11 (number)

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 [math]\displaystyle{ 2^{11}-1=2047 }[/math], 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 [math]\displaystyle{ \mathbb{Q}(\sqrt{-11}) }[/math] has the property of unique factorization and class number 1. 11 is the first prime repunit [math]\displaystyle{ R_{2} }[/math] 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 [math]\displaystyle{ K_{11} }[/math] 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 δp,r2 in [math]\displaystyle{ \Complex }[/math] where [math]\displaystyle{ q }[/math] is constrained to [math]\displaystyle{ q=2/(1-(p+r)/pr) }[/math], while three contain affine nodes and include infinite solutions, two in [math]\displaystyle{ \Complex }[/math], and one in [math]\displaystyle{ \Complex^2 }[/math].^[19]

22 regular complex apeirohedra of the form _p{a}_q{b}_r, where 21 exist in [math]\displaystyle{ \Complex^2 }[/math] and 1 in [math]\displaystyle{ \Complex^3 }[/math].^[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 [math]\displaystyle{ K_{11} }[/math] 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 [math]\displaystyle{ \mathrm{M}_{11} }[/math] is the smallest of twenty-six sporadic groups, defined as a sharply 4-transitive permutation group on eleven objects. It has order [math]\displaystyle{ 7920 =2^{4}\cdot3^{2}\cdot5\cdot11 = 8\cdot9\cdot10\cdot11 }[/math], 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 [math]\displaystyle{ \operatorname{S}(4,5,11) }[/math], with an induced action on unordered pairs of points that gives a rank 3 action on 55 points. Mathieu group [math]\displaystyle{ \mathrm{M}_{12} }[/math], on the other hand, is formed from the permutations of projective special linear group [math]\displaystyle{ \operatorname{PSL_2}(1,1) }[/math] with those of [math]\displaystyle{ (2,10)(3,4)(5,9)(6,7) }[/math]. It is the second-smallest sporadic group, and holds [math]\displaystyle{ \mathrm{M}_{11} }[/math] as a maximal subgroup and point stabilizer, with an order equal to [math]\displaystyle{ 95040 = 2^6\cdot3^3\cdot5\cdot11 = 8\cdot9\cdot10\cdot11\cdot12 }[/math], where 11 is also its largest prime factor, like [math]\displaystyle{ \mathrm{M}_{11} }[/math]. [math]\displaystyle{ \mathrm{M}_{12} }[/math] also centralizes an element of order 11 in the friendly giant [math]\displaystyle{ \mathrm {F}_{1} }[/math], 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 ([math]\displaystyle{ \mathrm{J}_2 }[/math], [math]\displaystyle{ \mathrm{J}_3 }[/math], [math]\displaystyle{ \mathrm{Ru} }[/math], [math]\displaystyle{ \mathrm{He} }[/math], and [math]\displaystyle{ \mathrm{Th} }[/math]). 11 is also not a prime factor of the order of the Tits group [math]\displaystyle{ \mathrm{T} }[/math], 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 [math]\displaystyle{ n }[/math] and [math]\displaystyle{ m }[/math] where [math]\displaystyle{ n!+1 = m^2 }[/math] are known; the largest pair (7, 71) satisfies [math]\displaystyle{ 5041 = 7!+1 }[/math]. 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 [math]\displaystyle{ 2p+1 }[/math], 11 is the third safe prime, from a [math]\displaystyle{ p }[/math] of 5,^[23] and the fourth Sophie Germain prime, which yields 23.^[24]

In decimal

11 is the smallest two-digit prime number. On the seven-segment display of a calculator, it is both a strobogrammatic prime and a dihedral prime.^[25]

Multiples of 11 by one-digit numbers yield palindromic numbers with matching double digits: 00, 11, 22, 33, 44, etc.

The sum of the first 11 non-zero positive integers, equivalently the 11th triangular number, is 66. On the other hand, the sum of the first 11 integers, from zero to ten, is 55.

The first four powers of 11 yield palindromic numbers: 11¹ = 11, 11² = 121, 11³ = 1331, and 11⁴ = 14641.

11 is the 11th index or member in the sequence of palindromic numbers, and 121, equal to [math]\displaystyle{ 11\times 11 }[/math], is the 22nd.^[26]

The factorial of 11, [math]\displaystyle{ 11!=39916800 }[/math], has about a 0.2% difference to the round number [math]\displaystyle{ 4\times 10^{7} }[/math], or 40 million. Among the first 100 factorials, the next closest to a round number is 96 ([math]\displaystyle{ 96! \approx 9.91678\times 10^{149} }[/math]), which is about 0.8% less than 10¹⁵⁰.^[27]

If a number is divisible by 11, reversing its digits will result in another multiple of 11. As long as no two adjacent digits of a number added together exceed 9, then multiplying the number by 11, reversing the digits of the product, and dividing that new number by 11 will yield a number that is the reverse of the original number; as in:

142,312 × 11 = 1,565,432 → 2,345,651 ÷ 11 = 213,241.

Divisibility tests

A simple test to determine whether an integer is divisible by 11 is to take every digit of the number in an odd position and add them, then take the remaining digits and add them. If the difference between the two sums is a multiple of 11, including 0, then the number is divisible by 11.^[28] For instance, with the number 65,637:

This technique also works with groups of digits rather than individual digits, so long as the number of digits in each group is odd, although not all groups have to have the same number of digits. If one uses three digits in each group, one gets from 65,637 the calculation,

Another test for divisibility is to separate a number into groups of two consecutive digits (adding a leading zero if there is an odd number of digits), and then add the numbers so formed; if the result is divisible by 11, the number is divisible by 11:

This also works by adding a trailing zero instead of a leading one, and with larger groups of digits, provided that each group has an even number of digits (not all groups have to have the same number of digits):

Multiplying 11

An easy way to multiply numbers by 11 in base 10 is:

If the number has:

• 1 digit, replicate the digit: 2 × 11 becomes 22.
• 2 digits, add the 2 digits and place the result in the middle: 47 × 11 becomes 4 (11) 7 or 4 (10+1) 7 or (4+1) 1 7 or 517.
• 3 digits, keep the first digit in its place for the result's first digit, add the first and second digits to form the result's second digit, add the second and third digits to form the result's third digit, and keep the third digit as the result's fourth digit. For any resulting numbers greater than 9, carry the 1 to the left.
  123 × 11 becomes 1 (1+2) (2+3) 3 or 1353.
  481 × 11 becomes 4 (4+8) (8+1) 1 or 4 (10+2) 9 1 or (4+1) 2 9 1 or 5291.
• 4 or more digits, follow the same pattern as for 3 digits.

List of basic calculations

In other bases

In duodecimal and higher bases (such as hexadecimal), 11 is represented as B, E, Z or ↋ (el), where 10 is A, T, W, X or ↊ (dek).

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.

Astronomy

• Apollo 11 was the first crewed spacecraft to land on the Moon.
• A sunspot cycle's periodicity is approximately 11 years.

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

• Grimes, James. "Eleven". Numberphile. Brady Haran. http://www.numberphile.com/videos/11.html. 

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