11 (number)

"Eleven" derives from the Old English ęndleofon, which is first attested in Bede's late 9th-century Ecclesiastical History of the English People.^[2][3] It has cognates in every Germanic language (for example, German elf), whose Proto-Germanic ancestor has been reconstructed as *ainalifa-,^[4] from the prefix *aina- (adjectival "one") and suffix *-lifa-, of uncertain meaning.^[3] It is sometimes compared with the Lithuanian vienúolika, though -lika is used as the suffix for all numbers from 11 to 19 (analogously to "-teen").^[3]

The Old English form has closer cognates in Old Frisian, Saxon, and Norse, whose ancestor has been reconstructed as *ainlifun. This was formerly thought to be derived from Proto-Germanic *tehun ("ten");^[3][5] it is now sometimes connected with *leikʷ- or *leip- ("left; remaining"), with the implicit meaning that "one is left" after counting to ten.^[3]

While 11 has its own name in Germanic languages such as English, German, or Swedish, and some Latin-based languages such as Spanish, Portuguese, and French, it is the first compound number in many other languages: Chinese 十一 shí yī, Korean 열하나 yeol hana or 십일 ship il.

11 is the fifth prime number, and the third super-prime. 11 forms a twin prime with 13,^[6] and sexy pair with 5 and 17. It is the first member of the second prime quadruplet (11, 13, 17, 19).^[7]

The first prime exponent that does not yield a Mersenne prime is 11, where ${\displaystyle 2^{11}-1=2047}$ , which is the first composite generalized Mersenne number. 11 is the first strong prime,^[8] such that for a prime ${\displaystyle p_{n}}$  there is ${\displaystyle p_{n}>{\tfrac {(prime_{(n-1)}+prime_{(n+1)})}{2}}}$ , and it is also the second good prime, whose square is greater than the product of any two prime numbers at the same number of positions before and after it in the sequence of prime numbers.^[9]

11 is the second member of the second pair (5, 11) of Brown numbers. Only three such pairs of numbers ${\displaystyle n}$  and ${\displaystyle m}$  where ${\displaystyle n!+1=m^{2}}$  are known. In abstract algebra, 11 is the fifth consecutive supersingular prime that divides the order of the largest sporadic group.^[10] Rows in Pascal's triangle can be seen as representation of powers of 11.^[11]

An 11-sided polygon is called a hendecagon, or undecagon. The complete graph ${\displaystyle K_{11}}$  has a total of 55 edges, which collectively represent the diagonals and sides of a hendecagon. A regular hendecagon cannot be constructed with a compass and straightedge alone, as 11 is not a product of distinct Fermat primes, and it is also the first polygon that is not able to be constructed with the aid of an angle trisector.^[12]

11 of 35 hexominoes can fold in a net to form a cube, while 11 of 66 octiamonds can fold into a regular octahedron. 11 appears as counts of uniform tessellations in various dimensions and spaces. There are 11 regular and semiregular convex uniform tilings in the Euclidean plane, which are dual to the 11 Laves tilings.^[13] 11 is also the number of regular complex apeirogons, which are tilings with polygons that have a countably infinite number of sides.^[14] Meanwhile, there are also 11 regular paracompact hyperbolic honeycombs with infinite facets and vertex figures in the third dimension.^[15] Outside of Euclidean geometry, 11 is the total number of regular hyperbolic honeycombs in the fourth dimension: 9 compact solutions are generated from regular 4-polytopes and regular star 4-polytopes, alongside 2 paracompact solutions.^[15]

In differential geometry, there are 11 orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the 3-variable Helmholtz equation can be solved using the separation of variables technique.

The Mathieu group ${\displaystyle \mathrm {M} _{11}}$  is the smallest of twenty-six sporadic groups, an algebraic structure defined as a permutation group on eleven objects. It has order ${\displaystyle 7920=2^{4}\cdot 3^{2}\cdot 5\cdot 11=8\cdot 9\cdot 10\cdot 11}$ , with 11 as its largest prime factor,. Its group action is the automorphism group of Steiner system ${\displaystyle \operatorname {S} (4,5,11)}$ , with an induced action on unordered pairs of points that gives a rank 3 action on 55 points. ${\displaystyle \mathrm {M} _{11}}$  is also the maximal subgroup Mathieu group ${\displaystyle \mathrm {M} _{12}}$ , which has an order of ${\displaystyle 95040=2^{6}\cdot 3^{3}\cdot 5\cdot 11=8\cdot 9\cdot 10\cdot 11\cdot 12}$ , where 11 is also its largest prime factor.^{[citation needed]}

In chemistry, Group 11 of the Periodic Table of the Elements (IUPAC numbering) consists of the three coinage metals copper, silver, and gold known from antiquity, and roentgenium, a recently synthesized superheavy element. 11 is the number of spacetime dimensions in M-theory.

Apollo 11 was the first crewed spacecraft to land on the Moon. In our solar system, the Sun has a sunspot cycle's periodicity that is approximately 11 years.

The interval of an octave plus a fourth is an 11th. A complete 11th chord has almost every note of a diatonic scale. Regarding musical instruments, there are 11 thumb keys on a bassoon, not counting the whisper key. (A few bassoons have a 12th thumb key.)

In sports, there are 11 players on an association football (soccer) team, 11 players on an American football team during play, 11 players on a cricket team on the field, and 11 players in a field hockey team. In the game of blackjack, an ace can count as either one or 11, whichever is more advantageous for the player.

The stylized maple leaf on the Flag of Canada has 11 points. The CA$ one-dollar loonie is in the shape of an 11-sided hendecagon, and clocks depicted on Canadian currency, like the Canadian 50-dollar bill, show 11:00.

Being one hour before 12:00, the eleventh hour means the last possible moment to take care of something, and often implies a situation of urgent danger or emergency (see Doomsday clock).

The number 11 (alongside its multiples 22 and 33) are master numbers in numerology, especially in New Age.^[16]

Grimes, James. "Eleven". Numberphile. Brady Haran. Archived from the original on 2017-10-15. Retrieved 2016-01-03.