# 7

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

7 (seven) is the natural number following 6 and preceding 8. It is a prime number, and is often considered lucky in Western culture, and is often seen as highly symbolic.

It is the first number whose pronunciation contains more than one syllable, not counting 0.

## Evolution of the Arabic digit

In the beginning, Indians wrote 7 more or less in one stroke as a curve that looks like an uppercase ⟨J⟩ vertically inverted. The western Ghubar Arabs' 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 Arabs developed the digit from a form that looked something like our 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.[1] 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 early 12th century the longer line diagonal was drawn in a rather semicircular transverse line.

On the seven-segment displays of pocket calculators and digital watches, 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,[2] and some in Britain and Ireland as well as Latin America, write 7 with a line in 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 for primary school in Russia, Ukraine, Bulgaria, Poland, other Slavic countries,[3] France, Italy, Belgium, Finland,[4] Romania, Germany, Greece,[5] and Hungary.[6][failed verification]

## Mathematics

Seven, the fourth prime number, is not only a Mersenne prime (since 23 − 1 = 7) but also a double Mersenne prime since the exponent, 3, is itself a Mersenne prime.[7] It is also a Newman–Shanks–Williams prime,[8] a Woodall prime,[9] a factorial prime,[10] a lucky prime,[11] a happy number (happy prime),[12] a safe prime (the only Mersenne safe prime), and the fourth Heegner number.[13]

• Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers. (See Lagrange's four-square theorem#Historical development.)
• Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree.
• 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 the only dimension, besides the familiar 3, in which a vector cross product can be defined.
• 7 is the lowest dimension of a known exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere.
• 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.[14] For example, 1/7 = 0.142857 142857... and 2/7 = 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.
Graph of the probability distribution of the sum of 2 six-sided dice

### Basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 15 25 50 100 1000
7 × x 7 14 21 28 35 42 49 56 63 70 105 175 350 700 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 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
7x 7 49 343 2401 16807 117649 823543 5764801 40353607 282475249
x7 1 128 2187 16384 78125 279936 823543 2097152 4782969 10000000
Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100
110 120 130 140 150 200 250 500 1000 10000 100000 1000000
x7 1 5 137 217 267 347 427 557 1017 1147 1307 1437 1567 2027
2157 2317 2447 2607 3037 4047 5057 13137 26267 411047 5643557 113333117

## Notes

1. 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
2. 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.
3. "Education writing numerals in grade 1." Archived 2008-10-02 at the Wayback Machine(Russian)
4. Elli Harju (August 6, 2015). ""Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?". Iltalehti (in Finnish).
5. "Μαθηματικά Α' Δημοτικού" [Mathematics for the First Grade] (PDF) (in Greek). Ministry of Education, Research, and Religions. p. 33. Retrieved May 7, 2018.
6. Weisstein, Eric W. "Double Mersenne Number". mathworld.wolfram.com. Retrieved 2020-08-06.
7. "Sloane's A088165 : NSW primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
8. "Sloane's A050918 : Woodall primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
9. "Sloane's A088054 : Factorial primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. "Sloane's A035497 : Happy primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 82
14. Weisstein, Eric W. "Heptagon". mathworld.wolfram.com. Retrieved 2020-08-25.
15. Weisstein, Eric W. "7". mathworld.wolfram.com. Retrieved 2020-08-07.
16. "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
17. 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...
18. Antoni, F. de; Lauro, N.; Rizzi, A. (2012-12-06). COMPSTAT: Proceedings in Computational Statistics, 7th Symposium held in Rome 1986. Springer Science & Business Media. p. 13. ISBN 978-3-642-46890-2. ...every catastrophe can be composed from the set of so called elementary catastrophes, which are of seven fundamental types.
19. Weisstein, Eric W. "Dice". mathworld.wolfram.com. Retrieved 2020-08-25.
20. "Millennium Problems | Clay Mathematics Institute". www.claymath.org. Retrieved 2020-08-25.
21. "Poincaré Conjecture | Clay Mathematics Institute". 2013-12-15. Archived from the original on 2013-12-15. Retrieved 2020-08-25.