7

This article is about the number. For the year, seeAD 7.For other uses, see7 (disambiguation)andNo. 7 (disambiguation).

Not to be confused with⁊.

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Cardinal	seven
Ordinal	7th (seventh)
Numeral system	septenary
Factorization	prime
Prime	4th
Divisors	1, 7
Greek numeral	Ζ´
Roman numeral	VII, vii
Greekprefix	hepta-/hept-
Latinprefix	septua-
Binary	1112
Ternary	213
Senary	116
Octal	78
Duodecimal	712
Hexadecimal	716
Greek numeral	Z,ζ
Amharic	፯
Arabic,Kurdish,Persian	٧
Sindhi,Urdu	۷
Bengali	৭
Chinese numeral	Bảy, thất
Devanāgarī	७
Telugu	౭
Tamil	௭
Hebrew	ז
Khmer	៧
Thai	๗
Kannada	೭
Malayalam	൭
Armenian	Է
Babylonian numeral	𒐛
Egyptian hieroglyph	𓐀
Morse code	_ _...

7(seven) is thenatural numberfollowing6and preceding8.It is the onlyprime numberpreceding acube.

As an early prime number in the series ofpositive integers,the number seven has greatly symbolic associations inreligion,mythology,superstitionandphilosophy.The sevenclassical planetsresulted in seven being the number of days in a week.7 is often consideredluckyinWestern cultureand is often seen as highly symbolic. Unlike Western culture, inVietnamese culture,the number seven is sometimes considered unlucky.

Evolution of the Arabic digit

For earlyBrahmi numerals,7 was written more or less in one stroke as a curve that looks like an uppercase⟨J⟩vertically inverted (ᒉ). The western Arab peoples' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arab peoples developed the digit from a form that looked something like 6 to one that looked like an uppercase V. Both modern Arab forms influenced the European form, a two-stroke form consisting of a horizontal upper stroke joined at its right to a stroke going down to the bottom left corner, a line that is slightly curved in some font variants. As is the case with the European digit, theChamandKhmer digitfor 7 also evolved to look like their digit 1, though in a different way, so they were also concerned with making their 7 more different. For the Khmer this often involved adding a horizontal line to the top of the digit.This is analogous to the horizontal stroke through the middle that is sometimes used inhandwritingin the Western world but which is almost never used incomputer fonts.This horizontal stroke is, however, important to distinguish the glyph for seven from the glyph foronein writing that uses a long upstroke in the glyph for 1. In some Greek dialects of the early 12th century the longer line diagonal was drawn in a rather semicircular transverse line.

Onseven-segment displays,7 is the digit with the most common graphic variation (1, 6 and 9 also have variant glyphs). Mostcalculatorsuse three line segments, but onSharp,Casio,and a few other brands of calculators, 7 is written with four line segments because in Japan, Korea and Taiwan 7 is written with a "hook" on the left, as ① in the following illustration.

While the shape of the character for the digit 7 has anascenderin most moderntypefaces,in typefaces withtext figuresthe character usually has adescender(⁊), as, for example, in.

Most people in Continental Europe,Indonesia,and some in Britain, Ireland, and Canada, as well as Latin America, write 7 with a line through the middle (7), sometimes with the top line crooked. The line through the middle is useful to clearly differentiate the digit from the digit one, as the two can appear similar when written in certain styles of handwriting. This form is used in official handwriting rules forprimary schoolin Russia, Ukraine, Bulgaria, Poland, other Slavic countries,France,Italy, Belgium, the Netherlands, Finland,Romania, Germany, Greece,and Hungary.

Mathematics

Seven, the fourth prime number, is not only aMersenne prime(since$2^{3}-1=7$) but also adouble Mersenne primesince the exponent, 3, is itself a Mersenne prime.It is also aNewman–Shanks–Williams prime,aWoodall prime,afactorial prime,aHarshad number,alucky prime,ahappy number(happy prime),asafe prime(the onlyMersenne safe prime), aLeyland prime of the second kindand the fourthHeegner number.

• Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers. (SeeLagrange's four-square theorem#Historical development.)
• Seven is thealiquot sumof one number, thecubic number8= 2³making it the base of the 7-aliquot tree; it is also, therefore, thesum-of-divisorsof only4,itsprime index.Furthermore, the smallest number with sevendivisorsis64= 8².
• The seventhtriangular numberis the secondperfect number28= 7 × 4,which precedes6.Indecimalrepresentation, thereciprocalof 7 repeats sixdigits(as 0.142857),whose sum whencyclingback to1is equal to 28. On the other hand, 7 is the number ofpartitionsof5,a valuenwhich yields the third perfect number496for 2^{n− 1}(2ⁿ− 1), by theEuclid-Euler theorem.
• 7 is the only numberDfor which the equation2ⁿ−D=x²has more than two solutions fornandxnatural.In particular, the equation2ⁿ− 7 =x²is known as theRamanujan–Nagell equation.
• There are 7frieze groupsin two dimensions, consisting ofsymmetriesof theplanewhose group oftranslationsisisomorphicto the group ofintegers.These are related to the17wallpaper groupswhose transformations andisometriesrepeat two-dimensional patterns in the plane.Theseventhindexedprime number is seventeen.
• As a consequence ofFermat's little theoremandEuler's criterion,all cubes are congruent to 0, 1, or 6 modulo 7.
• A seven-sided shape is aheptagon.Theregularn-gons forn⩽ 6 can be constructed bycompass and straightedgealone, which makes the heptagon the first regular polygon that cannot be directly constructed with these simple tools.Figurate numbersrepresenting heptagons are calledheptagonal numbers.7 is also acentered hexagonal number.

A heptagon inEuclidean spaceis unable to generateuniform tilingsalongside other polygons, like the regularpentagon.However, it is one of fourteen polygons that can fill aplane-vertex tiling,in its case only alongside a regulartriangleand a 42-sided polygon (3.7.42).This is also one of twenty-one such configurations from seventeen combinations of polygons, that features the largest and smallest polygons possible.

Otherwise, for any regularn-sided polygon, the maximum number of intersecting diagonals (other than through its center) is at most 7.Since the regular heptagon contains fourteendiagonals,the difference between its number of diagonals and its number of sides is seven; the heptagon is the only convex polygon to have a one-to-two ratio between the number of its sides and diagonals (as anyn-sided polygon forn≥ 3 sides, convex or concave, has⁠n(n– 3)/2⁠diagonals).

• InWythoff's kaleidoscopic constructions,seven distinct generator points that lie onmirroredges of a three-sidedSchwarz triangleare used to create mostuniform tilingsandpolyhedra;an eighth point lying on all three mirrors is technicallydegenerate,reserved to representsnubforms only.

Seven of eightsemiregular tilingsare Wythoffian (the only exception is theelongated triangular tiling), where there exist three tilings that areregular,all of which are Wythoffian.Seven of nine uniform colorings of the square tiling are also Wythoffian, and between thetriangular tilingandsquare tiling,there are sevennon-Wythoffianuniform colorings of a total twenty-one that belong to regular tilings (allhexagonal tilinguniform colorings are Wythoffian).

In two dimensions, there are precisely seven7-uniformKrotenheerdttilings, with no other suchk-uniform tilings fork> 7, and it is also the onlykfor which the count ofKrotenheerdttilings agrees withk.

• TheFano planeis the smallest possiblefinite projective planewith 7 points and 7 lines such that every line contains 3 points and 3 lines cross every point.Withgroup order168 = 2³·3·7, this plane holds 35 total triples of points where 7 are collinear and another 28 are non-collinear, whoseincidence graphis the 3-regularbipartateHeawood graphwith 14verticesand 21edges.This graphembedsin three dimensions as theSzilassi polyhedron,the simplesttoroidal polyhedronalongside itsdualwith 7 vertices, theCsászár polyhedron.
• In three-dimensional space there are sevencrystal systemsand fourteenBravais latticeswhich classify under sevenlattice systems,six of which are shared with the seven crystal systems.There are also collectively seventy-sevenWythoff symbolsthat represent alluniform figuresin three dimensions.

Graph of the probability distribution of the sum of two six-sided dice

• The seventh dimension is the only dimension aside from the familiar three where a vectorcross productcanbe defined.This is related to theoctonionsover theimaginary subspaceIm(O)in 7-space whosecommutatorbetween two octonions defines this vector product, wherein theFano planedescribes the multiplicativealgebraic structureof theunit octonions{e₀,e₁,e₂,..., e₇},withe₀anidentity element.

Also, the lowest known dimension for anexotic sphereis the seventh dimension, with a total of 28 differentiable structures; there may exist exotic smooth structures on thefour-dimensional sphere.

Inhyperbolic space,7 is the highest dimension for non-simplexhypercompactVinberg polytopesof rankn + 4mirrors, where there is one unique figure with elevenfacets.On the other hand, such figures with rankn + 3mirrors exist in dimensions 4, 5, 6 and 8;notin 7.Hypercompact polytopes with lowest possible rank ofn + 2mirrors exist up through the17thdimension, where there is a single solution as well.

• There are seven fundamental types ofcatastrophes.
• The positivedefinite quadraticinteger matrixrepresentative of alloddnumbers contains the set ofsevenintegers: {1, 3, 5,7,11, 15, 33} where seven is the middle indexed member.
• When rolling two standard six-sideddice,seven has a 6 in 6²(or⁠1/6⁠) probability of being rolled (1–6, 6–1, 2–5, 5–2, 3–4, or 4–3), the greatest of any number.The opposite sides of a standard six-sided dice always add to 7.
• TheMillennium Prize Problemsare seven problems inmathematicsthat were stated by theClay Mathematics Institutein 2000.Currently, six of the problems remainunsolved.

Basic calculations

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
7 ×x	7	14	21	28	35	42	49	56	63	70	77	84	91	98	105	112	119	126	133	140	147	154	161	168	175	350	700	7000

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
7 ÷x	7	3.5	2.3	1.75	1.4	1.16	1	0.875	0.7	0.7	0.63	0.583	0.538461	0.5	0.46
x÷ 7	0.142857	0.285714	0.428571	0.571428	0.714285	0.857142		1.142857	1.285714	1.428571	1.571428	1.714285	1.857142	2	2.142857

Exponentiation	1	2	3	4	5	6	7	8	9	10	11	12	13
7x	7	49	343	2401	16807	117649	823543	5764801	40353607	282475249	1977326743	13841287201	96889010407
x7	1	128	2187	16384	78125	279936	823543	2097152	4782969	10000000	19487171	35831808	62748517

Radix	1	5	10	15	20	25	50	75	100	125	150	200	250	500	1000	10000	100000	1000000
x7	1	5	137	217	267	347	1017	1357	2027	2367	3037	4047	5057	13137	26267	411047	5643557	113333117

In decimal

999,999divided by 7 is exactly142,857.Therefore, when avulgar fractionwith 7 in thedenominatoris converted to adecimalexpansion, the result has the same six-digitrepeating sequence after the decimal point, but the sequence can start with any of those six digits.For example,1/7 = 0.142857 142857...and2/7 = 0.285714 285714....

In fact, if one sorts the digits in the number 142,857 in ascending order, 124578, it is possible to know from which of the digits the decimal part of the number is going to begin with. The remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example, 628 ÷ 7 =⁠89+5/7⁠;here 5 is the remainder, and would correspond to number 7 in the ranking of the ascending sequence. So in this case,628 ÷ 7 = 89.714285.Another example,5238 ÷ 7 =⁠748+2/7⁠,hence the remainder is 2, and this corresponds to number 2 in the sequence. In this case,5238 ÷ 7 = 748.285714.

In science

• Sevencolors in a rainbow
• Sevencontinents
• Seven climes
• The neutralpH balance
• Number of notes in thediatonic scaleof Western music
• Number of spots most commonly found onladybugs
• Atomic number fornitrogen
• Number ofdiatomic molecules
• Seven basiccrystal systems

In psychology

• Seven, plus or minus twoas a model ofworking memory
• Sevenpsychological typescalled theSeven Raysin the teachings ofAlice A. Bailey
• In Western culture, seven is consistently listed as people's favorite number
• When guessing numbers 1–10, the number 7 is most likely to be picked
• Seven-year itch,a term that suggests that happiness in a marriage declines after around seven years

Classical antiquity

ThePythagoreansinvested particular numbers with unique spiritual properties. The number seven was considered to be particularly interesting because it consisted of the union of the physical (number4) with the spiritual (number3).In Pythagoreannumerologythe number 7 means spirituality.

References from classical antiquity to the number seven include:

• SevenClassical planetsand the derivativeSeven Heavens
• Seven Wonders of the Ancient World
• Sevenmetals of antiquity
• Seven days in the week
• Seven Seas
• Seven Sages
• Seven champions that fought Thebes
• Seven hills of RomeandSeven Kings of Rome
• Seven Sisters,the daughters ofAtlasalso known as thePleiades

Religion and mythology

Judaism

Main article:Significance of numbers in Judaism

The number seven forms a widespreadtypologicalpattern withinHebrew scripture,including:

• Seven days (more preciselyyom) of Creation, leading to the seventh day orSabbath(Genesis 1)
• Seven-fold vengeance visited on uponCainfor the killing ofAbel(Genesis 4:15)
• Seven pairs of every clean animal loaded onto the ark by Noah (Genesis 7:2)
• Seven years of plenty and seven years of famine in Pharaoh's dream (Genesis 41)
• Seventh son of Jacob,Gad,whose name means good luck (Genesis 46:16)
• Seven times bullock's blood is sprinkled before God (Leviticus 4:6)
• Seven nations God told theIsraelitesthey would displace when they entered the land ofIsrael(Deuteronomy 7:1)
• Seven days (de jure, but de facto eight days) of thePassoverfeast (Exodus 13:3–10)
• Seven-branchedcandelabrumorMenorah(Exodus 25)
• Seventrumpetsplayed by seven priests for seven days to bring down the walls of Jericho (Joshua 6:8)
• Seven things that are detestable to God (Proverbs 6:16–19)
• Seven Pillars of theHouse of Wisdom(Proverbs 9:1)
• Seven archangels in the deuterocanonicalBook of Tobit(12:15)

References to the number seven in Jewish knowledge and practice include:

• Seven divisions of the weekly readings oraliyahof theTorah
• Sevenaliyoton Shabbat
• Seven blessings recited under thechuppahduring a Jewish wedding ceremony
• Seven days of festive meals for a Jewish bride and groom after their wedding, known as Sheva Berachot or Seven Blessings
• SevenUshpizzinprayers to the Jewish patriarchs during the holiday ofSukkot

Christianity

Following the tradition of theHebrew Bible,theNew Testamentlikewise uses the number seven as part of atypologicalpattern:

Seven lampstands inThe Vision of John on PatmosbyJulius Schnorr von Carolsfeld,1860

• Seven loavesmultiplied into seven basketfulsof surplus (Matthew 15:32–37)
• Sevendemonswere driven out ofMary Magdalene(Luke 8:2)
• Seven last sayingsof Jesus on the cross
• Seven men of honest report, full of the Holy Ghost and wisdom (Acts 6:3)
• Seven Spirits of God,Seven ChurchesandSeven Sealsin theBook of Revelation

References to the number seven in Christian knowledge and practice include:

• Seven Gifts of the Holy Spirit
• Seven Corporal Acts of MercyandSeven Spiritual Acts of Mercy
• Seven deadly sins:lust, gluttony, greed, sloth, wrath, envy, and pride, and seven terraces of MountPurgatory
• Seven Virtues:chastity, temperance, charity, diligence, kindness, patience, and humility
• Seven JoysandSeven Sorrowsof the Virgin Mary
• Seven Sleepersof Christian myth
• SevenSacramentsin the Catholic Church (though some traditions assign a different number)

Islam

References to the number seven in Islamic knowledge and practice include:

• SevenayatinSurah al-Fatiha,the first chapter of the holyQur'an
• Sevencircumambulationsof Muslim pilgrims around theKaabainMeccaduring theHajjand theUmrah
• Seven walks betweenAl-Safa and Al-Marwahperformed Muslim pilgrims during theHajjand theUmrah
• Seven doors to hell (for heaven the number of doors is eight)
• Seven heavens(plural of sky) mentioned in Qur'an65:12
• Night Journey to the Seventh Heaven, (reported ascension to heaven to meet God)Isra' and Mi'rajinSurah Al-Isra'.
• Seventh daynaming ceremonyheld for babies
• Seven enunciators of divine revelation (nāṭiqs) according to the celebratedFatimidIsmailidignitaryNasir Khusraw
• Circle Seven Koran,the holy scripture of the Moorish Science Temple of America
• Seven earth as mentioned in the Quran
• Sevenchildren of Muhammad
• Seven years of abundance and seven of drought in Egypt during the time ofYusuf(Joseph) as mentioned in theQuran.

Hinduism

References to the number seven in Hindu knowledge and practice include:

• Seven worlds in the universe and seven seas in the world in Hindu cosmology
• Seven sages orSaptarishiand their seven wives or Sapta Matrka inHinduism
• SevenChakrasin eastern philosophy
• Seven stars in a constellation called "SaptharishiMandalam "in Indian astronomy
• Seven promises, orSaptapadi,and seven circumambulations around a fire at Hindu weddings
• Seven virgin goddesses orSaptha Kannimarworshipped in temples inTamil Nadu,India
• Seven hills atTirumalaknown as Yedu Kondalavadu inTelugu,or ezhu malaiyan inTamil,meaning "Sevenhills God"
• Seven steps taken by theBuddhaat birth
• Seven divine ancestresses of humankind inKhasimythology
• Seven octets orSaptakSwarasin Indian Music as the basis forRagascompositions
• Seven Social Sinslisted byMahatma Gandhi

Eastern tradition

Other references to the number seven in Eastern traditions include:

TheSeven Lucky GodsinJapanese mythology

• Seven Lucky Godsor gods of goodfortuneinJapanese mythology
• Seven-Branched Swordin Japanese mythology
• Seven Sages of the Bamboo Grovein China
• Seven minor symbols ofyangin Taoistyin-yang

Other references

Other references to the number seven in traditions from around the world include:

• The number seven had mystical and religious significance in Mesopotamian culture by the 22nd century BCE at the latest. This was likely because in the Sumeriansexagesimalnumber system, dividing by seven was the first division which resulted in infinitelyrepeating fractions.
• Seven palms in an EgyptianSacred Cubit
• Seven ranks inMithraism
• Seven hills of Istanbul
• Seven islands ofAtlantis
• SevenCherokeeclans
• Seven lives of cats inIranandGermanandRomancelanguage-speaking cultures
• Seven fingers on each hand, seven toes on each foot and seven pupils in each eye of the Irish epic heroCúchulainn
• Seventh sons will bewerewolvesinGalicianfolklore, or the son of a woman and a werewolf in other European folklores
• Seventh sons of a seventh sonwill be magicians with special powers of healing and clairvoyance in some cultures, or vampires in others
• Seven prominentlegendary monstersinGuaraní mythology
• Seven gateways traversed byInannaduring her descent into theunderworld
• Seven Wise Masters,a cycle of medieval stories
• Seven sistergoddessesorfatesinBaltic mythologycalled theDeivės Valdytojos.
• Seven legendary Cities of Gold, such asCibola,that the Spanish thought existed in South America
• Seven years spent byThomas the Rhymerin the faerie kingdom in the eponymous British folk tale
• Seven-year cycle in which theQueen of the Fairiespays a tithe toHell(or possiblyHel) in the tale ofTam Lin
• Seven Valleys,a text by the Prophet-FounderBahá'u'lláhin the Bahá'í faith
• Seven superuniverses in the cosmology ofUrantia
• Seven, the sacred number ofYemaya
• Seven holes representing eyes (سبع عيون) in an Assyrian evil eye bead – though occasionally two, and sometimes nine

See also

• Diatonic scale(7 notes)
• Seven colors in the rainbow
• Seven continents
• Seven liberal arts
• Seven Wonders of the Ancient World
• Seven days of theWeek
• Septenary (numeral system)
• Year Seven (School)
• Se7en (disambiguation)
• Sevens (disambiguation)
• One-seventh area triangle
• Z with stroke(Ƶ)
• List of highways numbered 7

References

• Wells, D.The Penguin Dictionary of Curious and Interesting NumbersLondon:Penguin Group(1987): 70–71