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{{Short description|Integer number 3}}
{{Hatnote|This article is about the number. For the years, see [[BC 3]] and [[AD 3]]. For other uses, see [[3 (disambiguation)]], [[III (disambiguation)]] and [[Number Three (disambiguation)]].}}
{{Distinguish|Ze (Cyrillic){{!}}З (ze)|Ezh{{!}}Ʒ (ezh)|Yogh{{!}}Ȝ (yogh)|ayin{{!}}ع (ʿayn)}}
{{technical reasons|:3|the keyboard symbols|List of emoticons}}
{{More footnotes needed|date=February 2024}}
{{Infobox number
|number=3
|numeral=[[Ternary numeral system|ternary]]
|factorization=[[prime number|prime]]
|prime=2nd
|divisor=1, 3
|roman =III, iii
|latin prefix=[[wikt:tre-|tre-]]/[[wikt:ter-|ter-]]
|lang1=[[Eastern Arabic numerals|Arabic]], [[Central Kurdish|Kurdish]], [[Persian language|Persian]], [[Sindhi language|Sindhi]], [[Urdu numerals|Urdu]]
|lang1 symbol={{resize|150%|٣}}
|lang2=[[Bengali language|Bengali]], [[Assamese language|Assamese]]
|lang2 symbol={{resize|150%|৩}}
|lang3=[[Chinese numeral|Chinese]]
|lang3 symbol=三，弎，叄
|lang4=[[Devanāgarī]]
|lang4 symbol={{resize|150%|३}}
|lang5=[[Ge'ez alphabet|Ge'ez]]
|lang5 symbol=፫
|lang6=[[Greek numerals|Greek]]
|lang6 symbol=γ (or Γ)
|lang7=[[Hebrew numerals|Hebrew]]
|lang7 symbol={{resize|150%|ג}}
|lang8=[[Japanese numerals|Japanese]]
|lang8 symbol=三/参
|lang9=[[Khmer numerals|Khmer]]
|lang9 symbol={{resize|150%|៣}}
|lang10=[[Armenian numerals|Armenian]]|lang10 symbol=Գ|lang11=[[Malayalam language|Malayalam]]
|lang11 symbol={{resize|150%|൩}}
|lang12=[[Tamil language|Tamil]]
|lang12 symbol={{resize|150%|௩}}
|lang13=[[Telugu language|Telugu]]
|lang13 symbol={{resize|150%|౩}}
|lang14=[[Kannada numerals|Kannada]]
|lang14 symbol={{resize|150%|೩}}
|lang15=[[Thai numerals|Thai]]
|lang15 symbol={{resize|150%|๓}}
|lang16=[[N'Ko script|N'Ko]]
|lang16 symbol={{resize|150%|߃}}
|lang17=[[Lao numerals|Lao]]
|lang17 symbol={{resize|150%|໓}}
|lang18=[[Georgian numerals|Georgian]]
|lang18 symbol={{resize|130%|Ⴂ/ⴂ/გ}} ([[Gani (letter)|Gani]])
|lang19=[[Babylonian cuneiform numerals|Babylonian numeral]]|lang19 symbol=𒐗|lang20=[[Maya numerals]]|lang20 symbol=•••|lang21=[[Morse code]]|lang21 symbol={{resize|150%|... _ _}}}}

'''3''' ('''three''') is a [[number]], [[numeral (linguistics)|numeral]] and [[numerical digit|digit]]. It is the [[natural number]] following [[2]] and preceding [[4]], and is the smallest odd [[prime number]] and the only prime preceding a square number. It has religious and cultural significance in many societies.<ref>{{Cite web |title=Merriam-Webster Dictionary |url=https://www.merriam-webster.com/dictionary/three |access-date=December 5, 2024 |website=Merriam-webster.com}}</ref>

==Evolution of the Arabic digit==
[[File:Evolution3glyph.png|x50px|left]]
The use of three lines to denote the number 3 occurred in many writing systems, including some (like Roman and [[Chinese numerals]]) that are still in use. That was also the original representation of 3 in the [[Brahmic numerals|Brahmic]] (Indian) numerical notation, its earliest forms aligned vertically.<ref name="Smith Karpinski 1911">{{cite book |last1=Smith |first1=David Eugene |last2=Karpinski |first2=Louis Charles |author1-link=David Eugene Smith |author2-link=Louis Charles Karpinski |title=The Hindu-Arabic numerals |date=1911 |publisher=Ginn and Company |location=Boston; London |pages=27–29, 40–41 |url=https://archive.org/details/hinduarabicnumer00smitrich/}}</ref>  However, during the [[Gupta Empire]] the sign was modified by the addition of a curve on each line. The [[Nāgarī script]] rotated the lines clockwise, so they appeared horizontally, and ended each line with a short downward stroke on the right. In cursive script, the three strokes were eventually connected to form a glyph resembling a {{angbr|3}} with an additional stroke at the bottom: '''३'''. 

The Indian digits spread to the [[Caliphate]] in the 9th century.  The bottom stroke was dropped around the 10th century in the western parts of the Caliphate, such as the [[Maghreb]] and [[Al-Andalus]], when a distinct variant ("Western Arabic") of the digit symbols developed, including modern Western 3.  In contrast, the Eastern Arabs retained and enlarged that stroke, rotating the digit once more to yield the modern ("Eastern") [[Arabic script|Arabic]] digit "'''٣'''".<ref>Georges Ifrah, ''The Universal History of Numbers: From Prehistory to the Invention of the Computer'' transl. David Bellos et al. London: The Harvill Press (1998): 393, Fig. 24.63</ref>

In most modern Western [[typeface]]s, the digit 3, like the other [[decimal digit]]s, has the height of a [[capital letter]], and sits on the [[baseline (typography)|baseline]].  In typefaces with [[text figures]], on the other hand, the glyph usually has the height of a [[lowercase letter]] "x" and a [[descender]]: "[[File:Text figures 036.svg|50px]]". In some [[French language|French]] text-figure typefaces, though, it has an [[ascender (typography)|ascender]] instead of a descender.

{{anchor|Flat-topped 3}}
A common graphic variant of the digit three has a flat top, similar to the letter [[Ʒ]] (ezh). This form, sometimes called a '''banker's 3''', can stop a forger from turning the 3 into an 8. It is found on [[Universal Product Code|UPC-A]] barcodes and [[standard 52-card deck]]s.{{citation needed|date=February 2025}}

== Mathematics ==
=== Divisibility rule ===
A [[natural number]] is [[divisible]] by 3 if the [[digital root|sum of its digits]] in [[base 10]] is also divisible by 3. This known as the [[divisibility rule]] of 3. Because of this, the reverse of any number that is divisible by three (or indeed, any [[permutation]] of its digits) is also divisible by three. This divisibility rule works in any [[positional notation|positional numeral system]] whose [[radix|base]] divided by three leaves a remainder of one (bases 4, 7, 10, etc.).{{Citation needed|date=February 2025}}

=== Properties ===
3 is the second smallest [[prime number]] and the first [[Parity (mathematics)|odd]] prime number. 3 is a [[twin prime]] with [[5]], and a [[cousin prime]] with [[7]].

A [[triangle]] is made of three [[Edge (geometry)|sides]]. It is the smallest non-self-intersecting [[polygon]] and the only polygon not to have proper [[diagonals]]. When doing quick estimates, 3 is a rough approximation of [[pi|{{pi}}]], 3.1415..., and a very rough approximation of [[E (mathematical constant)|''e'']], 2.71828...

3 is the first [[Mersenne prime]]. 3 is also the first of five known [[Fermat prime]]s. It is the second [[Fibonacci number|Fibonacci prime]] (and the second [[Lucas prime]]), the second [[Sophie Germain prime]], and the second [[factorial prime]]. 

3 is the second and only prime [[triangular number]],<ref>{{Cite web |title=A000217 - OEIS |url=https://oeis.org/A000217 |access-date=2024-11-28 |website=oeis.org}}</ref> and [[Gauss]] proved that every integer is the sum of at most 3 [[triangular numbers]].

Three is the only prime which is one less than a [[square number|perfect square]]. Any other number which is <math>n^2</math> − 1 for some integer <math>n</math> is not prime, since it is (<math>n</math> − 1)(<math>n</math> + 1). This is true for 3 as well (with <math>n</math> = 2), but in this case the smaller factor is 1. If <math>n</math> is greater than 2, both <math>n</math> − 1 and <math>n</math> + 1 are greater than 1 so their product is not prime.

=== Numeral systems ===
There is some evidence to suggest that early man may have used counting systems which consisted of "One, Two, Three" and thereafter "Many" to describe counting limits. Early peoples had a word to describe the quantities of one, two, and three but any quantity beyond was simply denoted as "Many". This is most likely based on the prevalence of this phenomenon among people in such disparate regions as the deep Amazon and Borneo jungles, where western civilization's explorers have historical records of their first encounters with these indigenous people.<ref>{{cite book | last1 = Gribbin | first1 = Mary  | last2 = Gribbin | first2 = John R. | last3 = Edney | first3 = Ralph | last4 = Halliday | first4 = Nicholas | title = Big numbers | publisher = Wizard | location = Cambridge | year = 2003 | isbn = 1840464313 }}</ref>

=== List of basic calculations ===
{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Multiplication]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
!21
!22
!23
!24
!25
!50
!100
!1000
!10000
|-
|'''3 × ''x'''''
|'''3'''
|[[6 (number)|6]]
|[[9 (number)|9]]
|[[12 (number)|12]]
|[[15 (number)|15]]
|[[18 (number)|18]]
|[[21 (number)|21]]
|[[24 (number)|24]]
|[[27 (number)|27]]
|[[30 (number)|30]]
|[[33 (number)|33]]
|[[36 (number)|36]]
|[[39 (number)|39]]
|[[42 (number)|42]]
|[[45 (number)|45]]
|[[48 (number)|48]]
|[[51 (number)|51]]
|[[54 (number)|54]]
|[[57 (number)|57]]
|[[60 (number)|60]]
|[[63 (number)|63]]
|[[66 (number)|66]]
|[[69 (number)|69]]
|[[72 (number)|72]]
|[[75 (number)|75]]
|[[150 (number)|150]]
|[[300 (number)|300]]
|[[3000 (number)|3000]]
|[[30000 (number)|30000]]
|}

{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Division (mathematics)|Division]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!width="5px"|
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
|-
|'''3 ÷ ''x'''''
|'''3'''
|1.5
|1
|0.75
|0.6
|0.5
|0.{{overline|428571}}
|0.375
|0.{{overline|3}}
|0.3
!
|0.{{overline|27}}
|0.25
|0.{{overline|230769}}
|0.2{{overline|142857}}
|0.2
|0.1875
|0.1{{overline|7647058823529411}}
|0.1{{overline|6}}
|0.1{{overline|57894736842105263}}
|0.15
|-
|'''''x'' ÷ 3'''
|0.{{overline|3}}
|0.{{overline|6}}
|1
|1.{{overline|3}}
|1.{{overline|6}}
|2
|2.{{overline|3}}
|2.{{overline|6}}
|'''3'''
|3.{{overline|3}}
!
|3.{{overline|6}}
|4
|4.{{overline|3}}
|4.{{overline|6}}
|5
|5.{{overline|3}}
|5.{{overline|6}}
|6
|6.{{overline|3}}
|6.{{overline|6}}
|}

{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!width="5px"|
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
|-
|'''3{{sup|''x''}}'''
|'''3'''
|9
|27
|[[81 (number)|81]]
|[[243 (number)|243]]
|729
|2187
|6561
|19683
|59049
!
|177147
|531441
|1594323
|4782969
|14348907
|43046721
|129140163
|387420489
|1162261467
|3486784401
|-
|'''''x''{{sup|3}}'''
|1
|[[8 (number)|8]]
|27
|[[64 (number)|64]]
|[[125 (number)|125]]
|[[216 (number)|216]]
|[[343 (number)|343]]
|[[512 (number)|512]]
|729
|[[1000 (number)|1000]]
!
|1331
|1728
|2197
|2744
|3375
|4096
|4913
|5832
|6859
|8000
|}

==Engineering==
*The [[triangle]], a [[polygon]] with three [[Edge (geometry)|edges]] and three [[Vertex (geometry)|vertices]], is the most stable physical shape. For this reason it is widely utilized in construction, engineering and design.<ref>"[https://web.archive.org/web/20120412203618/http://www.mathsinthecity.com/sites/most-stable-shape-triangle Most stable shape- triangle]". ''Maths in the city''. Retrieved February 23, 2015.</ref>

===Pseudoscience===
*Three is the symbolic representation for [[Mu (lost continent)|Mu]], [[Augustus Le Plongeon]]'s and [[James Churchward]]'s lost continent.<ref name="Three is for the Lost Continent of Mu">{{cite web |url=http://www.bibliotecapleyades.net/atlantida_mu/contmu/contmu09.htm |title=The Lost Continent of Mu – Symbols, Vignettes, Tableaux and Diagrams |first=James |last=Churchward |year=1931 |access-date=2016-03-15 |work=Biblioteca Pleyades |archive-date=2015-07-18 |archive-url=https://web.archive.org/web/20150718115213/http://www.bibliotecapleyades.net/atlantida_mu/contmu/contmu09.htm |url-status=live }}</ref>

==Religion and beliefs==
{{prose|section|date=October 2023}}{{more references|section|date=October 2023}}
{{See also|Triple deity}}
[[File:Triple Goddess Symbol.svg|thumb|right|Symbol of the Triple Goddess showing the waxing, full and waning Moon]]

Many world religions contain triple deities or concepts of trinity, including the [[Hindu]] [[Trimurti]] and [[Tridevi]], the [[Triglav (mythology)|Triglav]] (lit. "Three-headed one"), the chief god of the [[Slav]]s, the [[three Jewels]] of [[Buddhism]], the [[three Pure Ones]] of [[Taoism]], the [[Christianity|Christian]] [[Trinity|Holy Trinity]], and the [[Triple Goddess (Neopaganism)|Triple Goddess]] of [[Wicca]].

According to [[Pythagoras]] and the [[Pythagoreanism|Pythagorean]] school, the number 3, which they called ''triad'', is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself.<ref>{{citation |author=Priya Hemenway |title=Divine Proportion: Phi In Art, Nature, and Science |pages=53–54 |year=2005 |publisher=Sterling Publishing Company Inc. |isbn=1-4027-3522-7}}</ref>

[[File:Shield-Trinity-Scutum-Fidei-English.svg|thumb|The [[Shield of the Trinity]] is a diagram of the Christian doctrine of the Trinity.]]

===As a lucky or unlucky number===
Three ({{lang|zh|三}}, formal writing: {{lang|zh|叁}}, [[pinyin]] ''sān'', [[Cantonese]]: ''saam''<sup>1</sup>) is considered a [[numerology|good number]] in [[Chinese culture]] because it sounds like the word "alive" ({{lang|zh|生}} pinyin ''shēng'', Cantonese: ''saang''<sup>1</sup>), compared to [[4 (number)|four]] ({{lang|zh|四}}, pinyin: ''sì'', Cantonese: ''sei''<sup>1</sup>), which sounds like the word "death" ({{lang|zh|死}} pinyin ''sǐ'', Cantonese: ''sei''<sup>2</sup>).

The phrase "[[:wikt:Third time's the charm|Third time's the charm]]" refers to the superstition that after two failures in any endeavor, a third attempt is more likely to succeed.<ref>{{Cite web |title=Definition of THE THIRD TIME IS THE CHARM |url=https://www.merriam-webster.com/dictionary/the%20third%20time%20is%20the%20charm |access-date=2024-12-08 |website=www.merriam-webster.com |language=en}}</ref> However, some superstitions say the opposite, stating that [[luck]], especially bad luck, is often said to "come in threes".<ref>See "[http://www.encyclopedia.com/doc/1O214-bad.html bad] {{Webarchive|url=https://web.archive.org/web/20090302124523/http://www.encyclopedia.com/doc/1O214-bad.html |date=2009-03-02 }}" in the ''Oxford Dictionary of Phrase and Fable'', 2006, via Encyclopedia.com.</ref>

One such superstition, called "[[Three on a match|Three on a Match]]", says that it is unlucky to be the third person to light a cigarette from the same match or lighter. This superstition is sometimes asserted to have originated among soldiers in the trenches of the First World War when a sniper might see the first light, take aim on the second and fire on the third.<ref>{{Cite news |last=King |first=Stephen |date=1984-04-12 |title=1984, A BAD YEAR IF YOU FEAR FRIDAY THE 13TH |url=https://www.nytimes.com/1984/04/12/garden/1984-a-bad-year-if-you-fear-friday-the-13th.html |access-date=2025-02-06 |work=The New York Times |language=en-US |issn=0362-4331}}</ref><ref>{{Cite news |date=1935-12-07 |title=THREE CIGARETTES. |url=https://trove.nla.gov.au/newspaper/article/17211389 |access-date=2025-02-06 |work=Sydney Morning Herald}}</ref>

==See also==
{{Portal|Mathematics}}
*[[Cube (algebra)]] – (3 [[superscript]])
*[[Thrice]]
*[[Third (disambiguation)|Third]]
*[[Triad (disambiguation)|Triad]]
*[[Trio (disambiguation)|Trio]]
*[[Rule of three (disambiguation)|Rule of three]]
* [[Open-mid central unrounded vowel|ɜ]], {{unichar|025C}} also known as [[Open-mid central unrounded vowel|Reversed epsilon]]

==References==
{{Reflist|30em}}

*Wells, D. ''[[The Penguin Dictionary of Curious and Interesting Numbers]]'' London: Penguin Group. (1987): 46&ndash;48

==External links==
{{Wiktionary|three}}
{{Commons category}}
*[https://web.archive.org/web/20071023064015/http://threes.com/ Tricyclopedic Book of Threes] by Michael Eck
*[http://www.meddean.luc.edu/lumen/MedEd/GrossAnatomy/Threes.html Threes in Human Anatomy] by John A. McNulty
*{{cite web|last=Grime|first=James|title=3 is everywhere|url=http://www.numberphile.com/videos/three.html|work=Numberphile|publisher=[[Brady Haran]]|access-date=2013-04-13|archive-url=https://web.archive.org/web/20130514230849/http://www.numberphile.com/videos/three.html|archive-date=2013-05-14|url-status=dead}}
*[https://web.archive.org/web/20140201161634/http://numdic.com/3 The Number 3]
*[http://www.positiveintegers.org/3 The Positive Integer 3]
*[http://primes.utm.edu/curios/page.php/3.html Prime curiosities: 3]

{{Integers|zero}}
{{Authority control}}
{{DEFAULTSORT:3 (Number)}}
[[Category:Integers]]
[[Category:3 (number)| ]]