# 7

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Natural number

This article is about the number. For the year, see [AD 7](/source/AD_7). For other uses, see [7 (disambiguation)](/source/7_(disambiguation)) and [No. 7 (disambiguation)](/source/No._7_(disambiguation)).

Not to be confused with [⁊](/source/%E2%81%8A).

Natural number

← 6 7 8 → −1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinal seven Ordinal 7th (seventh) Numeral system septenary Factorization prime Prime 4th Divisors 1, 7 Greek numeral Ζ´ Roman numeral VII, vii Greek prefix hepta-/hept- Latin prefix septua-/sept- 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 七, 柒 Devanāgarī ७ Santali ᱗ 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 the [natural number](/source/Natural_number) following [6](/source/6) and preceding [8](/source/8). It is the only [prime number](/source/Prime_number) preceding a [cube](/source/Cube_(algebra)).

As an early prime number in the series of [positive integers](/source/Positive_integers), the number seven has symbolic associations in [religion](/source/Religion), [mythology](/source/Mythology), [superstition](/source/Superstition) and [philosophy](/source/Philosophy). The seven [classical planets](/source/Classical_planet) resulted in seven being the number of days in a week.[1] 7 is often considered [lucky](/source/Luck) in [Western culture](/source/Western_culture) and is often seen as highly symbolic.

## Evolution of the Arabic digit

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For early [Brahmi numerals](/source/Brahmi_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, the [Cham](/source/Cham_script#Numerals) and [Khmer digit](/source/Khmer_script#Numerals) 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.[2] This is analogous to the horizontal stroke through the middle that is sometimes used in [handwriting](/source/Handwriting) in the Western world but which is almost never used in [computer fonts](/source/Computer_fonts). This horizontal stroke is, however, important to distinguish the glyph for seven from the glyph for [one](/source/1_(number)) 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.

On [seven-segment displays](/source/Seven-segment_display), 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](/source/Sharp_Corporation) and [Casio](/source/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](/source/Schindler_Group) elevators in the United States and Canada installed or modernized from the late 1990s onwards usually use a 16 segment display and show the digit 7 in a manner more similar to that of handwriting.

While the shape of the character for the digit 7 has an [ascender](/source/Ascender_(typography)) in most modern [typefaces](/source/Typeface), in typefaces with [text figures](/source/Text_figures) the character usually has a [descender](/source/Descender), as, for example, in .

Most people in Continental Europe,[3] Indonesia,[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] and some in Britain, Ireland, Israel, Canada, and 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 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](/source/Primary_school) in Russia, Ukraine, Bulgaria, Poland, other Slavic countries,[4] France,[5] Italy, Belgium, the Netherlands, Finland,[6] Romania, Germany, Greece,[7] and Hungary.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

## In mathematics

Seven, the fourth prime number, is not only a [Mersenne prime](/source/Mersenne_prime) (since 2 3 − 1 = 7 {\displaystyle 2^{3}-1=7} ) but also a [double Mersenne prime](/source/Double_Mersenne_prime) since the exponent, 3, is itself a Mersenne prime.[8] It is also a [Newman–Shanks–Williams prime](/source/Newman%E2%80%93Shanks%E2%80%93Williams_prime),[9] a [Woodall prime](/source/Woodall_prime),[10] a [factorial prime](/source/Factorial_prime),[11] a [Harshad number](/source/Harshad_number), a [lucky prime](/source/Lucky_prime),[12] a [happy number](/source/Happy_number) (happy prime),[13] a [safe prime](/source/Safe_prime) (the only Mersenne safe prime), a [sexy prime](/source/Sexy_prime), a [Leyland number of the second kind](/source/Leyland_number#Leyland_number_of_the_second_kind)[14] and [Leyland prime of the second kind](/source/Leyland_number#Leyland_number_of_the_second_kind)[15] ( 2 5 − 5 2 {\displaystyle 2^{5}-5^{2}} ), and the fourth [Heegner number](/source/Heegner_number).[16] 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](/source/Heptagon).[17] The [regular](/source/Regular_polygon) *n*-gons for *n* ⩽ 6 can be constructed by [compass and straightedge](/source/Compass_and_straightedge) alone, which makes the heptagon the first regular polygon that cannot be directly constructed with these simple tools.[18]

7 is the only number *D* for which the equation 2*n* − *D* = *x*2 has more than two solutions for *n* and *x* [natural](/source/Natural_number). In particular, the equation 2*n* − 7 = *x*2 is known as the [Ramanujan–Nagell equation](/source/Ramanujan%E2%80%93Nagell_equation). 7 is one of seven numbers in the positive [definite quadratic](/source/Quadratic_form) [integer matrix](/source/Integer_matrix) representative of all [odd](/source/Parity_(mathematics)) numbers: {1, 3, 5, 7, 11, 15, 33}.[19][20]

There are 7 [frieze groups](/source/Frieze_group) in two dimensions, consisting of [symmetries](/source/Symmetry_group) of the [plane](/source/Plane_(geometry)) whose group of [translations](/source/Translation_(geometry)) is [isomorphic](/source/Isomorphic) to the group of [integers](/source/Integer).[21] These are related to the [17](/source/17_(number)) [wallpaper groups](/source/Wallpaper_group) whose transformations and [isometries](/source/Isometry) repeat two-dimensional patterns in the plane.[22][23]

A [heptagon](/source/Heptagon) in [Euclidean space](/source/Euclidean_space) is unable to generate [uniform tilings](/source/Uniform_tiling) alongside other polygons, like the regular [pentagon](/source/Pentagon). However, it is one of fourteen polygons that can fill a [plane-vertex tiling](/source/Euclidean_tilings_by_convex_regular_polygons#Plane-vertex_tilings), in its case only alongside a regular [triangle](/source/Equilateral_triangle) and a 42-sided polygon ([3.7.42](https://en.wikipedia.org/wiki/File:3.7.42_vertex.png)).[24][25] Otherwise, for any regular *n*-sided [polygon](/source/Polygon), the maximum number of intersecting diagonals (other than through its center) is at most 7.[26]

In two dimensions, there are precisely seven [7-uniform](/source/Euclidean_tilings_by_convex_regular_polygons#k-uniform_tilings) *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*.[27][28]

The [Fano plane](/source/Fano_plane), the smallest possible [finite projective plane](/source/Finite_projective_plane), has 7 points and 7 lines arranged such that every line contains 3 points and 3 lines cross every point.[29] This is related to other appearances of the number seven in relation to [exceptional objects](/source/Exceptional_object), like the fact that the [octonions](/source/Octonion) contain seven distinct square roots of −1, [seven-dimensional vectors](/source/Seven-dimensional_cross_product) have a [cross product](/source/Cross_product), and the number of [equiangular lines](/source/Equiangular_lines) possible in seven-dimensional space is anomalously large.[30][31][32]

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

The lowest known dimension for an [exotic sphere](/source/Exotic_sphere) is the seventh dimension.[33][34]

In [hyperbolic space](/source/Hyperbolic_space), 7 is the highest dimension for non-simplex [hypercompact *Vinberg polytopes*](/source/Coxeter%E2%80%93Dynkin_diagram#Hypercompact_Coxeter_groups_(Vinberg_polytopes)) of rank *n + 4* mirrors, where there is one unique figure with eleven [facets](/source/Facet_(geometry)). On the other hand, such figures with rank *n + 3* mirrors exist in dimensions 4, 5, 6 and 8; *not* in 7.[35]

There are seven fundamental types of [catastrophes](/source/Catastrophe_theory).[36]

When rolling two standard six-sided [dice](/source/Dice), seven has a 1 in 6 probability of being rolled, the greatest of any number.[37] The opposite sides of a standard six-sided die always add to 7.

The [Millennium Prize Problems](/source/Millennium_Prize_Problems) are seven problems in [mathematics](/source/Mathematics) that were stated by the [Clay Mathematics Institute](/source/Clay_Mathematics_Institute) in 2000.[38] Currently, six of the problems remain [unsolved](/source/Unsolved_problems_in_mathematics).[39]

### 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 2097152 4782969 10000000 19487171 35831808 62748517

#### Decimal calculations

[999,999](/source/999%2C999_(number)) divided by 7 is exactly [142,857](/source/142%2C857_(number)). Therefore, when a [vulgar fraction](/source/Vulgar_fraction) with 7 in the [denominator](/source/Denominator) is converted to a [decimal](/source/Decimal) expansion, the result has the same six-[digit](/source/Numerical_digit) repeating sequence after the decimal point, but the sequence can start with any of those six digits.[40] In [decimal](/source/Decimal) representation, the [reciprocal](/source/Multiplicative_inverse) of 7 repeats six [digits](/source/Numerical_digit) (as 0.142857),[41][42] whose sum when [cycling](/source/Cyclic_number#Relation_to_repeating_decimals) back to [1](/source/1) is equal to 28.

## In science

### In psychology

- In Western culture, seven is consistently listed as people's favorite number[43][44]

- When guessing numbers 1–10, the number 7 is most likely to be picked[45]

## Classical antiquity

["Number Seven" by William Sidney Gibson Read by Ruth Golding for LibriVox](https://en.wikipedia.org/wiki/File:Number_Seven_by_William_Sidney_Gibson_-_read_by_Ruth_Golding_for_LibriVox%27s_Short_Nonfiction_Collection_Vol._031_(2013).ogg)

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

*Problems playing this file? See [media help](https://en.wikipedia.org/wiki/Help:Media).*

The [Pythagoreans](/source/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](/source/4)) with the spiritual (number [3](/source/3)).[46] In Pythagorean [numerology](/source/Numerology) the number 7 means spirituality.

## Culture

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](/source/Sexagesimal) number system, dividing by seven was the first division which resulted in infinitely [repeating fractions](/source/Repeating_fraction).[47]

- [Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics)

Wikiquote has quotations related to ***[7 (number)](https://en.wikiquote.org/wiki/7_(number))***.

Wikimedia Commons has media related to [7 (number)](https://commons.wikimedia.org/wiki/Category:7_(number)).

Look up ***[seven](https://en.wiktionary.org/wiki/seven)*** in Wiktionary, the free dictionary.

## References

1. **[^](#cite_ref-1)** [Carl B. Boyer](/source/Carl_Benjamin_Boyer), *A History of Mathematics* (1968) p.52, 2nd edn.

1. **[^](#cite_ref-2)** 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

1. **[^](#cite_ref-3)** Eeva Törmänen (September 8, 2011). ["Aamulehti: Opetushallitus harkitsee numero 7 viivan palauttamista"](https://web.archive.org/web/20110917083226/http://www.tekniikkatalous.fi/viihde/aamulehti+opetushallitus+harkitsee+numero+7+viivan+palauttamista/a682831). *Tekniikka & Talous* (in Finnish). Archived from [the original](http://www.tekniikkatalous.fi/viihde/aamulehti+opetushallitus+harkitsee+numero+7+viivan+palauttamista/a682831) on September 17, 2011. Retrieved September 9, 2011.

1. **[^](#cite_ref-4)** ["Education writing numerals in grade 1."](http://www.adu.by/modules.php?name=News&file=article&sid=858) [Archived](https://web.archive.org/web/20081002092040/http://www.adu.by/modules.php?name=News&file=article&sid=858) 2008-10-02 at the [Wayback Machine](/source/Wayback_Machine)(Russian)

1. **[^](#cite_ref-5)** ["Example of teaching materials for pre-schoolers"](http://www.pour-enfants.fr/jeux-imprimer/apprendre/les-chiffres/ecrire-les-chiffres.png)(French)

1. **[^](#cite_ref-6)** Elli Harju (August 6, 2015). [""Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?"](https://www.iltalehti.fi/uutiset/a/2015080620139397). *Iltalehti* (in Finnish).

1. **[^](#cite_ref-7)** ["Μαθηματικά Α' Δημοτικού"](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) [Mathematics for the First Grade] (PDF) (in Greek). Ministry of Education, Research, and Religions. p. 33. Retrieved May 7, 2018.

1. **[^](#cite_ref-8)** Weisstein, Eric W. ["Double Mersenne Number"](https://mathworld.wolfram.com/DoubleMersenneNumber.html). *mathworld.wolfram.com*. Retrieved 2020-08-06.

1. **[^](#cite_ref-9)** ["Sloane's A088165 : NSW primes"](https://oeis.org/A088165). *The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-06-01.

1. **[^](#cite_ref-10)** ["Sloane's A050918 : Woodall primes"](https://oeis.org/A050918). *The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-06-01.

1. **[^](#cite_ref-11)** ["Sloane's A088054 : Factorial primes"](https://oeis.org/A088054). *The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-06-01.

1. **[^](#cite_ref-12)** ["Sloane's A031157 : Numbers that are both lucky and prime"](https://oeis.org/A031157). *The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-06-01.

1. **[^](#cite_ref-13)** ["Sloane's A035497 : Happy primes"](https://oeis.org/A035497). *The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-06-01.

1. **[^](#cite_ref-14)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.). ["Sequence A045575 (Leyland numbers of the second kind)"](https://oeis.org/A045575). *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*. OEIS Foundation.

1. **[^](#cite_ref-15)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.). ["Sequence A123206 (Leyland prime numbers of the second kind)"](https://oeis.org/A123206). *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*. OEIS Foundation.

1. **[^](#cite_ref-16)** ["Sloane's A003173 : Heegner numbers"](https://oeis.org/A003173). *The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-06-01.

1. **[^](#cite_ref-17)** Weisstein, Eric W. ["Heptagon"](https://mathworld.wolfram.com/Heptagon.html). *mathworld.wolfram.com*. Retrieved 2020-08-25.

1. **[^](#cite_ref-18)** Weisstein, Eric W. ["7"](https://mathworld.wolfram.com/7.html). *mathworld.wolfram.com*. Retrieved 2020-08-07.

1. **[^](#cite_ref-19)** Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". [*Number Theory Volume I: Tools and Diophantine Equations*](https://link.springer.com/book/10.1007/978-0-387-49923-9). [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics). Vol. 239 (1st ed.). [Springer](/source/Springer_Science%2BBusiness_Media). pp. 312–314. [doi](/source/Doi_(identifier)):[10.1007/978-0-387-49923-9](https://doi.org/10.1007%2F978-0-387-49923-9). [ISBN](/source/ISBN_(identifier)) [978-0-387-49922-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-49922-2). [OCLC](/source/OCLC_(identifier)) [493636622](https://search.worldcat.org/oclc/493636622). [Zbl](/source/Zbl_(identifier)) [1119.11001](https://zbmath.org/?format=complete&q=an:1119.11001).

1. **[^](#cite_ref-20)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.). ["Sequence A116582 (Numbers from Bhargava's 33 theorem.)"](https://oeis.org/A116582). *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*. OEIS Foundation. Retrieved 2024-02-03.

1. **[^](#cite_ref-21)** Heyden, Anders; Sparr, Gunnar; Nielsen, Mads; Johansen, Peter (2003-08-02). [*Computer Vision – ECCV 2002: 7th European Conference on Computer Vision, Copenhagen, Denmark, May 28–31, 2002. Proceedings. Part II*](https://books.google.com/books?id=4yCqCAAAQBAJ&q=seven+frieze+groups&pg=PA661). Springer. p. 661. [ISBN](/source/ISBN_(identifier)) [978-3-540-47967-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-47967-3). A frieze pattern can be classified into one of the 7 frieze groups...

1. **[^](#cite_ref-22)** [Grünbaum, Branko](/source/Branko_Gr%C3%BCnbaum); [Shephard, G. C.](/source/G.C._Shephard) (1987). "Section 1.4 Symmetry Groups of Tilings". [*Tilings and Patterns*](https://archive.org/details/isbn_0716711931). New York: W. H. Freeman and Company. pp. 40–45. [doi](/source/Doi_(identifier)):[10.2307/2323457](https://doi.org/10.2307%2F2323457). [ISBN](/source/ISBN_(identifier)) [0-7167-1193-1](https://en.wikipedia.org/wiki/Special:BookSources/0-7167-1193-1). [JSTOR](/source/JSTOR_(identifier)) [2323457](https://www.jstor.org/stable/2323457). [OCLC](/source/OCLC_(identifier)) [13092426](https://search.worldcat.org/oclc/13092426). [S2CID](/source/S2CID_(identifier)) [119730123](https://api.semanticscholar.org/CorpusID:119730123).

1. **[^](#cite_ref-23)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.). ["Sequence A004029 (Number of n-dimensional space groups.)"](https://oeis.org/A004029). *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*. OEIS Foundation. Retrieved 2023-01-30.

1. **[^](#cite_ref-24)** [Grünbaum, Branko](/source/Branko_Gr%C3%BCnbaum); [Shepard, Geoffrey](/source/G.C._Shephard) (November 1977). ["Tilings by Regular Polygons"](https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf) (PDF). *[Mathematics Magazine](/source/Mathematics_Magazine)*. **50** (5). Taylor & Francis, Ltd.: 231. [doi](/source/Doi_(identifier)):[10.2307/2689529](https://doi.org/10.2307%2F2689529). [JSTOR](/source/JSTOR_(identifier)) [2689529](https://www.jstor.org/stable/2689529). [S2CID](/source/S2CID_(identifier)) [123776612](https://api.semanticscholar.org/CorpusID:123776612). [Zbl](/source/Zbl_(identifier)) [0385.51006](https://zbmath.org/?format=complete&q=an:0385.51006). Archived from [the original](http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf) (PDF) on 2016-03-03. Retrieved 2023-01-09.

1. **[^](#cite_ref-25)** Jardine, Kevin. ["Shield - a 3.7.42 tiling"](http://gruze.org/tilings/3_7_42_shield). *Imperfect Congruence*. Retrieved 2023-01-09. 3.7.42 as a unit facet in an irregular tiling.

1. **[^](#cite_ref-26)** [Poonen, Bjorn](/source/Bjorn_Poonen); Rubinstein, Michael (1998). ["The Number of Intersection Points Made by the Diagonals of a Regular Polygon"](https://math.mit.edu/~poonen/papers/ngon.pdf) (PDF). *SIAM Journal on Discrete Mathematics*. **11** (1). Philadelphia: [Society for Industrial and Applied Mathematics](/source/Society_for_Industrial_and_Applied_Mathematics): 135–156. [arXiv](/source/ArXiv_(identifier)):[math/9508209](https://arxiv.org/abs/math/9508209). [doi](/source/Doi_(identifier)):[10.1137/S0895480195281246](https://doi.org/10.1137%2FS0895480195281246). [MR](/source/MR_(identifier)) [1612877](https://mathscinet.ams.org/mathscinet-getitem?mr=1612877). [S2CID](/source/S2CID_(identifier)) [8673508](https://api.semanticscholar.org/CorpusID:8673508). [Zbl](/source/Zbl_(identifier)) [0913.51005](https://zbmath.org/?format=complete&q=an:0913.51005).

1. **[^](#cite_ref-27)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.). ["Sequence A068600 (Number of n-uniform tilings having n different arrangements of polygons about their vertices.)"](https://oeis.org/A068600). *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*. OEIS Foundation. Retrieved 2023-01-09.

1. **[^](#cite_ref-28)** [Grünbaum, Branko](/source/Branko_Gr%C3%BCnbaum); [Shepard, Geoffrey](/source/G.C._Shephard) (November 1977). ["Tilings by Regular Polygons"](https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf) (PDF). *[Mathematics Magazine](/source/Mathematics_Magazine)*. **50** (5). Taylor & Francis, Ltd.: 236. [doi](/source/Doi_(identifier)):[10.2307/2689529](https://doi.org/10.2307%2F2689529). [JSTOR](/source/JSTOR_(identifier)) [2689529](https://www.jstor.org/stable/2689529). [S2CID](/source/S2CID_(identifier)) [123776612](https://api.semanticscholar.org/CorpusID:123776612). [Zbl](/source/Zbl_(identifier)) [0385.51006](https://zbmath.org/?format=complete&q=an:0385.51006). Archived from [the original](http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf) (PDF) on 2016-03-03. Retrieved 2023-01-09.

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1. **[^](#cite_ref-30)** [Massey, William S.](/source/William_S._Massey) (December 1983). ["Cross products of vectors in higher dimensional Euclidean spaces"](https://web.archive.org/web/20210226011747/https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf) (PDF). *The American Mathematical Monthly*. **90** (10). [Taylor & Francis, Ltd](/source/Taylor_%26_Francis%2C_Ltd): 697–701. [doi](/source/Doi_(identifier)):[10.2307/2323537](https://doi.org/10.2307%2F2323537). [JSTOR](/source/JSTOR_(identifier)) [2323537](https://www.jstor.org/stable/2323537). [S2CID](/source/S2CID_(identifier)) [43318100](https://api.semanticscholar.org/CorpusID:43318100). [Zbl](/source/Zbl_(identifier)) [0532.55011](https://zbmath.org/?format=complete&q=an:0532.55011). Archived from [the original](https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf) (PDF) on 2021-02-26. Retrieved 2023-02-23.

1. **[^](#cite_ref-31)** [Baez, John C.](/source/John_Baez) (2002). ["The Octonions"](http://math.ucr.edu/home/baez/octonions/). *Bulletin of the American Mathematical Society*. **39** (2). [American Mathematical Society](/source/American_Mathematical_Society): 152–153. [doi](/source/Doi_(identifier)):[10.1090/S0273-0979-01-00934-X](https://doi.org/10.1090%2FS0273-0979-01-00934-X). [MR](/source/MR_(identifier)) [1886087](https://mathscinet.ams.org/mathscinet-getitem?mr=1886087). [S2CID](/source/S2CID_(identifier)) [586512](https://api.semanticscholar.org/CorpusID:586512).

1. **[^](#cite_ref-32)** Stacey, Blake C. (2021). *A First Course in the Sporadic SICs*. Cham, Switzerland: Springer. pp. 2–4. [ISBN](/source/ISBN_(identifier)) [978-3-030-76104-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-76104-2). [OCLC](/source/OCLC_(identifier)) [1253477267](https://search.worldcat.org/oclc/1253477267).

1. **[^](#cite_ref-33)** Behrens, M.; Hill, M.; Hopkins, M. J.; Mahowald, M. (2020). ["Detecting exotic spheres in low dimensions using coker J"](https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12301). *Journal of the London Mathematical Society*. **101** (3). [London Mathematical Society](/source/London_Mathematical_Society): 1173. [arXiv](/source/ArXiv_(identifier)):[1708.06854](https://arxiv.org/abs/1708.06854). [doi](/source/Doi_(identifier)):[10.1112/jlms.12301](https://doi.org/10.1112%2Fjlms.12301). [MR](/source/MR_(identifier)) [4111938](https://mathscinet.ams.org/mathscinet-getitem?mr=4111938). [S2CID](/source/S2CID_(identifier)) [119170255](https://api.semanticscholar.org/CorpusID:119170255). [Zbl](/source/Zbl_(identifier)) [1460.55017](https://zbmath.org/?format=complete&q=an:1460.55017).

1. **[^](#cite_ref-34)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.). ["Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)"](https://oeis.org/A001676). *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*. OEIS Foundation. Retrieved 2023-02-23.

1. **[^](#cite_ref-35)** Tumarkin, Pavel; Felikson, Anna (2008). ["On *d*-dimensional compact hyperbolic Coxeter polytopes with *d + 4* facets"](https://www.ams.org/journals/mosc/2008-69-00/S0077-1554-08-00172-6/S0077-1554-08-00172-6.pdf) (PDF). *Transactions of the Moscow Mathematical Society*. **69**. Providence, R.I.: [American Mathematical Society](/source/American_Mathematical_Society) (Translation): 105–151. [doi](/source/Doi_(identifier)):[10.1090/S0077-1554-08-00172-6](https://doi.org/10.1090%2FS0077-1554-08-00172-6). [MR](/source/MR_(identifier)) [2549446](https://mathscinet.ams.org/mathscinet-getitem?mr=2549446). [S2CID](/source/S2CID_(identifier)) [37141102](https://api.semanticscholar.org/CorpusID:37141102). [Zbl](/source/Zbl_(identifier)) [1208.52012](https://zbmath.org/?format=complete&q=an:1208.52012).

1. **[^](#cite_ref-36)** Antoni, F. de; Lauro, N.; Rizzi, A. (2012-12-06). [*COMPSTAT: Proceedings in Computational Statistics, 7th Symposium held in Rome 1986*](https://books.google.com/books?id=3L_sCAAAQBAJ&q=seven+fundamental+types+of+catastrophes&pg=PA13). Springer Science & Business Media. p. 13. [ISBN](/source/ISBN_(identifier)) [978-3-642-46890-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-46890-2). ...every catastrophe can be composed from the set of so called elementary catastrophes, which are of seven fundamental types.

1. **[^](#cite_ref-37)** Weisstein, Eric W. ["Dice"](https://mathworld.wolfram.com/Dice.html). *mathworld.wolfram.com*. Retrieved 2020-08-25.

1. **[^](#cite_ref-38)** ["Millennium Problems | Clay Mathematics Institute"](http://www.claymath.org/millennium-problems). *www.claymath.org*. Retrieved 2020-08-25.

1. **[^](#cite_ref-39)** ["Poincaré Conjecture | Clay Mathematics Institute"](https://web.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture). 2013-12-15. Archived from [the original](http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture) on 2013-12-15. Retrieved 2020-08-25.

1. **[^](#cite_ref-40)** Bryan Bunch, *The Kingdom of Infinite Number*. New York: W. H. Freeman & Company (2000): 82

1. **[^](#cite_ref-41)** Wells, D. (1987). [*The Penguin Dictionary of Curious and Interesting Numbers*](https://archive.org/details/penguindictionar0000well_f3y1/mode/2up). London: [Penguin Books](/source/Penguin_Books). pp. 171–174. [ISBN](/source/ISBN_(identifier)) [0-14-008029-5](https://en.wikipedia.org/wiki/Special:BookSources/0-14-008029-5). [OCLC](/source/OCLC_(identifier)) [39262447](https://search.worldcat.org/oclc/39262447). [S2CID](/source/S2CID_(identifier)) [118329153](https://api.semanticscholar.org/CorpusID:118329153).

1. **[^](#cite_ref-42)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.). ["Sequence A060283 (Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's moved to end).)"](https://oeis.org/A060283). *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*. OEIS Foundation. Retrieved 2024-04-02.

1. **[^](#cite_ref-43)** Gonzalez, Robbie (4 December 2014). ["Why Do People Love The Number Seven?"](https://gizmodo.com/why-do-people-love-the-number-seven-so-much-1666353786). *Gizmodo*. Retrieved 20 February 2022.

1. **[^](#cite_ref-44)** Bellos, Alex. ["The World's Most Popular Numbers \[Excerpt\]"](https://www.scientificamerican.com/article/most-popular-numbers-grapes-of-math/). *Scientific American*. Retrieved 20 February 2022.

1. **[^](#cite_ref-45)** Kubovy, Michael; Psotka, Joseph (May 1976). ["The predominance of seven and the apparent spontaneity of numerical choices"](https://www.researchgate.net/publication/232582800). *Journal of Experimental Psychology: Human Perception and Performance*. **2** (2): 291–294. [doi](/source/Doi_(identifier)):[10.1037/0096-1523.2.2.291](https://doi.org/10.1037%2F0096-1523.2.2.291). Retrieved 20 February 2022.

1. **[^](#cite_ref-46)** ["Number symbolism – 7"](https://www.britannica.com/topic/number-symbolism/7). *Encyclopedia Britannica*.

1. **[^](#cite_ref-47)** Muroi, Kazuo (2014) [The Origin of the Mystical Number Seven in Mesopotamian Culture: Division by Seven in the Sexagesimal Number System](https://arxiv.org/pdf/1407.6246.pdf)

## Further reading

- Wells, D. *[The Penguin Dictionary of Curious and Interesting Numbers](/source/The_Penguin_Dictionary_of_Curious_and_Interesting_Numbers)* London: [Penguin Group](/source/Penguin_Group) (1987): 70–71

v t e Z {\displaystyle \mathbb {Z} } Integers −2, −1 0 to 199 0 to 99 100 to 199 0 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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 to 399 200 to 299 300 to 399 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 318 323 325 341 353 359 360 363 365 369 377 384 400 to 999 400s, 500s, and 600s 700s, 800s, and 900s 400 420 440 495 496 500 501 511 512 555 600 610 613 616 666 693 700 720 743 744 777 786 800 801 836 840 880 881 888 900 911 971 987 999 1000s and 10,000s 1000s 1000 1001 1023 1024 1089 1093 1105 1234 1289 1458 1510 1728 1729 1980 1987 2000 2520 3000 3511 4000 5000 5040 6000 6174 7000 7744 7825 8000 8128 8192 9000 9855 9999 10,000s 10,000 16,807 20,000 30,000 40,000 50,000 60,000 64,079 65,535 65,536 65,537 70,000 80,000 90,000 100,000s to 10,000,000,000,000s 100,000 142,857 144,000 1,000,000 10,000,000 43,112,609 100,000,000 1,000,000,000 2,147,483,647 4,294,967,295 10,000,000,000 100,000,000,000 1,000,000,000,000 10,000,000,000,000 Large numbers

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