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# 11 (number)

11 (eleven) is the natural number following 10 and preceding 12. It is the first number which cannot be represented by a human counting his or her eight fingers and two thumbs additively (although it can be represented in a variety of other ways using human bodily parts, such as counting additively with the twenty digits including the toes, using the thumb to count the finger phalanges of one hand additively up to twelve, or using the fingers in one hand to count up to fifteen binarily). Eleven is the smallest positive integer requiring three syllables in English, and it is also the largest prime number with a single-morpheme name in this language (although etymologically the word eleven originated from a Germanic compound *ainlif meaning "one left" ).

## In mathematics

Eleven 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 ten, such as duodecimal, hexadecimal, vigesimal and sexagesimal. 11 is the fourth Sophie Germain prime, the third safe prime, the fourth Lucas prime, and the first repunit 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.

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

Eleven is the Aliquot sum of one number, the discrete biprime 21 and is the base of the 11-aliquot tree.

As 11 is the smallest factor of the first eleven terms of the Euclid-Mullin sequence, it is the twelfth term.

An eleven-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.

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. For instance, if the number is 65,637 then (6 + 6 + 7) - (5 + 3) = 19 - 8 = 11, so 65,637 is divisible by eleven. 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 eleven.

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 eleven, the number is divisible by eleven. For instance, if the number is 65,637, 06 + 56 + 37 = 99, which is divisible by eleven, 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 eleven. 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 only integer that is not a Nivenmorphic number.

In base thirteen and higher bases (such as hexadecimal), eleven is represented as B, where ten is A. In duodecimal, however, eleven is sometimes represented as E and ten as T.

Multiplied by , eleven is a Heegner number.

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.

### 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 25 50 100 1000
$11 times x$ 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 231 242 275 550 1100 11000

Division 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15
$11 div x$ 11 5.5 $3.overline\left\{6\right\}$ 2.75 2.2 $1.8overline\left\{3\right\}$ $1.overline\left\{571428\right\}$ 1.375 $1.overline\left\{2\right\}$ 1.1
1 $0.91overline\left\{6\right\}$ $0.overline\left\{8\right\}4615overline\left\{3\right\}$ $0.7overline\left\{8\right\}5714overline\left\{2\right\}$ $0.7overline\left\{3\right\}$
$x div 11$ $0.overline\left\{09\right\}$ $0.overline\left\{18\right\}$ $0.overline\left\{27\right\}$ $0.overline\left\{36\right\}$ $0.overline\left\{45\right\}$ $0.overline\left\{54\right\}$ $0.overline\left\{63\right\}$ $0.overline\left\{72\right\}$ $0.overline\left\{81\right\}$ $0.overline\left\{90\right\}$
$1.overline\left\{09\right\}$ $1.overline\left\{18\right\}$ $1.overline\left\{27\right\}$ $1.overline\left\{36\right\}$ $1.overline\left\{45\right\}$

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
$x_\left\{11\right\}$ 1 5 $A_\left\{11\right\}$ $14_\left\{11\right\}$ $19_\left\{11\right\}$ $23_\left\{11\right\}$ $28_\left\{11\right\}$ $37_\left\{11\right\}$ $46_\left\{11\right\}$ $55_\left\{11\right\}$ $64_\left\{11\right\}$ $73_\left\{11\right\}$ $82_\left\{11\right\}$ $91_\left\{11\right\}$
$A0_\left\{11\right\}$ $AA_\left\{11\right\}$ $109_\left\{11\right\}$ $118_\left\{11\right\}$ $127_\left\{11\right\}$ $172_\left\{11\right\}$ $208_\left\{11\right\}$ $415_\left\{11\right\}$ $82A_\left\{11\right\}$ $7572_\left\{11\right\}$ $6914_\left\{11\right\}$ $623351_\left\{11\right\}$

### List of basic operations that make 11

$+$ $-$ $times$ $div$ $0 + 11$ $0 - \left(-11\right)$ N/A N/A $1 + 10$ $1 - \left(-10\right)$ $1 times 11$ $1 div 0.overline\left\{0\right\}overline\left\{9\right\}$ $2 + 9$ $2 - \left(-9\right)$ $2 times 5.5$ $2 div 0.overline\left\{1\right\}overline\left\{8\right\}$ $3 + 8$ $3 - \left(-8\right)$ $3 times 3.overline\left\{6\right\}$ $3 div 0.overline\left\{2\right\}overline\left\{7\right\}$ $4 + 7$ $4 - \left(-7\right)$ $4 times 2.75$ $4 div 0.overline\left\{3\right\}overline\left\{6\right\}$ $5 + 6$ $5 - \left(-6\right)$ $5 times 2.2$ $5 div 0.overline\left\{4\right\}overline\left\{5\right\}$ $6 + 5$ $6 - \left(-5\right)$ $6 times 1.8overline\left\{3\right\}$ $6 div 0.overline\left\{5\right\}overline\left\{4\right\}$ $7 + 4$ $7 - \left(-4\right)$ $7 times 1.overline\left\{5\right\}overline\left\{7\right\}overline\left\{1\right\}overline\left\{4\right\}overline\left\{2\right\}overline\left\{8\right\}$ $7 div 0.overline\left\{6\right\}overline\left\{3\right\}$ $8 + 3$ $8 - \left(-3\right)$ $8 times 1.375$ $8 div 0.overline\left\{7\right\}overline\left\{2\right\}$ $9 + 2$ $9 - \left(-2\right)$ $9 times 1.overline\left\{2\right\}$ $9 div 0.overline\left\{8\right\}overline\left\{1\right\}$ $10 + 1$ $10 - \left(-1\right)$ $10 times 1.1$ $10 div 0.overline\left\{9\right\}overline\left\{0\right\}$ $11 + 0$ $11 - 0$ $11 times 1$ $11 div 1$

## In music

• The interval of an octave and a fourth is an eleventh. A complete eleventh 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 twelfth thumb key.)
• Spinal Tap's amplifiers go up to eleven, every other bands' just go to ten, but it doesn't actually amplify the sound any greater.
• Maria Taylor's Debut solo album is entitled 11:11
• The All-American Rejects have a song on the CD Move Along that is called 11:11
• Andrew Bird has a song called 11:11
• The title of Bryan Adams' newest album is 11.
• The song Jimmy, by the band Tool mentions the number 11, in reference to a stroke suffered by lead singer Maynard James Keenan's mother when he was 11 years old; it also refers to the number of dimensions in M Theory and String Theory.

## In sports

There are eleven players on a soccer team on the field at a time as well as in a cricket team. Within a school or college, the phrases "the football XI" and "the cricket XI" generally refer 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 possibly others, in 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 eleven players in a field hockey team. The player wearing 11 will usually play on the left-hand side, as in soccer.

An American football team also has eleven players on the field at one time during play. 11 is also worn by quarterbacks, kickers, punter and wide receivers in American football's NFL. The only NFL team that has retired the #11 is the New York Giants, in honor of quarterback Phil Simms.

In rugby union the left wing wears the 11 shirt. Jonah Lomu wore the number when he played for the All Blacks as he played left wing (see rugby union above).

In rugby league, the 11 shirt is worn by a second-row forward.

In cricket, 11 is the number of players from the fielding (or bowling) side on the field of play at any one time. The eleventh batsman is usually the weakest batsman, at the end of the tail. He is primarily in the team for his bowling abilities.

The car number 11 was driven by Ayrton Senna as he won the 1988 Formula One World Championship. Also Darrell Waltrip and Cale Yarborough used the number when they won their NASCAR Winston Cup Series championships.

## Canada

• The stylized maple leaf on the Flag of Canada has eleven points.
• The Canadian one-dollar coin is a hendecagon, an eleven-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.

## Historical years

11 A.D., 11 B.C., 1911, 2011, etc.

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## References

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