11th Signal Brigade and Headquarters West Midlands

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 ', which is first attested in Bede's late 9th-century ''Ecclesiastical History of the English People''. It has cognates in every Germanic language (for example, German ), whose Proto-Germanic ancestor has been reconstructed as , from the prefix (adjectival "one") and suffix , of uncertain meaning. It is sometimes compared with the Lithuanian ', though ' is used as the suffix for all numbers from 11 to 19 (analogously to "-teen").

The Old English form has closer cognates in Old Frisian, Saxon, and Norse, whose ancestor has been reconstructed as . This was formerly thought to be derived from Proto-Germanic ("ten"); it is now sometimes connected with or ("left; remaining"), with the implicit meaning that "one is left" after counting to ten.''Oxford English Dictionary'', 1st ed. "eleven, ''adj.'' and ''n.''" Oxford University Press (Oxford), 1891.

In English, "eleven" is the only two-digit number that does not contain the letter T.

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: Italian , Chinese , Korean or .

In mathematics

11 is the fifth prime number, and the first two-digit numeric palindrome. The next prime number is 13, with which it comprises a twin prime. 11 is the first repunit prime, the first strong prime, the second unique prime, the second good prime, the third super-prime, and the fourth Lucas prime.

11 is the first prime number that is not an exponent for a Mersenne prime, as 2¹¹ − 1 = 2047, which is composite.

11 is a Heegner number, meaning that the ring of integers of the field $\mathbb(\sqrt)$ has the property of unique factorization. As a consequence, there exists at most one point on the elliptic curve ''x''³ = ''y''² + 11 that has positive-integer coordinates. In this case, this unique point is ( 15, 58).

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 rows of Pascal's Triangle can be seen as representation of the powers of 11.

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

An 11-sided polygon is called a hendecagon or undecagon. The complete graph ''K''₁₁ has a total of 55 edges, which collectively represent the diagonals and sides of a hendecagon.

The number 11 appears in tessellations in various dimensions and geometrical spaces; there are:

* 11 regular complex apeirogons, which are tilings with polygons that have a countably infinite number of sides. 8 solutions satisfy δ in $\Complex$ where ''q'' is constrained to , while 3 contain affine nodes and include infinite solutions, two in $\Complex$, and one in $\Complex^2$.
* 11 regular and semiregular convex uniform tilings in two-dimensional Euclidean space, which are dual to the 11 Laves tilings.
* 11 paracompact regular hyperbolic honeycombs with infinite facets and vertex figures in the third dimension (alongside 4 compact hyperbolic honeycombs).
* 11 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.

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 ''K''₁₁ and the 10-simplex, a regular polytope in 10 dimensions.

Mathieu group M₁₁ is the smallest sporadic group, defined as the sharply 4-transitive permutation group on 11 objects. It has order 7920 = 2⁴3²511 = 891011, with 11 as its largest prime factor, and a minimal faithful complex representation in 10 dimensions. Its group action is the automorphism group of Steiner system S(4,5,11), with an induced action on unordered pairs of points that gives a rank 3 action on 55 points. Mathieu group M₁₂, on the other hand, is formed from the permutations of projective special linear group PSL₂(11) with those of (2,10)(3,4)(5,9)(6,7). It is the second-smallest sporadic group, and holds M₁₁ as a maximal subgroup and point stabilizer, with an order equal to 95040 = 2⁶3³511 = 89101112, where 11 is also its largest prime factor, like M₁₁. M₁₂ also centralizes an element of order 11 in the friendly giant, and has an irreducible faithful complex representation in 11 dimensions.

Within safe and Sophie Germain primes of the form 2''p'' + 1, 11 is the third safe prime, from a ''p'' of 5, and the fourth Sophie Germain prime, which yields 23.

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.

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 11 x 11, is the 22nd.

The factorial of 11, 11 ! = 39916800, has about a 0.2% difference to the round number 4 x 10⁷, or 40 million. Among the first 100 factorials, the next closest to a round number is 96! ~ 9.91678 x 10¹⁴⁹, which is about 0.8% less than 10¹⁴⁹.

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. 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 base 13 and higher bases (such as hexadecimal), 11 is represented as B, where ten is A. In duodecimal, 11 is sometimes represented as E or ↋, and ten as T, X, or ↊.

In science

*11 is the atomic number of the element sodium.
*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.
*Messier object M11 is a magnitude 7.0 open cluster in the constellation Scutum, also known as the Wild Duck Cluster.
*The New General Catalogue object NGC 11 is a spiral galaxy

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