7

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

File:SevenGlyph.svg

For early Brahmi numerals, 7 was written more or less in one stroke as a curve that looks like an uppercase {{angbr|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, the Cham and Khmer digit for 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.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): 395, Fig. 24.67 This is analogous to the horizontal stroke through the middle that is sometimes used in handwriting in the Western world but which is almost never used in computer fonts. This horizontal stroke is, however, important to distinguish the glyph for seven from the glyph for one in 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.

File:Digital77.svg

On seven-segment displays, 7 is the digit with the most common graphic variation (1, 6 and 9 also have variant glyphs). Most devices use three line segments, but devices made by some Japanese companies such as Sharp and Casio, as well as in the Koreas and Taiwan, 7 is written with four line segments because in those countries, 7 is written with a "hook" on the left, as ① in the following illustration. Further segments can give further variation. For example, Schindler elevators in the United States and Canada installed or modernized from the late 1990s onwards usually use a sixteen segment display and show the digit 7 in a manner more similar to that of handwriting.

File:sevens.svg

While the shape of the character for the digit 7 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in 40px.

File:Hand Written 7.svgMost people in Continental Europe,{{Cite journal |title=Aamulehti: Opetushallitus harkitsee numero 7 viivan palauttamista |author=Eeva Törmänen |date=September 8, 2011 |journal=Tekniikka & Talous |url=http://www.tekniikkatalous.fi/viihde/aamulehti+opetushallitus+harkitsee+numero+7+viivan+palauttamista/a682831 |language=fi |access-date=September 9, 2011 |archive-url=https://web.archive.org/web/20110917083226/http://www.tekniikkatalous.fi/viihde/aamulehti+opetushallitus+harkitsee+numero+7+viivan+palauttamista/a682831 |archive-date=September 17, 2011 |url-status=dead }} Indonesia,{{citation needed|date=April 2024}} and some in Britain, Ireland, Israel, Canada, and Latin America, write 7 with a line through the middle ({{strikethrough|7}}), sometimes with the top line crooked. The line through the middle is useful to clearly differentiate that digit from the digit one, as they can appear similar when written in certain styles of handwriting. This form is used in official handwriting rules for primary school in Russia, Ukraine, Bulgaria, Poland, other Slavic countries,[http://www.adu.by/modules.php?name=News&file=article&sid=858 "Education writing numerals in grade 1."] {{webarchive|url=https://web.archive.org/web/20081002092040/http://www.adu.by/modules.php?name=News&file=article&sid=858 |date=2008-10-02 }}(Russian) France,[http://www.pour-enfants.fr/jeux-imprimer/apprendre/les-chiffres/ecrire-les-chiffres.png "Example of teaching materials for pre-schoolers"](French) Italy, Belgium, the Netherlands, Finland,{{Cite journal |title="Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin? |author=Elli Harju |date=August 6, 2015 |journal=Iltalehti |url=https://www.iltalehti.fi/uutiset/a/2015080620139397 |language=fi}} Romania, Germany, Greece,{{cite web |url=http://ebooks.edu.gr/modules/document/file.php/DSDIM-A102/%CE%94%CE%B9%CE%B4%CE%B1%CE%BA%CF%84%CE%B9%CE%BA%CF%8C%20%CE%A0%CE%B1%CE%BA%CE%AD%CF%84%CE%BF/%CE%92%CE%B9%CE%B2%CE%BB%CE%AF%CE%BF%20%CE%9C%CE%B1%CE%B8%CE%B7%CF%84%CE%AE/10-0007-02_Mathimatika_A-Dim_BM-1.pdf |title=Μαθηματικά Α' Δημοτικού |language=el |trans-title=Mathematics for the First Grade |publisher=Ministry of Education, Research, and Religions |access-date=May 7, 2018 |page=33}} and Hungary.{{citation needed|date=September 2021}}

Seven, the fourth prime number, is not only a Mersenne prime (since $2^3\; -\; 1\; =\; 7$) but also a double Mersenne prime since the exponent, 3, is itself a Mersenne prime.{{Cite web|last=Weisstein|first=Eric W.|title=Double Mersenne Number|url=https://mathworld.wolfram.com/DoubleMersenneNumber.html|access-date=2020-08-06|website=mathworld.wolfram.com}} It is also a Newman–Shanks–Williams prime,{{Cite web |url=https://oeis.org/A088165 |title=Sloane's A088165 : NSW primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}} a Woodall prime,{{Cite web |url=https://oeis.org/A050918 |title=Sloane's A050918 : Woodall primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}} a factorial prime,{{Cite web |url=https://oeis.org/A088054 |title=Sloane's A088054 : Factorial primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}} a Harshad number, a lucky prime,{{Cite web |url=https://oeis.org/A031157 |title=Sloane's A031157 : Numbers that are both lucky and prime |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}} a happy number (happy prime),{{Cite web |url=https://oeis.org/A035497 |title=Sloane's A035497 : Happy primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}} a safe prime (the only {{vanchor|Mersenne safe prime}}), a Leyland number of the second kind{{Cite OEIS|A045575|Leyland numbers of the second kind}} and Leyland prime of the second kind{{Cite OEIS|A123206|Leyland prime numbers of the second kind}} {{nowrap|($2^5-5^2$),}} and the fourth Heegner number.{{Cite web |url=https://oeis.org/A003173 |title=Sloane's A003173 : Heegner numbers |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}} Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers.

A seven-sided shape is a heptagon.{{Cite web |last=Weisstein |first=Eric W. |title=Heptagon |url=https://mathworld.wolfram.com/Heptagon.html |access-date=2020-08-25 |website=mathworld.wolfram.com}} The regular n-gons for n ⩽ 6 can be constructed by compass and straightedge alone, which makes the heptagon the first regular polygon that cannot be directly constructed with these simple tools.{{Cite web |last=Weisstein |first=Eric W. |title=7 |url=https://mathworld.wolfram.com/7.html |access-date=2020-08-07 |website=mathworld.wolfram.com}}

7 is the only number D for which the equation {{nowrap|1=2{{sup|n}} − D = x{{sup|2}}}} has more than two solutions for n and x natural. In particular, the equation {{nowrap|1=2{{sup|n}} − 7 = x{{sup|2}}}} is known as the Ramanujan–Nagell equation. 7 is one of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: {1, 3, 5, 7, 11, 15, 33}.{{Cite book |last1=Cohen |first1=Henri |url=https://link.springer.com/book/10.1007/978-0-387-49923-9 |title=Number Theory Volume I: Tools and Diophantine Equations |publisher=Springer |year=2007 |isbn=978-0-387-49922-2 |edition=1st |series=Graduate Texts in Mathematics |volume=239 |pages=312–314 |chapter=Consequences of the Hasse–Minkowski Theorem |doi=10.1007/978-0-387-49923-9 |oclc=493636622 |zbl=1119.11001}}{{Cite OEIS|A116582|Numbers from Bhargava's 33 theorem.|access-date=2024-02-03}}

There are 7 frieze groups in two dimensions, consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers.{{Cite book |last1=Heyden |first1=Anders |url=https://books.google.com/books?id=4yCqCAAAQBAJ&q=seven+frieze+groups&pg=PA661 |title=Computer Vision – ECCV 2002: 7th European Conference on Computer Vision, Copenhagen, Denmark, May 28–31, 2002. Proceedings. Part II |last2=Sparr |first2=Gunnar |last3=Nielsen |first3=Mads |last4=Johansen |first4=Peter |date=2003-08-02 |publisher=Springer |isbn=978-3-540-47967-3 |pages=661 |quote=A frieze pattern can be classified into one of the 7 frieze groups...}} These are related to the 17 wallpaper groups whose transformations and isometries repeat two-dimensional patterns in the plane.{{Cite book |author1=Grünbaum, Branko |author-link=Branko Grünbaum |author2=Shephard, G. C. |author2-link=G.C. Shephard |url-access=registration |url=https://archive.org/details/isbn_0716711931 |title=Tilings and Patterns |chapter=Section 1.4 Symmetry Groups of Tilings |publisher=W. H. Freeman and Company |location=New York |year=1987 |pages=40–45 |doi=10.2307/2323457 |jstor=2323457 |isbn=0-7167-1193-1 |oclc=13092426 |s2cid=119730123 }}{{Cite OEIS |A004029 |Number of n-dimensional space groups. |access-date=2023-01-30 }}

A heptagon in Euclidean space is unable to generate uniform tilings alongside other polygons, like the regular pentagon. However, it is one of fourteen polygons that can fill a plane-vertex tiling, in its case only alongside a regular triangle and a 42-sided polygon (3.7.42).{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=Mathematics Magazine |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd. |page=231 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 |archive-date=2016-03-03 |access-date=2023-01-09 |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 }}{{Cite web |last=Jardine |first=Kevin |url=http://gruze.org/tilings/3_7_42_shield|title=Shield - a 3.7.42 tiling |website=Imperfect Congruence |access-date=2023-01-09 }} 3.7.42 as a unit facet in an irregular tiling. Otherwise, for any regular n-sided polygon, the maximum number of intersecting diagonals (other than through its center) is at most 7.{{Cite journal |last1=Poonen |first1=Bjorn |author1-link=Bjorn Poonen |last2=Rubinstein |first2=Michael |title=The Number of Intersection Points Made by the Diagonals of a Regular Polygon |url=https://math.mit.edu/~poonen/papers/ngon.pdf |journal=SIAM Journal on Discrete Mathematics |volume=11 |issue=1 |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |year=1998 |pages=135–156 |doi=10.1137/S0895480195281246 |arxiv=math/9508209 |mr=1612877 |zbl=0913.51005 |s2cid=8673508 }}

In two dimensions, there are precisely seven 7-uniform Krotenheerdt tilings, with no other such k-uniform tilings for k > 7, and it is also the only k for which the count of Krotenheerdt tilings agrees with k.{{Cite OEIS |A068600 |Number of n-uniform tilings having n different arrangements of polygons about their vertices. |access-date=2023-01-09 }}{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=Mathematics Magazine |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd. |page=236 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 |archive-date=2016-03-03 |access-date=2023-01-09 |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 }}

The Fano plane, the smallest possible finite projective plane, has 7 points and 7 lines arranged such that every line contains 3 points and 3 lines cross every point.{{Cite book |first1=Tomaž |last1=Pisanski |first2=Brigitte |last2=Servatius |author1-link=Tomaž Pisanski |author2-link=Brigitte Servatius |title=Configurations from a Graphical Viewpoint |chapter=Section 1.1: Hexagrammum Mysticum |chapter-url=https://link.springer.com/chapter/10.1007/978-0-8176-8364-1_5 |edition=1 |publisher=Birkhäuser |series=Birkhäuser Advanced Texts |location=Boston, MA |year=2013 |pages=5–6 |isbn=978-0-8176-8363-4 |oclc=811773514 |doi=10.1007/978-0-8176-8364-1 |zbl=1277.05001 }} This is related to other appearances of the number seven in relation to exceptional objects, like the fact that the octonions contain seven distinct square roots of −1, seven-dimensional vectors have a cross product, and the number of equiangular lines possible in seven-dimensional space is anomalously large.{{Cite journal |url=https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf |title=Cross products of vectors in higher dimensional Euclidean spaces |first=William S. |last=Massey |author-link=William S. Massey |journal=The American Mathematical Monthly |volume=90 |issue=10 |publisher=Taylor & Francis, Ltd |date=December 1983 |pages=697–701 |doi=10.2307/2323537 |jstor=2323537 |s2cid=43318100 |zbl=0532.55011 |access-date=2023-02-23 |archive-date=2021-02-26 |archive-url=https://web.archive.org/web/20210226011747/https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf |url-status=dead }}{{Cite journal |last1=Baez |first1=John C. |author-link=John Baez |url=http://math.ucr.edu/home/baez/octonions/ |title=The Octonions |journal=Bulletin of the American Mathematical Society |volume=39 |issue=2 |publisher=American Mathematical Society |pages=152–153 |year=2002 |doi=10.1090/S0273-0979-01-00934-X |mr=1886087|s2cid=586512 |doi-access=free }}{{Cite book|last=Stacey |first=Blake C. |title=A First Course in the Sporadic SICs |date=2021 |publisher=Springer |isbn=978-3-030-76104-2 |location=Cham, Switzerland |pages=2–4 |oclc=1253477267}}File:Dice Distribution (bar).svg

The lowest known dimension for an exotic sphere is the seventh dimension.{{Cite journal |last1=Behrens |first1=M. |last2=Hill |first2=M. |last3=Hopkins |first3=M. J. |last4=Mahowald |first4=M. |date=2020 |title=Detecting exotic spheres in low dimensions using coker J |url=https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12301 |journal=Journal of the London Mathematical Society |publisher=London Mathematical Society |volume=101 |issue=3 |pages=1173 |arxiv=1708.06854 |doi=10.1112/jlms.12301 |mr=4111938 |s2cid=119170255 |zbl=1460.55017}}{{Cite OEIS|A001676|Number of h-cobordism classes of smooth homotopy n-spheres.|access-date=2023-02-23}}

In hyperbolic space, 7 is the highest dimension for non-simplex hypercompact Vinberg polytopes of rank n + 4 mirrors, where there is one unique figure with eleven facets. On the other hand, such figures with rank n + 3 mirrors exist in dimensions 4, 5, 6 and 8; not in 7.{{Cite journal |last1=Tumarkin |first1=Pavel |last2=Felikson |first2=Anna |url=https://www.ams.org/journals/mosc/2008-69-00/S0077-1554-08-00172-6/S0077-1554-08-00172-6.pdf |title=On d-dimensional compact hyperbolic Coxeter polytopes with d + 4 facets |journal=Transactions of the Moscow Mathematical Society |volume=69 |publisher=American Mathematical Society (Translation) |location=Providence, R.I. |year=2008 |pages=105–151 |doi= 10.1090/S0077-1554-08-00172-6 |doi-access=free |mr=2549446 |s2cid=37141102 |zbl=1208.52012 }}

There are seven fundamental types of catastrophes.{{Cite book|last1=Antoni|first1=F. de|url=https://books.google.com/books?id=3L_sCAAAQBAJ&q=seven+fundamental+types+of+catastrophes&pg=PA13|title=COMPSTAT: Proceedings in Computational Statistics, 7th Symposium held in Rome 1986|last2=Lauro|first2=N.|last3=Rizzi|first3=A.|date=2012-12-06|publisher=Springer Science & Business Media|isbn=978-3-642-46890-2|pages=13|quote=...every catastrophe can be composed from the set of so called elementary catastrophes, which are of seven fundamental types.}}

When rolling two standard six-sided dice, seven has a 1 in 6 probability of being rolled, the greatest of any number.{{Cite web|last=Weisstein|first=Eric W.|title=Dice|url=https://mathworld.wolfram.com/Dice.html|access-date=2020-08-25|website=mathworld.wolfram.com}} The opposite sides of a standard six-sided die always add to 7.

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.{{Cite web |title=Millennium Problems {{!}} Clay Mathematics Institute |url=http://www.claymath.org/millennium-problems |access-date=2020-08-25 |website=www.claymath.org}} Currently, six of the problems remain unsolved.{{Cite web |date=2013-12-15 |title=Poincaré Conjecture {{!}} Clay Mathematics Institute |url=http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-url=https://web.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-date=2013-12-15 |access-date=2020-08-25}}

{{num|999,999}} divided by 7 is exactly {{num|142,857}}. Therefore, when a vulgar fraction with 7 in the denominator is converted to a decimal expansion, the result has the same six-digit repeating sequence after the decimal point, but the sequence can start with any of those six digits.Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 82 In decimal representation, the reciprocal of 7 repeats six digits (as 0.{{overline|142857}}),{{Cite book |last=Wells |first=D. |url=https://archive.org/details/penguindictionar0000well_f3y1/mode/2up |title=The Penguin Dictionary of Curious and Interesting Numbers |publisher=Penguin Books |year=1987 |isbn=0-14-008029-5 |location=London |pages=171–174 |oclc=39262447 |url-access=registration |s2cid=118329153}}{{Cite OEIS|A060283|Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's moved to end).|access-date=2024-04-02}} whose sum when cycling back to 1 is equal to 28.

• Seven, plus or minus two as a model of working memory
• In Western culture, seven is consistently listed as people's favorite number{{cite web |last1=Gonzalez |first1=Robbie |title=Why Do People Love The Number Seven? |url=https://gizmodo.com/why-do-people-love-the-number-seven-so-much-1666353786 |website=Gizmodo |date=4 December 2014 |access-date=20 February 2022 }}{{cite web |last1=Bellos |first1=Alex |title=The World's Most Popular Numbers [Excerpt] |url=https://www.scientificamerican.com/article/most-popular-numbers-grapes-of-math/ |website=Scientific American |access-date=20 February 2022}}
• When guessing numbers 1–10, the number 7 is most likely to be picked{{cite journal |last1=Kubovy |first1=Michael |last2=Psotka |first2=Joseph |title=The predominance of seven and the apparent spontaneity of numerical choices. |journal=Journal of Experimental Psychology: Human Perception and Performance |date=May 1976 |volume=2 |issue=2 |pages=291–294 |doi=10.1037/0096-1523.2.2.291 |url=https://www.researchgate.net/publication/232582800 |access-date=20 February 2022}}
• Seven-year itch, a term that suggests that happiness in a marriage declines after around seven years

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| title = {{center|"Number Seven"
by William Sidney Gibson
Read by Ruth Golding for LibriVox}}

| description = {{center|Audio 00:15:59 ([https://archive.org/stream/householdwords13dick#page/454/mode/2up full text])}}

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The Pythagoreans invested particular numbers with unique spiritual properties. The number seven was considered to be particularly interesting because it consisted of the union of the physical (number 4) with the spiritual (number 3).{{Cite web|url=https://www.britannica.com/topic/number-symbolism/7|title=Number symbolism – 7}} In Pythagorean numerology the number 7 means spirituality.

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 Sumerian sexagesimal number system, dividing by seven was the first division which resulted in infinitely repeating fractions.Muroi, Kazuo (2014) [https://arxiv.org/pdf/1407.6246.pdf The Origin of the Mystical Number Seven in Mesopotamian Culture: Division by Seven in the Sexagesimal Number System]

{{Portal|Mathematics}}

{{Wikiquote|7 (number)}}

{{Commons category}}

{{Wiktionary|seven}}

• Diatonic scale (7 notes)
• Seven colors in the rainbow
• Seven continents
• Seven liberal arts
• Seven sacraments (disambiguation)
• Seven virtues
• Seven deadly sins
• Seven Wonders of the Ancient World
• New 7 Wonders of Nature
• Seven Kings of Rome
• Seven Laws of Noah
• Seven Archangels
• Seven trumpets
• Seven Summits
• Seven heavens
• Seven seals
• Seven Seas
• Seven bowls
• Seven necessities
• Seven Sisters (disambiguation)
• Seven days of the Week
• Septenary (numeral system)
• Year Seven (School)
• Se7en (disambiguation)
• Sevens (disambiguation)
• One-seventh area triangle
• Seven Dwarfs

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{{Reflist}}

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

{{Integers|zero}}

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Category:Integers

Category:7 (number)