# 11 (number)

 ← 10 11 12 →
Cardinaleleven
Ordinal11th
(eleventh)
Factorizationprime
Prime5th
Divisors1, 11
Greek numeralΙΑ´
Roman numeralXI
Greek prefixhendeca-/hendeka-
Latin prefixundeca-
Binary10112
Ternary1023
Senary156
Octal138
DuodecimalB12
Bangla১১
Hebrew numeralיא
Devanagari numerals११
Malayalam൰൧
Tamil numeralsகக
Telugu౧౧

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 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 ùndici, 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. The next prime number is 13, with which it comprises a twin prime.[6] 11 is the first repunit prime (it is the first repunit of any kind),[7] the first strong prime,[8] the second unique prime,[9] the second good prime,[10] the third super-prime, the fourth Lucas prime,[11] and the fifth supersingular prime.[12]

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 ${\displaystyle \mathbb {Q} ({\sqrt {-11}})}$ has the property of unique factorization. As a consequence, there exists at most one point on the elliptic curve x3 = y2 + 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.[13]

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 K11 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.[14]

11 and some of its multiples appear as counts of uniform tessellations in various dimensions and spaces; there are:

22 edge-to-edge uniform tilings with convex and star polygons, and 33 uniform tilings with zizgzag apeirogons that alternate between two angles.[16][17]
22 regular complex apeirohedra of the form p{a}q{b}r, where 21 exist in ${\displaystyle \mathbb {C} ^{2}}$ and 1 in ${\displaystyle \mathbb {C} ^{3}}$.[19]
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.[20]

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 K11 and the 10-simplex, a regular polytope in 10 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.[12] Only five of twenty-six sporadic groups do not contain 11 as a prime factor that divides their group order (J2, J3, Ru, He, and Th). 11 is also not a prime factor of the order of the Tits group T, which is sometimes categorized as non-strict group of Lie type, or sporadic group.

In particular, Mathieu group M11 is the smallest sporadic group, defined as the sharply 4-transitive permutation group on 11 objects. It has order 7920 = 24 · 32 ·· 11 = 8 ·· 10 · 11, 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 = 26 · 33 ·· 11 = 8 ·· 10 · 11 · 12, 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 2p + 1, 11 is the third safe prime, from a p of 5,[21] and the fourth Sophie Germain prime, which yields 23.[22]

### 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.[23]

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.[24]

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.[25]

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.[26] For instance, with the number 65,637:

(6 + 6 + 7) - (5 + 3) = 19 - 8 = 11, so 65,637 is divisible by 11.

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,

(065) - 637 = -572, which is divisible by 11.

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:

06 + 56 + 37 = 99, which 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):

65 + 63 + 70 = 198, which is divisible by 11.

#### 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

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 50 100 1000
11 × x 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 275 550 1100 11000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
11 ÷ x 11 5.5 3.6 2.75 2.2 1.83 1.571428 1.375 1.2 1.1 1 0.916 0.846153 0.7857142 0.73
x ÷ 11 0.09 0.18 0.27 0.36 0.45 0.54 0.63 0.72 0.81 0.90 1 1.09 1.18 1.27 1.36
Exponentiation 1 2 3 4 5 6 7 8 9 10 11
11x 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937424601 285311670611
x11 1 2048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609 100000000000 285311670611

### 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 ↊.

Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100
110 120 130 140 150 200 250 500 1000 10000 100000 1000000
x11 1 5 A11 1411 1911 2311 2811 3711 4611 5511 6411 7311 8211 9111
A011 AA11 10911 11811 12711 17211 20811 41511 82A11 757211 6914A11 62335111

## In religion and spirituality

### Christianity

After Judas Iscariot was disgraced, Jesus's remaining apostles were sometimes called "the Eleven" (Mark 16:11; Luke 24:9 and 24:33), even after Matthias was added to bring the number back to 12, as in Acts 2:14:[28] Peter stood up with the eleven (New International Version). The New Living Translation says Peter stepped forward with the eleven other apostles, making clear that the number of apostles was now 12.

Saint Ursula is said to have been martyred in the 3rd or 4th century in Cologne with a number of companions, whose reported number "varies from five to eleven".[29] A legend that Ursula died with 11,000 virgin companions[30] has been thought to appear from misreading XI. M. V. (Latin abbreviation for "Eleven martyr virgins") as "Eleven thousand virgins".

### Babylonian

In the Enûma Eliš the goddess Tiamat creates 11 monsters to avenge the death of her husband, Apsû.

### Mysticism

The number 11 (alongside its multiples 22 and 33) are master numbers in numerology, especially in New Age.[31] In astrology, Aquarius is the 11th astrological sign of the Zodiac.[32]

## References

1. ^ Bede, Eccl. Hist., Bk. V, Ch. xviii.
2. ^ Specifically, in the line jjvjv ðæt rice hæfde endleofan wintra.[1]
3. Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
4. ^ Kroonen, Guus (2013). Etymological Dictionary of Proto-Germanic. Leiden: Brill. p. 11f. ISBN 978-90-04-18340-7.
5. ^ Dantzig, Tobias (1930), Number: The Language of Science.
6. ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-22.
7. ^ "Sloane's A004022 : Primes of the form (10^n - 1)/9". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
8. ^ Sloane, N. J. A. (ed.). "Sequence A051634 (Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-10.
9. ^ "Sloane's A040017 : Unique period primes (no other prime has same period as 1/p) in order (periods are given in A051627)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2018-11-20.
10. ^ "Sloane's A028388 : Good primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ "Sloane's A005479 : Prime Lucas numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. ^ a b Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes: primes dividing order of Monster simple group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-22.
13. ^ Mueller, Francis J. (1965). "More on Pascal's Triangle and powers of 11". The Mathematics Teacher. 58 (5): 425–428. doi:10.5951/MT.58.5.0425. JSTOR 27957164.
14. ^
15. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 233. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
16. ^ Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.5 Tilings Using Star Polygons". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 82–89. ISBN 0-7167-1193-1. OCLC 13092426. S2CID 119730123.
17. ^ Grünbaum, Branko; Miller, J. C. P.; Shephard, G. C. (1981). "Uniform Tilings with Hollow Tiles". The Geometric Vein: The Coxeter Festschrift. New York: Springer-Verlag. pp. 47–48. doi:10.1007/978-1-4612-5648-9_3. ISBN 978-1-4612-5650-2. MR 0661769. OCLC 7597141.{{cite book}}: CS1 maint: date and year (link)
18. ^ Coxeter, H.S.M. (1991). "11.6 Apeirogons". Regular Complex Polytopes (2 ed.). London: Cambridge University Press. p. 111, 112. ISBN 978-0521394901. MR 1119304. OCLC 21562167. S2CID 116900933.
19. ^ Coxeter, H.S.M. (1991). "12.8 Cycles of Honeycombs". Regular Complex Polytopes (2 ed.). London: Cambridge University Press. p. 138-140. ISBN 978-0521394901. MR 1119304. OCLC 21562167. S2CID 116900933.
20. ^ a b Coxeter, H. S. M. (1956). "Regular Honeycombs in Hyperbolic Space" (PDF). Proceedings of the International Congress of Mathematicians (1954). Amsterdam: North-Holland Publishing Co. 3: 167–168. MR 0087114. S2CID 18079488. Zbl 0073.36603. Archived from the original (PDF) on 2015-04-02.
21. ^ "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
22. ^ "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
23. ^ "Sloane's A134996 : Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2020-12-17.
24. ^ Sloane, N. J. A. (ed.). "Sequence A002113 (Palindromes in base 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-11.
25. ^ "List of first 100 factorial numbers". Encyclopedia of Online Integer Sequences (OEIS). Retrieved August 30, 2022.
26. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1.
27. ^ photos., Robert Erdmann, Bob Erdmann, Robert, Bob, Erdmann, NGC, IC, Astronomy, Deep-Sky, Database, Galaxy, Galaxies, Dreyer, Herschel, telescope, Corwin, Skiff, Buta, Archinal, Cragin, Ling, Gottlieb, Deep, Sky, Space, Catalog, Catalogs, pictures. "The NGC / IC Project - Home of the Historically Corrected New General Catalogue (HCNGC) since 1993". ngcicproject.org. Archived from the original on 2013-01-15. Retrieved 2011-06-20.
28. ^
29. ^ Ursulines of the Roman Union, Province of Southern Africa, St. Ursula and Companions Archived 2016-03-19 at the Wayback Machine, accessed 10 July 2016
30. ^ Four scenes from the life of St Ursula, accessed 10 July 2016
31. ^ Sharp, Damian (2001). Simple Numerology: A Simple Wisdom book (A Simple Wisdom Book series). Red Wheel. p. 7. ISBN 978-1573245609.
32. ^ "Aquarius". Oxford Dictionaries. n.d. Archived from the original on August 9, 2018. Retrieved June 27, 2022.
33. ^ The Eleven - Grateful Dead | Song Info | AllMusic, retrieved 2020-08-10
34. ^ Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. Archived from the original on 2012-04-04. Retrieved 2012-09-06.
35. ^ Eleven - Come | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
36. ^ Eleven - Incognito | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
37. ^ Eleven - Martina McBride | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
38. ^ Eleven - 22-Pistepirkko | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
39. ^ Eleven - Eleven | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
40. ^ Eleven - Harry Connick, Jr. | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
41. ^ Eleven - Tina Arena | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
42. ^
43. ^ Eleven - Reamonn | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
44. ^ Eleven - Wagon Cookin' | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
45. ^ Eleven - Mr. Fogg | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
46. ^ Eleven - The Birdland Big Band, Tommy Igoe | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
47. ^ Eleven - Pearl Django | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
48. ^ Eleven - Daniel Pena, Daniel Peña | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
49. ^ Eleven - The Knux | User Reviews | AllMusic, retrieved 2020-08-10
50. ^ Eleven - Igor Lumpert & Innertextures | User Reviews | AllMusic, retrieved 2020-08-10
51. ^ 11 - The Smithereens| Songs, Reviews, Credits | AllMusic, retrieved 2022-12-04
52. ^ ESMD, US Census Bureau Classification Development Branch. "US Census Bureau Site North American Industry Classification System main page". census.gov.
53. ^ "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20.