1.
10 (number)
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10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十
2.
12 (number)
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12 is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a highly composite number, divisible by 2,3,4. It is central to systems of counting, including the Western calendar and units of time. The word twelve is the largest number with a name in English. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance and it derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels Book of John. It has cognates in every Germanic language, whose Proto-Germanic ancestor has been reconstructed as *twaliƀi, from *twa and suffix *-lif- or *-liƀ- of uncertain meaning. It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19, every other Indo-European language instead uses a form of two+ten, such as the Latin duōdecim. The usual ordinal form is twelfth but dozenth or duodecimal is also used in some contexts, similarly, a group of twelve things is usually a dozen but may also be referred to as a duodecad. The adjective referring to a group of twelve is duodecuple, as with eleven, the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan, with the implicit meaning that two is left after having already counted to ten. The Lithuanian suffix is also considered to share a similar development, the suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten. While, as mentioned above,12 has its own name in Germanic languages such as English and German, it is a number in many other languages, e. g. Italian dodici. In Germany, according to an old tradition, the numbers 0 through 12 were spelt out, the Duden now calls this tradition outdated and no longer valid, but many writers still follow it. Another system spells out all numbers written in one or two words, Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1,2,3,4,6 and 12. Twelve is also a composite number, the next one being twenty-four. Twelve is also a highly composite number, the next one being sixty. It is the first composite number of the form p2q, a square-prime,12 has an aliquot sum of 16. Accordingly,12 is the first abundant number and demonstrates an 8-member aliquot sequence,12 is the 3rd composite number in the 3-aliquot tree, the only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant, Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number
3.
9 (number)
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9 is the natural number following 8 and preceding 10. In the NATO phonetic alphabet, the digit 9 is called Niner, five-digit produce PLU codes that begin with 9 are organic. Common terminal digit in psychological pricing, Nine is a number that appears often in Indian Culture and mythology. Nine influencers are attested in Indian astrology, in the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements, Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. Navaratri is a festival dedicated to the nine forms of Durga. Navaratna, meaning 9 jewels may also refer to Navaratnas - accomplished courtiers, Navratan - a kind of dish, according to Yoga, the human body has nine doors - two eyes, two ears, the mouth, two nostrils, and the openings for defecation and procreation. In Indian aesthetics, there are nine kinds of Rasa, Nine is considered a good number in Chinese culture because it sounds the same as the word long-lasting. Nine is strongly associated with the Chinese dragon, a symbol of magic, there are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales –81 yang and 36 yin, all three numbers are multiples of 9 as well as having the same digital root of 9. The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City, the name of the area called Kowloon in Hong Kong literally means, nine dragons. The nine-dotted line delimits certain island claims by China in the South China Sea, the nine-rank system was a civil service nomination system used during certain Chinese dynasties. 9 Points of the Heart / Heart Master Channels in Traditional Chinese Medicine, the nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. The Ennead is a group of nine Egyptian deities, who, in versions of the Osiris myth. The Nine Worthies are nine historical, or semi-legendary figures who, in Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil. The nine Muses in Greek mythology are Calliope, Clio, Erato, Euterpe, Melpomene, Polyhymnia, Terpsichore, Thalia and it takes nine days to fall from heaven to earth, and nine more to fall from earth to Tartarus—a place of torment in the underworld. Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo, according to Georges Ifrah, the origin of the 9 integers can be attributed to ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0. In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot, the Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, as time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller
4.
15 (number)
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15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name, in spoken English, the numbers 15 and 50 are often confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed,15 /fɪfˈtiːn/ vs 50 /ˈfɪfti/, however, in dates such as 1500 or when contrasting numbers in the teens, the stress generally shifts to the first syllable,15 /ˈfɪftiːn/. In a 24-hour clock, the hour is in conventional language called three or three oclock. A composite number, its divisors being 1,3 and 5. A repdigit in binary and quaternary, in hexadecimal, as well as all higher bases,15 is represented as F. the 4th discrete semiprime and the first member of the discrete semiprime family. It is thus the first odd discrete semiprime, the number proceeding 15,14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21, the smallest number that can be factorized using Shors quantum algorithm. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet, the aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members. 15 is the composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree, fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33. 15 and 16 form a Ruth-Aaron pair under the definition in which repeated prime factors are counted as often as they occur. There are 15 solutions to Známs problem of length 7, if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems. Group 15 of the table are sometimes known as the pnictogens. 15 Madadgar is designated as a number in Pakistan, for mobile phones, similar to the international GSM emergency number 112, if 112 is used in Pakistan. 112 can be used in an emergency if the phone is locked. The Hanbali Sunni madhab states that the age of fifteen of a solar or lunar calendar is when ones taklif begins and is the stage whereby one has his deeds recorded. In the Hebrew numbering system, the number 15 is not written according to the method, with the letters that represent 10 and 5
5.
17 (number)
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17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution
6.
19 (number)
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19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
7.
20 (number)
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20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants
8.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
9.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
10.
30 (number)
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30 is the natural number following 29 and preceding 31. Thirty is the sum of the first four squares, which makes it a square pyramidal number and it is a primorial and is the smallest Giuga number. 30 is the smallest sphenic number, and the smallest of the form 2 ×3 × r,30 has an aliquot sum of 42, the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises Thirty has but one number for which it is the aliquot sum, adding up some subsets of its divisors gives 30, hence 30 is a semiperfect number. 30 is the largest number such that all smaller than itself. A polygon with thirty sides is called a triacontagon, the icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a graph with 30 vertices. The atomic number of zinc is 30 Messier object M30, a magnitude 8, the duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series began on June 19,1803 BC. The duration of Saros series 30 was 1316.2 years, Thirty is, Used to indicate the end of a newspaper story, a copy editors typographical notation. S. Judas Iscariot betrayed Jesus for 30 pieces of silver, one of the rallying-cries of the 1960s student/youth protest movement was the slogan, Dont trust anyone over thirty. In Franz Kafkas novel The Trial Joseph wakes up on the morning of his birthday to find himself under arrest for an unspecified crime. After making many attempts to find the nature of the crime or the name of his accuser. The number of uprights that formed the Sarsen Circle at Stonehenge, western Christianitys most prolific 20th century essayist, F. W. Also in that essay Boreham writes It was said of Keats, in tennis, the number 30 represents the second point gained in a game. Under NCAA rules for basketball, the offensive team has 30 seconds to attempt a shot. As of 2012, three of the four major leagues in the United States and Canada have 30 teams each. The California Angels baseball team retired the number in honor of its most notable wearer, Nolan Ryan, the San Francisco Giants extended the same honor to Orlando Cepeda
11.
40 (number)
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Despite being related to the word four, the modern spelling of 40 is forty. The archaic form fourty is now considered a misspelling, the modern spelling possibly reflects a pronunciation change due to the horse–hoarse merger. Forty is a number, an octagonal number, and as the sum of the first four pentagonal numbers. Adding up some subsets of its divisors gives 40, hence 40 is a semiperfect number, given 40, the Mertens function returns 0. 40 is the smallest number n with exactly 9 solutions to the equation φ = n, Forty is the number of n-queens problem solutions for n =7. Since 402 +1 =1601 is prime,40 is a Størmer number,40 is a repdigit in base 3 and a Harshad number in base 10. Negative forty is the temperature at which the Fahrenheit and Celsius scales correspond. It is referred to as either minus forty or forty below, the planet Venus forms a pentagram in the night sky every eight years with it returning to its original point every 40 years with a 40-day regression. The duration of Saros series 40 was 1280.1 years, lunar eclipse series which began on -1387 February 12 and ended on -71 April 12. The duration of Saros series 40 was 1316.2 years, the number 40 is used in Jewish, Christian, Islamic, and other Middle Eastern traditions to represent a large, approximate number, similar to umpteen. In the Hebrew Bible, forty is often used for periods, forty days or forty years. Rain fell for forty days and forty nights during the Flood, spies explored the land of Israel for forty days. The Hebrew people lived in the Sinai desert for forty years and this period of years represents the time it takes for a new generation to arise. Moses life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, and his return to lead his people out, several Jewish leaders and kings are said to have ruled for forty years, that is, a generation. Examples include Eli, Saul, David, and Solomon, goliath challenged the Israelites twice a day for forty days before David defeated him. He went up on the day of Tammuz to beg forgiveness for the peoples sin. He went up on the first day of Elul and came down on the day of Tishrei. A mikvah consists of 40 seah of water 40 lashes is one of the punishments meted out by the Sanhedrin, One of the prerequisites for a man to study Kabbalah is that he is forty years old
12.
60 (number)
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60 is the natural number following 59 and preceding 61. Being three times 20, it is called three score in older literature. It is a number, with divisors 1,2,3,4,5,6,10,12,15,20,30. Because it is the sum of its divisors, it is a unitary perfect number. Being ten times a number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6 and it is the smallest number with exactly 12 divisors. It is the sum of a pair of twin primes and the sum of four consecutive primes and it is adjacent to two primes. It is the smallest number that is the sum of two odd primes in six ways, the smallest non-solvable group has order 60. There are four Archimedean solids with 60 vertices, the icosahedron, the rhombicosidodecahedron, the snub dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs, there are also two Archimedean solids with 60 edges, the snub cube and the icosidodecahedron. The skeleton of the forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares, in geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, a number system with base 60 is called sexagesimal. It is the smallest positive integer that is written only the smallest. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule and this ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 is an isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz
13.
80 (number)
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80 is the natural number following 79 and preceding 81. 80 is, the sum of Eulers totient function φ over the first sixteen integers, a semiperfect number, since adding up some subsets of its divisors gives 80. Palindromic in bases 3,6,9,15,19 and 39, a repdigit in bases 3,9,15,19 and 39. A Harshad number in bases 2,3,4,5,6,7,9,10,11,13,15 and 16 The Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes. Every solvable configuration of the Fifteen puzzle can be solved in no more than 80 single-tile moves, the atomic number of mercury According to Exodus 7,7, Moses was 80 years old when he initially spoke to Pharaoh on behalf of his people. Today,80 years of age is the age limit for cardinals to vote in papal elections. Jerry Rice wore the number 80 for the majority of his NFL career
14.
90 (number)
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90 is the natural number preceded by 89 and followed by 91. In English speech, the numbers 90 and 19 are often confused, when carefully enunciated, they differ in which syllable is stressed,19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in such as 1999, and when contrasting numbers in the teens and when counting, such as 17,18,19. 90 is, a perfect number because it is the sum of its unitary divisors. A semiperfect number because it is equal to the sum of a subset of its divisors, a Perrin number, preceded in the sequence by 39,51,68. Palindromic and a repdigit in bases 14,17,29, a Harshad number since 90 is divisible by the sum of its base 10 digits. In normal space, the angles of a rectangle measure 90 degrees each. Also, in a triangle, the angle opposing the hypotenuse measures 90 degrees. Thus, an angle measuring 90 degrees is called a right angle, ninety is, the atomic number of thorium, an actinide. As an atomic weight,90 identifies an isotope of strontium, the latitude in degrees of the North and the South geographical poles. NFL, New York Jets Dennis Byrds #90 is retired +90 is the code for international direct dial phone calls to Turkey,90 is the code for the French département Belfort
15.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
16.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
17.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
18.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
19.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
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Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
21.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
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Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
23.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
24.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
25.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
26.
Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
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Senary
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
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Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
29.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
30.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
31.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
32.
Base 36
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
33.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
34.
Old English
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Old English or Anglo-Saxon is the earliest historical form of the English language, spoken in England and southern and eastern Scotland in the early Middle Ages. It was brought to Great Britain by Anglo-Saxon settlers probably in the mid 5th century, Old English developed from a set of Anglo-Frisian or North Sea Germanic dialects originally spoken by Germanic tribes traditionally known as the Angles, Saxons, and Jutes. As the Anglo-Saxons became dominant in England, their language replaced the languages of Roman Britain, Common Brittonic, a Celtic language, Old English had four main dialects, associated with particular Anglo-Saxon kingdoms, Mercian, Northumbrian, Kentish and West Saxon. It was West Saxon that formed the basis for the standard of the later Old English period, although the dominant forms of Middle. The speech of eastern and northern parts of England was subject to strong Old Norse influence due to Scandinavian rule, Old English is one of the West Germanic languages, and its closest relatives are Old Frisian and Old Saxon. Like other old Germanic languages, it is different from Modern English. Old English grammar is similar to that of modern German, nouns, adjectives, pronouns, and verbs have many inflectional endings and forms. The oldest Old English inscriptions were using a runic system. Old English was not static, and its usage covered a period of 700 years, from the Anglo-Saxon settlement of Britain in the 5th century to the late 11th century, some time after the Norman invasion. While indicating that the establishment of dates is a process, Albert Baugh dates Old English from 450 to 1150, a period of full inflections. Perhaps around 85 per cent of Old English words are no longer in use, Old English is a West Germanic language, developing out of Ingvaeonic dialects from the 5th century. It came to be spoken over most of the territory of the Anglo-Saxon kingdoms which became the Kingdom of England and this included most of present-day England, as well as part of what is now southeastern Scotland, which for several centuries belonged to the Anglo-Saxon kingdom of Northumbria. Other parts of the island – Wales and most of Scotland – continued to use Celtic languages, Norse was also widely spoken in the parts of England which fell under Danish law. Anglo-Saxon literacy developed after Christianisation in the late 7th century, the oldest surviving text of Old English literature is Cædmons Hymn, composed between 658 and 680. There is a corpus of runic inscriptions from the 5th to 7th centuries. The Old English Latin alphabet was introduced around the 9th century, with the unification of the Anglo-Saxon kingdoms by Alfred the Great in the later 9th century, the language of government and literature became standardised around the West Saxon dialect. In Old English, typical of the development of literature, poetry arose before prose, a later literary standard, dating from the later 10th century, arose under the influence of Bishop Æthelwold of Winchester, and was followed by such writers as the prolific Ælfric of Eynsham. This form of the language is known as the Winchester standard and it is considered to represent the classical form of Old English
35.
Bede
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He is well known as an author and scholar, and his most famous work, Ecclesiastical History of the English People gained him the title The Father of English History. Bedes monastery had access to a library which included works by Eusebius, Orosius. Almost everything that is known of Bedes life is contained in the last chapter of his Ecclesiastical History of the English People, a history of the church in England. It was completed in about 731, and Bede implies that he was then in his fifty-ninth year, a minor source of information is the letter by his disciple Cuthbert which relates Bedes death. Bede, in the Historia, gives his birthplace as on the lands of this monastery, Bede says nothing of his origins, but his connections with men of noble ancestry suggest that his own family was well-to-do. Bedes first abbot was Benedict Biscop, and the names Biscop, Bedes name reflects West Saxon Bīeda. It is an Anglo-Saxon short name formed on the root of bēodan to bid, the name also occurs in the Anglo-Saxon Chronicle, s. a. 501, as Bieda, one of the sons of the Saxon founder of Portsmouth, the Liber Vitae of Durham Cathedral names two priests with this name, one of whom is presumably Bede himself. Some manuscripts of the Life of Cuthbert, one of Bedes works, mention that Cuthberts own priest was named Bede, at the age of seven, Bede was sent to the monastery of Monkwearmouth by his family to be educated by Benedict Biscop and later by Ceolfrith. Bede does not say whether it was intended at that point that he would be a monk. Monkwearmouths sister monastery at Jarrow was founded by Ceolfrith in 682, in 686, plague broke out at Jarrow. The two managed to do the service of the liturgy until others could be trained. The young boy was almost certainly Bede, who would have been about 14, when Bede was about 17 years old, Adomnán, the abbot of Iona Abbey, visited Monkwearmouth and Jarrow. Bede would probably have met the abbot during this visit, in about 692, in Bedes nineteenth year, Bede was ordained a deacon by his diocesan bishop, John, who was bishop of Hexham. There might have been minor orders ranking below a deacon, in Bedes thirtieth year, he became a priest, with the ordination again performed by Bishop John. In about 701 Bede wrote his first works, the De Arte Metrica and De Schematibus et Tropis and he continued to write for the rest of his life, eventually completing over 60 books, most of which have survived. Not all his output can be dated, and Bede may have worked on some texts over a period of many years. His last-surviving work is a letter to Ecgbert of York, a former student, Bede may also have worked on one of the Latin bibles that were copied at Jarrow, one of which is now held by the Laurentian Library in Florence
36.
Ecclesiastical History of the English People
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It is believed to have been completed in 731 when Bede was approximately 59 years old. The Historia ecclesiastica gentis Anglorum, or An Ecclesiastical History of the English People is Bedes best-known work, the first of the five books begins with some geographical background and then sketches the history of England, beginning with Caesars invasion in 55 BC. The second book begins with the death of Gregory the Great in 604, and follows the progress of Christianity in Kent. These encountered a setback when Penda, the king of Mercia. The setback was temporary, and the book recounts the growth of Christianity in Northumbria under kings Oswald. The climax of the book is the account of the Council of Whitby. The fourth book begins with the consecration of Theodore as Archbishop of Canterbury, the fifth book brings the story up to Bedes day, and includes an account of missionary work in Frisia, and of the conflict with the British church over the correct dating of Easter. Bede wrote a preface for the work, in which he dedicates it to Ceolwulf, the preface mentions that Ceolwulf received an earlier draft of the book, presumably Ceolwulf knew enough Latin to understand it, and he may even have been able to read it. Divided into five books, the Historia covers the history of England, ecclesiastical and political and this is impressive, nevertheless, the Historia, like other historical writing from this period has a lower degree of objectivity than modern historical writings. It seems to be a mixture of fact, legend and literature, the monastery at Jarrow had an excellent library. Both Benedict Biscop and Ceolfrith had acquired books from the Continent, for the period prior to Augustines arrival in 597, Bede drew on earlier writers, including Orosius, Eutropius, Pliny, and Solinus. He used Constantiuss Life of Germanus as a source for Germanuss visits to Britain, Bedes account of the invasion of the Anglo-Saxons is drawn largely from Gildass De Excidio et Conquestu Britanniae. Bede would also have been familiar with more recent accounts such as Eddius Stephanuss Life of Wilfrid and he also drew on Josephuss Antiquities, and the works of Cassiodorus, and there was a copy of the Liber Pontificalis in Bedes monastery. Bede also had correspondents who supplied him with material, almost all of Bedes information regarding Augustine is taken from these letters, which includes the Libellus responsionum, as chapter 27 of book 1 is often known. Bede also mentions an Abbot Esi as a source for the affairs of the East Anglian church, the historian Walter Goffart argues that Bede based the structure of the Historia on three works, using them as the framework around which the three main sections of the work were structured. For the early part of the work, up until the Gregorian mission, the second section, detailing the Gregorian mission of Augustine of Canterbury was framed on the anonymous Life of Gregory the Great written at Whitby. The last section, detailing events after the Gregorian mission, Goffart asserts were modelled on Stephen of Ripons Life of Wilfrid, the History of the English Church and People has a clear polemical and didactic purpose. Bede sets out not just to tell the story of the English, in political terms he is a partisan of his native Northumbria, amplifying its role in English history over and above that of Mercia, its great southern rival
37.
Germanic language
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It is the third most spoken Indo-European subdivision, behind Italic and Indo-Iranian, and ahead of Balto-Slavic languages. Limburgish varieties have roughly 1.3 million speakers along the Dutch–Belgian–German border, the main North Germanic languages are Norwegian, Danish, Swedish, Icelandic, and Faroese, which have a combined total of about 20 million speakers. The East Germanic branch included Gothic, Burgundian, and Vandalic, the last to die off was Crimean Gothic, spoken in the late 18th century in some isolated areas of Crimea. The total number of Germanic languages throughout history is unknown, as some of them—especially East Germanic languages—disappeared during or after the Migration Period. Proto-Germanic, along all of its descendants, is characterized by a number of unique linguistic features. Early varieties of Germanic enter history with the Germanic tribes moving south from Scandinavia in the 2nd century BC, to settle in the area of todays northern Germany, furthermore, it is the de facto language of the United Kingdom, the United States and Australia. It is also a language in Nicaragua and Malaysia. German is a language of Austria, Belgium, Germany, Liechtenstein, Luxembourg and Switzerland and has regional status in Italy, Poland, Namibia. German also continues to be spoken as a minority language by immigrant communities in North America, South America, Central America, Mexico, a German dialect, Pennsylvania Dutch, is still present amongst Anabaptist populations in Pennsylvania in the United States. Dutch is a language of Aruba, Belgium, Curaçao. The Netherlands also colonised Indonesia, but Dutch was scrapped as a language after Indonesian independence. Dutch was until 1925 an official language in South Africa, but evolved in and was replaced by Afrikaans, Afrikaans is one of the 11 official languages in South Africa and is a lingua franca of Namibia. It is used in other Southern African nations as well, low German is a collection of sometimes very diverse dialects spoken in the northeast of the Netherlands and northern Germany. Scots is spoken in Lowland Scotland and parts of Ulster, frisian is spoken among half a million people who live on the southern fringes of the North Sea in the Netherlands, Germany, and Denmark. Luxembourgish is mainly spoken in the Grand Duchy of Luxembourg, though it extends into small parts of Belgium, France. Limburgish varieties are spoken in the Limburg and Rhineland regions, along the Dutch–Belgian–German border, Swedish is also one of the two official languages in Finland, along with Finnish, and the only official language in the Åland Islands. Danish is also spoken natively by the Danish minority in the German state of Schleswig-Holstein, Norwegian is the official language of Norway. Icelandic is the language of Iceland, and is spoken by a significant minority in the Faroe Islands
38.
Proto-Germanic language
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Proto-Germanic is the reconstructed proto-language of the Germanic branch of the Indo-European languages. The end of the Common Germanic period is reached with the beginning of the Migration Period in the fourth century, the Proto-Germanic language is not directly attested by any coherent surviving texts, it has been reconstructed using the comparative method. Fragmentary direct attestation exists of Common Germanic in early runic inscriptions, the Proto-Germanic language developed in southern Scandinavia, the Urheimat of the Germanic tribes. Proto-Germanic developed out of pre-Proto-Germanic during the Pre-Roman Iron Age of Northern Europe, Proto-Germanic itself was likely spoken after c. Early Germanic expansion in the Pre-Roman Iron Age placed Proto-Germanic speakers in contact with the Continental Celtic La Tène horizon, a number of Celtic loanwords in Proto-Germanic have been identified. By the 1st century AD, Germanic expansion reaches the Danube and the Upper Rhine in the south, at about the same time, extending east of the Vistula, Germanic speakers come into contact with early Slavic cultures, as reflected in early Germanic loans in Proto-Slavic. By the 3rd century, LPGmc speakers had expanded over significant distance, the period marks the breakup of Late Proto-Germanic and the beginning of the Germanic migrations. The earliest coherent text in Proto-Norse become available c.400 in runic inscriptions, the delineation of Late Common Germanic from Proto-Norse about then is largely a matter of convention. Early West Germanic becomes available in the 5th century with the Frankish Bergakker inscription, between the two points, many sound changes occurred. Phylogeny as applied to historical linguistics involves the descent of languages. The Germanic languages form a tree with Proto-Germanic at its root that is a branch of the Indo-European tree, borrowing of lexical items from contact languages makes the relative position of the Germanic branch within Indo-European less clear than the positions of the other branches of Indo-European. In the course of the development of linguistics, various solutions have been proposed, none certain. In the evolutionary history of a family, philologists consider a genetic tree model appropriate only if communities do not remain in effective contact as their languages diverge. The internal diversification of West Germanic developed in an especially non-treelike manner, Proto-Germanic is generally agreed to have begun about 500 BC. Its hypothetical ancestor between the end of Proto-Indo-European and 500 BC is termed Pre-Proto-Germanic, whether it is to be included under a wider meaning of Proto-Germanic is a matter of usage. The fixation of the led to sound changes in unstressed syllables. For Lehmann, the boundary was the dropping of final -a or -e in unstressed syllables, for example, post-PIE *wóyd-e > Gothic wait. Antonsen agreed with Lehmann about the boundary but later found runic evidence that the -a was not dropped, ékwakraz … wraita, I, Wakraz
39.
1 (number)
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1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few
40.
Lithuanian language
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Lithuanian is the official state language of Lithuania and is recognized as one of the official languages of the European Union. There are about 2.9 million native Lithuanian speakers in Lithuania, Lithuanian is a Baltic language, related to Latvian. It is written in a Latin alphabet, Lithuanian is often said to be the most conservative living Indo-European language, retaining many features of Proto-Indo-European now lost in other Indo-European languages. Anyone wishing to hear how Indo-Europeans spoke should come and listen to a Lithuanian peasant, among Indo-European languages, Lithuanian is extraordinarily conservative, retaining many archaic features otherwise found only in ancient languages such as Sanskrit or Ancient Greek. For this reason, it is one of the most important sources in the reconstruction of the Proto-Indo-European language despite its late attestation, the Proto-Balto-Slavic languages branched off directly from Proto-Indo-European, then branched into Proto-Baltic and Proto-Slavic. Proto-Baltic branched off into Proto-West Baltic and Proto-East Baltic, according to some glottochronological speculations, the Eastern Baltic languages split from the Western Baltic ones between AD400 and AD600. The Greek geographer Ptolemy had already written of two Baltic tribe/nations by name, the Galindai and Sudinoi in the 2nd century AD, the differentiation between Lithuanian and Latvian started after AD800, for a long period, they could be considered dialects of a single language. At a minimum, transitional dialects existed until the 14th or 15th century, also, the 13th- and 14th-century occupation of the western part of the Daugava basin by the German Sword Brethren had a significant influence on the languages independent development. The earliest surviving written Lithuanian text is a translation dating from about 1503–1525 of the Lords Prayer, the Hail Mary, printed books existed after 1547, but the level of literacy among Lithuanians was low through the 18th century, and books were not commonly available. Brought into the country by book smugglers despite the threat of prison sentences. Jonas Jablonskis made significant contributions to the formation of the standard Lithuanian language and his proposal for Standard Lithuanian was based on his native Western Aukštaitijan dialect with some features of the eastern Prussian Lithuanians dialect spoken in Lithuania Minor. These dialects had preserved archaic phonetics mostly intact due to the influence of the neighbouring Old Prussian language, Lithuanian has been the official language of Lithuania since 1918. During the Soviet era, it was used in official discourse along with Russian, Lithuanian is one of two living Baltic languages, along with Latvian. An earlier Baltic language, Old Prussian, was extinct by the 18th century, such an opinion was first represented by the likes of August Schleicher, and to a certain extent, Antoine Meillet. Endzelīns thought that the similarity between Baltic and Slavic was explicable through language contact while Schleicher, Meillet and others argued for a kinship between the two families. An attempt to reconcile the opposing stances was made by Jan Michał Rozwadowski and he proposed that the two language groups were indeed a unity after the division of Indo-European, but also suggested that after the two had divided into separate entities, they had posterior contact. The genetic kinship view is augmented by the fact that Proto-Balto-Slavic is easily reconstructible with important proofs in historic prosody, vyacheslav Ivanov and Vladimir Toporov believed in the unity of Balto-Slavic, but not in the unity of Baltic. In the 1960s, they proposed a new division, that into East-Baltic, West-Baltic, the Ivanov–Toporov theory is gaining ground among students of comparative-historic grammar of Indo-European language, and seems to be replacing the previous two stances in most PIE textbooks
41.
Old Frisian
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Old Frisian is a West Germanic language spoken between the 8th and 16th centuries in the area between the Rhine and Weser on the European North Sea coast. The Frisian settlers on the coast of South Jutland also spoke Old Frisian, the language of the earlier inhabitants of the region between the Zuiderzee and Ems River is attested in only a few personal names and place-names. Old Frisian evolved into Middle Frisian, spoken from the 16th to the 19th century, in the early Middle Ages, Frisia stretched from the area around Bruges, in what is now Belgium, to the Weser River in northern Germany. At the time, the Frisian language was spoken along the entire southern North Sea coast and this region is referred to as Greater Frisia or Frisia Magna, and many of the areas within it still treasure their Frisian heritage. However, by 1300, their territory had been pushed back to the Zuiderzee, hence, a close relationship exists between Old Frisian and Old English. Generally, Old Frisian phonologically resembles Old English, in particular, it shares the palatalisation of velar consonants also found in Old English. For example, whereas the closely related Old Saxon and Old Dutch retain the velar in dag, Old Frisian has dei, when followed by front vowels the Germanic /k/ changed to a /tʃ/ sound. The Old Frisian for church was tzirke or tzerke, in Old English it was ċiriċe, while Old Saxon, another feature shared between the two is Anglo-Frisian brightening, which fronted a to e under some circumstances. In unstressed syllables, o merges into a, and i into e as in Old English, the old Germanic diphthongs *ai and *au become ē/ā and ā, respectively, in Old Frisian, as in ēn/ān from Proto-Germanic *ainaz, and brād from *braudą. In comparison, these diphthongs become ā and ēa in Old English, the diphthong *eu generally becomes ia, and Germanic *iu is retained. These diphthongs initially began with a i, but the stress later shifts to the second component, giving to iā. For example, thiād and liūde from Proto-Germanic *þeudō and *liudīz, between vowels, h generally disappears, as in Old English and Old Dutch. Word-initial h- on the hand is retained. Some of the texts that are preserved from this period are from the 12th or 13th centuries, generally, all these texts are restricted to legal writings. These runic writings however usually consist of no more than inscriptions of a single or few words, amsterdam and Philadelphia, John Benjamins,2009. They show a degree of linguistic uniformity. Westeremden yew-stick Fon Alra Fresena Fridome Hunsigo MSS H1, H2, Ten Commandements,17 petitiones Londriucht Thet Freske Riim Skeltana Riucht law code
42.
Old Saxon
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Old Saxon, also known as Old Low German, is a Germanic language and the earliest recorded form of Low German. It belongs to the West Germanic branch and is most closely related to the Anglo-Frisian languages and it is documented from the 8th century until the 12th century, when it evolved into Middle Low German. It was spoken on the north-west coast of Germany and in the Netherlands by Saxon peoples and it is close enough to Old Anglo-Frisian that it partially participates in the Ingvaeonic nasal spirant law, it is also closely related to Old Dutch. The grammar of Old Saxon was fully inflected with five grammatical cases, the dual forms occurred in the first and second persons only and referred to groups of two. For a long time, Old Saxon and Old Dutch were not distinguished, there are also various differences in their phonological evolutions, Old Saxon being considered as an Ingvaeonic language whereas Old Dutch is an Istvaeonic language. Old Saxon probably evolved primarily from Ingvaeonic dialects in the West Germanic branch of Proto-Germanic in the 5th century. However, it seems that some Middle Dutch took the Old Saxon a-stem ending from some Middle Low German dialects, however,1150 marks the inceptive period of profuse Low German writing wherein the language is patently different from Old Saxon. One of the most striking differences between Middle Low German and Old Saxon is in a feature of speech known as vowel reduction, while round vowels in word-final syllables were rather frequent in Old Saxon, in Middle Low German, such are leveled to a schwa. Thus, such Old Saxon words like gisprekan or dagô became gespreken and daghe, Old Saxon did not participate in the High German consonant shift, and thus preserves stop consonants p, t, k that have been shifted in Old High German to various fricatives and affricates. The Germanic diphthongs ai, au consistently develop into long vowels ē, ō, whereas in Old High German they appear either as ei, ou or ē, ō depending on the following consonant. Old Saxon, alone of the West Germanic languages except for Frisian, consistently preserves Germanic -j- after a consonant, Germanic umlaut, when it occurs with short a, is inconsistent, e. g. hebbean or habbian to have. This feature was carried over into the descendant-language of Old Saxon, Middle Low German, apart from the e, however, the umlaut is not marked in writing. The table below lists the consonants of Old Saxon, phonemes written in parentheses represent allophones and are not independent phonemes. Notes, The voiceless spirants /f/, /θ/, and /s/ gain voiced allophones when between vowels and this change is only faithfully reflected in writing for. The other two continued to be written as before. Beginning in the later Old Saxon period, stops became devoiced word-finally as well, notably, geminated /v/ gave /bb/, and geminated /ɣ/ probably gave /ɡɡ/. Germanic *h is retained as in these positions and thus merges with devoiced /ɣ/, notes, Long vowels were rare in unstressed syllables and mostly occurred due to suffixation or compounding. Notes, The closing diphthongs /ei/ and /ou/ sometimes occur in texts, probably under the influence of Franconian or High German dialects, the situation for the front opening diphthongs is somewhat unclear in some texts
43.
Old Norse
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Old Norse was a North Germanic language that was spoken by inhabitants of Scandinavia and inhabitants of their overseas settlements during about the 9th to 13th centuries. These dates, however, are not absolute, since written Old Norse is found well into the 15th century, Old Norse was divided into three dialects, Old West Norse, Old East Norse and Old Gutnish. Old West and East Norse formed a continuum, with no clear geographical boundary between them. For example, Old East Norse traits were found in eastern Norway, although Old Norwegian is classified as Old West Norse, most speakers spoke Old East Norse in what is present day Denmark and Sweden. Old Gutnish, the more obscure dialectal branch, is included in the Old East Norse dialect due to geographical associations. It developed its own features and shared in changes to both other branches. The 12th century Icelandic Gray Goose Laws state that Swedes, Norwegians, Icelanders and Danes spoke the same language, another term used, used especially commonly with reference to West Norse, was norrœnt mál. In some instances the term Old Norse refers specifically to Old West Norse, the Old East Norse dialect was spoken in Denmark, Sweden, settlements in Kievan Rus, eastern England, and Danish settlements in Normandy. The Old Gutnish dialect was spoken in Gotland and in settlements in the East. In the 11th century, Old Norse was the most widely spoken European language, in Kievan Rus, it survived the longest in Veliky Novgorod, probably lasting into the 13th century there. Norwegian is descended from Old West Norse, but over the centuries it has heavily influenced by East Norse. Old Norse also had an influence on English dialects and Lowland Scots and it also influenced the development of the Norman language, and through it and to a smaller extent, that of modern French. Various other languages, which are not closely related, have heavily influenced by Norse, particularly the Norman dialects, Scottish Gaelic. The current Finnish and Estonian words for Sweden are Ruotsi and Rootsi, of the modern languages, Icelandic is the closest to Old Norse. Written modern Icelandic derives from the Old Norse phonemic writing system, contemporary Icelandic-speakers can read Old Norse, which varies slightly in spelling as well as semantics and word order. However, pronunciation, particularly of the phonemes, has changed at least as much as in the other North Germanic languages. Faroese retains many similarities but is influenced by Danish, Norwegian, although Swedish, Danish and the Norwegian languages have diverged the most, they still retain asymmetric mutual intelligibility. Speakers of modern Swedish, Norwegian and Danish can mostly understand each other without studying their neighboring languages, the languages are also sufficiently similar in writing that they can mostly be understood across borders
44.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
45.
Radix
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base
46.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
47.
Hendecagon
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In geometry, a hendecagon or 11-gon is an eleven-sided polygon. A regular hendecagon is represented by Schläfli symbol, a regular hendecagon has internal angles of 147.27 degrees. The area of a regular hendecagon with side length a is given by A =114 a 2 cot π11 ≃9.36564 a 2, as 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector and it can, however, be constructed via neusis construction. Close approximations to the regular hendecagon can be constructed, however, for instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long. The following construction description is given by T, the regular hendecagon has Dih11 symmetry, order 22. Since 11 is a number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z11. These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r22 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g11 subgroup has no degrees of freedom but can seen as directed edges. The Canadian dollar coin, the loonie, is similar to, but not exactly, the cross-section of a loonie is actually a Reuleaux hendecagon. Anthony dollar has a hendecagonal outline along the inside of its edges
48.
Multiplication
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Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
49.
Coordinate systems
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space, depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems, another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis, for a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
50.
Helmholtz equation
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In mathematics, the Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation ∇2 A + k 2 A =0 where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude. The Helmholtz equation often arises in the study of problems involving partial differential equations in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, for example, consider the wave equation u =0. Separation of variables begins by assuming that the function u is in fact separable. Substituting this form into the equation, and then simplifying, we obtain the following equation. Notice the expression on the left-hand side depends only on r, as a result, this equation is valid in the general case if and only if both sides of the equation are equal to a constant value. Rearranging the first equation, we obtain the Helmholtz equation, ∇2 A + k 2 A = A =0 and we now have Helmholtzs equation for the spatial variable r and a second-order ordinary differential equation in time. The solution in time will be a combination of sine and cosine functions, with angular frequency of ω. Alternatively, integral transforms, such as the Laplace or Fourier transform, are used to transform a hyperbolic PDE into a form of the Helmholtz equation. Because of its relationship to the equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology. The solution to the spatial Helmholtz equation A =0 can be obtained for simple geometries using separation of variables, the two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieus differential equation, the solvable shapes all correspond to shapes whose dynamical billiard table is integrable, that is, not chaotic. When the motion on a billiard table is chaotic, then no closed form solutions to the Helmholtz equation are known. The study of systems is known as quantum chaos, as the Helmholtz equation. An interesting situation happens with a shape where about half of the solutions are integrable, a simple shape where this happens is with the regular hexagon. Another simple shape where this happens is with an L shape made by reflecting a square down, if the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form A r r +1 r A r +1 r 2 A θ θ + k 2 A =0 and we may impose the boundary condition that A vanish if r = a, thus A =0. The method of separation of variables leads to solutions of the form A = R Θ
51.
Separation of variables
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Suppose a differential equation can be written in the form d d x f = g h which we can write more simply by letting y = f, d y d x = g h. As long as h ≠0, we can rearrange terms to obtain, d y h = g d x, dx can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of dx as a differential is somewhat advanced, some who dislike Leibnizs notation may prefer to write this as 1 h d y d x = g, but that fails to make it quite as obvious why this is called separation of variables. If one can evaluate the two integrals, one can find a solution to the differential equation, observe that this process effectively allows us to treat the derivative d y d x as a fraction which can be separated. This allows us to solve differential equations more conveniently, as demonstrated in the example below. Separation of variables may be used to solve this differential equation, then we have P0 = K1 + A e 0 Noting that e 0 =1, and solving for A we get A = K − P0 P0. Thus, and − λ here is the eigenvalue for both operators, and T and X are corresponding eigenfunctions. We will now show that solutions for X for values of λ ≤0 cannot occur, then there exist real numbers B, C such that X = B e − λ x + C e − − λ x. From we get and therefore B =0 = C which implies u is identically 0, then there exist real numbers B, C such that X = B x + C. From we conclude in the manner as in 1 that u is identically 0. Therefore, it must be the case that λ >0, then there exist real numbers A, B, C such that T = A e − λ α t, and X = B sin + C cos . From we get C =0 and that for some integer n, λ = n π L. This solves the equation in the special case that the dependence of u has the special form of. In general, the sum of solutions to which satisfy the boundary conditions also satisfies, hence a complete solution can be given as u = ∑ n =1 ∞ D n sin n π x L exp , where Dn are coefficients determined by initial condition. Given the initial condition u | t =0 = f and this is the sine series expansion of f. Multiplying both sides with sin n π x L and integrating over result in D n =2 L ∫0 L f sin n π x L d x. This method requires that the eigenfunctions of x, here n =1 ∞, are orthogonal, in general this is guaranteed by Sturm-Liouville theory. Suppose the equation is nonhomogeneous, with the condition the same as
52.
11-cell
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In mathematics, the 11-cell is a self-dual abstract regular 4-polytope. It has 11 vertices,55 edges and 55 faces and its symmetry group is the projective special linear group L2, so it has 660 symmetries. It was discovered in 1977 by Branko Grünbaum, who constructed it by pasting hemi-icosahedra together and it was independently discovered by H. S. M. Coxeter in 1984, who studied its structure and symmetry in greater depth. Orthographic projection of 10-simplex with 11 vertices,55 edges, the abstract 11-cell contains the same number of vertices and edges as the 10-dimensional 10-simplex, and contains 1/3 of its 165 faces. Thus it can be drawn as a figure in 11-space, although then its hemi-icosahedral cells are skew. 57-cell Icosahedral honeycomb - regular hyperbolic honeycomb with same Schläfli symbol, peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press,2002. A Symmetrical Arrangement of Eleven hemi-Icosahedra, Annals of Discrete Mathematics 20 pp103–114,2007 ISAMA paper, Hyperseeing the Regular Hendecachoron, Carlo H. Séquin & Jaron Lanier, Also Isama 2007, Texas A&m hyper-Seeing the Regular Hendeca-choron
53.
Hexominoes
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A hexomino is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. The name of type of figure is formed with the prefix hex-. When rotations and reflections are not considered to be distinct shapes, when reflections are considered distinct, there are 60 one-sided hexominoes. When rotations are also considered distinct, there are 216 fixed hexominoes, the figure shows all possible free hexominoes, coloured according to their symmetry groups,20 hexominoes have no symmetry. Their symmetry group consists only of the identity mapping,6 hexominoes have an axis of mirror symmetry parallel to the gridlines. Their symmetry group has two elements, the identity and a reflection in a parallel to the sides of the squares. 2 hexominoes have an axis of symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection,5 hexominoes have point symmetry, also known as rotational symmetry of order 2. Their symmetry group has two elements, the identity and the 180° rotation,2 hexominoes have two axes of mirror symmetry, both parallel to the gridlines. Their symmetry group has four elements and it is the dihedral group of order 2, also known as the Klein four-group. This results in 20 ×8 + ×4 +2 ×2 =216 fixed hexominoes, each of the 35 hexominoes satisfies the Conway criterion, hence every hexomino is capable of tiling the plane. Although a complete set of 35 hexominoes has a total of 210 squares, a simple way to demonstrate that such a packing of hexominoes is not possible is via a parity argument. If the hexominoes are placed on a pattern, then 11 of the hexominoes will cover an even number of black squares and 24 of the hexominoes will cover an odd number of black squares. Overall, a number of black squares will be covered in any arrangement. However, any rectangle of 210 squares will have 105 black squares and 105 white squares, however, there are other simple figures of 210 squares that can be packed with the hexominoes. For example, a 15 ×15 square with a 3 ×5 rectangle removed from the centre has 210 squares, with checkerboard colouring, it has 106 white and 104 black squares, so parity does not prevent a packing, and a packing is indeed possible. It is also possible for two sets of pieces to fit a rectangle of size 420, or for the set of 60 one-sided hexominoes to fit a rectangle of size 360, a polyhedral net for the cube is necessarily a hexomino, with 11 hexominoes actually being nets. They appear on the right, again coloured according to their symmetry groups
54.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
55.
Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Associativity of addition and multiplication For all a, b, and c in F, the following hold, a + = + c. Commutativity of addition and multiplication For all a and b in F, the following hold, a + b = b + a. Existence of additive and multiplicative identity elements There exists an element of F, called the identity element and denoted by 0, such that for all a in F. Likewise, there is an element, called the identity element and denoted by 1, such that for all a in F. To exclude the trivial ring, the identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 =1. In other words, subtraction and division operations exist, distributivity of multiplication over addition For all a, b and c in F, the following equality holds, a · = +. A simple example of a field is the field of numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers. The additive inverse of such a fraction is simply −a/b, to see the latter, note that b a ⋅ a b = b a a b =1. In addition to number systems such as the rationals, there are other. The following example is a field consisting of four elements called O, I, A and B, the notation is chosen such that O plays the role of the additive identity element, and I is the multiplicative identity. One can check that all field axioms are satisfied, for example, A · = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity. This field is called a field with four elements
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66 (number)
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66 is the natural number following 65 and preceding 67. Usages of this include,66 is, a sphenic number. A semiperfect number, being a multiple of a perfect number, an Erdős–Woods number, since it is possible to find sequences of 66 consecutive integers such that each inner member shares a factor with either the first or the last member. Palindromic and a repdigit in bases 10,21 and 32 a Harshad number in bases 2,4,5,6,7,8,11,12,13 and 16. Messier object Spiral Galaxy M66, a magnitude 10.0 galaxy in the constellation Leo, the New General Catalogue object NGC66, a peculiar barred spiral galaxy in the constellation Cetus. The Saros number of the solar eclipse series began on 12 March -756. The duration of Saros series 66 was 1298.1 years, the Saros number of the lunar eclipse series which began on 12 August -671 and ended on 27 January 826. The duration of Saros series 66 was 1496.5 years, the atomic number of dysprosium, a lanthanide. 66 megahertz is a common divisor for the front side bus speed, overall central processing unit speed, and base bus speed. On a Core 2 CPU, and a Core 2 motherboard, the FSB is 1066 MHz, the speed is usually 666.67 MHz. The designation of the historic U. S. Route 66, the designation of US Interstate 66, a freeway that runs from Virginia to Washington, D. C. The designation of 36 US state and 2 territorial highways, phillips 66, a brand of gasoline and service station in the United States. The total number of chapters in the Bible book of Isaiah, the number of verses in Chapter 3 of the book of Lamentations in the Old Testament. The total number of books in the Protestant edition of the Bible combined, in Abjad numerals, The Name Of Allah numeric value is 66. The Green Bay Packers of the National Football League retired jersey number 66 for linebacker Ray Nitschke, the last Packer to wear number 66 before it was retired in 1983 to honor Nitschke was offensive tackle Larry Pfohl, who is better known to professional wrestling fans as Lex Luger. Jersey number 66 was retired by the Pittsburgh Penguins of the National Hockey League in honor of Mario Lemieux, sixty Six is a 2006 British movie about a bar mitzvah in London on the day of the 1966 World Cup final. In the Star Wars movie series, Order 66 is an order to the clone troopers to kill the Jedi commanding them. Route 66 was a popular US television series on CBS from 1960 to 1964, in the video game Fullmetal Alchemist, elusive villain Barry the Chopper is imprisoned in cell number 66, which later becomes his alias when battling the brothers at Laboratory Five
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88 (number)
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88 is the natural number following 87 and preceding 89. An Erdős–Woods number, since it is possible to find sequences of 88 consecutive integers such that each member shares a factor with either the first or the last member. A palindromic number in bases 5,10,21, and 43, a repdigit in bases 10,21 and 43. The atomic number of the element radium, the number of constellations in the sky as defined by the International Astronomical Union. Messier object M88, a magnitude 11.0 spiral galaxy in the constellation Coma Berenices, the New General Catalogue object NGC88, a spiral galaxy in the constellation Phoenix, and a member of Roberts Quartet. Space Shuttle Mission 88, launched and completed in December,1998, the Saros number of the solar eclipse series which began on -246 October 6 and ended on 1233 March. The duration of Saros series 88 was 1478.4 years, further, the Saros number of the lunar eclipse series which began on 38 July and ended on 1336 August. The duration of Saros series 88 was 1298.1 years, approximately the number of days it takes Mercury to complete its orbit. Number 88 symbolizes fortune and good luck in Chinese culture, since the word 8 sounds similar to the word fā, the number 8 is considered to be the luckiest number in Chinese culture, and prices in Chinese supermarkets often contain many 8s. The shape of the Chinese character for 8 implies that a person will have a great, wide future as the character starts narrow, the Chinese government has been auctioning auto license plates containing many 8s for tens of thousands of dollars. The 2008 Beijing Olympics opened on 8/8/08 at 8 p. m, in amateur radio,88 is used as shorthand for love and kisses when signing a message or ending an exchange. It is used in word, Morse code, and in various digital modes. It is considered more intimate than 73, which means best regards. The two may be used together, sometimes either expression is pluralized by appending an -s. These number codes originate with the 92 Code adopted by Western Union in 1859, neo-Nazis use the number 88 as an abbreviation for the Nazi salute Heil Hitler. The letter H is eighth in the alphabet, whereby 88 becomes HH, often, this number is associated with the number 14, e. g. 14/88, 14-88, or 1488, this number symbolizes the Fourteen Words coined by David Lane, a prominent white nationalist. Example uses of 88 include the song 88 Rock n Roll Band by Landser, currently, the Michigan-based ANP uses 14 in its domain name and 88 as part of a radio sign-off. Professional golfer Kathy Whitworth, throughout her career won 88 LPGA Tour tournaments