== Rational numbers ==
{{main|Rational number}}
A rational number is any number that can be expressed as the [[quotient]] or [[fraction (mathematics)|fraction]] {{math|''p''/''q''}} of two [[integer]]s, a [[numerator]] {{math|''p''}} and a non-zero [[denominator]] {{math|''q''}}.<ref name="Rosen">{{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}}</ref> Since {{math|''q''}} may be equal to&nbsp;1, every integer is trivially a rational number. The [[set (mathematics)|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]] <math>\mathbb{Q}</math>, Unicode {{unichar|211A|DOUBLE-STRUCK CAPITAL Q}});<ref>{{cite web|url=http://searchdatacenter.techtarget.com/definition/Mathematical-Symbols|title=Mathematical Symbols|last1=Rouse|first1=Margaret|access-date=1 April 2015}}</ref> it was thus denoted in 1895 by [[Giuseppe Peano]] after ''[[wikt:quoziente|quoziente]]'', Italian for "[[quotient]]".

Rational numbers such as 0.12 can be represented in [[Infinity|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)]].

{{sticky header}}
{|class="wikitable sortable sticky-header"
|+ Table of notable rational numbers
|-
! Decimal expansion !! Fraction
!Notability
|-
| 1.0
| rowspan="2" style="text-align:center;" |{{sfrac|1|1}}
| rowspan="2" |One is the multiplicative identity. One is a rational number, as it is equal to 1/1.
|-
|1
|-
|  −0.083 333...
| style="text-align:center;"|{{sfrac|−|1|12}}
|The value assigned to the series [[1 + 2 + 3 + 4 + ⋯|1+2+3...]] by [[zeta function regularization]] and [[Ramanujan summation]].
|- 
| 0.5
| style="text-align:center;"|{{sfrac|1|2}}
| [[One half]] occurs commonly in mathematical equations and in real world proportions. One half appears in the formula for the area of a triangle: {{sfrac|2}} × base × perpendicular height and in the formulae for [[figurate numbers]], such as [[triangular number]]s and [[pentagonal number]]s.
|-
|3.142 857...
| style="text-align:center;"|{{sfrac|22|7}}
|A widely used approximation for the number <math>\pi</math>. It can be [[Proof that 22/7 exceeds π|proven]] that this number exceeds <math>\pi</math>.
|-
|0.166 666...
| style="text-align:center;"|{{sfrac|1|6}}
|One sixth. Often appears in mathematical equations, such as in the [[List of mathematical series|sum of squares of the integers]] and in the solution to the Basel problem.
|}

== Real numbers ==
[[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.

=== Algebraic numbers ===
{{main|Algebraic number}}
{{sticky header}}
{|class="wikitable sortable sticky-header"
|-
!Name
! Expression !! Decimal expansion !! Notability
|-
|[[Golden ratio conjugate]] (<math>\Phi</math>)
| style="text-align:center;" | <math>\frac{\sqrt{5}-1}{2}</math>
|{{val|0.618033988749894848204586834366}}
|[[Multiplicative inverse|Reciprocal]] of (and one less than) the [[golden ratio]].
|-
|[[Twelfth root of two]]
| style="text-align:center;" | <math>\sqrt[12]{2}</math>
|{{val|1.059463094359295264561825294946}}
|Proportion between the frequencies of adjacent [[semitone]]s in the [[12 tone equal temperament]] scale.
|-
|[[Cube root]] of two
| style="text-align:center;" | <math>\sqrt[3]{2}</math>
|{{val|1.259921049894873164767210607278}}
|Length of the edge of a [[cube]] with volume two. See [[doubling the cube]] for the significance of this number.
|-
|[[Conway constant#Basic properties|Conway's constant]]
| style="text-align:center;" | (cannot be written as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots)
|{{val|1.303577269034296391257099112153}}
|Defined as the unique positive real root of a certain polynomial of degree 71. The limit ratio between subsequent numbers in the binary [[Look-and-say sequence]] ({{OEIS2C|A014715}}).
|-
|[[Plastic ratio]]
| style="text-align:center;" |<math>\sqrt[3]{\frac{1}{2} +\frac{1}{6} \sqrt{\frac{23}{3}}} +\sqrt[3]{\frac{1}{2} -\frac{1}{6} \sqrt{\frac{23}{3}}}</math>
|{{val|1.324717957244746025960908854478}}
|The only real solution of <math>x^3 = x + 1</math>.({{OEIS2C|A060006}}) The limit ratio between subsequent numbers in the [[Plastic ratio#Van der Laan sequence|Van der Laan sequence]]. ({{OEIS2C|A182097}}) 
|-
|[[Square root of two]] 
| style="text-align:center;" | <math>\sqrt{2}</math>
|{{val|1.414213562373095048801688724210}}
|{{sqrt|2}} = 2 sin 45° = 2 cos 45°  [[Square root of two]] a.k.a. [[Pythagoras' constant]].  Ratio of [[diagonal]] to side length in a [[Square (geometry)|square]].  Proportion between the sides of [[paper size]]s in the [[ISO 216]] series (originally [[DIN]] 476 series).
|-
|[[Supergolden ratio]]
| style="text-align:center;" |<math>\dfrac{1 +\sqrt[3]{\dfrac{29 +3\sqrt{3 \cdot 31}}{2}} +\sqrt[3]{\dfrac{29 -3\sqrt{3 \cdot 31}}{2}}}{3}</math>
|{{val|1.465571231876768026656731225220}}
|The only real solution of <math>x^3 = x^2 + 1</math>.({{OEIS2C|A092526}}) The limit ratio between subsequent numbers in [[Supergolden ratio#Narayana sequence|Narayana's cows sequence]]. ({{OEIS2C|A000930}})
|-
|[[Triangular number#Triangular roots and tests for triangular numbers|Triangular root]] of 2
| style="text-align:center;" | <math>\frac{\sqrt{17}-1}{2}</math>
|{{val|1.561552812808830274910704927987}}
| 
|-
|[[Golden ratio]] (φ)
| style="text-align:center;" | <math>\frac{\sqrt{5}+1}{2}</math>
| {{val|1.618033988749894848204586834366}}
|The larger of the two real roots of ''x''{{sup|2}} = ''x'' + 1.
|-
|[[Square root of three]] 
| style="text-align:center;" | <math>\sqrt{3}</math>
|{{val|1.732050807568877293527446341506}}
|{{sqrt|3}} = 2 sin 60° = 2 cos 30° . A.k.a. ''[[vesica piscis|the measure of the fish]]'' or Theodorus' constant. Length of the [[space diagonal]] of a [[cube]] with edge length 1. [[Altitude (triangle)|Altitude]] of an [[equilateral triangle]] with side length 2. Altitude of a [[hexagon|regular hexagon]] with side length 1 and diagonal length 2.
|-
|[[Tribonacci numbers|Tribonacci constant]]
| style="text-align:center;" |<math>\frac{1 +\sqrt[3]{19 +3\sqrt{3 \cdot 11}} +\sqrt[3]{19 -3\sqrt{3 \cdot 11}}}{3}</math>
|{{val|1.839286755214161132551852564653}}
|The only real solution of <math>x^3 = x^2 + x + 1</math>.({{OEIS2C|A058265}}) The limit ratio between subsequent numbers in the [[Generalizations of Fibonacci numbers#Tribonacci numbers|Tribonacci sequence]].({{OEIS2C|A000073}}) Appears in the volume and coordinates of the [[snub cube]] and some related polyhedra.
|-
|[[Supersilver ratio]]
| style="text-align:center;" |<math>\dfrac{2 +\sqrt[3]{\dfrac{43 +3\sqrt{3 \cdot 59}}{2}} +\sqrt[3]{\dfrac{43 -3\sqrt{3 \cdot 59}}{2}}}{3}</math>
|{{val|2.20556943040059031170202861778}}
|The only real solution of <math>x^3 = 2x^2 + 1</math>.({{OEIS2C|A356035}}) The limit ratio between subsequent numbers in the [[Supersilver ratio#Third-order Pell sequences|third-order Pell sequence]]. ({{OEIS2C|A008998}})
|-
|[[Square root of five]]
| style="text-align:center;" | <math>\sqrt{5}</math>
|{{val|2.236067977499789696409173668731}}
|Length of the [[diagonal]] of a 1 × 2 [[rectangle]].
|-
|[[Silver ratio]] (δ{{sub|S}})
| style="text-align:center;" | <math>\sqrt{2}+1</math>
|{{val|2.414213562373095048801688724210}}
|The larger of the two real roots of ''x''{{sup|2}} = 2''x'' + 1.<br> Altitude of a [[octagon|regular octagon]] with side length 1.
|-
|[[Bronze ratio]] (S{{sub|3}})
| style="text-align:center;" | <math>\frac{\sqrt{13}+3}{2}</math>
|{{val|3.302775637731994646559610633735}}
|The larger of the two real roots of ''x''{{sup|2}} = 3''x'' + 1.
|}

=== Transcendental numbers ===
{{main|Transcendental number}}
{{sticky header}}
{|class="wikitable sortable sticky-header"
|-
!Name
!Symbol
or

Formula
!Decimal expansion
!Notes and notability
|-
|[[Gelfond's constant]]
|<math>e^{\pi}</math>
|{{val|23.14069263277925}}...
|
|-
|[[Ramanujan's constant]]
|<math>e^{\pi\sqrt{163}}</math>
|{{val|262537412640768743.99999999999925}}...
|
|-
|[[Gaussian integral]]
|<math>\sqrt{\pi}</math>
|{{val|1.772453850905516}}...
|
|-
|[[Komornik–Loreti constant]]
|<math>q</math>
|{{val|1.787231650}}...
|
|-
|[[Universal parabolic constant]]
|<math>P_2</math>
|{{val|2.29558714939}}...
|
|-
|[[Gelfond–Schneider constant]]
|<math>2^{\sqrt{2}}</math>
|{{val|2.665144143}}...
|
|-
|[[E (mathematical constant)|Euler's number]]
|<math>e</math>
|{{val|2.718281828459045235360287471352662497757247}}...
|Raising e to the power of <math>i</math>{{pi}} will result in <math>-1</math>.
|-
|[[Pi]]
|<math>
\pi</math>
|{{val|3.141592653589793238462643383279502884197169399375}}...
|Pi is a constant irrational number that is the result of dividing the circumference of a circle by its diameter.
|-
|[[Super Root|Super square-root]] of 2
|<math display="inline">\sqrt{2}_s</math><ref>{{Citation|last=Lipscombe|first=Trevor Davis|title=Super Powers: Calculate Squares, Square Roots, Cube Roots, and More|date=2021-05-06|url=http://dx.doi.org/10.1093/oso/9780198852650.003.0010|work=Quick(er) Calculations|pages=103–124|publisher=Oxford University Press|doi=10.1093/oso/9780198852650.003.0010|isbn=978-0-19-885265-0|access-date=2021-10-28}}</ref>
|{{val|1.559610469}}...<ref>{{cite web|url=http://www.qbyte.org/puzzles/p029s.html|title=Nick's Mathematical Puzzles: Solution 29|archive-url=https://web.archive.org/web/20111018184029/http://www.qbyte.org/puzzles/p029s.html|archive-date=2011-10-18|url-status=live}}</ref>
|
|-
|[[Liouville constant]]
|<math display="inline">L</math>
|{{val|0.110001000000000000000001000}}...
|
|-
|[[Champernowne constant]]
|<math display="inline">C_{10}</math>
|{{val|0.12345678910111213141516}}...
|This constant contains every number string inside it, as its decimals are just every number in order. (1,2,3,etc.)
|-
|[[Prouhet–Thue–Morse constant]]
|<math display="inline">\tau</math>
|{{val|0.412454033640}}...
|
|-
|[[Omega constant]]
|<math>\Omega</math>
|{{val|0.5671432904097838729999686622}}...
|
|-
|[[Cahen's constant]]
|<math display="inline">C
</math>
|{{val|0.64341054629}}...
|
|-
|[[Natural logarithm of 2]]
|ln 2
|{{val|0.693147180559945309417232121458}}
|
|-
|[[Lemniscate constant]]
|<math display="inline">\varpi</math>
|{{val|2.622057554292119810464839589891}}...
|The ratio of the perimeter of [[Lemniscate of Bernoulli|Bernoulli's lemniscate]] to its diameter. 
|-
|[[Tau (mathematical constant)|Tau]]
|<math>\tau=2\pi</math>
|{{val|6.283185307179586476925286766559}}...
|The ratio of the [[circumference]] to a [[radius]], and the number of [[radian]]s in a complete circle;<ref>"The Penguin Dictionary of Curious and Interesting Numbers" by [[David G. Wells|David Wells]], page 69</ref><ref>Sequence {{OEIS2C|A019692}}.</ref> 2 <math>\times</math> {{pi}}
|}


=== Irrational but not known to be transcendental===
Some numbers are known to be [[irrational number]]s, but have not been proven to be transcendental. This differs from the algebraic numbers, which are known not to be transcendental.
{{sticky header}}
{|class="wikitable sortable sticky-header"
|-
!Name
!Decimal expansion
!Proof of irrationality
!Reference of unknown transcendentality
|-
|[[Riemann zeta function|ζ]](3), also known as [[Apéry's constant]]
|{{val|1.202056903159594285399738161511449990764986292}}
|<ref name="Apery-1979">See {{harvnb|Apéry|1979}}.</ref>
|<ref>"The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33</ref>
|-
|[[Erdős–Borwein constant]], E
|{{val|1.606695152415291763}}...
|<ref>{{citation|last=Erdős|first=P.|title=On arithmetical properties of Lambert series|url=http://www.renyi.hu/~p_erdos/1948-04.pdf|journal=J. Indian Math. Soc. |series=New Series|volume=12|pages=63–66|year=1948|mr=0029405|author-link=Paul Erdős}}</ref><ref>{{citation|last=Borwein|first=Peter B.|title=On the irrationality of certain series|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|volume=112|issue=1|pages=141–146|year=1992|doi=10.1017/S030500410007081X|bibcode=1992MPCPS.112..141B|mr=1162938|author-link=Peter Borwein|citeseerx=10.1.1.867.5919|s2cid=123705311 }}</ref>
|{{Citation needed|date=July 2019}}
|-
|[[Copeland–Erdős constant]]
|{{val|0.235711131719232931374143}}...
|Can be proven with [[Dirichlet's theorem on arithmetic progressions]] or [[Bertrand's postulate]] (Hardy and Wright, p.&nbsp;113) or [[Olivier Ramaré|Ramare's theorem]] that every even integer is a sum of at most six primes. It also follows directly from its normality.
|{{Citation needed|date=July 2019}}
|-
|[[Prime constant]], ρ
|{{val|0.414682509851111660248109622}}...
|Proof of the number's irrationality is given at [[prime constant]].
|{{Citation needed|date=July 2019}}
|-
|[[Reciprocal Fibonacci constant]], ψ
|{{val|3.359885666243177553172011302918927179688905133731}}...
| <ref>André-Jeannin, Richard; 'Irrationalité de la somme des inverses de certaines suites récurrentes.'; ''Comptes Rendus de l'Académie des Sciences - Series I - Mathematics'', vol. 308, issue 19 (1989), pp. 539-541.</ref><ref>S. Kato, 'Irrationality of reciprocal sums of Fibonacci numbers', Master's thesis, Keio Univ. 1996</ref>
|<ref>Duverney, Daniel, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa; '[https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/62370/1/1060-10.pdf Transcendence  of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers]';</ref>
|}

=== Real but not known to be irrational, nor transcendental ===
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.
{{sticky header}}
{|class="wikitable sortable sticky-header"
|-
!Name and symbol
!Decimal expansion
!Notes
|-
|[[Euler–Mascheroni constant]], γ
|{{val|0.577215664901532860606512090082}}...<ref>{{Cite web|title=A001620 - OEIS|url=https://oeis.org/A001620|access-date=2020-10-14|website=oeis.org}}</ref>
|Believed to be transcendental but not proven to be so. However, it was shown that at least one of <math>\gamma</math> and the Euler-Gompertz constant <math>\delta</math> is transcendental.<ref name=":4">{{Cite journal|last=Rivoal|first=Tanguy|date=2012|title=On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant|url=https://projecteuclid.org/euclid.mmj/1339011525|journal=[[Michigan Mathematical Journal]]|language=EN|volume=61|issue=2|pages=239–254|doi=10.1307/mmj/1339011525|issn=0026-2285|doi-access=free}}</ref><ref name=":5">{{Cite journal|last=Lagarias|first=Jeffrey C.|date=2013-07-19|title=Euler's constant: Euler's work and modern developments|journal=Bulletin of the American Mathematical Society|volume=50|issue=4|pages=527–628|doi=10.1090/S0273-0979-2013-01423-X|arxiv=1303.1856|issn=0273-0979|doi-access=free}}</ref> It was also shown that all but at most one number in an infinite list containing <math>\frac{\gamma}{4}</math> have to be transcendental.<ref>{{Cite journal|last1=Murty|first1=M. Ram|last2=Saradha|first2=N.|date=2010-12-01|title=Euler–Lehmer constants and a conjecture of Erdös|url=http://www.sciencedirect.com/science/article/pii/S0022314X10001836|journal=Journal of Number Theory|language=en|volume=130|issue=12|pages=2671–2682|doi=10.1016/j.jnt.2010.07.004|issn=0022-314X|citeseerx=10.1.1.261.753}}</ref><ref>{{Cite journal|last1=Murty|first1=M. Ram|last2=Zaytseva|first2=Anastasia|date=2013-01-01|title=Transcendence of Generalized Euler Constants|url=https://www.tandfonline.com/doi/abs/10.4169/amer.math.monthly.120.01.048|journal=The American Mathematical Monthly|volume=120|issue=1|pages=48–54|doi=10.4169/amer.math.monthly.120.01.048|s2cid=20495981|issn=0002-9890}}</ref>
|-
|[[Gompertz constant|Euler–Gompertz constant]], δ
|0.596 347 362 323 194 074 341 078 499 369...<ref>{{Cite web|title=A073003 - OEIS|url=https://oeis.org/A073003|access-date=2020-10-14|website=oeis.org}}</ref>
|It was shown that at least one of the Euler-Mascheroni constant <math>\gamma</math> and the Euler-Gompertz constant <math>\delta</math> is transcendental.<ref name=":4" /><ref name=":5" />
|-
|[[Catalan's constant]], G
|{{val|0.915965594177219015054603514932384110774}}...
|It is not known whether this number is irrational.<ref>{{citation|last=Nesterenko|first=Yu. V.|title=On Catalan's constant|date=January 2016|journal=Proceedings of the Steklov Institute of Mathematics|volume=292|issue=1|pages=153–170|doi=10.1134/s0081543816010107|s2cid=124903059}}</ref>
|-
|[[Khinchin's constant]], K<sub>0</sub>
|{{val|2.685452001}}...<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/KhinchinsConstant.html|title = Khinchin's Constant}}</ref>
|It is not known whether this number is irrational.<ref>{{MathWorld|urlname=KhinchinsConstant|title=Khinchin's constant}}</ref>
|-
|1st [[Feigenbaum constant]], δ
|4.6692...
|Both Feigenbaum constants are believed to be [[Transcendental number|transcendental]], although they have not been proven to be so.<ref name=Briggs>{{Cite thesis
|first=Keith
|last=Briggs
|url=http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf
|publisher=[[University of Melbourne]]
|year=1997
|degree=PhD
|title=Feigenbaum scaling in discrete dynamical systems
}}</ref>
|-
|2nd [[Feigenbaum constant]], α
|2.5029...
|Both Feigenbaum constants are believed to be [[Transcendental number|transcendental]], although they have not been proven to be so.<ref name=Briggs/>
|-
|[[Glaisher–Kinkelin constant]], A
|{{val|1.28242712}}...
|
|-
|[[Backhouse's constant]]
|{{val|1.456074948}}...
|
|-
|[[Fransén–Robinson constant]], F
|{{val|2.8077702420}}...
|
|-
|[[Lévy's constant]],β 
|1.18656 91104 15625 45282...
|
|-
|[[Mills' constant]], A
|{{val|1.30637788386308069046}}...
|It is not known whether this number is irrational.{{harv|Finch|2003}}
|-
|[[Ramanujan–Soldner constant]], μ
|{{val|1.451369234883381050283968485892027449493}}...
|
|-
|[[Sierpiński's constant]], K
|{{val|2.5849817595792532170658936}}...
|
|-
|[[Totient summatory constant]]
|{{val|1.339784}}...<ref>{{OEIS2C|A065483}}</ref>
|
|-
|[[Double exponential function#Doubly exponential sequences|Vardi's constant]], E
|{{val|1.264084735305}}...
|
|-
|[[Somos' quadratic recurrence constant]], σ
|{{val|1.661687949633594121296}}...
|
|-
|[[Niven's constant]], C
|{{val|1.705211}}...
|
|-
|[[Brun's constant]], B<sub>2</sub>
|{{val|1.902160583104}}...
|The irrationality of this number would be a consequence of the truth of the infinitude of [[twin prime]]s. 
|-
|[[Landau's totient constant]]
|{{val|1.943596}}...<ref>{{OEIS2C|A082695}}</ref>
|
|-
|[[Brun's constant|Brun's constant for prime quadruplets]], B<sub>4</sub>
|{{val|0.8705883800}}...
|
|-
|[[Random Fibonacci sequence|Viswanath's constant]]
|{{val|1.1319882487943}}...
|
|-
|[[Khinchin–Lévy constant]]
|{{val|1.1865691104}}...<ref>{{Cite web|url=http://mathworld.wolfram.com/LevyConstant.html|title=Lévy Constant}}</ref>
|This number represents the probability that three random numbers have no [[common factor]] greater than 1.<ref name="David Wells page 29">"The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.</ref>
|-
|[[Landau–Ramanujan constant]]
|{{val|0.76422365358922066299069873125}}...
|
|-
|[[Fresnel integral|C(1)]]
|{{val|0.77989340037682282947420641365}}...
|
|-
|[[Riemann–Siegel formula|Z(1)]]
|{{val|-0.736305462867317734677899828925614672}}...
|
|-
|[[Heath-Brown–Moroz constant]], C
|{{val|0.001317641}}...
|
|-
|[[Kepler–Bouwkamp constant]],K'
|{{val|0.1149420448}}...
|
|-
|[[MRB constant]],S
|{{val|0.187859}}...
|It is not known whether this number is irrational.
|-
|[[Meissel–Mertens constant]], M
|{{val|0.2614972128476427837554268386086958590516}}...
|
|-
|[[Bernstein's constant]], β
|{{val|0.2801694990}}...
|
|-
|[[Gauss–Kuzmin–Wirsing constant]], λ<sub>1</sub>
|{{val|0.3036630029}}...<ref>{{mathworld|urlname=Gauss-Kuzmin-WirsingConstant|title=Gauss–Kuzmin–Wirsing Constant}}</ref>
|
|-
|[[Hafner–Sarnak–McCurley constant]],σ
|{{val|0.3532363719}}...
|
|-
|[[Artin's conjecture on primitive roots|Artin's constant]],C{{Sub|Artin}}
|{{val|0.3739558136}}...
|
|-
|[[Fresnel integral|S(1)]]
|{{val|0.438259147390354766076756696625152}}...
|
|-
|[[Dawson integral|F(1)]]
|{{val|0.538079506912768419136387420407556}}...
|
|-
|[[Stephens' constant]]
|{{val|0.575959}}...<ref>{{OEIS2C|A065478}}</ref>
|
|-
|[[Golomb–Dickman constant]], λ
|{{val|0.62432998854355087099293638310083724}}...
|
|-
|[[Twin prime conjecture#First Hardy–Littlewood conjecture|Twin prime constant]], C<sub>2</sub>
|{{val|0.660161815846869573927812110014}}...
|
|-
|[[Feller–Tornier constant]]
|{{val|0.661317}}...<ref>{{OEIS2C|A065493}}</ref>
|
|-
|[[Laplace limit]], ε
|{{val|0.6627434193}}...<ref>{{Cite web|url=http://mathworld.wolfram.com/LaplaceLimit.html|title=Laplace Limit}}</ref>
|
|-
|[[Embree–Trefethen constant]]
|{{val|0.70258}}...
|
|}

=== Numbers not known with high precision ===
{{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|Berry–Esseen Theorem]]: 0.4097 < ''C'' < 0.4748
* [[De Bruijn–Newman constant]]: 0 ≤ Λ ≤ 0.2
* [[Chaitin's constant]]s Ω, which are transcendental and provably impossible to compute.
* [[Bloch's theorem (complex variables)#Bloch's constant|Bloch's constant]] (also [[Landau's constants|2nd Landau's constant]]): 0.4332 < ''B'' < 0.4719
* [[Landau's constants|1st Landau's constant]]: 0.5 < ''L'' < 0.5433
* [[Landau's constants|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 ==
{{main|Hypercomplex number}}
[[Hypercomplex number]] is a term for an [[element (mathematics)|element]] of a unital [[algebra over a field|algebra]] over the [[field (mathematics)|field]] of [[real number]]s. The [[complex number]]s are often symbolised by a boldface {{Math|'''C'''}} (or [[blackboard bold]] <math>\mathbb{\Complex}</math>, Unicode {{Unichar|2102|DOUBLE-STRUCK CAPITAL C}}), while the set of [[quaternion]]s is denoted by a boldface {{Math|'''H'''}} (or [[blackboard bold]] <math>\mathbb{H}</math>, Unicode {{Unichar|210D|DOUBLE-STRUCK CAPITAL H}}).

=== Algebraic complex numbers ===
* [[Imaginary unit]]: <math display="inline">i=\sqrt{-1}</math>
* ''n''th [[roots of unity]]: <math display="inline">\xi_{n}^{k}=\cos\bigl(2\pi\frac{k}{n}\bigr)+i\sin\bigl(2\pi\frac{k}{n}\bigr)</math>, while <math display="inline">0 \leq k \leq n-10</math>, [[greatest common divisor|GCD]](''k'', ''n'') = 1

===Other hypercomplex numbers===
* The [[quaternion]]s
* The [[octonion]]s
* The [[sedenion]]s
* The [[trigintaduonion]]s
* The [[dual number]]s (with an [[infinitesimal]])

== Transfinite numbers ==
{{main|Transfinite number}}
[[Transfinite numbers]] are numbers that are "[[Infinity|infinite]]" in the sense that they are larger than all [[finite set|finite]] numbers, yet not necessarily [[absolutely infinite]].
* [[Aleph-null]]: ℵ<sub>0</sub>, the smallest infinite cardinal, and the cardinality of <math>\mathbb{N}</math>, the set of [[natural number]]s
* [[Aleph-one]]: ℵ<sub>1</sub>, the cardinality of ω<sub>1</sub>, the set of all countable ordinal numbers
* [[Beth-one]]: <math>\beth_1</math> or <math>\mathfrak c</math>, the [[cardinality of the continuum]] 2{{sup|ℵ<sub>0</sub>}}
* [[First infinite ordinal|Omega]]: ω, the smallest [[infinite ordinal]]

== Numbers representing physical quantities ==
{{main|Physical constant|List of physical constants}}
Physical quantities that appear in the universe are often described using [[physical constant]]s.
* [[Avogadro constant]]: {{physconst|NA|symbol=yes}}
* [[Mass of the electron|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|Speed of light in vacuum]]: {{physconst|c|symbol=yes}}
* [[Vacuum electric permittivity]]: {{physconst|eps0|symbol=yes}}

== Numbers representing geographical and astronomical distances ==
* [[Equator#Exact length|{{Val|6378.137}}]], the average equatorial radius of Earth in [[kilometers]] (following [[GRS 80]] and [[WGS 84]] standards).
* [[Equator#Exact length|{{Val|40075.0167}}]], the length of the [[Equator]] in kilometers (following GRS 80 and WGS 84 standards).
* [[Lunar distance (astronomy)|{{Val|384399}}]], 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|{{Val|149597870700}}]], the average distance between the Earth and the Sun or [[Astronomical Unit]] (AU), in meters.
* [[Light-year|{{Val|9460730472580800}}]], one [[light-year]], the distance travelled by light in one [[Julian year (astronomy)|Julian year]], in meters.
* [[Parsec|{{Val|30856775814913673}}]], the distance of one [[parsec]], another astronomical unit, in whole meters.

== Numbers without specific values ==
{{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 name]]s, or when precision is unnecessary or undesirable. One technical term for such words is "non-numerical vague quantifier".<ref>[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 }}</ref> Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".<ref>[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"]</ref>

== Named numbers ==
*[[Hardy–Ramanujan number]], 1729
*[[Kaprekar's constant]], [[6174]]
*[[Eddington number]], ~10<sup>80</sup>
*[[Googol]], 10<sup>100</sup>
*[[Shannon number]]
*[[Centillion]], 10<sup>303</sup>
*[[Skewes's number]]
*[[Googolplex]], 10<sup>(10<sup>100</sup>)</sup>
*[[Mega (number)|Mega]]/Circle(2)
*[[Moser's number]]
*[[Graham's number]]
*[[Kruskal's tree theorem#TREE function|TREE(3)]]
*[[Friedman's SSCG function|SSCG(3)]]
*[[Rayo's number]]

== See also ==
{{Portal|Mathematics}}
<!-- Please keep entries in alphabetical order & add a short description [[WP:SEEALSO]] -->
{{div col|colwidth=20em|small=yes}}
* [[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]]
{{div col end}}
<!-- please keep entries in alphabetical order -->

== References ==
{{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 <math>\zeta(2)</math> et <math>\zeta(3)</math>
 | year = 1979
 | journal = Astérisque
 | volume = 61
 | pages = 11–13
}}.

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

== External links ==
* [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)

{{DEFAULTSORT:Numbers}}
[[Category:Number-related lists| ]]
[[Category:Mathematical tables]]
[[Category:Numeral systems| ]]