It is the first number which cannot be represented by a human counting their eight fingers and two thumbs additively. 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" ).
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 eleven, 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 eleven 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). Eleven 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 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.
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. 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.
If a number is divisible by 11, so is its reverse.
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.
Eleven 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.
Eleven of the thirty-five hexominoes can be folded to form cubes. Eleven of the sixty-six octiamonds can be folded to form octahedra.
In the Online Encyclopedia of Integer Sequences, the partition function is (sequence A000041 in OEIS). The fraction of elements in this sequence that are multiples of eleven is far higher than the one-eleventh that 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. Eleven is the smallest such number, the only such number that is prime, and the only such number that is not a multiple of three.
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