# 11 (number)

(Redirected from Eleven)
"Ⅺ" redirects here. For other uses, see XI (disambiguation) and xi (disambiguation).
 ← 10 11 12 →
Cardinal eleven
Ordinal 11th
(eleventh)
Factorization prime
Prime 5th
Divisors 1, 11
Roman numeral XI
Greek prefix hendeca-/hendeka-
Latin prefix undeca-
Binary 10112
Ternary 1023
Quaternary 234
Quinary 215
Senary 156
Octal 138
Duodecimal B12
Vigesimal B20
Base 36 B36

11 (eleven or /iˈlɛvɛn/) is the natural number following 10 and preceding 12. 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, 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 mathematics

11 is the 5th smallest prime number. It is the smallest two-digit prime number in the decimal base; as well as, of course, in undecimal (where it is the smallest two-digit number). It is also the smallest three-digit prime in ternary, and the smallest four-digit prime in binary, but a single-digit prime in bases larger than 11, such as duodecimal, hexadecimal, vigesimal and sexagesimal. 11 is the fourth Sophie Germain prime,[5] the third safe prime,[6] the fourth Lucas prime,[7] the first repunit prime,[8] and the second good prime.[9] 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. The next prime is 13, with which it comprises a twin prime. 11 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. Displayed on a calculator, 11 is a strobogrammatic prime and a dihedral prime because it reads the same whether the calculator is turned upside down or reflected on a mirror, or both.

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.)

Because it has a reciprocal of unique period length among primes, 11 is the second unique prime.[10] 11 goes into 99 exactly 9 times, so vulgar fractions with 11 in the denominator have two digit repeating sequences in their decimal expansions. Multiples of 11 by one-digit numbers all have matching double digits: 00 (=0), 11, 22, 33, 44, etc. Bob Dorough, in his Schoolhouse Rock song "The Good Eleven", called them "Double-digit doogies" (soft g). 11 is the Aliquot sum of one number, the discrete semiprime 21 and is the base of the 11-aliquot tree.

As 11 is the smallest factor of the first 11 terms of the Euclid–Mullin sequence, it is the 12th term.[11]

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

In both base 6 and base 8, the smallest prime with a composite sum of digits is 11.

Any number b + 1 is written as "11b" in base b, so 11 is trivially a palindrome in base 10. However 11 is a strictly non-palindromic number.[12] It is the only palindromic prime with an even number of digits.

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.[13] 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 10, 11 is the smallest integer that is not a Nivenmorphic number.

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.

11 is a Størmer number,[14] a Heegner number,[15] and a Mills prime.[16]

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.

The partition numbers (sequence A000041 in the OEIS) contain much more multiples of 11 than the one-eleventh one would expect.

According to David A. Klarner, a leading researcher and contributor to the study of polyominoes, it is possible to cut a rectangle into an odd number of congruent, non-rectangular polyominoes. 11 is the smallest such number, the only such number that is prime, and the only such number that is not a multiple of three.

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.)

### 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 21 22 23 24 25 50 100 1000
11 × x 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 231 242 253 264 275 550 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 12 13
11x 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937421601 285311670611 3138428376721 34522712143931
x11 1 2048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609 100000000000 285311670611 743008370688 1792160394037
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

௧௧ Tamil Malayalam Telugu

## 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".[17] A legend that Ursula died with eleven thousand virgin companions [18] has been thought to appear from misreading XI. M. V. (Latin abbreviation for "Eleven martyr virgins") as "Eleven thousand virgins".

### Thelema

11 is a spiritually significant number in Thelema.[citation needed]

### Babylonian

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

## In computing

• 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.

## References

1. ^ Bede, Eccl. Hist., Bk. V, Ch. xviii.
2. ^ Specifically, in the line Osred ðæt rice hæfde endleofan wintra.[1]
3. Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
4. ^ Dantzig, Tobias (1930), Number: The Language of Science.
5. ^ "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
6. ^ "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
7. ^ "Sloane's A005479 : Prime Lucas numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
8. ^ "Sloane's A004022 : Primes of the form (10^n - 1)/9". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
9. ^ "Sloane's A028388 : Good primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. ^ "Sloane's A040017 : Unique period primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ "Sloane's A000945 : Euclid-Mullin sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. ^ "Sloane's A016038 : Strictly non-palindromic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1.
14. ^ "Sloane's A005528 : Størmer numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
15. ^ "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
16. ^ "Sloane's A051254 : Mills primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
17. ^ Ursulines of the Roman Union, Province of Southern Africa, St. Ursula and Companions, accessed 10 July 2016
18. ^ Four scenes from the life of St Ursula, accessed 10 July 2016
19. ^ Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. Archived from the original on 2012-04-04. Retrieved 2012-09-06.
20. ^ "Cavs Announce Zydrunas Ilgauskas' Jersey (#11) to be Retired". THE OFFICIAL SITE OF THE CLEVELAND CAVALIERS.
21. ^ "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20.