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 requiring three syllables and the largest prime number with a single-morpheme name.

 ← 10 11 12 →
Cardinal eleven
Ordinal 11th
(eleventh)
Factorization prime
Prime 5th
Divisors 1, 11
Greek numeral ΙΑ´
Roman numeral XI
Binary 10112
Ternary 1023
Quaternary 234
Quinary 215
Senary 156
Octal 138
Duodecimal B12
Vigesimal B20
Base 36 B36

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, 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]

In languages

Grammar

While, as mentioned above, 11 has its own name in Germanic languages such as English, German, and Swedish. 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), Chinese 十一 shí yī, Korean 열한 yeol han.

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.

There are 11 regular and semiregular convex uniform tilings in the second dimension, and 11 planigons that correspond to these 11 regular and semiregular tilings.

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.[6] 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.

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,[7] the third safe prime,[8] the fourth Lucas prime,[9] the first repunit prime,[10] and the second good prime.[11] 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 ${\displaystyle \mathbb {Q} ({\sqrt {-11}})}$ 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

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
11x 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937421601
x11 1 2048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609 100000000000
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 numeral systems

११ Devanagari Tamil Malayalam Telugu Bangla

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:[13] 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".[14] A legend that Ursula died with eleven thousand virgin companions[15] 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 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 other fields

• 11:11
• 11:11 (numerology)

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. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1.
7. ^ "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
8. ^ "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
9. ^ "Sloane's A005479 : Prime Lucas numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. ^ "Sloane's A004022 : Primes of the form (10^n - 1)/9". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ "Sloane's A028388 : Good primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. ^ http://www.ngcicproject.org/
13. ^ http://bible.cc/acts/2-14.htm
14. ^ Ursulines of the Roman Union, Province of Southern Africa, St. Ursula and Companions, accessed 10 July 2016
15. ^ Four scenes from the life of St Ursula, accessed 10 July 2016
16. ^ Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. Archived from the original on 2012-04-04. Retrieved 2012-09-06.
17. ^ https://www.census.gov/cgi-bin/sssd/naics/naicsrch?code=11&search=2012%20NAICS%20Search
18. ^ "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20.