11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.
Name
"Eleven" derives from the Old Englishęndleofon, 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 Lithuanianvienúolika, though -lika is used as the suffix for all numbers from 11 to 19 (analogously to "-teen").[3]
The Old English form has closer cognates in Old Frisian, Saxon, and Norse, whose ancestor has been reconstructed as *ainlifun. This was formerly thought to be 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 counting to ten.[3]
In languages
While 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: Chinese 十一shí yī, Korean 열하나yeol hana or 십일ship il.
11 is the first even-digited palindrome (an integer and its reverse being halves of another integer) and the only palindromic prime among them; all such palindromes are multiples of 11. (For this reason, palindromic primes beyond 3-digit skip to 5-digit, then 7-digit, ad infinitum.)
The rows of Pascal's triangle can be seen as representation of the powers of 11.[14]
A regular hendecagon cannot be constructed with a compass and straightedge alone, as 11 is not a product of distinct Fermat primes, and it is also the first polygon that is not able to be constructed with the aid of an angle trisector.[15]
22 edge-to-edge uniform tilings with convex and star polygons, and 33 uniform tilings with zizgzag apeirogons that alternate between two angles.[17][18]
11 regular complex apeirogons, which are tilings with polygons that have a countably infinite number of sides. 8 solutions of the form p{q}r satisfy δp,r 2 in where is constrained to , while three contain affine nodes and include infinite solutions, two in , and one in .[19]
22 regular complex apeirohedra of the form p{a}q{b}r, where 21 exist in and 1 in .[20]
The first eleven prime numbers (from 2 through 31) are consecutive supersingular primes that divide the order of the friendly giant, with the remaining four supersingular primes (41, 47, 59, and 71) lying between five non-supersingular primes.[13] Only five of twenty-six sporadic groups do not contain 11 as a prime factor that divides their group order (, , , , and ). 11 is also not a prime factor of the order of the Tits group, which is sometimes categorized as non-strict group of Lie type, or sporadic group.
11 is the second member of the second pair (5, 11) of Brown numbers. Only three such pairs of numbers and where are known; the largest pair (7, 71) satisfies . In this last pair 5040 is the factorial of 7, which is divisible by all integers less than 13 with the exception of 11. The members of the first pair (4,5) multiply to 20 — the prime index of 71— that is also eleventh composite number.[22]
Multiples of 11 by one-digit numbers yield palindromic numbers with matching double digits: 00, 11, 22, 33, 44, etc.
The sum of the first 11 non-zero positive integers, equivalently the 11th triangular number, is 66. On the other hand, the sum of the first 11 integers, from zero to ten, is 55.
The first four powers of 11 yield palindromic numbers: 111 = 11, 112 = 121, 113 = 1331, and 114 = 14641.
11 is the 11th index or member in the sequence of palindromic numbers, and 121, equal to , is the 22nd.[26]
The factorial of 11, , has about a 0.2% difference to the round number, or 40 million. Among the first 100 factorials, the next closest to a round number is 96 (), which is about 0.8% less than 10150.[27]
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; as in:
A simple test to determine whether an integer is divisible by 11 is to take every digit of the number in an odd position and add them, then take the remaining digits and add them. If the difference between the two sums is a multiple of 11, including 0, then the number is divisible by 11.[28] For instance, with the number 65,637:
(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. 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 the numbers so formed; if the result is divisible by 11, the number is divisible by 11:
06 + 56 + 37 = 99, which is divisible by 11.
This also works by adding a trailing zero instead of a leading one, and with larger groups of digits, provided that each group has an even number of digits (not all groups have to have the same number of digits):
65 + 63 + 70 = 198, which is divisible by 11.
Multiplying 11
An easy way to multiply numbers by 11 in base 10 is:
If the number has:
1 digit, replicate the digit: 2 × 11 becomes 22.
2 digits, add the 2 digits and place the result in the middle: 47 × 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 to form the result's second digit, add the second and third digits 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. 123 × 11 becomes 1 (1+2) (2+3) 3 or 1353. 481 × 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.
List of basic calculations
In other bases
In duodecimal and higher bases (such as hexadecimal), 11 is represented as B, E, Z or ↋ (el), where 10 is A, T, W, X or ↊ (dek).
Being one hour before 12:00, the eleventh hour means the last possible moment to take care of something, and often implies a situation of urgent danger or emergency (see Doomsday clock).
"{11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer)."
^"Sloane's A004022: Primes of the form (10^n - 1)/9". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^"Sloane's A040017: Unique period primes (no other prime has same period as 1/p) in order (periods are given in A051627)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2018-11-20.
^Mueller, Francis J. (1965). "More on Pascal's Triangle and powers of 11". The Mathematics Teacher. 58 (5): 425–428. doi:10.5951/MT.58.5.0425. JSTOR 27957164.
^Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.5 Tilings Using Star Polygons". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 82–89. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
^ a bCoxeter, H. S. M. (1956). "Regular Honeycombs in Hyperbolic Space" (PDF). Proceedings of the International Congress of Mathematicians (1954). 3. Amsterdam: North-Holland Publishing Co.: 167–168. MR 0087114. S2CID 18079488. Zbl 0073.36603. Archived from the original (PDF) on 2015-04-02.
^"Sloane's A005385: Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^"Sloane's A005384: Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^"Sloane's A134996: Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2020-12-17.