11 (number)

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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
Hexadecimal B16
Vigesimal B20
Base 36 B36

11 (eleven Listeni/ɨˈlɛvɨn/ 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. Its etymology originates from a Germanic compound ainlif meaning "one left".[1]).

In mathematics[edit]

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, the third safe prime, the fourth Lucas prime, the first repunit prime, and the second good prime. 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 x 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. 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.

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.

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.[2] 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.

11 is a Størmer number, a Heegner number, and a Mills prime.

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.

The partition numbers (sequence A000041 in 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 n power is the nth row of Pascal's Triangle. (This works for any base)

List of basic calculations[edit]

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 \times 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 \div x 11 5.5 3.\overline{6} 2.75 2.2 1.8\overline{3} 1.\overline{571428} 1.375 1.\overline{2} 1.1
1 0.91\overline{6} 0.\overline{8}4615\overline{3} 0.7\overline{8}5714\overline{2} 0.7\overline{3}
x \div 11 0.\overline{09} 0.\overline{18} 0.\overline{27} 0.\overline{36} 0.\overline{45} 0.\overline{54} 0.\overline{63} 0.\overline{72} 0.\overline{81} 0.\overline{90}
1 1.\overline{09} 1.\overline{18} 1.\overline{27} 1.\overline{36}
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
11 ^ x\, 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937421601 285311670611 3138428376721 34522712143931
x ^ {11}\, 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
x_{11} \ 1 5 A_{11} \ 14_{11} \ 19_{11} \ 23_{11} \ 28_{11} \ 37_{11} \ 46_{11} \ 55_{11} \ 64_{11} \ 73_{11} \ 82_{11} \ 91_{11} \
A0_{11} \ AA_{11} \ 109_{11} \ 118_{11} \ 127_{11} \ 172_{11} \ 208_{11} \ 415_{11} \ 82A_{11} \ 7572_{11} \ 6914_{11} \ 623351_{11} \

In science[edit]

Astronomy[edit]

  • The Saros number of the solar eclipse series which began on -2511 December 26 and ended on -1158 March 18. The duration of Saros series 11 was 1352.2 years, and it contained 76 solar eclipses.
  • The Saros number of the lunar eclipse series which began on -2389 June 19 and ended on -1037 September 8. The duration of Saros series 11 was 1352.2 years, and it contained 76 lunar eclipses.

In religion[edit]

After Judas Iscariot was disgraced, the remaining apostles of Jesus were sometimes described as "the Eleven"; this occurred even after Matthias was added to bring the number to 12, as in Acts 2:14.

The legend of the martyrdom of the Eleven Thousand Virgins has been thought to appear from misreading XI. M. V. (Latin abbreviation for "Eleven martyr virgins") as "Eleven thousand virgins".

11 is a spiritually significant number in Thelema.

In music[edit]

  • The interval of an octave and a fourth is an 11th. A complete 11th chord has almost every note of a diatonic scale.
  • The number of thumb keys on a bassoon, not counting the whisper key. (A few bassoons have a 12th thumb key.)
  • In the mockumentary This Is Spinal Tap, Spinal Tap's amplifiers go up to eleven.
  • In Igor Stravinsky's The Rite of Spring, there are 11 consecutive repetitions of the same chord.
  • In Tool's song Jimmy, and in Negativland's song Time Zones the number 11 is heard numerous times in the lyrics.
  • "Eleven pipers piping" is the gift on the 11th day of Christmas in the carol "The Twelve Days of Christmas"
  • The Eleven is a song by The Grateful Dead.
  • Eleven Records is the record label of Jason Webley, and many of Webley's works feature the number 11.[3]

In sports[edit]

  • There are 11 players on a soccer team on the field at a time as well as in a cricket team. Within a school or college, the phrase first eleven (or first XI) - often "first football XI" and "first cricket XI" - generally refers to the first (best) team currently playing. Other teams are often referred to as "the second XI" etc.
  • Also in soccer, in the German language (and others like Italian - "gli undici metri" -, countries that predominantly use the metric system) a penalty kick is referred to as "Elfmeter" because the penalty spot is approximately 11m (precisely 12 yards) from the goal line. Historically, in the Pyramid formation that position names are taken from, a left wing-forward in football wears number 11. In the modern game, especially using the 4-4-2 formation, it is worn by a left-sided midfielder. Less commonly a striker will wear the shirt.
  • 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 11 shirt.
  • In cricket, the 11th batsman is usually the weakest batsman, at the end of the tail. He is primarily in the team for his bowling abilities.

In the military[edit]

In computing[edit]

In Canada[edit]

  • The stylized maple leaf on the Flag of Canada has 11 points.
  • The Canadian one-dollar coin is a hendecagon, an 11-sided polygon.
  • Clocks depicted on Canadian currency, for example 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 being represented separately in each province, as well as at the federal level.

In other fields[edit]

See also[edit]

References[edit]

  1. ^ "eleven". Online Etymology Dictionary. Retrieved 2011-11-11. 
  2. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1. 
  3. ^ Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. Retrieved 2012-09-06. 
  4. ^ "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20.