11 (number)

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 and the largest prime number with a single-morpheme name.

← 10 11 12 →
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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	10112
Ternary	1023
Octal	138
Duodecimal	B12
Hexadecimal	B16
Hebrew numeral	יא

Name

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]

In languages

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 열한 yeol han.

In mathematics

11 is a prime number. It is the smallest two-digit prime number in the decimal base.

The next prime is 13, with which it comprises a twin prime.

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.

An 11-sided polygon is called a hendecagon or undecagon.

There are 11 regular and semiregular convex uniform tilings in the second dimension, and 11 planigons that correspond to these 11 regular and semiregular tilings.

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.[6] 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:

• 1 digit - Replicate the digit (so 2 × 11 becomes 22).
• 2 digits - Add the 2 digits together and place the result in the middle (so 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 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 × 11 becomes 1 (1+2) (2+3) 3 or 1353. Example 2: 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.

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 and ten as T or X.

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.

11 of the thirty-five hexominoes can be folded to form cubes. 11 of the sixty-six octiamonds can be folded to form octahedra.

11 is the fourth Sophie Germain prime,[7] the third safe prime,[8] the fourth Lucas prime,[9] the first repunit prime,[10] the second good prime,[11] and the second unique prime.[12] 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.

11 raised to the nth power is the nth row of Pascal's Triangle. (This works for any base, but the number eleven must be changed to the number represented as 11 in that base; for example, in duodecimal this must be done using thirteen.)

11 is a Heegner number, meaning that the ring of integers of the field ${\displaystyle \mathbb {Q} ({\sqrt {-11}})}$ has the property of unique factorization.

One consequence of this is that there exists at most one point on the elliptic curve x³ = y² + 11 that has positive-integer coordinates. In this case, this unique point is (15, 58).

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^x	11	121	1331	14641	161051	1771561	19487171	214358881	2357947691	25937421601
x^11	1	2048	177147	4194304	48828125	362797056	1977326743	8589934592	31381059609	100000000000

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
x11	1	5	A11	1411	1911	2311	2811	3711	4611	5511	6411	7311	8211	9111
	A011	AA11	10911	11811	12711	17211	20811	41511	82A11	757211	6914A11	62335111

In numeral systems

११	Devanagari
கக	Tamil
൧൧	Malayalam
౧౧	Telugu
১১	Bangla

In science

• 11 is the atomic number of the element sodium.
• 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.
• The number of spacetime dimensions in M-theory.

Astronomy

• Apollo 11 was the first manned spacecraft to land on the Moon.
• The approximate periodicity of a sunspot cycle is 11 years.
• Messier object M11, a magnitude 7.0 open cluster in the constellation Scutum, also known as the Wild Duck Cluster.
• The New General Catalogue object[13] NGC 11, a spiral galaxy in the constellation Andromeda.
• The 11th moon of Jupiter is Himalia.

In religion

Christianity

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:[14] 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".[15] A legend that Ursula died with eleven thousand virgin companions[16] has been thought to appear from misreading XI. M. V. (Latin abbreviation for "Eleven martyr virgins") as "Eleven thousand virgins".

Babylonian

In the Enûma Eliš the goddess Tiamat creates eleven monsters to take revenge for the death of her husband, Apsû.

In music

• The interval of an octave and a fourth is an 11th. A complete 11th 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 12th thumb key.)
• In the mockumentary This Is Spinal Tap, Spinal Tap's amplifiers go up to eleven.
• In Igor Stravinsky's The Rite of Spring, there are 11 consecutive repetitions of the same chord.
• In Tool's song "Jimmy" and in Negativland's song "Time Zones", the number 11 is heard numerous times in the lyrics.
• "Eleven pipers piping" is the gift on the 11th day of Christmas in the carol "The Twelve Days of Christmas."
• In Green Grow the Rushes, O, Eleven is for "the eleven who went to heaven."
• "The Eleven" is a song by The Grateful Dead.[17]
• In "Time Enough For Rocking When We're Old" by The Magnetic Fields, a lyric references "when our pheromones go up to eleven."
• Eleven Records is the record label of Jason Webley, and many of Webley's works feature the number 11.[18]
• Eleven is an album by:
  • Come[19]
  • incognito[20]
  • Martina McBride[21]
  • 22-Pistepirkko[22]
  • Eleven[23]
  • Harry Connick, Jr.[24]
  • Tina Arena[25]
  • Jeff Lorber and Mike Stern[26]
  • Reamonn[27]
  • Wagon Cookin'[28]
  • Mr. Fogg[29]
  • The Birdland Big Band[30]
  • Pearl Django[31]
  • Daniel Peña[32]
  • The Knux[33]
  • Igor Lumpert[34]

In sports

• There are 11 players on an association football (soccer) team on the field at a time.
• An American football team also has 11 players on the field at one time during play. #11 is worn by quarterbacks, kickers, punter and wide receivers in American football's NFL.
• There are 11 players on a bandy team on the ice at a time.
• In cricket, a team has 11 players on the field. The 11th player is usually the weakest batsman, at the tail-end. He is primarily in the team for his bowling abilities.
• There are 11 players in a field hockey team. The player wearing 11 will usually play on the left-hand side, as in soccer.
• In NASCAR Racing, Denny Hamlin won in his 11th race of the closest Daytona 500 in No. 11 car for a margin of 0.01 seconds.
• In most rugby league competitions (but not the Super League, which uses static squad numbering), one of the starting second-row forwards wears the number 11.
• In rugby union, the starting left wing wears the number 11 shirt.

In the military

• The number of guns in a gun salute to U.S. Army, Air Force and Marine Corps Brigadier Generals, and to Navy and Coast Guard Rear Admirals Lower Half.
• The Military Occupational Specialty (MOS) designator given to US Army Infantry Officer as well as to enlisted personnel (AKA 11 MOS Series, or 11B, 11C, 11D, 11H, 11M, etc.)
• The number of General Orders for Sentries in the Marine Corps and United States Navy.
• A page in the Service Record Book of an enlisted Marine for writing down disciplinary actions.
• World War I ended with an Armistice on November 11, 1918, which went into effect at 11:00 am—the 11th hour on the 11th day of the 11th month of the year. Armistice Day is still observed on November 11 of each year, although it is now called Veterans Day in the United States and Remembrance Day in the Commonwealth of Nations and parts of Europe.

In computing

• In Mozilla Firefox, Opera, Konqueror for KDE, Google Chrome and Internet Explorer for Windows, the function key F11 key toggles full screen viewing mode. In macOS, F11 hides all open windows.
• The windowing system for Unix computers is known as X11.
• Computers of the PDP-11 series from Digital Equipment Corporation were informally referred to as "elevens".

In Canada

• The stylized maple leaf on the Flag of Canada has 11 points.
• The loonie is a hendecagon, an 11-sided polygon.
• Clocks depicted on Canadian currency, like the Canadian fifty-dollar bill, show 11:00.

In other fields

• Sector 11[35] in the North American Industry Classification System is the code for Agriculture, Forestry, Fishing and Hunting industries.
• Being only 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).
• In Astrology, Aquarius is the 11th astrological sign of the Zodiac.
• In Basque, hamaika ("eleven") has the double meaning of "infinite", probably from amaigabe, "endless", as in Hamaika aldiz etortzeko esan dizut! ("I told you infinite/eleven times to come!").
• English-speaking surveyors have developed several slang terms for 11 to distinguish it from its rhyme "seven": "punk," "top," & "railroad"[36]
• American Airlines flight 11, a Boston-Los Angeles flight which crashed into the North Tower of the World Trade Center in New York City, New York after being hijacked by terrorists on September 11, 2001.
• The number 11 bus is a low-cost way of sightseeing in London.
• In the game of blackjack, an Ace can be counted as either one or 11, whichever is more advantageous for the player.
• 11 is the number of the French department Aude.
• Three films – Ben-Hur (1959), Titanic (1997), and The Lord of the Rings: The Return of the King (2003) – have each won 11 Academy Awards, including Best Picture of their respective years.
• Ocean's Eleven is the name of two American films.
• In the anime series Code Geass, Japan is known as Area 11 of the Brittanian Empire.
• Eleven is the name of a character in the 2016 Netflix original series Stranger Things, Presented by Millie Bobby Brown.

See also

• 11:11
• 11:11 (numerology)
• XI

References

1. Bede, Eccl. Hist., Bk. V, Ch. xviii.
2. Specifically, in the line jjvjv ðæt rice hæfde endleofan wintra.[1]
3. Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
4. Kroonen, Guus (2013). Etymological Dictionary of Proto-Germanic. Leiden: Brill. p. 11f. ISBN 978-90-04-18340-7.
5. Dantzig, Tobias (1930), Number: The Language of Science.
6. Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1.
7. "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
8. "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
9. "Sloane's A005479 : Prime Lucas numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. "Sloane's A004022 : Primes of the form (10^n - 1)/9". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. "Sloane's A028388 : Good primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. "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.
13. photos., Robert Erdmann, Bob Erdmann, Robert, Bob, Erdmann, NGC, IC, Astronomy, Deep-Sky, Database, Galaxy, Galaxies, Dreyer, Herschel, telescope, Corwin, Skiff, Buta, Archinal, Cragin, Ling, Gottlieb, Deep, Sky, Space, Catalog, Catalogs, pictures. "The NGC / IC Project - Home of the Historically Corrected New General Catalogue (HCNGC) since 1993". ngcicproject.org. Archived from the original on 2012-12-18. Retrieved 2011-06-20.
14. "Acts 2:14 Then Peter stood up with the eleven, lifted up his voice, and addressed the crowd: "Men of Judea and all who dwell in Jerusalem, let this be known to you, and listen carefully to my words". bible.cc.
15. Ursulines of the Roman Union, Province of Southern Africa, St. Ursula and Companions Archived 2016-03-19 at the Wayback Machine, accessed 10 July 2016
16. Four scenes from the life of St Ursula, accessed 10 July 2016
17. The Eleven - Grateful Dead | Song Info | AllMusic, retrieved 2020-08-10
18. Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. Archived from the original on 2012-04-04. Retrieved 2012-09-06.
19. Eleven - Come | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
20. Eleven - Incognito | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
21. Eleven - Martina McBride | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
22. Eleven - 22-Pistepirkko | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
23. Eleven - Eleven | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
24. Eleven - Harry Connick, Jr. | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
25. Eleven - Tina Arena | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
26. Eleven - Jeff Lorber, The Jeff Lorber Fusion, Mike Stern | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
27. Eleven - Reamonn | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
28. Eleven - Wagon Cookin' | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
29. Eleven - Mr. Fogg | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
30. Eleven - The Birdland Big Band, Tommy Igoe | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
31. Eleven - Pearl Django | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
32. Eleven - Daniel Pena, Daniel Peña | Songs, Reviews, Credits | AllMusic, retrieved 2020-08-10
33. Eleven - The Knux | User Reviews | AllMusic, retrieved 2020-08-10
34. Eleven - Igor Lumpert & Innertextures | User Reviews | AllMusic, retrieved 2020-08-10
35. ESMD, US Census Bureau Classification Development Branch. "US Census Bureau Site North American Industry Classification System main page". census.gov.
36. "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20.

External links

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