11 (number)

For other uses, see 11 (disambiguation).

← 10 11 12 →
← 10 11 12 13 14 15 16 17 18 19 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	eleven
Ordinal	11th (eleventh)
Numeral system	undecimal
Factorization	prime
Prime	5th
Divisors	1, 11
Greek numeral	ΙΑ´
Roman numeral	XI
Greek prefix	hendeca-/hendeka-
Latin prefix	undeca-
Binary	10112
Ternary	1023
Senary	156
Octal	138
Duodecimal	B12
Hexadecimal	B16
Bangla	১১
Hebrew numeral	י"א
Devanagari numerals	११
Malayalam	൰൧
Tamil numerals	கக
Telugu	౧౧
Babylonian numeral	𒌋𒐕

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 English ęndleofon, which is first attested in Bede's late 9th-century Ecclesiastical History of the English People. It has cognates in every Germanic language (for example, German elf), whose Proto-Germanic ancestor has been reconstructed as *ainalifa-, from the prefix *aina- (adjectival "one") and suffix *-lifa-, of uncertain meaning. It is sometimes compared with the Lithuanian vienúolika, though -lika is used as the suffix for all numbers from 11 to 19 (analogously to "-teen").

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"); it is now sometimes connected with *leikʷ- or *leip- ("left; remaining"), with the implicit meaning that "one is left" after counting to ten.

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 열하나 yeol hana or 십일 ship il.

Mathematics

11 is the fifth prime number, and the third super-prime. 11 forms a twin prime with 13, and sexy pair with 5 and 17. It is the first member of the second prime quadruplet (11, 13, 17, 19).

Arithmetic and algebraic properties arising from the number or it's use

The first prime exponent that does not yield a Mersenne prime is 11, where there is $2^{11}-1=2047$, which is the first composite generalized Mersenne number. 11 is the first strong prime, such that for a prime $p_{n}$ there is $p_{n}>{\tfrac {(prime_{(n-1)}+prime_{(n+1)})}{2}}$, and it is also the second good prime, whose square is greater than the product of any two prime numbers at the same number of positions before and after it in the sequence of prime numbers.

11 is the second member of the second pair (5, 11) of Brown numbers. Only three such pairs of numbers $n$ and $m$ where $n!+1=m^{2}$ are known; the largest pair (7, 71) satisfies $5041=7!+1$. In this last pair, the factorial of 7 = 5040 is divisible by all integers less than 13, with the exception of 11.

More consequentially, 5 is the fifth Heegner number, meaning that the ring of integers of the field $\mathbb {Q} ({\sqrt {-11}})$ has the property of unique factorization and class number 1. In abstract algebra, 11 is the fifth consecutive supersingular prime that divides the order of the largest sporadic group.

Rows in Pascal's triangle can be seen as representation of powers of 11.

In geometric relationships and concepts

Copper engraving of a hendecagon, by Anton Ernst Burkhard von Birckenstein (1698)

An 11-sided polygon is called a hendecagon, or undecagon. The complete graph $K_{11}$ has a total of 55 edges, which collectively represent the diagonals and sides of a hendecagon. 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.

11 of 35 hexominoes can fold in a net to form a cube, while 11 of 66 octiamonds can fold into a regular octahedron. 11 appears as counts of uniform tessellations in various dimensions and spaces. There are 11 regular and semiregular convex uniform tilings in the Euclidean plane, which are dual to the 11 Laves tilings. 11 is also the number of regular complex apeirogons, which are tilings with polygons that have a countably infinite number of sides. Meanwhile, there are also 11 regular paracompact hyperbolic honeycombs with infinite facets and vertex figures in the third dimension. Outside of Euclidean geometry, 11 is the total number of regular hyperbolic honeycombs in the fourth dimension: 9 compact solutions are generated from regular 4-polytopes and regular star 4-polytopes, alongside 2 paracompact solutions.

In differential geometry, 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.

Use inside abstract algebra

Mathieu group $\mathrm {M} _{11}$ is the smallest of twenty-six sporadic groups, an algebraic structure defined as a sharply 4-transitive permutation group on eleven objects. It has order $7920=2^{4}\cdot 3^{2}\cdot 5\cdot 11=8\cdot 9\cdot 10\cdot 11$, with 11 as its largest prime factor, and a minimal faithful complex representation in ten dimensions. Its group action is the automorphism group of Steiner system $\operatorname {S} (4,5,11)$, with an induced action on unordered pairs of points that gives a rank 3 action on 55 points. Mathieu group $\mathrm {M} _{12}$, on the other hand, holds $\mathrm {M} _{11}$ as a maximal subgroup, with an order equal to $95040=2^{6}\cdot 3^{3}\cdot 5\cdot 11=8\cdot 9\cdot 10\cdot 11\cdot 12$, where 11 is also its largest prime factor. $\mathrm {M} _{12}$ centralizes an element of order 11 in the the largest sporadic group, $\mathrm {F} _{1}$. $\mathrm {M} _{12}$ holds an irreducible faithful complex representation in eleven dimensions.

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		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
11x	11	121	1331	14641	161051	1771561	19487171	214358881	2357947691	25937424601	285311670611
x11	1	2048	177147	4194304	48828125	362797056	1977326743	8589934592	31381059609	100000000000	285311670611

As an integer in decimal

11 is the smallest palindromic number and more specifically repunit, as well as the first two-digit prime number, which makes it the only two-digit palindromic and repunit prime; all other two-digit palindromes are multiples of 11 (00, 11, 22, 33, 44, etc; for this reason, palindromic primes beyond 3-digit skip to 5-digit, then 7-digit, ad infinitum). On the seven-segment display of a calculator, 11 is both a strobogrammatic prime and a dihedral prime. The first four powers of 11 yield palindromic numbers as well: 11¹ = 11, 11² = 121, 11³ = 1331, and 11⁴ = 14641.

11 is the second unique prime in base ten.

Base-10 divisibility tests

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:

142,312 × 11 = 1,565,432 → 2,345,651 ÷ 11 = 213,241.

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. 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 by 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.

Science

In chemistry, Group 11 of the Periodic Table of the Elements (IUPAC numbering) consists of the three coinage metals copper, silver, and gold known from antiquity, and roentgenium, a recently synthesized superheavy element. 11 is the number of spacetime dimensions in M-theory.

In astronomy

Apollo 11 was the first crewed spacecraft to land on the Moon. In our solar system, the Sun has a sunspot cycle's periodicity that is approximately 11 years.

Music

See also: Eleven (disambiguation) § Music

The interval of an octave plus a fourth is an 11th. A complete 11th chord has almost every note of a diatonic scale. Regarding musical instruments, there are 11 thumb keys on a bassoon, not counting the whisper key. (A few bassoons have a 12th thumb key.)

Sports and games

In sports, there are 11 players on an association football (soccer) team, 11 players on an American football team during play, 11 players on a cricket team on the field, and 11 players in a field hockey team. In the game of blackjack, an ace can count as either one or 11, whichever is more advantageous for the player.

Cultural references

In Canada

The stylized maple leaf on the Flag of Canada has 11 points. The CA$ one-dollar loonie is in the shape of an 11-sided hendecagon, and clocks depicted on Canadian currency, like the Canadian 50-dollar bill, show 11:00.

"Eleventh hour"

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).

Mysticism

The number 11 (alongside its multiples 22 and 33) are master numbers in numerology, especially in New Age.

External links

Grimes, James. "Eleven". Numberphile. Brady Haran. Archived from the original on 2017-10-15. Retrieved 2016-01-03.