# 7

7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube.

 ← 6 7 8 →
Cardinalseven
Ordinal7th
(seventh)
Numeral systemseptenary
Factorizationprime
Prime4th
Divisors1, 7
Greek numeralΖ´
Roman numeralVII, vii
Greek prefixhepta-/hept-
Latin prefixseptua-
Binary1112
Ternary213
Senary116
Octal78
Duodecimal712
Greek numeralZ, ζ
Amharic
Arabic, Kurdish, Persian٧
Sindhi, Urdu۷
Bengali
Chinese numeral七, 柒
Devanāgarī
Telugu
Tamil
Hebrewז
Khmer
Thai
Malayalam
ArmenianԷ
Babylonian numeral𒐛
Egyptian hieroglyph𓐀
Morse code_ _...

As an early prime number in the series of positive integers, the number seven has greatly symbolic associations in religion, mythology, superstition and philosophy. The seven classical planets resulted in seven being the number of days in a week.[1] 7 is often considered lucky in Western culture and is often seen as highly symbolic. Unlike Western culture, in Vietnamese culture, the number seven is sometimes considered unlucky.[citation needed]

## Evolution of the Arabic digit

For early 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 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.[2] 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.

On seven-segment displays, 7 is the digit with the most common graphic variation (1, 6 and 9 also have variant glyphs). Most calculators use three line segments, but on Sharp, 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 an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender (⁊), as, for example, in  .

Most people in Continental Europe,[3] Indonesia,[citation needed] 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 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,[4] France,[5] Italy, Belgium, the Netherlands, Finland,[6] Romania, Germany, Greece,[7] and Hungary.[citation needed]

## In mathematics

Seven, the fourth prime number, is not only a Mersenne prime (since ${\displaystyle 2^{3}-1=7}$ ) but also a double Mersenne prime since the exponent, 3, is itself a Mersenne prime.[8] It is also a Newman–Shanks–Williams prime,[9] a Woodall prime,[10] a factorial prime,[11] a Harshad number, a lucky prime,[12] a happy number (happy prime),[13] a safe prime (the only Mersenne safe prime), a Leyland prime of the second kind and the fourth Heegner number.[14] 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.[15] 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.[16]

7 is the only number D for which the equation 2nD = x2 has more than two solutions for n and x natural. In particular, the equation 2n − 7 = x2 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}.[17][18]

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.[19] These are related to the 17 wallpaper groups whose transformations and isometries repeat two-dimensional patterns in the plane.[20][21]

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).[22][23] This is also one of twenty-one such configurations from seventeen combinations of polygons, that features the largest and smallest polygons possible.[24][25] Otherwise, for any regular n-sided 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 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, 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.[29] 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.[30][31][32]

The lowest known dimension for an exotic sphere is the seventh dimension.[33][34]

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.[35]

There are seven fundamental types of catastrophes.[36]

When rolling two standard six-sided 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 are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.[38] Currently, six of the problems remain unsolved.[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 823543 2097152 4782969 10000000 19487171 35831808 62748517

#### In decimal

In decimal representation, the reciprocal of 7 repeats six digits (as 0.142857),[40][41] whose sum when cycling back to 1 is equal to 28.

999,999 divided by 7 is exactly 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.[42]

## Classical antiquity

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).[46] In Pythagorean numerology the number 7 means spirituality.

References from classical antiquity to the number seven include:

## Religion and mythology

### Judaism

The number seven forms a widespread typological pattern within Hebrew scripture, including:

• Seven days (more precisely yom) of Creation, leading to the seventh day or Sabbath (Genesis 1)
• Seven-fold vengeance visited on upon Cain for the killing of Abel (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 the Israelites they would displace when they entered the land of Israel (Deuteronomy 7:1)
• Seven days (de jure, but de facto eight days) of the Passover feast (Exodus 13:3–10)
• Seven-branched candelabrum or Menorah (Exodus 25)
• Seven trumpets played 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 the House of Wisdom (Proverbs 9:1)
• Seven archangels in the deuterocanonical Book of Tobit (12:15)

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

• Seven divisions of the weekly readings or aliyah of the Torah
• Seven aliyot on Shabbat
• Seven blessings recited under the chuppah during 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
• Seven Ushpizzin prayers to the Jewish patriarchs during the holiday of Sukkot

### Christianity

Following the tradition of the Hebrew Bible, the New Testament likewise uses the number seven as part of a typological pattern:

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

### Islam

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

### Hinduism

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

Other references to the number seven in Eastern traditions include:

### Other references

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

## Notes

1. ^ Carl B. Boyer, A History of Mathematics (1968) p.52, 2nd edn.
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
3. ^ Eeva Törmänen (September 8, 2011). "Aamulehti: Opetushallitus harkitsee numero 7 viivan palauttamista". Tekniikka & Talous (in Finnish). Archived from the original on September 17, 2011. Retrieved September 9, 2011.
4. ^ "Education writing numerals in grade 1." Archived 2008-10-02 at the Wayback Machine(Russian)
5. ^
6. ^ Elli Harju (August 6, 2015). ""Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?". Iltalehti (in Finnish).
7. ^ "Μαθηματικά Α' Δημοτικού" [Mathematics for the First Grade] (PDF) (in Greek). Ministry of Education, Research, and Religions. p. 33. Retrieved May 7, 2018.
8. ^ Weisstein, Eric W. "Double Mersenne Number". mathworld.wolfram.com. Retrieved 2020-08-06.
9. ^ "Sloane's A088165 : NSW primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. ^ "Sloane's A050918 : Woodall primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ "Sloane's A088054 : Factorial primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. ^ "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. ^ "Sloane's A035497 : Happy primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
14. ^ "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
15. ^ Weisstein, Eric W. "Heptagon". mathworld.wolfram.com. Retrieved 2020-08-25.
16. ^ Weisstein, Eric W. "7". mathworld.wolfram.com. Retrieved 2020-08-07.
17. ^ Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. Vol. 239 (1st ed.). Springer. pp. 312–314. doi:10.1007/978-0-387-49923-9. ISBN 978-0-387-49922-2. OCLC 493636622. Zbl 1119.11001.
18. ^ Sloane, N. J. A. (ed.). "Sequence A116582 (Numbers from Bhargava's 33 theorem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-03.
19. ^ 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. Springer. p. 661. ISBN 978-3-540-47967-3. A frieze pattern can be classified into one of the 7 frieze groups...
20. ^ Grünbaum, Branko; Shephard, G. C. (1987). "Section 1.4 Symmetry Groups of Tilings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 40–45. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
21. ^ Sloane, N. J. A. (ed.). "Sequence A004029 (Number of n-dimensional space groups.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-30.
22. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 231. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
23. ^ Jardine, Kevin. "Shield - a 3.7.42 tiling". Imperfect Congruence. Retrieved 2023-01-09. 3.7.42 as a unit facet in an irregular tiling.
24. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 229–230. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
25. ^ Dallas, Elmslie William (1855). "Part II. (VII): Of the Circle, with its Inscribed and Circumscribed Figures − Equal Division and the Construction of Polygons". The Elements of Plane Practical Geometry. London: John W. Parker & Son, West Strand. p. 134.
"...It will thus be found that, including the employment of the same figures, there are seventeen different combinations of regular polygons by which this may be effected; namely, —
When three polygons are employed, there are ten ways; viz., 6,6,63.7.423,8,243,9,183,10,153,12,124,5,204,6,124,8,85,5,10.
With four polygons there are four ways, viz., 4,4,4,43,3,4,123,3,6,63,4,4,6.
With five polygons there are two ways, viz., 3,3,3,4,43,3,3,3,6.
With six polygons one way — all equilateral triangles [ 3.3.3.3.3.3 ]."
Note: the only four other configurations from the same combinations of polygons are: 3.4.3.12, (3.6)2, 3.4.6.4, and 3.3.4.3.4.
26. ^ Poonen, Bjorn; Rubinstein, Michael (1998). "The Number of Intersection Points Made by the Diagonals of a Regular Polygon" (PDF). SIAM Journal on Discrete Mathematics. 11 (1). Philadelphia: Society for Industrial and Applied Mathematics: 135–156. arXiv:math/9508209. doi:10.1137/S0895480195281246. MR 1612877. S2CID 8673508. Zbl 0913.51005.
27. ^ Sloane, N. J. A. (ed.). "Sequence A068600 (Number of n-uniform tilings having n different arrangements of polygons about their vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-09.
28. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 236. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
29. ^ Pisanski, Tomaž; Servatius, Brigitte (2013). "Section 1.1: Hexagrammum Mysticum". Configurations from a Graphical Viewpoint. Birkhäuser Advanced Texts (1 ed.). Boston, MA: Birkhäuser. pp. 5–6. doi:10.1007/978-0-8176-8364-1. ISBN 978-0-8176-8363-4. OCLC 811773514. Zbl 1277.05001.
30. ^ Massey, William S. (December 1983). "Cross products of vectors in higher dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10). Taylor & Francis, Ltd: 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Zbl 0532.55011. Archived from the original (PDF) on 2021-02-26. Retrieved 2023-02-23.
31. ^ Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2). American Mathematical Society: 152–153. doi:10.1090/S0273-0979-01-00934-X. MR 1886087. S2CID 586512.
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34. ^ Sloane, N. J. A. (ed.). "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-23.
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