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11 (number)


Source: http://en.wikipedia.org/wiki/11_(number)
Updated: 2017-07-31T19:21Z
← 101112 →
Cardinaleleven
Ordinal11th
(eleventh)
Factorizationprime
Prime5th
Divisors1, 11
Roman numeralXI
Greek prefixhendeca-/hendeka-
Latin prefixundeca-
Binary10112
Ternary1023
Quaternary234
Quinary215
Senary156
Octal138
DuodecimalB12
HexadecimalB16
VigesimalB20
Base 36B36

11 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 and the largest prime number with a single-morpheme name.

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 *ainlif, from the prefix *aino- (adjectival "one") and suffix *-lif- of uncertain meaning.[3] It is sometimes compared with the Lithuanian vënólika, 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][4] it is now sometimes connected with *leiq or *leip ("left; remaining"), with the implicit meaning that "one is left" after having already counted to ten.[3]

In languages

Grammar

While, as mentioned above, 11 has its own name in Germanic languages such as English and German, it is the first compound number in many other languages, e.g. Italian ùndici (but in Spanish and Portuguese, 16, and in French, 17 is the first compound number), Japanese 十一 jūichi.

In mathematics

11 is a prime number. It is the smallest two-digit prime number in the decimal base.

The next prime is 13, with which it comprises a twin prime.

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.

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

In base 10, there is a simple test to determine if an integer is divisible by 11: 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.[5] 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).

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

  • 1 digit - Replicate the digit (so 2 x 11 becomes 22).
  • 2 digits - Add the 2 digits together and place the result in the middle (so 47 x 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 x 11 becomes 1 (1+2) (2+3) 3 or 1353. Example 2: 481 x 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.

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 and ten as T or X.

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.

See also 11-cell.

11 of the thirty-five hexominoes can be folded to form cubes. 11 of the sixty-six octiamonds can be folded to form octahedra.

11 is the fourth Sophie Germain prime,[6] the third safe prime,[7] the fourth Lucas prime,[8] the first repunit prime,[9] and the second good prime.[10] Although it is necessary for n to be prime for 2n − 1 to be a Mersenne prime, the converse is not true: 211 − 1 = 2047 which is 23 × 89.

11 raised to the nth power is the nth row of Pascal's Triangle. (This works for any base, but the number eleven must be changed to the number represented as 11 in that base; for example, in duodecimal this must be done using thirteen.)

11 is a Heegner number, meaning that the ring of integers of the field \mathbb{Q}(\sqrt{-11}) has the property of unique factorization.

One 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).

List of basic calculations

Multiplication123456789101112131415161718192025501001000
11 × x112233445566778899110121132143154165176187198209220275550110011000
Division12345678910
1112131415
11 ÷ x115.53.62.752.21.831.5714281.3751.21.1
10.9160.8461530.78571420.73
x ÷ 110.090.180.270.360.450.540.630.720.810.90
11.091.181.271.36
Exponentiation12345678910
11x11121133114641161051177156119487171214358881235794769125937421601
x11120481771474194304488281253627970561977326743858993459231381059609100000000000
Radix151015202530405060708090100
1101201301401502002505001000100001000001000000
x1115A1114111911231128113711461155116411731182119111
A011AA1110911118111271117211208114151182A117572116914A1162335111

In numeral systems

௧௧Tamil
൧൧Malayalam
౧౧Telugu
১১Bangla

In science

Astronomy

In religion

Christianity

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: 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".[11] A legend that Ursula died with eleven thousand virgin companions [12] 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 eleven monsters to take revenge for the death of her husband, Apsû.

In music

In sports

  • There are 11 players on an association football (soccer) team on the field at a time
  • An American football team also has 11 players on the field at one time during play. #11 is worn by quarterbacks, kickers, punter and wide receivers in American football's NFL.
  • There are 11 players on a bandy team on the ice at a time
  • In cricket, a team has 11 players on the field. The 11th player is usually the weakest batsman, at the tail-end. He is primarily in the team for his bowling abilities.
  • There are 11 players in a field hockey team. The player wearing 11 will usually play on the left-hand side, as in soccer.
  • In most rugby league competitions (but not the European Super League, which uses static squad numbering), one of the starting second-row forwards wears the number 11.
  • In rugby union, the starting left wing wears the number 11 shirt.

In the military

In computing

In Canada

  • The stylized maple leaf on the Flag of Canada has 11 points.
  • The loonie is a hendecagon, an 11-sided polygon.
  • Clocks depicted on Canadian currency, like the Canadian fifty-dollar bill, show 11:00.
  • Eleven denominations of Canadian currency are produced in large quantities.
  • Due to Canada's federal nature, eleven legally distinct Crowns effectively exist in the country, with the Monarch represented separately in each province, and at the federal level.

In other fields

See also

References

  1. ^ Bede, Eccl. Hist., Bk. V, Ch. xviii.
  2. ^ Specifically, in the line Osred 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. ^ Dantzig, Tobias (1930), Number: The Language of Science .
  5. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1. 
  6. ^ "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 
  7. ^ "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 
  8. ^ "Sloane's A005479 : Prime Lucas numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 
  9. ^ "Sloane's A004022 : Primes of the form (10^n - 1)/9". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 
  10. ^ "Sloane's A028388 : Good primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 
  11. ^ Ursulines of the Roman Union, Province of Southern Africa, St. Ursula and Companions, accessed 10 July 2016
  12. ^ Four scenes from the life of St Ursula, accessed 10 July 2016
  13. ^ Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. Archived from the original on 2012-04-04. Retrieved 2012-09-06. 
  14. ^ "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20. 

External links

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