Editing 4

{{Short description|Integer number 4}}
{{Hatnote|This article is about the number. For the years, see [[BC 4]] and [[4 AD]]. For other uses, see [[4 (disambiguation)]], [[IV (disambiguation)]] and [[Number Four (disambiguation)|Number 4]].}} {{distinguish|Cuatrillo}}
{{Use dmy dates|date=April 2022}} 
{{Infobox number
|number=4
|numeral=[[Quaternary numeral system|quaternary]]
|divisor=1, 2, 4
|roman={{ubl|IV ([[subtractive notation]])|IIII ([[additive notation]])}}
|greek prefix=[[wikt:tetra-|tetra-]]
|latin prefix=[[wikt:quadri-|quadri-]]/[[wikt:quadr-|quadr-]]
|lang1=[[Armenian numerals|Armenian]]|lang1 symbol=Դ|lang2=[[Eastern Arabic numerals|Arabic]], [[Central Kurdish|Kurdish]]
|lang2 symbol={{resize|150%|٤}}
|lang3=[[Persian language|Persian]], [[Sindhi language|Sindhi]]
|lang3 symbol={{resize|150%|{{lang|sd|۴}}}}
|lang4=[[Shahmukhi alphabet|Shahmukhi]], [[Urdu numerals|Urdu]]
|lang4 symbol={{resize|150%|{{lang|ur|{{nq|۴}}}}}}
|lang5=[[Ge'ez alphabet|Ge'ez]]
|lang5 symbol={{resize|150%|፬}}
|lang6=[[Bengali language|Bengali]], [[Assamese language|Assamese]]
|lang6 symbol={{resize|150%|৪}}
|lang7=[[Chinese numeral]]
|lang7 symbol=四，亖，肆
|lang9=[[Devanāgarī|Devanagari]]
|lang9 symbol={{resize|150%|४}}
|lang10=[[Santali language|Santali]]
|lang10 symbol={{resize|150%|᱔}}
|lang11=[[Telugu language|Telugu]]
|lang11 symbol={{resize|150%|౪}}
|lang12=[[Malayalam]]
|lang12 symbol={{resize|150%|൪}}
|lang13=[[Tamil language|Tamil]]
|lang13 symbol={{resize|150%|௪}}
|lang14=[[Hebrew (language)|Hebrew]]
|lang14 symbol={{resize|150%|ד}}
|lang15=[[Khmer numerals|Khmer]]
|lang15 symbol={{resize|150%|៤}}
|lang16=[[Thai numerals|Thai]]
|lang16 symbol={{resize|150%|๔}}
|lang17=[[Kannada language|Kannada]]
|lang17 symbol={{resize|150%|೪}}
|lang18=[[Burmese language|Burmese]]
|lang18 symbol={{resize|150%|၄}}
|lang19=[[Babylonian cuneiform numerals|Babylonian numeral]]|lang19 symbol=𒐘|lang20=[[Egyptian numerals|Egyptian hieroglyph]],  [[counting rods|Chinese counting rod]]|lang20 symbol={{!}}{{!}}{{!}}{{!}}|lang21=[[Maya numerals]]|lang21 symbol=••••|lang22=[[Morse code]]|lang22 symbol={{resize|150%|.... _}}}}
'''4''' ('''four''') is a [[number]], [[numeral (linguistics)|numeral]] and [[numerical digit|digit]]. It is the [[natural number]] following [[3]] and preceding [[5]]. It is a [[square number]], the smallest [[semiprime]] and [[composite number]], and is [[tetraphobia|considered unlucky]] in many East Asian cultures.

==Evolution of the Hindu-Arabic digit==

{{More citations needed section|date=May 2024}}

[[File:Evolution4glyph.png|x50px|left]]
[[File:Vier.jpg|thumb|120px|left|Two modern handwritten fours]]
[[File:Algund Dominikanerinnenklosterkirche Portal bezeichnet 1481 (cropped).jpg|right|thumb|Sculpted date "1481" in the Convent church of Maria Steinach in [[Algund]], [[South Tirol]], [[Italy]]. The upward loop signifies the number 4.]]
[[Brahmic numerals]] represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The [[Shunga Empire|Shunga]] would add a horizontal line on top of the digit, and the [[Northern Satraps|Kshatrapa]] and [[Pallava dynasty|Pallava]] evolved the digit to a point where the speed of writing was a secondary concern. The [[Arab]]s' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross.<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): 394, Fig. 24.64</ref>

While the shape of the character for the digit 4 has an [[ascender (typography)|ascender]] in most modern [[typeface]]s, in typefaces with [[text figures]] the glyph usually has a [[descender]], as, for example, in [[File:TextFigs148.svg|40px]].

On the [[seven-segment display]]s of pocket calculators and digital watches, as well as certain [[optical character recognition]] fonts, 4 is seen with an open top: [[File:Seven-segment 4.svg|x25px]].<ref>{{Cite web|date=2019-04-22|title=Seven Segment Displays (7-Segment) {{!}} Pinout, Types and Applications|url=https://www.electronicshub.org/seven-segment-displays/|access-date=2020-07-28|website=Electronics Hub|language=en-US|archive-date=28 July 2020|archive-url=https://web.archive.org/web/20200728014108/https://www.electronicshub.org/seven-segment-displays/|url-status=live}}</ref>

[[Television station]]s that operate on [[channel 4 (disambiguation)|channel 4]] have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the [[Canadian Aboriginal syllabics]] letter ᔦ. The [[magnetic ink character recognition]] "CMC-7" font also uses this variety of "4".<ref>{{Cite news|date=2017-02-02|title=Battle of the MICR Fonts: Which Is Better, E13B or CMC7? - Digital Check|language=en-US|work=Digital Check|url=https://www.digitalcheck.com/battle-micr-fonts-better-e13b-cmc7/|access-date=2020-07-28|archive-date=3 August 2020|archive-url=https://web.archive.org/web/20200803161254/https://www.digitalcheck.com/battle-micr-fonts-better-e13b-cmc7/|url-status=live}}</ref>

==Mathematics ==
There are four elementary arithmetic [[Operation (mathematics)|operations]] in mathematics: [[addition]] ('''+'''), [[subtraction]] ('''−'''), [[multiplication]] ('''×'''), and [[Division (mathematics)|division]] ('''÷''').<ref>{{Cite web |last=Tiwari |first=Arvind Kumar |date=2023 |title=What are the four basic mathematical operations, and what do they mean? |url=https://www.quora.com/What-are-the-four-basic-mathematical-operations-and-what-do-they-mean |access-date=30 September 2024 |website=Quora}}</ref>

[[Lagrange's four-square theorem]] states that every positive integer can be written as the sum of at most four [[square number|square]]s.<ref>{{Citation|last=Spencer|first=Joel|title=Four Squares with Few Squares|year=1996|work=Number Theory: New York Seminar 1991–1995|pages=295–297|editor-last=Chudnovsky|editor-first=David V.|place=New York, NY|publisher=Springer US| language=en|doi=10.1007/978-1-4612-2418-1_22|isbn=978-1-4612-2418-1|editor2-last=Chudnovsky|editor2-first=Gregory V.|editor3-last=Nathanson|editor3-first=Melvyn B.}}</ref><ref>{{Cite book|last=Peterson|first=Ivars|url=https://books.google.com/books?id=4gWSAraVhtAC&q=7+for+instance+cannot+be+written+as+the+sum+of+three+squares.&pg=PA95|title=Mathematical Treks: From Surreal Numbers to Magic Circles|date=2002|publisher=MAA|isbn=978-0-88385-537-9|pages=95|language=en|quote=7 is an example of an integer that can't be written as the sum of three squares.}}</ref> Four is one of four [[Harshad number|all-Harshad number]]s. Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. <math>4x=y^{2}-z^{2}</math>.

A four-sided plane figure is a [[quadrilateral]] or quadrangle, sometimes also called a ''tetragon''. It can be further classified as a [[rectangle]] or ''oblong'', [[kite]], [[rhombus]], and [[square]].

Four is the highest degree general [[polynomial equation]] for which there is a [[solution in radicals]].<ref>{{Cite book| last=Bajnok|first=Béla|url=https://books.google.com/books?id=cNFzKnvxXoAC&q=Abel%E2%80%93Ruffini+theorem&pg=PT78|title=An Invitation to Abstract Mathematics|date=2013-05-13|publisher=Springer Science & Business Media| isbn=978-1-4614-6636-9| language=en|quote=There is no algebraic formula for the roots of the general polynomial of degrees 5 or higher.}}</ref>

Four is the only square number <math>I=i\times i</math> where <math>I - 1</math> is a prime number.

The [[four-color theorem]] states that a [[planar graph]] (or, equivalently, a flat [[map]] of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors.<ref>{{cite book |first=Bryan |last=Bunch |title=The Kingdom of Infinite Number |location=New York |publisher=W. H. Freeman & Company |year=2000 |page=48}}</ref> Three colors are not, in general, sufficient to guarantee this.<ref>{{Cite book|last=Ben-Menahem|first=Ari|url=https://books.google.com/books?id=9tUrarQYhKMC&q=three+colors+map+not+enough&pg=PA2147|title=Historical Encyclopedia of Natural and Mathematical Sciences|date=2009-03-06|publisher=Springer Science & Business Media| isbn=978-3-540-68831-0|pages=2147|language=en|quote=(i.e. That there are maps for which three colors are not sufficient)}}</ref> The largest planar [[complete graph]] has four vertices.<ref>{{Cite book|last=Molitierno|first=Jason J.| url=https://books.google.com/books?id=2kvNBQAAQBAJ&q=largest+planar+complete+graph+has+four+vertices&pg=PA197| title=Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs|date=2016-04-19|publisher=CRC Press| isbn=978-1-4398-6339-8|pages=197|language=en|quote=... The complete graph on the largest number of vertices that is planar is K4 and that a(K4) equals 4.}}</ref>

A solid figure with four faces as well as four vertices is a [[tetrahedron]], which is the smallest possible number of faces and vertices a [[polyhedron]] can have.<ref>{{Cite book|last1=Grossnickle|first1=Foster Earl|url=https://books.google.com/books?id=Q2474oSAsc4C&q=4+is+the+smallest+possible+number+of+faces+(as+well+as+vertices)+of+a+polyhedron.|title=Discovering Meanings in Elementary School Mathematics|last2=Reckzeh|first2=John|date=1968|publisher=Holt, Rinehart and Winston|pages=337|isbn=9780030676451|language=en|quote=...the smallest possible number of faces that a polyhedron may have is four}}</ref> The regular tetrahedron, also called a 3-[[simplex]], is the simplest [[Platonic solid]].<ref>{{Cite book|last1=Grossnickle|first1=Foster Earl|url=https://books.google.com/books?id=Q2474oSAsc4C&q=4+is+the+smallest+possible+number+of+faces+(as+well+as+vertices)+of+a+polyhedron.|title=Discovering Meanings in Elementary School Mathematics|last2=Reckzeh|first2=John|date=1968|publisher=Holt, Rinehart and Winston|pages=337|isbn=9780030676451|language=en|quote=...face of the platonic solid. The simplest of these shapes is the tetrahedron...}}</ref> It has four [[regular triangle]]s as faces that are themselves at [[self-dual polytope|dual positions]] with the vertices of another tetrahedron.<ref>{{Cite book|last1=Hilbert|first1=David|url=https://books.google.com/books?id=7WY5AAAAQBAJ&q=self-dual+regular+polyhedron&pg=PA143|title=Geometry and the Imagination|last2=Cohn-Vossen |first2=Stephan |date=1999|publisher=American Mathematical Soc.|isbn=978-0-8218-1998-2|pages=143|language=en|quote=...the tetrahedron plays an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs...}}</ref>

The smallest non-[[cyclic group]] has four elements; it is the [[Klein four-group]].<ref>{{Cite book|first=Jeremy|last=Horne|url=https://books.google.com/books?id=ZfYoDwAAQBAJ&pg=PA299|title=Philosophical Perceptions on Logic and Order|date=2017-05-19|publisher=IGI Global|isbn=978-1-5225-2444-1|pages=299|language=en|quote=The Klein four-group is the smallest noncyclic group,...|access-date=31 October 2022|archive-date=31 October 2022|archive-url=https://web.archive.org/web/20221031005437/https://books.google.com/books?id=ZfYoDwAAQBAJ&pg=PA299|url-status=live}}</ref> ''A{{sub|n}}'' [[alternating group]]s are not [[simple group|simple]] for values <math>n</math> ≤ <math>4</math>.

There are four [[Hopf fibration]]s of [[hypersphere]]s:

<math display=block>
\begin{align}
S^0 & \hookrightarrow S^1 \to S^1, \\
S^1 & \hookrightarrow S^3 \to S^2, \\
S^3 & \hookrightarrow S^7 \to S^4, \\
S^7 & \hookrightarrow S^{15}\to S^8. \\
\end{align}</math>

They are defined as locally trivial [[fibration]]s that map <math>f : S^{2n-1} \rightarrow S^{n}</math> for values of <math>n=2,4,8</math> (aside from the trivial fibration mapping between two [[Point (geometry)|points]] and a [[circle]]).<ref>{{Cite book |last=Shokurov |first= A.V. |editor=Michiel Hazewinkel |editor-link=Michiel Hazewinkel |chapter=Hopf fibration |title=Encyclopedia of Mathematics |publisher=[[European Mathematical Society]] |location=Helsinki |chapter-url=https://encyclopediaofmath.org/wiki/Hopf_fibration |year=2002 |isbn=1402006098 |oclc=1013220521 |access-date=2023-04-30 |archive-date=1 May 2023 |archive-url=https://web.archive.org/web/20230501005558/https://encyclopediaofmath.org/wiki/Hopf_fibration |url-status=live }}</ref>

In [[Knuth's up-arrow notation]], <math>2+2=2\times2=2^{2}=2\uparrow\uparrow 2=2\uparrow\uparrow\uparrow2=\;...\; = 4</math>, and so forth, for any number of up arrows.<ref>{{Cite book |last=Hodges |first=Andrew |url=https://books.google.com/books?id=HOcpgfiDu40C&q=2+%E2%86%91%E2%86%91+2&pg=PA249 |title=One to Nine: The Inner Life of Numbers |date=2008-05-17 |publisher=W. W. Norton & Company |isbn=978-0-393-06863-4 |pages=249 |language=en |quote=2 ↑↑ ...  ↑↑ 2 is always 4}}</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
|-
|'''4 × ''x'''''
|'''4'''
|{{num|8}}
|{{num|12}}
|{{num|16}}
|{{num|20}}
|{{num|24}}
|{{num|28}}
|{{num|32}}
|{{num|36}}
|{{num|40}}
|{{num|44}}
|{{num|48}}
|{{num|52}}
|{{num|56}}
|{{num|60}}
|{{num|64}}
|{{num|68}}
|{{num|72}}
|{{num|76}}
|{{num|80}}
|{{num|84}}
|{{num|88}}
|{{num|92}}
|{{num|96}}
|{{num|100}}
|{{num|200}}
|{{num|400}}
|{{num|4000}}
|}

{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Division (mathematics)|Division]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
|-
|'''4 ÷ ''x'''''
|'''4'''
|[[2]]
|1.{{overline|3}}
|1
|0.8
|0.{{overline|6}}
|0.{{overline|571428}}
|0.5
|0.{{overline|4}}
|0.4
|0.{{overline|36}}
|0.{{overline|3}}
|0.{{overline|307692}}
|0.{{overline|285714}}
|0.2{{overline|6}}
|0.25
|-
|'''''x'' ÷ 4'''
|0.25
|0.5
|0.75
|1
|1.25
|1.5
|1.75
|2
|2.25
|2.5
|2.75
|3
|3.25
|3.5
|3.75
|4
|}

{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
|-
|'''4{{sup|''x''}}'''
|4
|{{num|16}}
|{{num|64}}
|[[256]]
|1024
|4096
|16384
|65536
|262144
|1048576
|4194304
|16777216
|67108864
|268435456
|1073741824
|4294967296
|-
|'''''x''{{sup|4}}'''
|1
|{{num|16}}
|81
|256
|625
|1296
|2401
|4096
|6561
|10000
|14641
|20736
|28561
|38416
|50625
|65536
|}

==In culture==
*Four is the sacred number of the [[Zia (New Mexico)|Zia]], an indigenous tribe located in the U.S. state of [[New Mexico]].<ref>{{Cite book|url=https://books.google.com/books?id=GZpLAQAAMAAJ&q=Four+is+the+sacred+number+of+the+Zia|title=Bulletin - State Department of Education|date=1955|publisher=Department of Education|pages=151|language=en|quote=Four was a sacred number of Zia}}</ref>
*The Chinese, the Koreans, and the Japanese are [[tetraphobia|superstitious about the number four]] because it is a [[homonym]] for "death" in their languages.<ref>{{Cite book|last=Lachenmeyer|first=Nathaniel|url=https://books.google.com/books?id=sDXJ1s0YNAgC&q=+homonym+for+%22death%22|title=13: The Story of the World's Most Notorious Superstition|date=2005|publisher=Penguin Group (USA) Incorporated|isbn=978-0-452-28496-8|pages=187|language=en|quote=In Chinese, Japanese, and Korean, the word for four is, unfortunately, an exact homonym for death}}</ref>

==In logic and philosophy==
[[File:Mugs of tea viewed from above.jpg|thumb|Four mugs]]
*The symbolic meanings of the number four are linked to those of the cross and the square. "Almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. Its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself". Where lines of latitude and longitude intersect, they divide the earth into four proportions. Throughout the world kings and chieftains have been called "lord of the four suns" or "lord of the four quarters of the earth",<ref>Chevalier, Jean and Gheerbrant, Alain (1994), ''The Dictionary of Symbols''. The quote beginning "Almost from prehistoric times..." is on p. 402.</ref> which is understood to refer to the extent of their powers both territorially and in terms of total control of their subjects' doings. 
*The [[Square of Opposition]], in both its Aristotelian version and its [[Square of Opposition#Modern squares of opposition|Boolean version]], consists of four forms: A ("All ''S'' is ''R''"), I ("Some ''S'' is ''R''"), E ("No ''S'' is ''R''"), and O ("Some ''S'' is not ''R''").

==In technology==
*In [[internet slang]], "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you".
*In [[Leet]]speak, "4" may be used to replace the letter "A".

==Other groups of four==
*Approximately four weeks (4 times 7 days) to a lunar month ([[synodic month]] = 29.54 days). Thus the number four is universally an integral part of primitive sacred calendars.

==References==
{{Reflist}}
*Wells, D. ''[[The Penguin Dictionary of Curious and Interesting Numbers]]'' London: Penguin Group. (1987): 55–58

==External links==
{{Wiktionary|four}}
{{Commons category}}
*[http://www.marijn.org/everything-is-4 Marijn.Org on Why is everything four?]
*[https://web.archive.org/web/20180303094710/http://www.samuel-beckett.net/Penelope/four_symbolism.html A few thoughts on the number four], by Penelope Merritt at samuel-beckett.net
*[http://www.numdic.com/4 The Number 4]
*[http://www.positiveintegers.org/4 The Positive Integer 4]
*[http://primes.utm.edu/curios/page.php/4.html Prime curiosities: 4]

{{Integers|zero}}
{{Authority control}}

[[Category:Integers]]
[[Category:4 (number)| ]]