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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.

Evolution of the glyph

In the beginning, various Hindus 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 character more rectilinear. The eastern Arabs developed the character from a 6 lookalike into an uppercase V lookalike. Both modern Arab forms influenced the European form, a two-stroke character consisting of a horizontal upper line joined at its right to a line going down to the bottom left corner, a line that is slightly curved in some font variants. As is the case with the European glyph, the Cham and Khmer glyph for 7 also evolved to look like their glyph for 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 above the glyph.^[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 writings that use 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 number with the most common glyph 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 as ① in the illustration to the right.

While the shape of the 7 character 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 character from the number one, as the two can appear similar when written in certain styles of handwriting. This glyph 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. It is also a Newman–Shanks–Williams prime,^[7] a Woodall prime,^[8] a factorial prime,^[9] a lucky prime,^[10] a happy number (happy prime),^[11] a safe prime (the only Mersenne safe prime), and the fourth Heegner number.^[12]

• 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 2ⁿ − D = 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.^[13] 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.

• A seven-sided shape is a heptagon. The regular n-gons for n ≤ 6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons (including seven) are called heptagonal numbers. Seven is also a centered hexagonal number.^[14]
• Seven is the first integer reciprocal (multiplicative inverse) with infinitely repeating sexagesimal representation.
• There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers.
• There are seven fundamental types of catastrophes.

Graph of the probability distribution of the sum of 2 six-sided dice

• When rolling two standard six-sided dice, seven has a 6 in 6² (or 1/6) probability of being rolled (1–6, 6–1, 2–5, 5–2, 3–4, or 4–3), the greatest of any number.
• The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved.
• 7 is the last digit of Graham's number.

Basic calculations

See also

• Septenary (numeral system)
• Septenary (Theosophy)
• Seven climes
• Year Seven (School)
• Se7en (disambiguation)
• Sevens (disambiguation)
• Luck
• One-seventh area triangle
• Diatonic scale

Notes

References

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