11 (number)

Summary

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

← 10 11 12 →
Cardinaleleven
Ordinal11th
(eleventh)
Factorizationprime
Prime5th
Divisors1, 11
Greek numeralΙΑ´
Roman numeralXI
Greek prefixhendeca-/hendeka-
Latin prefixundeca-
Binary10112
Ternary1023
Octal138
DuodecimalB12
HexadecimalB16
Bangla১১
Hebrew numeralיא
Devanagari numerals११
Malayalam൰൧
Tamil numeralsகக
Telugu౧౧

NameEdit

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, although -lika is used as the suffix for all numbers from 11 to 19 (analogous to "-teen").[3]

The Old English form has closer cognates in Old Frisian, Saxon, and Norse, whose ancestor has been reconstructed as *ainlifun. This has formerly been considered 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 having already counted to ten.[3]

Eleven is the only two-digit number that, when spelt in English, does not contain the letter T.

In languagesEdit

While, as mentioned above, 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 mathematicsEdit

11 is a prime number. The next prime is 13, with which it comprises a twin prime. 11 is the first repunit prime,[6] the first strong prime,[7] the second unique prime,[8] the second good prime,[9] and the fourth Lucas prime.[10]

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   has the property of unique factorization. A consequence of this is that 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.[11]

11 of the thirty-five hexominoes can be folded to form (i.e., can serve as a net for) a cube. 11 of the sixty-six octiamonds can be folded to form a regular octahedron.

An 11-sided polygon is called a hendecagon or undecagon.

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

The 11-cell is a self-dual abstract 4-polytope. It is notable for the fact that it is a universal polytope: it is the only abstract polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.

Mathieu group M11 is the smallest sporadic group, defined as the sharply 4-transitive permutation group on 11 objects. 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. It has a minimal faithful complex representation in 10 dimensions.

Within safe and Sophie Germain primes of the form 2p + 1, 11 is the third safe prime, from a p equal to 5,[14] and the fourth Sophie Germain prime, which yields 23.[15]

In DecimalEdit

It is the smallest two-digit prime number. On the seven-segment display of a calculator, 11 is both a strobogrammatic prime and a dihedral prime.[16]

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. (For example: 142,312 × 11 = 1,565,432 → 2,345,651 ÷ 11 = 213,241.)

Multiples of 11 by one-digit numbers all have matching double digits: 00 (=0), 11, 22, 33, 44, etc.

11 is the 11th palindromic number, and 121, equal to 11 x 11, is the 22nd palindromic number.[17]

Divisibility TestsEdit

A simple test to determine if an integer is divisible by 11 is to take every digit of the number located in odd position and add them up, then take the remaining digits and add them up. If the difference between the two sums is a multiple of 11, including 0, then the number is divisible by 11.[18] For instance, if the number is 65,637 then (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. For instance, 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 up the numbers so formed; if the result is divisible by 11, the number is divisible by 11. For instance, if the number is 65,637, 06 + 56 + 37 = 99, which is divisible by 11, so 65,637 is divisible by eleven. This also works by adding a trailing zero instead of a leading one: 65 + 63 + 70 = 198, which is divisible by 11. This also works with larger groups of digits, providing that each group has an even number of digits (not all groups have to have the same number of digits).

Multiplying 11Edit

An easy way of multiplying numbers by 11 in base 10 is: If the number has:

  • 1 digit - Replicate the digit (so 2 × 11 becomes 22).
  • 2 digits - Add the 2 digits together and place the result in the middle (so 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 together to form the result's second digit, add the second and third digits together 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. Example 1: 123 × 11 becomes 1 (1+2) (2+3) 3 or 1353. Example 2: 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 calculationsEdit

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 25937421601 285311670611
x11 1 2048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609 100000000000 285311670611

In Other BasesEdit

In base 13 and higher bases (such as hexadecimal), 11 is represented as B, where ten is A. In duodecimal, however, 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 scienceEdit

AstronomyEdit

In religion and spiritualityEdit

ChristianityEdit

After Judas Iscariot was disgraced, the remaining apostles of Jesus were sometimes described as "the Eleven" (Mark 16:11; Luke 24:9 and 24:33); this occurred even after Matthias was added to bring the number to twelve, as in Acts 2:14:[20] 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 twelve.

Saint Ursula is said to have been martyred in the third or fourth century in Cologne with a number of companions, whose reported number "varies from five to eleven".[21] A legend that Ursula died with eleven thousand virgin companions[22] has been thought to appear from misreading XI. M. V. (Latin abbreviation for "Eleven martyr virgins") as "Eleven thousand virgins".

BabylonianEdit

In the Enûma Eliš the goddess Tiamat creates eleven monsters to take revenge for the death of her husband, Apsû.

MysticismEdit

The number 11 (alongside its multiples 22, and 33) are master numbers in numerology, especially in New Age.[23] In astrology, Aquarius is the 11th astrological sign of the Zodiac.[24]

In musicEdit

In sportsEdit

In the militaryEdit

In computingEdit

In CanadaEdit

In other fieldsEdit

See alsoEdit

ReferencesEdit

  1. ^ Bede, Eccl. Hist., Bk. V, Ch. xviii.
  2. ^ Specifically, in the line jjvjv ðæt rice hæfde endleofan wintra.[1]
  3. ^ a b c d e 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's A004022 : Primes of the form (10^n - 1)/9". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  7. ^ 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.
  8. ^ "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.
  9. ^ "Sloane's A028388 : Good primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  10. ^ "Sloane's A005479 : Prime Lucas numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  11. ^ 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.
  12. ^ Coxeter, H.S.M. (1991), Regular Complex Polytopes, Table VI. The regular honeycombs, Cambridge University Press, p. 111, 136, ISBN 0-521-39490-2
  13. ^ H.S.M. Coxeter (1956). "Regular Honeycombs in Hyperbolic Space". Penn State. p. 168. Retrieved 2022-06-15.
  14. ^ "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  15. ^ "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  16. ^ "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.
  17. ^ Sloane, N. J. A. (ed.). "Sequence A002113 (Palindromes in base 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-11.
  18. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1.
  19. ^ 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.
  20. ^ "Acts 2:14 Then Peter stood up with the eleven, lifted up his voice, and addressed the crowd: "Men of Judea and all who dwell in Jerusalem, let this be known to you, and listen carefully to my words". bible.cc.
  21. ^ 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
  22. ^ Four scenes from the life of St Ursula, accessed 10 July 2016
  23. ^ Sharp, Damian (2001). Simple Numerology: A Simple Wisdom book (A Simple Wisdom Book series). Red Wheel. p. 7. ISBN 978-1573245609.
  24. ^ "Aquarius". Oxford Dictionaries. n.d. Retrieved June 27, 2022.
  25. ^ The Eleven - Grateful Dead | Song Info | AllMusic, retrieved 2020-08-10
  26. ^ Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. Archived from the original on 2012-04-04. Retrieved 2012-09-06.
  27. ^ Eleven - Come | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  28. ^ Eleven - Incognito | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  29. ^ Eleven - Martina McBride | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  30. ^ Eleven - 22-Pistepirkko | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  31. ^ Eleven - Eleven | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  32. ^ Eleven - Harry Connick, Jr. | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  33. ^ Eleven - Tina Arena | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  34. ^ Eleven - Jeff Lorber, The Jeff Lorber Fusion, Mike Stern | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  35. ^ Eleven - Reamonn | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  36. ^ Eleven - Wagon Cookin' | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  37. ^ Eleven - Mr. Fogg | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  38. ^ Eleven - The Birdland Big Band, Tommy Igoe | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  39. ^ Eleven - Pearl Django | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  40. ^ Eleven - Daniel Pena, Daniel Peña | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
  41. ^ Eleven - The Knux | User Reviews | AllMusic, retrieved 2020-08-10
  42. ^ Eleven - Igor Lumpert & Innertextures | User Reviews | AllMusic, retrieved 2020-08-10
  43. ^ ESMD, US Census Bureau Classification Development Branch. "US Census Bureau Site North American Industry Classification System main page". census.gov.
  44. ^ "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20.

External linksEdit

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