View source for 5

{{Short description|Integer number 5}}
{{Hatnote|This article is about the number. For the years, see [[5 BC]] and [[0s|AD 5]]. For other uses, see [[5 (disambiguation)]], [[Number Five (disambiguation)]], and [[The Five (disambiguation)]].}}
{{infobox number
|number=5
|numeral=[[quinary]]
|prime=3rd
|divisor=1, 5
|roman =V, v
|greek prefix=[[Wiktionary:penta-|penta-]]/[[Wiktionary:pent-|pent-]]
|latin prefix=[[Wiktionary:quinque-|quinque-]]/[[Wiktionary:quinqu-|quinqu-]]/[[Wiktionary:quint-|quint-]]
|lang1=[[Greek numeral|Greek]]
|lang1 symbol=ε (or Ε)
|lang2=[[Eastern Arabic numerals|Arabic]], [[Central Kurdish|Kurdish]]
|lang2 symbol={{resize|150%|٥}}
|lang3=[[Persian language|Persian]], [[Sindhi language|Sindhi]], [[Urdu numerals|Urdu]]
|lang3 symbol={{resize|150%|۵}}
|lang4=[[Ge'ez script|Ge'ez]]
|lang4 symbol=፭
|lang5=[[Bengali language|Bengali]]
|lang5 symbol={{resize|150%|৫}}
|lang6=[[Kannada language|Kannada]]
|lang6 symbol={{resize|150%|೫}}
|lang7=[[Punjabi language|Punjabi]]
|lang7 symbol={{resize|150%|੫}}
|lang8=[[Chinese numeral]]
|lang8 symbol=五
|lang9=[[Armenian numerals|Armenian]]|lang9 symbol=Ե|lang10=[[Devanāgarī]]
|lang10 symbol={{resize|150%|५}}
|lang11=[[Hebrew language|Hebrew]]
|lang11 symbol={{resize|150%|ה}}
|lang12=[[Khmer numerals|Khmer]]
|lang12 symbol=៥
|lang13=[[Indian numerals#Telugu numerals and their names|Telugu]]
|lang13 symbol={{resize|150%|౫}}
|lang14=[[Indian numerals#Malayalam numerals and their names|Malayalam]]
|lang14 symbol={{resize|150%|൫}}
|lang15=[[Indian numerals#Tamil numerals and their names|Tamil]]
|lang15 symbol={{resize|150%|௫}}
|lang16=[[Thai numerals|Thai]]
|lang16 symbol=๕|lang17=[[Babylonian cuneiform numerals|Babylonian numeral]]|lang17 symbol=𒐙|lang18=[[Egyptian numerals|Egyptian hieroglyph]],  [[counting rods|Chinese counting rod]]|lang18 symbol={{!}}{{!}}{{!}}{{!}}{{!}}|lang19=[[Maya numerals]]|lang19 symbol=𝋥|lang20=[[Morse code]]|lang20 symbol={{resize|150%|.....}}
|lang21=[[ASCII|ASCII value]]
|lang21 symbol=[[Enquiry character|ENQ]]
|cardinal=five|ordinal=5th
(fifth)}} 

'''5''' ('''five''') is a [[number]], [[numeral (linguistics)|numeral]] and [[numerical digit|digit]]. It is the [[natural number]], and [[cardinal number]], following [[4]] and preceding [[6]], and is a [[prime number]]. 

Humans, and many other animals, have 5 [[Digit (anatomy)|digits]] on their [[Limb (anatomy)|limbs]].

== Mathematics ==
[[File:Pythagoras' Special Triples.svg|right|upright=0.65|thumb|The first [[Pythagorean triple]] ]]

5 is a [[Fermat prime]], a [[Mersenne prime]] exponent, as well as a [[Fibonacci number]]. 5 is the first [[congruent number]], as well as the length of the [[hypotenuse]] of the smallest integer-sided [[right triangle]], making part of the smallest [[Pythagorean triple]] ([[3]], [[4]], 5).<ref>{{Cite OEIS|1=A003273 |2=Congruent numbers|access-date=2016-06-01}}</ref> 

5 is the first [[safe prime]]<ref>{{Cite OEIS|A005385|Safe primes p: (p-1)/2 is also prime|access-date=2023-02-14}}</ref> and the first [[good prime]].<ref>{{Cite OEIS|A028388|Good primes|access-date=2016-06-01}}</ref> 11 forms the first pair of [[sexy prime]]s with 5.<ref>{{Cite OEIS|A023201|Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)|access-date=2023-01-14}}</ref> 5 is the second [[Fermat number|Fermat prime]], of a total of five known Fermat primes.<ref>{{Cite OEIS|A019434|Fermat primes|access-date=2022-07-21}}</ref> 5 is also the first of three known [[Wilson prime]]s (5, 13, 563).<ref name="WilsP">{{Cite OEIS|A007540|Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).|access-date=2023-09-06}}</ref> 

=== Geometry ===
A shape with five sides is called a [[pentagon]]. The pentagon is the first [[regular polygon]] that does not [[Tessellation|tile]] the [[Plane (geometry)|plane]] with copies of itself. It is the largest [[Face (geometry)|face]] any of the five regular three-dimensional regular [[Platonic solid]] can have. 

A [[conic]] is determined using [[Five points determine a conic|five points in the same way]] that two points are needed to determine a [[Line (geometry)|line]].<ref name="conic">{{Cite journal |first=A. C. |last=Dixon |author-link=Alfred Cardew Dixon |title=The Conic through Five Given Points |journal=The Mathematical Gazette |volume=4 |number=70 |date=March 1908 |pages=228–230 |publisher=The Mathematical Association |jstor=3605147 |doi=10.2307/3605147 |s2cid=125356690 |url=https://zenodo.org/record/2014634 }}</ref> A [[pentagram]], or five-pointed [[Polygram (geometry)|polygram]], is a [[star polygon]] constructed by connecting some non-adjacent of a [[regular pentagon]] as [[Star polygon#Regular star polygon|self-intersecting edges]].<ref>{{Cite OEIS|A307681|Difference between the number of sides and the number of diagonals of a convex n-gon.}}</ref> The internal geometry of the pentagon and pentagram (represented by its [[Schläfli symbol]] {{math|1={5/2<nowiki>}</nowiki> }}) appears prominently in [[Penrose tilings]]. Pentagrams are [[Facet (geometry)|facet]]s inside [[Kepler–Poinsot polyhedron|Kepler–Poinsot star polyhedra]] and [[Regular 4-polytope#Regular star (Schläfli–Hess) 4-polytopes|Schläfli–Hess star polychora]].

There are five regular [[Platonic solid]]s the [[tetrahedron]], the [[cube]], the [[octahedron]], the [[dodecahedron]], and the [[icosahedron]].<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 61</ref>

The [[Hadwiger–Nelson problem|chromatic number]] of the [[Plane (geometry)|plane]] is the minimum number of [[Graph coloring|colors]] required to color the plane such that no pair of points at a distance of 1 has the same color.<ref>{{Cite journal |last=de Grey |first=Aubrey D.N.J. |author-link=Aubrey de Grey |year=2018 |title=The Chromatic Number of the Plane is At Least 5 |journal=[[Geombinatorics]] |volume=28 |pages=5–18 |arxiv=1804.02385 |mr=3820926 |s2cid=119273214}}</ref> Five is a lower depending for the chromatic number of the plane, but this may depend on the choice of [[Axiom of choice|set-theoretical axioms]]:<ref>{{Cite journal |last1=Exoo |first1=Geoffrey |last2=Ismailescu |first2=Dan |year=2020 |title=The Chromatic Number of the Plane is At Least 5: A New Proof |journal=[[Discrete & Computational Geometry]] |location=New York, NY |publisher=[[Springer Science+Business Media|Springer]] |volume=64 |pages=216–226 |arxiv=1805.00157 |doi=10.1007/s00454-019-00058-1 |mr=4110534 |s2cid=119266055 |zbl=1445.05040}}</ref>

The plane contains a total of five [[Bravais lattice]]s, or arrays of [[Point (geometry)|points]] defined by discrete [[Translation (geometry)|translation]] operations. [[Euclidean tilings by convex regular polygons|Uniform tilings]] of the plane, are generated from combinations of only five regular polygons.<ref>{{Cite journal |last1=Grünbaum |first1=Branko |author-link=Branko Grünbaum |last2=Shepard |first2=Geoffrey |author-link2=G.C. Shephard |date=November 1977 |title=Tilings by Regular Polygons |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=[[Mathematics Magazine]] |publisher=Taylor & Francis, Ltd. |volume=50 |issue=5 |pages=227–236 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 |archive-date=2016-03-03 |access-date=2023-01-18 |archive-url=https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |url-status=dead }}</ref>

=== Higher dimensional geometry ===
A [[5-cell|hypertetrahedron]], or 5-cell, is the 4 dimensional analogue of the [[tetrahedron]]. It has five vertices. Its orthographic projection is [[Homomorphism|homomorphic]] to the group ''K''<sub>5.</sub><ref name="RegPoly3">{{Cite book |author=H. S. M. Coxeter |author-link=Harold Scott MacDonald Coxeter |title=[[Regular Polytopes (book)|Regular Polytopes]] |publisher=[[Dover Publications, Inc.]] |year=1973 |isbn=978-0-486-61480-9 |edition=3rd |location=New York |pages=1–368}}</ref>{{rp|p.120}}

There are five fundamental [[Uniform 4-polytope#Convex uniform 4-polytopes|mirror symmetry point group families in 4-dimensions]]. There are also 5 [[Coxeter-Dynkin diagram#Compact|compact hyperbolic Coxeter groups]], or [[Uniform 4-polytope#Prismatic uniform 4-polytopes|4-prisms]], of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.<ref>{{Cite book |last1=McMullen |first1=Peter |author1-link=Peter McMullen |url=https://archive.org/details/abstractregularp0000mcmu |title=Abstract Regular Polytopes |last2=Schulte |first2=Egon |author2-link=Egon Schulte |publisher=Cambridge University Press |year=2002 |isbn=0-521-81496-0 |series=Encyclopedia of Mathematics and its Applications |volume=92 |location=Cambridge |pages=162–164 |doi=10.1017/CBO9780511546686 |mr=1965665 |url-access=registration |s2cid=115688843}}</ref>

[[File:Schlegel wireframe 5-cell.png|upright|left|thumb|The four-dimensional [[5-cell]] is the simplest regular [[polychoron]].]]

=== Algebra ===
[[File:Magic Square Lo Shu.svg|upright=0.55|right|thumb|The smallest non-trivial [[magic square]]]]5 is the value of the central [[Magic square#Properties of magic squares|cell]] of the first non-trivial [[magic square|normal magic square]], called the [[Luoshu Square|''Luoshu'' square]]. All [[integer]]s <math>n \ge 34</math> can be expressed as the sum of five non-zero [[Square number|squares]].<ref>{{Cite book |last1=Niven |first1=Ivan |author1-link=Ivan M. Niven |title=An Introduction to the Theory of Numbers |last2=Zuckerman |first2=Herbert S. |last3=Montgomery |first3=Hugh L. |author3-link=Hugh Lowell Montgomery |publisher=[[Wiley (publisher)|John Wiley]] |year=1980 |isbn=978-0-19-853171-5 |edition=5th |location=New York, NY |pages=144, 145}}</ref><ref>{{Cite OEIS|A047701|All positive numbers that are not the sum of 5 nonzero squares.|access-date=2023-09-20}}

:Only twelve integers up to [[33 (number)|33]] cannot be expressed as the sum of five non-zero squares: {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33} where 2, 3 and 7 are the only such primes without an expression.</ref> There are five countably infinite [[Ramsey class]]es of [[permutation]]s.<ref name="ramsey">{{Cite journal |last1=Böttcher |first1=Julia |author1-link=Julia Böttcher |last2=Foniok |first2=Jan |year=2013 |title=Ramsey Properties of Permutations |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p2 |journal=The Electronic Journal of Combinatorics |volume=20 |issue=1 |page=P2 |arxiv=1103.5686v2 |doi=10.37236/2978 |s2cid=17184541 |zbl=1267.05284}}</ref>{{rp|p.4}}

5 is [[conjecture]]d to be the only [[Parity (mathematics)|odd]], [[untouchable number]]; if this is the case, then five will be the only odd prime number that is not the base of an [[Aliquot sequence|aliquot]] tree.<ref>{{Cite journal |last1=Pomerance |first1=Carl |last2=Yang |first2=Hee-Sung |date=14 June 2012 |title=On Untouchable Numbers and Related Problems |url=https://math.dartmouth.edu/~carlp/uupaper3.pdf |publisher=[[Dartmouth College]] |page=1 |s2cid=30344483 |website=math.dartmouth.edu}} 2010 Mathematics Subject Classification. 11A25, 11Y70, 11Y16.</ref>[[File:SporadicGroups.png|thumb|300px|This diagram shows the [[subquotient]] relations of the twenty-six '''sporadic groups'''; the five [[Mathieu group]]s form the simplest class (colored red [[File:EllipseSubqR.svg]]). ]]Every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; [[Harald Andres Helfgott|Helfgott]] has provided a proof of this<ref>{{Cite book |last=Helfgott |first=Harald Andres |title=Seoul [[International Congress of Mathematicians]] Proceedings |date=2014 |publisher=Kyung Moon SA |isbn=978-89-6105-805-6 |editor-last=Jang |editor-first=Sun Young |volume=2 |location=Seoul, KOR |pages=391–418 |chapter=The ternary Goldbach problem |oclc=913564239 |chapter-url=https://www.imj-prg.fr/wp-content/uploads/2020/prix/helfgott2014.pdf}}</ref> (also known as the [[Goldbach's weak conjecture|odd Goldbach conjecture]]) that is already widely acknowledged by mathematicians as it still undergoes [[peer-review]]. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).<ref>{{Cite journal |last=Tao |first=Terence |date=March 2014 |title=Every odd number greater than 1 has a representation is the sum of at most five primes |url=https://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02733-0/S0025-5718-2013-02733-0.pdf |journal=Mathematics of Computation |volume=83 |pages=997–1038 |doi=10.1090/S0025-5718-2013-02733-0 |mr=3143702 |s2cid=2618958 |number=286}}</ref>{{Unsolved|mathematics|Is 5 the only odd, untouchable number?}}
=== Group theory ===
In [[graph theory]], all [[Graph theory|graphs]] with four or fewer vertices are [[Planar graph|planar]], however, there is a graph with five vertices that is not: ''K''<sub>5</sub>, the [[complete graph]] with five vertices. By [[Kuratowski's theorem]], a finite graph is planar [[iff|if and only if]] it does not contain a subgraph that is a subdivision of , or ''K''<sub>3,3</sub>, the [[utility graph]].<ref>{{Cite journal |last=Burnstein |first=Michael|title=Kuratowski-Pontrjagin theorem on planar graphs |journal=[[Journal of Combinatorial Theory]] | series=Series B |volume=24 |issue=2 |year=1978 |pages=228–232 |doi= 10.1016/0095-8956(78)90024-2 |doi-access=free }}</ref>

There are five complex [[exceptional Lie algebra]]s. The five [[Mathieu groups]] constitute the [[Sporadic group#First generation (5 groups): the Mathieu groups|first generation]] in the [[Sporadic groups#Happy Family|happy family]] of [[sporadic groups]]. These are also the first five sporadic groups [[Classification of finite simple groups#Timeline of the proof|to have been described]].<ref name="griess">{{Cite book |author=Robert L. Griess, Jr. |author-link=Robert Griess |title=Twelve Sporadic Groups |publisher=Springer-Verlag |year=1998 |isbn=978-3-540-62778-4 |series=Springer Monographs in Mathematics |location=Berlin |pages=1−169 |doi=10.1007/978-3-662-03516-0 |mr=1707296 |zbl=0908.20007 |s2cid=116914446}}</ref>{{rp|p.54}} A [[Centralizer and normalizer|centralizer]] of an element of order 5 inside the [[Monster group|largest sporadic group]] <math>\mathrm {F_1}</math> arises from the product between [[Harada–Norton group|Harada–Norton]] sporadic group <math>\mathrm{HN}</math> and a group of order 5.<ref>{{Cite journal |last1=Lux |first1=Klaus |last2=Noeske |first2=Felix |last3=Ryba |first3=Alexander J. E. |year=2008 |title=The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2 |journal=[[Journal of Algebra]] |location=Amsterdam |publisher=[[Elsevier]] |volume=319 |issue=1 |pages=320–335 |doi=10.1016/j.jalgebra.2007.03.046 |mr=2378074 |s2cid=120706746 |zbl=1135.20007 |doi-access=free}}</ref><ref>{{Cite journal |last=Wilson |first=Robert A. |author-link=Robert Arnott Wilson |year=2009 |title=The odd local subgroups of the Monster |journal=Journal of Australian Mathematical Society (Series A) |location=Cambridge |publisher=[[Cambridge University Press]] |volume=44 |issue=1 |pages=12–13 |doi=10.1017/S1446788700031323 |mr=914399 |s2cid=123184319 |zbl=0636.20014 |doi-access=free}}</ref>

== List of basic calculations ==
{| class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Multiplication]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
|-
|'''5 × ''x'''''
|'''5'''
|[[10 (number)|10]]
|[[15 (number)|15]]
|[[20 (number)|20]]
|[[25 (number)|25]]
|[[30 (number)|30]]
|[[35 (number)|35]]
|[[40 (number)|40]]
|[[45 (number)|45]]
|[[50 (number)|50]]
|[[55 (number)|55]]
|[[60 (number)|60]]
|[[65 (number)|65]]
|[[70 (number)|70]]
|[[75 (number)|75]]
|[[80 (number)|80]]
|[[85 (number)|85]]
|[[90 (number)|90]]
|[[95 (number)|95]]
|[[100 (number)|100]]
|}

{| class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Division (mathematics)|Division]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
|-
|'''5 ÷ ''x'''''
|'''5'''
|2.5
|1.{{overline|6}}
|1.25
|rowspan="2"|[[1]]
|0.8{{overline|3}}
|0.{{overline|714285}}
|0.625
|0.{{overline|5}}
|0.5
|0.{{overline|45}}
|0.41{{overline|6}}
|0.{{overline|384615}}
|0.3{{overline|571428}}
|0.{{overline|3}}
|-
|'''''x'' ÷ 5'''
|0.2
|0.4
|0.6
|0.8
|1.2
|1.4
|1.6
|1.8
|[[2]]
|2.2
|2.4
|2.6
|2.8
|[[3]]
|}

{| class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
|-
|'''5{{sup|''x''}}'''
|'''5'''
|[[25 (number)|25]]
|[[125 (number)|125]]
|[[625 (number)|625]]
|rowspan="2"|3125
|15625
|78125
|390625
|1953125
|9765625
|48828125
|244140625
|1220703125
|6103515625
|30517578125
|-
|'''''x''{{sup|5}}'''
|[[1]]
|[[32 (number)|32]]
|[[243 (number)|243]]
|[[1024 (number)|1024]]
|7776
|16807
|32768
|59049
|100000
|161051
|248832
|371293
|537824
|759375
|}

== Evolution of the Arabic digit ==

[[File:Evolution5glyph.png|x45px|left]]

The evolution of the modern Western digit for the numeral for five is traced back to the [[Brahmi numerals|Indian system]] of numerals, where on some earlier versions, the numeral bore resemblance to variations of the number four, rather than "5" (as it is represented today). The [[Kushan Empire|Kushana]] and [[Gupta Empire|Gupta]] empires in what is now [[India]] had among themselves several forms that bear no resemblance to the modern digit. Later on, Arabic traditions transformed the digit in several ways, producing forms that were still similar to the numeral for four, with similarities to the numeral for three; yet, still unlike the modern five.<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. 24.65</ref> It was from those digits that Europeans finally came up with the modern 5 (represented in writings by Dürer, for example). 

While the shape of the character for the digit 5 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:Text figures 256.svg|45px]].

[[File:Seven-segment 5.svg|x50px|left]]
On the [[seven-segment display]] of a calculator and digital clock, it is often represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice versa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number. This makes it often indistinguishable from the letter S. Higher segment displays may sometimes may make use of a diagonal for one of the two.

== Other fields ==

=== Religion ===

==== Islam ====
The [[Five Pillars of Islam]].<ref>{{Cite web|title=PBS – Islam: Empire of Faith – Faith – Five Pillars|url=https://www.pbs.org/empires/islam/faithpillars.html|access-date=2020-08-03|website=www.pbs.org}}</ref> The [[Five-pointed star|five-pointed]] [[Simple polygon|simple]] star ☆ is one of the five used in Islamic [[Girih tiles]].<ref>{{Cite journal |last=Sarhangi |first=Reza |year=2012 |title=Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs |url=https://link.springer.com/content/pdf/10.1007/s00004-012-0117-5.pdf |journal=Nexus Network Journal |volume=14 |issue=2 |page=350 |doi=10.1007/s00004-012-0117-5 |s2cid=124558613 |doi-access=free}}</ref>

=== Mysticism ===
==== Gnosticism ====
The number five was an important symbolic number in [[Manichaeism]], with heavenly beings, concepts, and others often grouped in sets of five.{{Citation needed|date=November 2024}}

The [[pentagram]], or five-pointed star, bears mystic significance in various belief systems including [[Baháʼí Faith|Baháʼí]], [[Christianity]], [[Freemasonry]], [[Satanism]], [[Taoism]], [[Thelema]], and [[Wicca]].

=== Miscellaneous ===
[[File:5 playing cards.jpg|thumb|The fives of all four suits in [[playing card]]s]]
*"Give me five" is a common phrase used preceding a [[high five]].

== See also ==
{{Portal|Mathematics}}

* [[5 (disambiguation)]]
{{clear}}

== References ==
{{Reflist}}

=== Further reading ===
*{{Cite book |last=Wells |first=D. |title=The Penguin Dictionary of Curious and Interesting Numbers |title-link=The Penguin Dictionary of Curious and Interesting Numbers |location=London, UK |publisher=[[Penguin Group]] |date=1987 |pages=58–67}}

== External links ==
*[http://primes.utm.edu/curios/page.php/5.html Prime curiosities: 5]
*{{Commons category-inline}}

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

{{DEFAULTSORT:5 (Number)}}
[[Category:Integers]]
[[Category:5 (number)]]