11 (number)
| ||||
---|---|---|---|---|
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 | 1011_{2} | |||
Ternary | 102_{3} | |||
Quaternary | 23_{4} | |||
Quinary | 21_{5} | |||
Senary | 15_{6} | |||
Octal | 13_{8} | |||
Duodecimal | B_{12} | |||
Hexadecimal | B_{16} | |||
Vigesimal | B_{20} | |||
Base 36 | B_{36} | |||
Bangla | ১১ | |||
Hebrew numeral | יא | |||
Devanagari numerals | ११ | |||
Malayalam | ൰൧ | |||
Tamil numerals | கக | |||
Telugu | ౧౧ |
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.^{[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, 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 열하나 yeol hana or 십일 ship il.
In mathematics
Eleven is the fifth prime number, and the first two-digit numeric palindrome in decimal. It forms a twin prime with 13,^{[6]} and it is the first member of the second prime quadruplet (11, 13, 17, 19).^{[7]}. The eleventh prime number is the first prime exponent that does not yield a Mersenne prime, where [math]\displaystyle{ 2^{11}-1=2047 }[/math], which is composite. On the other hand, the eleventh prime number 31 is the third Mersenne prime, while the thirty-first prime number 127 is not only a Mersenne prime but also the second double Mersenne prime. 11 is also the fifth Heegner number, meaning that the ring of integers of the field [math]\displaystyle{ \mathbb{Q}(\sqrt{-11}) }[/math] has the property of unique factorization and class number 1. 11 is the first prime repunit [math]\displaystyle{ R_{2} }[/math] in decimal (and simply, the first repunit),^{[8]} as well as the second unique prime in base ten.^{[9]} It is the first strong prime,^{[10]} the second good prime,^{[11]} the third super-prime, the fourth Lucas prime,^{[12]} and the fifth consecutive supersingular prime.^{[13]}
The rows of Pascal's triangle can be seen as representation of the powers of 11.^{[14]}
11 of 35 hexominoes can fold in a net to form a cube, while 11 of 66 octiamonds can fold into a regular octahedron.
An 11-sided polygon is called a hendecagon, or undecagon. The complete graph [math]\displaystyle{ K_{11} }[/math] 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.^{[15]}
11 and some of its multiples appear 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.^{[16]}
- 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,r2 in [math]\displaystyle{ \Complex }[/math] where [math]\displaystyle{ q }[/math] is constrained to [math]\displaystyle{ q=2/(1-(p+r)/pr) }[/math], while three contain affine nodes and include infinite solutions, two in [math]\displaystyle{ \Complex }[/math], and one in [math]\displaystyle{ \Complex^2 }[/math].^{[19]}
- 22 regular complex apeirohedra of the form _{p}{a}_{q}{b}_{r}, where 21 exist in [math]\displaystyle{ \Complex^2 }[/math] and 1 in [math]\displaystyle{ \Complex^3 }[/math].^{[20]}
- 11 regular paracompact hyperbolic honeycombs with infinite facets and vertex figures in the third dimension.^{[21]}
- 11 total 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.^{[21]}
- 55 uniform Euclidean 4-honeycombs exist in the fourth dimension, and 66 uniform Euclidean 5-honeycombs exist in the fifth dimension.
The 11-cell is a self-dual abstract 4-polytope with 11 vertices, 55 edges, 55 triangular faces, and 11 hemi-icosahedral cells. It is universal in the sense that it is the only abstract polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures. The 11-cell contains the same number of vertices and edges as the complete graph [math]\displaystyle{ K_{11} }[/math] and the 10-simplex, a regular polytope in 10 dimensions.
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.
Mathieu group [math]\displaystyle{ \mathrm{M}_{11} }[/math] is the smallest of twenty-six sporadic groups, defined as a sharply 4-transitive permutation group on eleven objects. It has order [math]\displaystyle{ 7920 =2^{4}\cdot3^{2}\cdot5\cdot11 = 8\cdot9\cdot10\cdot11 }[/math], 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 [math]\displaystyle{ \operatorname{S}(4,5,11) }[/math], with an induced action on unordered pairs of points that gives a rank 3 action on 55 points. Mathieu group [math]\displaystyle{ \mathrm{M}_{12} }[/math], on the other hand, is formed from the permutations of projective special linear group [math]\displaystyle{ \operatorname{PSL_2}(1,1) }[/math] with those of [math]\displaystyle{ (2,10)(3,4)(5,9)(6,7) }[/math]. It is the second-smallest sporadic group, and holds [math]\displaystyle{ \mathrm{M}_{11} }[/math] as a maximal subgroup and point stabilizer, with an order equal to [math]\displaystyle{ 95040 = 2^6\cdot3^3\cdot5\cdot11 = 8\cdot9\cdot10\cdot11\cdot12 }[/math], where 11 is also its largest prime factor, like [math]\displaystyle{ \mathrm{M}_{11} }[/math]. [math]\displaystyle{ \mathrm{M}_{12} }[/math] also centralizes an element of order 11 in the friendly giant [math]\displaystyle{ \mathrm {F}_{1} }[/math], the largest sporadic group, and holds an irreducible faithful complex representation in eleven dimensions.
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 ([math]\displaystyle{ \mathrm{J}_2 }[/math], [math]\displaystyle{ \mathrm{J}_3 }[/math], [math]\displaystyle{ \mathrm{Ru} }[/math], [math]\displaystyle{ \mathrm{He} }[/math], and [math]\displaystyle{ \mathrm{Th} }[/math]). 11 is also not a prime factor of the order of the Tits group [math]\displaystyle{ \mathrm{T} }[/math], which is sometimes categorized as non-strict group of Lie type, or sporadic group. In addition, there are eleven sporadic groups whose standard generators do not hold further conditions that yield a semi-presentation (or set of relations characterizing a set of generators of a group [math]\displaystyle{ \mathrm {G} }[/math] up to automorphism).^{[22]}
Within safe and Sophie Germain primes of the form [math]\displaystyle{ 2p+1 }[/math], 11 is the third safe prime, from a [math]\displaystyle{ p }[/math] of 5,^{[23]} and the fourth Sophie Germain prime, which yields 23.^{[24]}
In decimal
11 is the smallest two-digit prime number. On the seven-segment display of a calculator, it is both a strobogrammatic prime and a dihedral prime.^{[25]}
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: 11^{1} = 11, 11^{2} = 121, 11^{3} = 1331, and 11^{4} = 14641.
11 is the 11th index or member in the sequence of palindromic numbers, and 121, equal to [math]\displaystyle{ 11\times 11 }[/math], is the 22nd.^{[26]}
The factorial of 11, [math]\displaystyle{ 11!=39916800 }[/math], has about a 0.2% difference to the round number [math]\displaystyle{ 4\times 10^{7} }[/math], or 40 million. Among the first 100 factorials, the next closest to a round number is 96 ([math]\displaystyle{ 96! \approx 9.91678\times 10^{149} }[/math]), which is about 0.8% less than 10^{149}.^{[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:
- 142,312 × 11 = 1,565,432 → 2,345,651 ÷ 11 = 213,241.
Divisibility tests
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:
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,
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:
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):
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
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 | 25937424601 | 285311670611 |
x^{11} | 1 | 2048 | 177147 | 4194304 | 48828125 | 362797056 | 1977326743 | 8589934592 | 31381059609 | 100000000000 | 285311670611 |
In other bases
In base 13 and higher bases (such as hexadecimal), 11 is represented as B, where ten is A. In duodecimal, 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} |
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 crewed spacecraft to land on the Moon.
- A sunspot cycle's periodicity is approximately 11 years.
- Messier object M11 is a magnitude 7.0 open cluster in the constellation Scutum, also known as the Wild Duck Cluster.
- The New General Catalogue object^{[29]} NGC 11 is a spiral galaxy in the constellation Andromeda.
- The 11th moon of Jupiter is Himalia.
In religion and spirituality
Christianity
After Judas Iscariot was disgraced, Jesus's remaining apostles were sometimes called "the Eleven" (Mark 16:11; Luke 24:9 and 24:33), even after Matthias was added to bring the number back to 12, as in Acts 2:14:^{[30]} 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 12.
Saint Ursula is said to have been martyred in the 3rd or 4th century in Cologne with a number of companions, whose reported number "varies from five to eleven".^{[31]} A legend that Ursula died with 11,000 virgin companions^{[32]} 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 11 monsters to avenge the death of her husband, Apsû.
Mysticism
The number 11 (alongside its multiples 22 and 33) are master numbers in numerology, especially in New Age.^{[33]} In astrology, Aquarius is the 11th astrological sign of the Zodiac.^{[34]}
In music
- The interval of an octave plus a fourth is an 11th. A complete 11th chord has almost every note of a diatonic scale.
- There are 11 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 appears repeatedly 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.^{[35]}
- 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.^{[36]}
- Eleven is the title of albums by:
- Come^{[37]}
- incognito^{[38]}
- Martina McBride^{[39]}
- 22-Pistepirkko^{[40]}
- Eleven^{[41]}
- Harry Connick Jr.^{[42]}
- Tina Arena^{[43]}
- Jeff Lorber and Mike Stern^{[44]}
- Reamonn^{[45]}
- Wagon Cookin'^{[46]}
- Mr. Fogg^{[47]}
- The Birdland Big Band^{[48]}
- Pearl Django^{[49]}
- Daniel Peña^{[50]}
- The Knux^{[51]}
- Igor Lumpert^{[52]}
- The Smithereens^{[53]}
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, punters and wide receivers as well as running backs, defensive backs, safeties and linebackers 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 side, as in soccer.
- 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 number 11.
In the military
- The number of guns in a gun salute to U.S. Army, United States 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 U.S. 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 X11.
- Computers of the PDP-11 series from Digital Equipment Corporation were informally called "elevens".
- Windows 11
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 50-dollar bill, show 11:00.
In other fields
- Sector 11^{[54]} in the North American Industry Classification System is the code for Agriculture, Forestry, Fishing and Hunting industries.
- 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).
- In Basque, hamaika ("eleven") has the double meaning of "infinite", probably from Basque 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", including "punk," "top," & "railroad".^{[55]}
- American Airlines Flight 11, a Boston-Los Angeles flight, crashed into the North Tower of the World Trade Center in New York City after terrorists hijacked it on September 11, 2001.
- The number 11 bus is a low-cost way to sightsee in London.
- In the game of blackjack, an ace can count 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.
- 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, portrayed by Millie Bobby Brown.
- Eleven is a British television production company.
See also
- 11:11
- 11:11 (numerology)
- XI
References
- ↑ Bede, Eccl. Hist., Bk. V, Ch. xviii.
- ↑ Specifically, in the line jjvjv ðæt rice hæfde endleofan wintra.^{[1]}
- ↑ ^{3.0} ^{3.1} ^{3.2} ^{3.3} ^{3.4} Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
- ↑ Kroonen, Guus (2013). Etymological Dictionary of Proto-Germanic. Leiden: Brill. p. 11f. ISBN 978-90-04-18340-7.
- ↑ Dantzig, Tobias (1930), Number: The Language of Science.
- ↑ Sloane, N. J. A., ed. "Sequence A001359 (Lesser of twin primes.)". OEIS Foundation. https://oeis.org/A001359. Retrieved 2023-01-22.
- ↑ Sloane, N. J. A., ed. "Sequence A136162 (List of prime quadruplets {p, p+2, p+6, p+8}.)". OEIS Foundation. https://oeis.org/A136162. Retrieved 2023-03-02.
- "{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". OEIS Foundation. https://oeis.org/A004022.
- ↑ "Sloane's A040017 : Unique period primes (no other prime has same period as 1/p) in order (periods are given in A051627)". OEIS Foundation. https://oeis.org/search?q=unique+prime.
- ↑ Sloane, N. J. A., ed. "Sequence A051634 (Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2)". OEIS Foundation. https://oeis.org/A051634. Retrieved 2022-08-10.
- ↑ "Sloane's A028388 : Good primes". OEIS Foundation. https://oeis.org/A028388.
- ↑ "Sloane's A005479 : Prime Lucas numbers". OEIS Foundation. https://oeis.org/A005479.
- ↑ ^{13.0} ^{13.1} Sloane, N. J. A., ed. "Sequence A002267 (The 15 supersingular primes: primes dividing order of Monster simple group.)". OEIS Foundation. https://oeis.org/A002267. Retrieved 2023-01-22.
- ↑ 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. https://www.jstor.org/stable/27957164.
- ↑ Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly (Taylor & Francis, Ltd) 95 (3): 191–194. doi:10.2307/2323624. https://www.tandfonline.com/doi/abs/10.1080/00029890.1988.11971989?journalCode=uamm20.
- ↑ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons". Mathematics Magazine (Taylor & Francis, Ltd.) 50 (5): 233. doi:10.2307/2689529. http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf.
- ↑ 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. OCLC 13092426. https://archive.org/details/isbn_0716711931.
- ↑ Grünbaum, Branko; Miller, J. C. P.; Shephard, G. C. (1981). "Uniform Tilings with Hollow Tiles". in Davis, Chandler; Grünbaum, Branko; Sherk, F. A.. The Geometric Vein: The Coxeter Festschrift. New York: Springer-Verlag. pp. 47–48. doi:10.1007/978-1-4612-5648-9_3. ISBN 978-1-4612-5650-2. OCLC 7597141. https://archive.org/details/geometricveincox0000unse.
- ↑ Coxeter, H.S.M. (1991). "11.6 Apeirogons". Regular Complex Polytopes (2 ed.). London: Cambridge University Press. p. 111, 112. doi:10.2307/3617711. ISBN 978-0521394901. OCLC 21562167.
- ↑ Coxeter, H.S.M. (1991). "12.8 Cycles of Honeycombs". Regular Complex Polytopes (2 ed.). London: Cambridge University Press. p. 138-140. doi:10.2307/3617711. ISBN 978-0521394901. OCLC 21562167.
- ↑ ^{21.0} ^{21.1} Coxeter, H. S. M. (1956). "Regular Honeycombs in Hyperbolic Space". Proceedings of the International Congress of Mathematicians (1954) (Amsterdam: North-Holland Publishing Co.) 3: 167–168. http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf.
- ↑ Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations". The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249). Cambridge, U.K: Cambridge University Press. p. 261–273. doi:10.1017/CBO9780511565830.024. ISBN 9780511565830. OCLC 726827806. https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/ASGRweb.pdf.
- ↑ "Sloane's A005385 : Safe primes". OEIS Foundation. https://oeis.org/A005385.
- ↑ "Sloane's A005384 : Sophie Germain primes". OEIS Foundation. https://oeis.org/A005384.
- ↑ "Sloane's A134996 : Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes.". OEIS Foundation. https://oeis.org/A134996.
- ↑ Sloane, N. J. A., ed. "Sequence A002113 (Palindromes in base 10.)". OEIS Foundation. https://oeis.org/A002113. Retrieved 2022-08-11.
- ↑ "List of first 100 factorial numbers". https://oeis.org/A000142/b000142.txt.
- ↑ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 47. ISBN 978-1-84800-000-1.
- ↑ 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". http://www.ngcicproject.org/.
- ↑ "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.". http://bible.cc/acts/2-14.htm.
- ↑ Ursulines of the Roman Union, Province of Southern Africa, St. Ursula and Companions , accessed 10 July 2016
- ↑ Four scenes from the life of St Ursula, accessed 10 July 2016
- ↑ Sharp, Damian (2001) (in English). Simple Numerology: A Simple Wisdom book (A Simple Wisdom Book series). Red Wheel. p. 7. ISBN 978-1573245609.
- ↑ "Aquarius". Oxford Dictionaries. n.d.. https://en.oxforddictionaries.com/definition/aquarius. Retrieved June 27, 2022.
- ↑ (in en-us) The Eleven - Grateful Dead | Song Info | AllMusic, https://www.allmusic.com/song/the-eleven-mt0012693977, retrieved 2020-08-10
- ↑ Corazon, Billy (July 1, 2009). "Imaginary Interview: Jason Webley". Three Imaginary Girls. http://www.threeimaginarygirls.com/features/2009jul/imaginaryinterviewjasonwebley.
- ↑ (in en-us) Eleven - Come | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0000095300, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Incognito | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0000353339, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Martina McBride | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002215185, retrieved 2020-08-10
- ↑ (in en-us) Eleven - 22-Pistepirkko | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0001045033, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Eleven | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0000621201, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Harry Connick, Jr. | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0000616960, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Tina Arena | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002893078, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Jeff Lorber, The Jeff Lorber Fusion, Mike Stern | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0003309756, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Reamonn | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002042439, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Wagon Cookin' | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002221693, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Mr. Fogg | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002343774, retrieved 2020-08-10
- ↑ (in en-us) Eleven - The Birdland Big Band, Tommy Igoe | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002352968, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Pearl Django | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002383195, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Daniel Pena, Daniel Peña | Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/eleven-mw0002714071, retrieved 2020-08-10
- ↑ (in en-us) Eleven - The Knux | User Reviews | AllMusic, https://www.allmusic.com/album/eleven-mw0002991318, retrieved 2020-08-10
- ↑ (in en-us) Eleven - Igor Lumpert & Innertextures | User Reviews | AllMusic, https://www.allmusic.com/album/eleven-mw0003156790, retrieved 2020-08-10
- ↑ (in en-us) 11 - The Smithereens| Songs, Reviews, Credits | AllMusic, https://www.allmusic.com/album/11-mw0000653755, retrieved 2022-12-04
- ↑ ESMD, US Census Bureau Classification Development Branch. "US Census Bureau Site North American Industry Classification System main page". https://www.census.gov/cgi-bin/sssd/naics/naicsrch?code=11&search=2012+NAICS+Search.
- ↑ "Surveying Units and Terms". Directlinesoftware.com. 2012-07-30. http://www.directlinesoftware.com/survey.htm.
External links
- Grimes, James. "Eleven". Numberphile. Brady Haran. http://www.numberphile.com/videos/11.html.
Original source: https://en.wikipedia.org/wiki/11 (number).
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