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.
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Cardinal | eleven | |||
Ordinal | 11th (eleventh) | |||
Factorization | prime | |||
Prime | 5th | |||
Divisors | 1, 11 | |||
Greek numeral | ΙΑ´ | |||
Roman numeral | XI | |||
Greek prefix | hendeca-/hendeka- | |||
Latin prefix | undeca- | |||
Binary | 1011_{2} | |||
Ternary | 102_{3} | |||
Octal | 13_{8} | |||
Duodecimal | B_{12} | |||
Hexadecimal | B_{16} | |||
Bangla | ১১ | |||
Hebrew numeral | יא | |||
Devanagari numerals | ११ | |||
Malayalam | ൰൧ | |||
Tamil numerals | கக | |||
Telugu | ౧౧ |
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]}
Eleven is the only two-digit number that, when spelt in English, does not contain the letter T.
While, as mentioned above, 11 has its own name in Germanic languages such as English, German, or Swedish, and some Latin based languages such as Spanish, Portuguese, and French, it is the first compound number in many other languages: Italian ùndici , Chinese 十一 shí yī, Korean 열하나 yeol hana or 십일 ship il.
11 is a prime number. The next prime is 13, with which it comprises a twin prime. 11 is the first repunit prime,^{[6]} the first strong prime,^{[7]} the second unique prime,^{[8]} the second good prime,^{[9]} and the fourth Lucas prime.^{[10]}
11 is the first prime number that is not an exponent for a Mersenne prime, as 2^{11} − 1 = 2047, which is composite.
11 is a Heegner number, meaning that the ring of integers of the field has the property of unique factorization. A consequence of this is that there exists at most one point on the elliptic curve x^{3} = y^{2} + 11 that has positive-integer coordinates. In this case, this unique point is (15, 58).
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.
The rows of Pascal's Triangle can be seen as representation of the powers of 11.^{[11]}
11 of the thirty-five hexominoes can be folded to form (i.e., can serve as a net for) a cube. 11 of the sixty-six octiamonds can be folded to form a regular octahedron.
An 11-sided polygon is called a hendecagon or undecagon.
The number 11 appears in tessellations in various dimensions and geometrical spaces; there are:
The 11-cell is a self-dual abstract 4-polytope. It is notable for the fact that it is a universal polytope: it is the only abstract polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.
Mathieu group M_{11} is the smallest sporadic group, defined as the sharply 4-transitive permutation group on 11 objects. Its group action is the automorphism group of Steiner system S(4,5,11), with an induced action on unordered pairs of points that gives a rank 3 action on 55 points. It has a minimal faithful complex representation in 10 dimensions.
Within safe and Sophie Germain primes of the form 2p + 1, 11 is the third safe prime, from a p equal to 5,^{[14]} and the fourth Sophie Germain prime, which yields 23.^{[15]}
It is the smallest two-digit prime number. On the seven-segment display of a calculator, 11 is both a strobogrammatic prime and a dihedral prime.^{[16]}
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.
11 is the 11^{th} palindromic number, and 121, equal to 11 x 11, is the 22^{nd} palindromic number.^{[17]}
A simple test to determine if an integer is divisible by 11 is to 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.^{[18]} 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:
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 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|
11^{x} | 11 | 121 | 1331 | 14641 | 161051 | 1771561 | 19487171 | 214358881 | 2357947691 | 25937421601 | 285311670611 |
x^{11} | 1 | 2048 | 177147 | 4194304 | 48828125 | 362797056 | 1977326743 | 8589934592 | 31381059609 | 100000000000 | 285311670611 |
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 or ↋, and ten as T, X, or ↊.
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} | 6914A_{11} | 623351_{11} |
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:^{[20]} 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".^{[21]} A legend that Ursula died with eleven thousand virgin companions^{[22]} has been thought to appear from misreading XI. M. V. (Latin abbreviation for "Eleven martyr virgins") as "Eleven thousand virgins".
In the Enûma Eliš the goddess Tiamat creates eleven monsters to take revenge for the death of her husband, Apsû.
The number 11 (alongside its multiples 22, and 33) are master numbers in numerology, especially in New Age.^{[23]} In astrology, Aquarius is the 11th astrological sign of the Zodiac.^{[24]}