11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.

Name

"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]

In languages

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.

In mathematics

Eleven is the fifth prime number, and the first two-digit numeric palindrome in decimal. It forms a twin prime with 13,^[6] and it is the first member of the second prime quadruplet (11, 13, 17, 19).^[7] 11 is the first prime exponent that does not yield a Mersenne prime, where ${\displaystyle 2^{11}-1=2047}$, which is composite. On the other hand, the eleventh prime number 31 is the third Mersenne prime, while the thirty-first prime number 127 is not only a Mersenne prime but also the second double Mersenne prime. 11 is also the fifth Heegner number, meaning that the ring of integers of the field ${\displaystyle \mathbb {Q} ({\sqrt {-11}})}$ has the property of unique factorization and class number 1. 11 is the first prime repunit ${\displaystyle R_{2}}$ in decimal (and simply, the first repunit),^[8] as well as the second unique prime in base ten.^[9] It is the first strong prime,^[10] the second good prime,^[11] the third super-prime, the fourth Lucas prime,^[12] and the fifth consecutive supersingular prime.^[13]

The rows of Pascal's triangle can be seen as representation of the powers of 11.^[14]

11 of 35 hexominoes can fold in a net to form a cube, while 11 of 66 octiamonds can fold into a regular octahedron.

Copper engraving of a hendecagon, by Anton Ernst Burkhard von Birckenstein (1698)

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.^[15]

11 and some of its multiples appear 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.^[16]

22 edge-to-edge uniform tilings with convex and star polygons, and 33 uniform tilings with zizgzag apeirogons that alternate between two angles.^[17][18]

• 11 regular complex apeirogons, which are tilings with polygons that have a countably infinite number of sides. 8 solutions of the form _p{q}_r satisfy δ^p,r
  ₂ in ${\displaystyle \mathbb {C} }$ where ${\displaystyle q}$ is constrained to ${\displaystyle q=2/(1-(p+r)/pr)}$, while three contain affine nodes and include infinite solutions, two in ${\displaystyle \mathbb {C} }$, and one in ${\displaystyle \mathbb {C} ^{2}}$.^[19]

22 regular complex apeirohedra of the form _p{a}_q{b}_r, where 21 exist in ${\displaystyle \mathbb {C} ^{2}}$ and 1 in ${\displaystyle \mathbb {C} ^{3}}$.^[20]

• 11 regular paracompact hyperbolic honeycombs with infinite facets and vertex figures in the third dimension.^[21]

11 total 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.^[21]

• 55 uniform Euclidean 4-honeycombs exist in the fourth dimension, and 66 uniform Euclidean 5-honeycombs exist in the fifth dimension.

The 11-cell is a self-dual abstract 4-polytope with 11 vertices, 55 edges, 55 triangular faces, and 11 hemi-icosahedral cells. It is universal in the sense that it is the only abstract polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures. The 11-cell contains the same number of vertices and edges as the complete graph ${\displaystyle K_{11}}$ and the 10-simplex, a regular polytope in 10 dimensions.

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.

Mathieu group ${\displaystyle \mathrm {M} _{11}}$ is the smallest of twenty-six sporadic groups, defined as a sharply 4-transitive 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, and a minimal faithful complex representation in ten dimensions. 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. Mathieu group ${\displaystyle \mathrm {M} _{12}}$, on the other hand, is formed from the permutations of projective special linear group ${\displaystyle \operatorname {PSL_{2}} (1,1)}$ with those of ${\displaystyle (2,10)(3,4)(5,9)(6,7)}$. It is the second-smallest sporadic group, and holds ${\displaystyle \mathrm {M} _{11}}$ as a maximal subgroup and point stabilizer, with an order equal to ${\displaystyle 95040=2^{6}\cdot 3^{3}\cdot 5\cdot 11=8\cdot 9\cdot 10\cdot 11\cdot 12}$, where 11 is also its largest prime factor, like ${\displaystyle \mathrm {M} _{11}}$. ${\displaystyle \mathrm {M} _{12}}$ also centralizes an element of order 11 in the friendly giant ${\displaystyle \mathrm {F} _{1}}$, the largest sporadic group, and holds an irreducible faithful complex representation in eleven dimensions.

The first eleven prime numbers (from 2 through 31) are consecutive supersingular primes that divide the order of the friendly giant, with the remaining four supersingular primes (41, 47, 59, and 71) lying between five non-supersingular primes.^[13] Only five of twenty-six sporadic groups do not contain 11 as a prime factor that divides their group order (${\displaystyle \mathrm {J} _{2}}$, ${\displaystyle \mathrm {J} _{3}}$, ${\displaystyle \mathrm {Ru} }$, ${\displaystyle \mathrm {He} }$, and ${\displaystyle \mathrm {Th} }$). 11 is also not a prime factor of the order of the Tits group ${\displaystyle \mathrm {T} }$, which is sometimes categorized as non-strict group of Lie type, or sporadic group.

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; the largest pair (7, 71) satisfies ${\displaystyle 5041=7!+1}$. In this last pair 5040 is the factorial of 7, which is divisible by all integers less than 13 with the exception of 11. The members of the first pair (4,5) multiply to 20 — the prime index of 71— that is also eleventh composite number.^[22]

Within safe and Sophie Germain primes of the form ${\displaystyle 2p+1}$, 11 is the third safe prime, from a ${\displaystyle p}$ of 5,^[23] and the fourth Sophie Germain prime, which yields 23.^[24]

In science

• 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.
• The number of spacetime dimensions in M-theory.

In music

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

In mysticism

The number 11 (alongside its multiples 22 and 33) are master numbers in numerology, especially in New Age.^[29]

In Canada

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.

In other fields

• 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).
• 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.

See also

• 11:11 (numerology)
• XI

References

External links

Look up eleven in Wiktionary, the free dictionary.

Wikimedia Commons has media related to 11 (number).

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