1.
38 (number)
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38 is the natural number following 37 and preceding 39. 38 is the 11th distinct semiprime and the 7th in the family and it is the initial member of the third distinct semiprime pair. 38 has a sum of 22 which is itself a distinct semiprime In fact 38 is the first number to be at the head of a chain of four distinct semiprimes in its 8-member aliquot sequence. 38 is the 8th member of the 7-aliquot tree, −1 yields 523022617466601111760007224100074291199999999, which is the 16th factorial prime. There is no answer to the equation φ =38, making 38 a nontotient,38 is the sum of the squares of the first three primes. 37 and 38 are the first pair of positive integers not divisible by any of their digits. 38 is the largest even number which cannot be written as the sum of two odd composite numbers, there are only two normal magic hexagons, order 1 and order 3. The sum of row of an order 3 magic hexagon is 38. The duration of Saros series 38 was 1298.1 years, the lunar eclipse series which began on -1408 April 16 and ended on -111 June 3. The duration of Saros series 38 was 1298.1 years, the New General Catalogue object NGC38, a spiral galaxy in the constellation Pisces Thirty-eight is also, The 38th parallel north is the pre-Korean War boundary between North Korea and South Korea
2.
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
3.
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
4.
31 (number)
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31 is the natural number following 30 and preceding 32. As a Mersenne prime,31 is related to the perfect number 496,31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a triangular number, the lowest prime centered pentagonal number. For the Steiner tree problem,31 is the number of possible Steiner topologies for Steiner trees with 4 terminals, at 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31,31 is a repdigit in base 5, and base 2. The numbers 31,331,3331,33331,333331,3333331, for a time it was thought that every number of the form 3w1 would be prime. Here,31 divides every fifteenth number in 3w1, the atomic number of gallium Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is visible to the naked eye in a modestly dark sky. The New General Catalogue object NGC31, a galaxy in the constellation Phoenix The Saros number of the solar eclipse series which began on -1805 January 31. The duration of Saros series 31 was 1316.2 years, the Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, the jersey number 31 has been retired by several North American sports teams in honor of past playing greats, In Major League Baseball, The San Diego Padres, for Dave Winfield. The Chicago Cubs, for Ferguson Jenkins and Greg Maddux, the Atlanta Braves, also for Maddux. The New York Mets, for Mike Piazza, in the NBA, The Boston Celtics, for Cedric Maxwell. The Indiana Pacers, for Reggie Miller, in the NHL, The Edmonton Oilers, for Grant Fuhr. The New York Islanders, for Billy Smith, in the NFL, The Atlanta Falcons, for William Andrews. The New Orleans Saints, for Jim Taylor, NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company. In ice hockey goaltenders often wear the number 31, in football the number 31 has been retired by Queens Park Rangers F. C.31 from the Prime Pages
5.
34 (number)
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34 is the natural number following 33 and preceding 35. 34 is the ninth distinct semiprime and has four divisors including one and its neighbors,33 and 35, also are distinct semiprimes, having four divisors each, and 34 is the smallest number to be surrounded by numbers with the same number of divisors as it has. It is also in the first cluster of three distinct semiprimes, being within 33,34,35, the next cluster of semiprimes is 85,86,87. It is the ninth Fibonacci number and a companion Pell number, since it is an odd-indexed Fibonacci number,34 is a Markov number, appearing in solutions with other Fibonacci numbers, such as, etc. This number is the constant of a 4 by 4 normal magic square. It has the sum,20, in the following descending sequence 34,20,22,14,10,8,7,1. There is no solution to the equation φ =34, making 34 a nontotient, nor is there a solution to the equation x − φ =34, making 34 a noncototient. The atomic number of selenium One of the numbers in physics. Messier object M34, a magnitude 6, the duration of Saros series 34 was 1532.5 years, and it contained 86 solar eclipses. The Saros number of the lunar eclipse series began on 1633 BC May. The duration of Saros series 34 was 1298.1 years, the Minnesota Twins, for Hall of Famer Kirby Puckett. The Oakland Athletics and Milwaukee Brewers, both for Hall of Famer Rollie Fingers, the Boston Red Sox have announced they will retire the number for David Ortiz in 2017. Additionally, the Los Angeles Dodgers have not issued the number since the departure of Fernando Valenzuela following the 1990 season, under current team policy, Valenzuelas number is not eligible for retirement because he is not in the Hall of Fame. In the NBA, The Houston Rockets, for Hall of Famer Hakeem Olajuwon, the Los Angeles Lakers retired the number for Hall of Famer Shaquille ONeal on April 2,2013. In the NFL, The Chicago Bears, for Hall of Famer Walter Sweetness Payton, the Houston Oilers, for Hall of Famer Earl Campbell. The franchise continues to honor the number in its current incarnation as the Tennessee Titans, in the NCAA, The Auburn University Tigers, for Hall of Famer Bo Jackson. In The Count of Monte Cristo, Number 34 is how Edmond Dantès is referred to during his imprisonment in the Château dIf.34 from the Prime Pages
6.
35 (number)
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35 is the natural number following 34 and preceding 36. 35 is the sum of the first five numbers, making it a tetrahedral number. 35 is the number of ways that three things can be selected from a set of seven unique things also known as the combination of seven things taken three at a time,35 is a centered cube number, a pentagonal number and a pentatope number. 35 is a highly cototient number, since there are solutions to the equation x − φ =35 than there are for any other integers below it except 1. There are 35 free hexominoes, the polyominoes made from six squares, since the greatest prime factor of 352 +1 =1226 is 613, which is obviously more than 35 twice,35 is a Størmer number. 35 is a semiprime, the tenth, and the first with 5 as the lowest non-unitary factor. The aliquot sum of 35 is 13 this being the composite number with such an aliquot sum. 35 is the last member of the first triple cluster of semiprimes 33,34,35, the second such triple discrete semiprime cluster is 85,86,87. 35 is the highest number one can count to on ones fingers using base 6, the Chicago White Sox, for 2014 Hall of Fame inductee Frank Thomas. The San Diego Padres, for Randy Jones, in the NBA, The Boston Celtics, for Reggie Lewis. The Indiana Pacers, for Roger Brown, the Utah Jazz, for Darrell Griffith. The Golden State Warriors, for Kevin Durant In the NHL, The Chicago Blackhawks, in MotoGP,35 is the rider number of British rider, Cal Crutchlow. 35 mm film is the film gauge most commonly used for both analog photography and motion pictures The minimum age of candidates for election to the United States or Irish Presidency. 35 is used as a slang term throughout North America to denote failure, hardship, or self-defeat
7.
37 (number)
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37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37
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.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
19.
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
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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
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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
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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
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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
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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
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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
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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
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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
30.
F26A graph
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In the mathematical field of graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges. It has chromatic number 2, chromatic index 3, diameter 5, radius 5 and it is also a 3-vertex-connected and 3-edge-connected graph. The F26A graph is Hamiltonian and can be described by the LCF notation 13, the automorphism group of the F26A graph is a group of order 78. It acts transitively on the vertices, on the edges, therefore the F26A graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex, according to the Foster census, the F26A graph is the only cubic symmetric graph on 26 vertices. It is also a Cayley graph for the dihedral group D26, generated by a, ab, and ab4, where, D26 = ⟨ a, b | a 2 = b 13 =1, a b a = b −1 ⟩. The F26A graph is the smallest cubic graph where the group acts regularly on arcs. The characteristic polynomial of the F26A graph is equal to 6, the F26A graph can be embedded as a chiral regular map in the torus, with 13 hexagonal faces
31.
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
32.
3 (number)
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3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
33.
Mertens function
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In number theory, the Mertens function is defined for all positive integers n as M = ∑ k =1 n μ where μ is the Möbius function. The function is named in honour of Franz Mertens and this definition can be extended to positive real numbers as follows, M = M. Less formally, M is the count of square-free integers up to x that have a number of prime factors. Because the Möbius function only takes the values −1,0, and +1, the Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture was proven false in 1985 by Andrew Odlyzko, however, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M, namely M = O. Since high values for M grow at least as fast as the root of x. Here, O refers to Big O notation, the true rate of growth of M is not known. An unpublished conjecture of Steve Gonek states that 0 < lim sup x → ∞ | M | x 5 /4 < ∞, probabilistic evidence towards this conjecture is given by Nathan Ng. Using the Euler product one finds that 1 ζ = ∏ p = ∑ n =1 ∞ μ n s where ζ is the Riemann zeta function and the product is taken over primes. Then, using this Dirichlet series with Perrons formula, one obtains,12 π i ∫ c − i ∞ c + i ∞ x s s ζ d s = M where c >1. Conversely, one has the Mellin transform 1 ζ = s ∫1 ∞ M x s +1 d x which holds for R e >1. A curious relation given by Mertens himself involving the second Chebyshev function is ψ = M log + M log + M log + ⋯. Assuming that there are not multiple non-trivial roots of ζ we have the formula by the residue theorem. Weyl conjectured that the Mertens function satisfied the approximate functional-differential equation y 2 − ∑ r =1 N B2 r. Another formula for the Mertens function is M = ∑ a ∈ F n e 2 π i a where F n is the Farey sequence of order n and this formula is used in the proof of the Franel–Landau theorem. M is the determinant of the n × n Redheffer matrix, using sieve methods similar to those used in prime counting, the Mertens function has been computed for all integers up to an increasing range of x. The Mertens function for all values up to x may be computed in O time. Combinatorial based algorithms can compute isolated values of M in O time, see A084237 for values of M at powers of 10
34.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone
35.
Semiprime
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In mathematics, a semiprime is a natural number that is the product of two prime numbers. The semiprimes less than 100 are 4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94, and 95. Semiprimes that are not perfect squares are called discrete, or distinct, by definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its factors are 1,2,13. The total number of prime factors Ω for a n is two, by definition. A semiprime is either a square of a prime or square-free, the square of any prime number is a semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a number is a semiprime without knowing the two factors. A composite n non-divisible by primes ≤ n 3 is semiprime, various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits. These are considered novelties, since their construction method might prove vulnerable to factorization, for a semiprime n = pq the value of Eulers totient function is particularly simple when p and q are distinct, φ = = p q − +1 = n − +1. If otherwise p and q are the same, φ = φ = p = p2 − p = n − p and these methods rely on the fact that finding two large primes and multiplying them together is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of specific large semiprimes, the most recent such challenge closed in 2007. In practical cryptography, it is not sufficient to choose just any semiprime, the factors p and q of n should both be very large, around the same order of magnitude as the square root of n, this makes trial division and Pollards rho algorithm impractical. At the same time they should not be too close together, or else the number can be quickly factored by Fermats factorization method. The number may also be chosen so that none of p −1, p +1, q −1, or q +1 are smooth numbers, protecting against Pollards p −1 algorithm or Williams p +1 algorithm. However, these checks cannot take future algorithms or secret algorithms into account, in 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. It consisted of 1679 binary digits intended to be interpreted as a 23×73 bitmap image, the number 1679 = 23×73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns. Chens theorem Weisstein, Eric W. Semiprime
36.
Symmetric graph
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In other words, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of adjacent vertices. Such a graph is also called 1-arc-transitive or flag-transitive. By definition, a graph without isolated vertices must also be vertex transitive. Since the definition above maps one edge to another, a graph must also be edge transitive. However, an edge-transitive graph need not be symmetric, since a—b might map to c—d, semi-symmetric graphs, for example, are edge-transitive and regular, but not vertex-transitive. Every connected symmetric graph must thus be both vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree, however, for even degree, there exist connected graphs which are vertex-transitive and edge-transitive, but not symmetric. The smallest connected half-transitive graph is Holts graph, with degree 4 and 27 vertices, confusingly, some authors use the term symmetric graph to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above, a distance-transitive graph is one where instead of considering pairs of adjacent vertices, the definition covers two pairs of vertices, each the same distance apart. Such graphs are symmetric, by definition. A t-arc is defined to be a sequence of t+1 vertices, a t-transitive graph is a graph such that the automorphism group acts transitively on t-arcs, but not on -arcs. Since 1-arcs are simply edges, every graph of degree 3 or more must be t-transitive for some t. The cube is 2-transitive, for example, combining the symmetry condition with the restriction that graphs be cubic yields quite a strong condition, and such graphs are rare enough to be listed. The Foster census and its extensions provide such lists, the Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 the then current Foster census was published in book form. The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices, Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance-transitive graphs listed above, together with the Foster graph, the Rado graph forms an example of a symmetric graph with infinitely many vertices and infinite degree. The vertex-connectivity of a graph is always equal to the degree d. In contrast, for graphs in general, the vertex-connectivity is bounded below by 2/3. A t-transitive graph of degree 3 or more has girth at least 2, however, there are no finite t-transitive graphs of degree 3 or more for t ≥8
37.
Atomic number
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The atomic number or proton number of a chemical element is the number of protons found in the nucleus of an atom of that element. It is identical to the number of the nucleus. The atomic number identifies a chemical element. In an uncharged atom, the number is also equal to the number of electrons. The atomic number Z, should not be confused with the mass number A and this number of neutrons, N, completes the weight, A = Z + N. Atoms with the atomic number Z but different neutron numbers N. Historically, it was these atomic weights of elements that were the quantities measurable by chemists in the 19th century. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge, loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, and so they can be numbered in order. Dmitri Mendeleev claimed that he arranged his first periodic tables in order of atomic weight, however, in consideration of the elements observed chemical properties, he changed the order slightly and placed tellurium ahead of iodine. This placement is consistent with the practice of ordering the elements by proton number, Z. A simple numbering based on periodic table position was never entirely satisfactory and this central charge would thus be approximately half the atomic weight. This proved eventually to be the case, the experimental position improved dramatically after research by Henry Moseley in 1913. To do this, Moseley measured the wavelengths of the innermost photon transitions produced by the elements from aluminum to gold used as a series of movable anodic targets inside an x-ray tube. The square root of the frequency of these photons increased from one target to the next in an arithmetic progression and this led to the conclusion that the atomic number does closely correspond to the calculated electric charge of the nucleus, i. e. the element number Z. Among other things, Moseley demonstrated that the series must have 15 members—no fewer. After Moseleys death in 1915, the numbers of all known elements from hydrogen to uranium were examined by his method. There were seven elements which were not found and therefore identified as still undiscovered, from 1918 to 1947, all seven of these missing elements were discovered. By this time the first four transuranium elements had also been discovered, in 1915 the reason for nuclear charge being quantized in units of Z, which were now recognized to be the same as the element number, was not understood
38.
Yttrium
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Yttrium is a chemical element with symbol Y and atomic number 39. It is a transition metal chemically similar to the lanthanides and has often been classified as a rare earth element. Yttrium is almost always found in combination with elements in rare earth minerals. 89Y is the stable isotope, and the only isotope found in the Earths crust. In 1787, Carl Axel Arrhenius found a new mineral near Ytterby in Sweden and named it ytterbite, johan Gadolin discovered yttriums oxide in Arrhenius sample in 1789, and Anders Gustaf Ekeberg named the new oxide yttria. Elemental yttrium was first isolated in 1828 by Friedrich Wöhler, the most important uses of yttrium are LEDs and phosphors, particularly the red phosphors in television set cathode ray tube displays. Yttrium has no biological role. Exposure to yttrium compounds can cause disease in humans. Yttrium is a soft, silver-metallic, lustrous and highly crystalline transition metal in group 3, yttrium is the first d-block element in the fifth period. The pure element is relatively stable in air in bulk form and this film can reach a thickness of 10 µm when yttrium is heated to 750 °C in water vapor. When finely divided, however, yttrium is very unstable in air, yttrium nitride is formed when the metal is heated to 1000 °C in nitrogen.5 to 67.5, placing it between the lanthanides gadolinium and erbium. It often also falls in the range for reaction order. Yttrium is so close in size to the yttrium group of heavy lanthanide ions that in solution. Even though the lanthanides are one row farther down the table than yttrium. As a trivalent transition metal, yttrium forms various inorganic compounds, generally in the state of +3. A good example is yttrium oxide, also known as yttria, yttrium forms a water-insoluble fluoride, hydroxide, and oxalate, but its bromide, chloride, iodide, nitrate and sulfate are all soluble in water. The Y3+ ion is colorless in solution because of the absence of electrons in the d and f electron shells, water readily reacts with yttrium and its compounds to form Y 2O3. Concentrated nitric and hydrofluoric acids do not rapidly attack yttrium, with halogens, yttrium forms trihalides such as yttrium fluoride, yttrium chloride, and yttrium bromide at temperatures above roughly 200 °C
39.
Messier object
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The Messier objects are a set of over 100 astronomical objects first listed by French astronomer Charles Messier in 1771. The number of objects in the lists he published reached 103, a similar list had been published in 1654 by Giovanni Hodierna, but attracted attention only recently and was probably not known to Messier. The first edition covered 45 objects numbered M1 to M45, the first such addition came from Nicolas Camille Flammarion in 1921, who added Messier 104 after finding a note Messier made in a copy of the 1781 edition of the catalogue. M105 to M107 were added by Helen Sawyer Hogg in 1947, M108 and M109 by Owen Gingerich in 1960, M102 was observed by Méchain, who communicated his notes to Messier. Méchain later concluded that this object was simply a re-observation of M101, though sources suggest that the object Méchain observed was the galaxy NGC5866. Messiers final catalogue was included in the Connaissance des Temps for 1784 and these objects are still known by their Messier number from this list. Messier lived and did his work at the Hôtel de Cluny. The list he compiled contains only objects found in the sky area he could observe and he did not observe or list objects visible only from farther south, such as the Large and Small Magellanic Clouds. A summary of the astrophysics of each Messier object can be found in the Concise Catalog of Deep-sky Objects, in early spring, astronomers sometimes gather for Messier marathons, when all of the objects can be viewed over a single night
40.
Open Cluster M39
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Open Cluster M39 is an open cluster in the constellation Cygnus. It was discovered by Charles Messier in 1764, M39 is at a distance of about 800 light-years away from Earth. Its age is estimated to be from 200 to 300 million years and it is located at Right Ascension 21 hours,32.2 minutes, and Declination +48 degrees 26 minutes. It has a magnitude of 5.5 Messier 39, SEDS Messier pages Messier 39 on WikiSky, DSS2, SDSS, GALEX, IRAS, Hydrogen α, X-Ray, Astrophoto, Sky Map, Articles and images
41.
Visual magnitude
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The apparent magnitude of a celestial object is a number that is a measure of its brightness as seen by an observer on Earth. The brighter an object appears, the lower its magnitude value, the Sun, at apparent magnitude of −27, is the brightest object in the sky. It is adjusted to the value it would have in the absence of the atmosphere, furthermore, the magnitude scale is logarithmic, a difference of one in magnitude corresponds to a change in brightness by a factor of 5√100, or about 2.512. The measurement of apparent magnitudes or brightnesses of celestial objects is known as photometry, apparent magnitudes are used to quantify the brightness of sources at ultraviolet, visible, and infrared wavelengths. An apparent magnitude is measured in a specific passband corresponding to some photometric system such as the UBV system. In standard astronomical notation, an apparent magnitude in the V filter band would be denoted either as mV or often simply as V, the scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes. The brightest stars in the sky were said to be of first magnitude, whereas the faintest were of sixth magnitude. Each grade of magnitude was considered twice the brightness of the following grade and this rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest, and is generally believed to have originated with Hipparchus. This implies that a star of magnitude m is 2.512 times as bright as a star of magnitude m +1 and this figure, the fifth root of 100, became known as Pogsons Ratio. The zero point of Pogsons scale was defined by assigning Polaris a magnitude of exactly 2. However, with the advent of infrared astronomy it was revealed that Vegas radiation includes an Infrared excess presumably due to a disk consisting of dust at warm temperatures. At shorter wavelengths, there is negligible emission from dust at these temperatures, however, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the scale was extrapolated to all wavelengths on the basis of the black body radiation curve for an ideal stellar surface at 11000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance for the zero magnitude point, with the modern magnitude systems, brightness over a very wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30, astronomers have developed other photometric zeropoint systems as alternatives to the Vega system. The AB magnitude zeropoint is defined such that an objects AB, the dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Since an increase of 5 magnitudes corresponds to a decrease in brightness by a factor of exactly 100, each magnitude increase implies a decrease in brightness by the factor 5√100 ≈2.512. Inverting the above formula, a magnitude difference m1 − m2 = Δm implies a brightness factor of F2 F1 =100 Δ m 5 =100.4 Δ m ≈2.512 Δ m
42.
Open cluster
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An open cluster is a group of up to a few thousand stars that were formed from the same giant molecular cloud and have roughly the same age. More than 1,100 open clusters have been discovered within the Milky Way Galaxy and they are loosely bound by mutual gravitational attraction and become disrupted by close encounters with other clusters and clouds of gas as they orbit the galactic center. This can result in a migration to the body of the galaxy. Open clusters generally survive for a few hundred years, with the most massive ones surviving for a few billion years. In contrast, the more massive clusters of stars exert a stronger gravitational attraction on their members. Open clusters have been only in spiral and irregular galaxies. Young open clusters may not be contained within the cloud from which they formed. Over time, radiation pressure from the cluster will disperse the molecular cloud, typically, about 10% of the mass of a gas cloud will coalesce into stars before radiation pressure drives the rest of the gas away. Open clusters are key objects in the study of stellar evolution, because the cluster members are of similar age and chemical composition, their properties are more easily determined than they are for isolated stars. A number of clusters, such as the Pleiades, Hyades or the Alpha Persei Cluster are visible with the naked eye. Some others, such as the Double Cluster, are barely perceptible without instruments, the Wild Duck Cluster, M11, is an example. The prominent open cluster the Pleiades has been recognized as a group of stars since antiquity, while the Hyades forms part of Taurus, other open clusters were noted by early astronomers as unresolved fuzzy patches of light. The Roman astronomer Ptolemy mentions the Praesepe, the Double Cluster in Perseus, however, it would require the invention of the telescope to resolve these nebulae into their constituent stars. Indeed, in 1603 Johann Bayer gave three of these clusters designations as if they were single stars, the first person to use a telescope to observe the night sky and record his observations was the Italian scientist Galileo Galilei in 1609. When he turned the telescope toward some of the nebulous patches recorded by Ptolemy, he found they were not a single star, for Praesepe, he found more than 40 stars. Where previously observers had noted only 6-7 stars in the Pleiades, in his 1610 treatise Sidereus Nuncius, Galileo Galilei wrote, the galaxy is nothing else but a mass of innumerable stars planted together in clusters. Influenced by Galileos work, the Sicilian astronomer Giovanni Hodierna became possibly the first astronomer to use a telescope to find previously undiscovered open clusters, in 1654, he identified the objects now designated Messier 41, Messier 47, NGC2362 and NGC2451. Between 1774–1781, French astronomer Charles Messier published a catalogue of objects that had a nebulous appearance similar to comets
43.
Constellation
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A constellation is formally defined as a region of the celestial sphere, with boundaries laid down by the International Astronomical Union. The constellation areas mostly had their origins in Western-traditional patterns of stars from which the constellations take their names, in 1922, the International Astronomical Union officially recognized the 88 modern constellations, which cover the entire sky. They began as the 48 classical Greek constellations laid down by Ptolemy in the Almagest, Constellations in the far southern sky are late 16th- and mid 18th-century constructions. 12 of the 88 constellations compose the zodiac signs, though the positions of the constellations only loosely match the dates assigned to them in astrology. The term constellation can refer to the stars within the boundaries of that constellation. Notable groupings of stars that do not form a constellation are called asterisms, when astronomers say something is “in” a given constellation they mean it is within those official boundaries. Any given point in a coordinate system can unambiguously be assigned to a single constellation. Many astronomical naming systems give the constellation in which an object is found along with a designation in order to convey a rough idea in which part of the sky it is located. For example, the Flamsteed designation for bright stars consists of a number, the word constellation seems to come from the Late Latin term cōnstellātiō, which can be translated as set of stars, and came into use in English during the 14th century. It also denotes 88 named groups of stars in the shape of stellar-patterns, the Ancient Greek word for constellation was ἄστρον. Colloquial usage does not draw a distinction between constellation in the sense of an asterism and constellation in the sense of an area surrounding an asterism. The modern system of constellations used in astronomy employs the latter concept, the term circumpolar constellation is used for any constellation that, from a particular latitude on Earth, never sets below the horizon. From the North Pole or South Pole, all constellations south or north of the equator are circumpolar constellations. In the equatorial or temperate latitudes, the term equatorial constellation has sometimes been used for constellations that lie to the opposite the circumpolar constellations. They generally include all constellations that intersect the celestial equator or part of the zodiac, usually the only thing the stars in a constellation have in common is that they appear near each other in the sky when viewed from the Earth. In galactic space, the stars of a constellation usually lie at a variety of distances, since stars also travel on their own orbits through the Milky Way, the star patterns of the constellations change slowly over time. After tens to hundreds of thousands of years, their familiar outlines will become unrecognisable, the terms chosen for the constellation themselves, together with the appearance of a constellation, may reveal where and when its constellation makers lived. The earliest direct evidence for the constellations comes from inscribed stones and it seems that the bulk of the Mesopotamian constellations were created within a relatively short interval from around 1300 to 1000 BC
44.
Cygnus (constellation)
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Cygnus /ˈsɪɡnəs/ is a northern constellation lying on the plane of the Milky Way, deriving its name from the Latinized Greek word for swan. The swan is one of the most recognizable constellations of the summer and autumn. Cygnus was among the 48 constellations listed by the 2nd century astronomer Ptolemy, Cygnus contains Deneb, one of the brightest stars in the night sky and one corner of the Summer Triangle, as well as some notable X-ray sources and the giant stellar association of Cygnus OB2. One of the stars of this association, NML Cygni, is one of the largest stars currently known. The constellation is home to Cygnus X-1, a distant X-ray binary containing a supergiant. Many star systems in Cygnus have known planets as a result of the Kepler Mission observing one patch of the sky, in Greek mythology, Cygnus has been identified with several different legendary swans. The Greeks also associated this constellation with the story of Phaethon, the son of Helios the sun god. Phaethon, however, was unable to control the reins, forcing Zeus to destroy the chariot with a thunderbolt, according to the myth, Phaethons brother, Cycnus, grieved bitterly and spent many days diving into the river to collect Phaethons bones to give him a proper burial. The gods were so touched by Cycnuss devotion to his brother that they turned him into a swan, in Ovids Metamorphoses, there are three people named Cygnus, all of whom are transformed into swans. Alongside Cycnus, noted above, he mentions a boy from Tempe who commits suicide when Phyllius refuses to him a tamed bull that he demands. He also mentions a son of Neptune who is a warrior in the Trojan War who is eventually defeated by Achilles. In Polynesia, Cygnus was often recognized as a separate constellation, in Tonga it was called Tuula-lupe, and in the Tuamotus it was called Fanui-tai. Deneb was also given a name. The name Deneb comes from the Arabic name dhaneb, meaning tail, from the phrase Dhanab ad-Dajājah, in New Zealand it was called Mara-tea, in the Society Islands it was called Pirae-tea or Taurua-i-te-haapa-raa-manu, and in the Tuamotus it was called Fanui-raro. Beta Cygni was named in New Zealand, it was likely called Whetu-kaupo, Gamma Cygni was called Fanui-runga in the Tuamotus. The three-letter abbreviation for the constellation, as adopted by the IAU in 1922, is Cyg, the official constellation boundaries, as set by Eugène Delporte in 1930, are defined as a polygon of 28 segments. In the equatorial coordinate system, the right ascension coordinates of these borders lie between 19h 07. 3m and 22h 02. 3m, while the coordinates are between 27. 73° and 61. 36°. Covering 804 square degrees and around 1. 9% of the night sky, Cygnus culminates at midnight on 29 June, and is most visible in the evening from the early summer to mid-autumn in the Northern Hemisphere
45.
New General Catalogue
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The NGC contains 7,840 objects, known as the NGC objects. It is one of the largest comprehensive catalogues, as it includes all types of space objects and is not confined to, for example. Dreyer also published two supplements to the NGC in 1895 and 1908, known as the Index Catalogues, describing a further 5,386 astronomical objects. Objects in the sky of the southern hemisphere are catalogued somewhat less thoroughly, the Revised New General Catalogue and Index Catalogue was compiled in 2009 by Wolfgang Steinicke. The original New General Catalogue was compiled during the 1880s by John Louis Emil Dreyer using observations from William Herschel and his son John, Dreyer had already published a supplement to Herschels General Catalogue of Nebulae and Clusters, containing about 1,000 new objects. In 1886, he suggested building a second supplement to the General Catalogue and this led to the publication of the New General Catalogue in the Memoirs of the Royal Astronomical Society in 1888. Assembling the NGC was a challenge, as Dreyer had to deal with many contradicting and unclear reports, while he did check some himself, the sheer number of objects meant Dreyer had to accept them as published by others for the purpose of his compilation. Dreyer was a careful transcriber and made few errors himself, and he was very thorough in his referencing, which allowed future astronomers to review the original references and publish corrections to the original NGC. The first major update to the NGC is the Index Catalogue of Nebulae and Clusters of Stars and it serves as a supplement to the NGC, and contains an additional 5,386 objects, collectively known as the IC objects. It summarizes the discoveries of galaxies, clusters and nebulae between 1888 and 1907, most of them made possible by photography, a list of corrections to the IC was published in 1912. The Revised New Catalogue of Nonstellar Astronomical Objects was compiled by Jack W. Sulentic and William G. Tifft in the early 1970s, and was published in 1973, as an update to the NGC. However, because the update had to be completed in just three summers, it failed to incorporate several previously-published corrections to the NGC data, and even introduced new errors. NGC2000.0 is a 1988 compilation of the NGC and IC made by Roger W. Sinnott and it incorporates several corrections and errata made by astronomers over the years. However, it too ignored the original publications and favoured modern corrections, the NGC/IC Project is a collaboration formed in 1993. It aims to identify all NGC and IC objects, and collect images, the Revised New General Catalogue and Index Catalogue is a compilation made by Wolfgang Steinicke in 2009. It is considered one of the most comprehensive and authoritative treatments of the NGC, messier object Catalogue of Nebulae and Clusters of Stars The Interactive NGC Catalog Online Adventures in Deep Space, Challenging Observing Projects for Amateur Astronomers
46.
Spiral galaxy
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A spiral galaxy is a type of galaxy originally described by Edwin Hubble in his 1936 work The Realm of the Nebulae and, as such, forms part of the Hubble sequence. Spiral galaxies consist of a flat, rotating disk containing stars, gas and dust, and these are surrounded by a much fainter halo of stars, many of which reside in globular clusters. Spiral galaxies are named for the structures that extend from the center into the galactic disc. The spiral arms are sites of ongoing star formation and are brighter than the surrounding disc because of the young, hot OB stars that inhabit them. Roughly two-thirds of all spirals are observed to have a component in the form of a bar-like structure, extending from the central bulge. Our own Milky Way has recently confirmed to be a barred spiral. The most convincing evidence for its existence comes from a recent survey, performed by the Spitzer Space Telescope, together with irregular galaxies, spiral galaxies make up approximately 60% of galaxies in the local Universe. They are mostly found in low-density regions and are rare in the centers of galaxy clusters, Spiral arms are regions of stars that extend from the center of spiral and barred spiral galaxies. These long, thin regions resemble a spiral and thus give spiral galaxies their name, naturally, different classifications of spiral galaxies have distinct arm-structures. Sc and SBc galaxies, for instance, have very loose arms, whereas Sa, either way, spiral arms contain many young, blue stars, which make the arms so bright. A bulge is a huge, tightly packed group of stars, the term commonly refers to the central group of stars found in most spiral galaxies. Using the Hubble classification, the bulge of Sa galaxies is usually composed of Population II stars, further, the bulge of Sa and SBa galaxies tends to be large. In contrast, the bulges of Sc and SBc galaxies are much smaller and are composed of young, some bulges have similar properties to those of elliptical galaxies, others simply appear as higher density centers of disks, with properties similar to disk galaxies. Many bulges are thought to host a supermassive black hole at their centers, such black holes have never been directly observed, but many indirect proofs exist. In our own galaxy, for instance, the object called Sagittarius A* is believed to be a black hole. There is a correlation between the mass of the black hole and the velocity dispersion of the stars in the bulge. However, some stars inhabit a spheroidal halo or galactic spheroid, the orbital behaviour of these stars is disputed, but they may describe retrograde and/or highly inclined orbits, or not move in regular orbits at all. The galactic halo also contains many globular clusters, due to their irregular movement around the center of the galaxy—if they do so at all—these stars often display unusually high proper motion
47.
Andromeda (constellation)
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Andromeda is one of the 48 constellations listed by the 2nd-century Greco-Roman astronomer Ptolemy and remains one of the 88 modern constellations. Located north of the equator, it is named for Andromeda, daughter of Cassiopeia, in the Greek myth. Andromeda is most prominent during autumn evenings in the Northern Hemisphere, because of its northern declination, Andromeda is visible only north of 40° south latitude, for observers farther south it lies below the horizon. It is one of the largest constellations, with an area of 722 square degrees. This is over 1,400 times the size of the moon, 55% of the size of the largest constellation, Hydra. Its brightest star, Alpha Andromedae, is a star that has also been counted as a part of Pegasus, while Gamma Andromedae is a colorful binary. Only marginally dimmer than Alpha, Beta Andromedae is a red giant, the constellations most obvious deep-sky object is the naked-eye Andromeda Galaxy, the closest spiral galaxy to the Milky Way and one of the brightest Messier objects. Several fainter galaxies, including M31s companions M110 and M32, as well as the more distant NGC891, the Blue Snowball Nebula, a planetary nebula, is visible in a telescope as a blue circular object. Andromeda is the location of the radiant for the Andromedids, a meteor shower that occurs in November. The uranography of Andromeda has its roots most firmly in the Greek tradition, the stars that make up Pisces and the middle portion of modern Andromeda formed a constellation representing a fertility goddess, sometimes named as Anunitum or the Lady of the Heavens. Andromeda is known as the Chained Lady or the Chained Woman in English and it was known as Mulier Catenata in Latin and al-Marat al Musalsalah in Arabic. Offended at her remark, the nymphs petitioned Poseidon to punish Cassiopeia for her insolence, Andromedas panicked father, Cepheus, was told by the Oracle of Ammon that the only way to save his kingdom was to sacrifice his daughter to Cetus. Perseus and Andromeda then married, the myth recounts that the couple had nine children together – seven sons, after Andromedas death Athena placed her in the sky as a constellation, to honor her. Several of the neighboring constellations also represent characters in the Perseus myth and it is connected with the constellation Pegasus. Andromeda was one of the original 48 constellations formulated by Ptolemy in his 2nd-century Almagest, in which it was defined as a specific pattern of stars. She is typically depicted with α Andromedae as her head, ο and λ Andromedae as her chains, and δ, π, μ, Β, however, there is no universal depiction of Andromeda and the stars used to represent her body, head, and chains. Arab astronomers were aware of Ptolemys constellations, but they included a second constellation representing a fish at Andromedas feet, several stars from Andromeda and most of the stars in Lacerta were combined in 1787 by German astronomer Johann Bode to form Frederici Honores. It was designed to honor King Frederick II of Prussia, in 1922, the IAU defined its recommended three-letter abbreviation, And
48.
Torah
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The Torah is the central reference of Judaism. It has a range of meanings and it can most specifically mean the first five books of the twenty-four books of the Tanakh, and it usually includes the rabbinic commentaries. In rabbinic literature the word Torah denotes both the five books and the Oral Torah, the Oral Torah consists of interpretations and amplifications which according to rabbinic tradition have been handed down from generation to generation and are now embodied in the Talmud and Midrash. According to the Midrash, the Torah was created prior to the creation of the world, traditionally, the words of the Torah are written on a scroll by a scribe in Hebrew. A Torah portion is read publicly at least once every three days in the presence of a congregation, reading the Torah publicly is one of the bases for Jewish communal life. The word Torah in Hebrew is derived from the root ירה, the meaning of the word is therefore teaching, doctrine, or instruction, the commonly accepted law gives a wrong impression. Other translational contexts in the English language include custom, theory, guidance, the earliest name for the first part of the Bible seems to have been The Torah of Moses. This title, however, is neither in the Torah itself. It appears in Joshua and Kings, but it cannot be said to refer there to the entire corpus, in contrast, there is every likelihood that its use in the post-Exilic works was intended to be comprehensive. Other early titles were The Book of Moses and The Book of the Torah, Christian scholars usually refer to the first five books of the Hebrew Bible as the Pentateuch, a term first used in the Hellenistic Judaism of Alexandria, meaning five books, or as the Law. The Torah starts from the beginning of Gods creating the world, through the beginnings of the people of Israel, their descent into Egypt, and it ends with the death of Moses, just before the people of Israel cross to the promised land of Canaan. Interspersed in the narrative are the teachings given explicitly or implicitly embedded in the narrative. This is followed by the story of the three patriarchs, Joseph and the four matriarchs, God gives to the patriarchs a promise of the land of Canaan, but at the end of Genesis the sons of Jacob end up leaving Canaan for Egypt due to a regional famine. They had heard there was a grain storage and distribution facility in Egypt. Exodus begins the story of Gods revelation to his people of Israel through Moses, Moses receives the Torah from God, and teaches His laws and Covenant to the people of Israel. It also talks about the first violation of the covenant when the Golden Calf was constructed, Exodus includes the instructions on building the Tabernacle and concludes with its actual construction. Leviticus begins with instructions to the Israelites on how to use the Tabernacle, leviticus 26 provides a detailed list of rewards for following Gods commandments and a detailed list of punishments for not following them. Numbers tells how Israel consolidated itself as a community at Sinai, set out from Sinai to move towards Canaan, even Moses sins and is told he would not live to enter the land
49.
Sanhedrin
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The Sanhedrin was an assembly of twenty-three to seventy-one men appointed in every city in the Land of Israel. Judges in ancient Israel were the leaders and Teachers of the nation of Israel. The Mishnah arrives at the number based on an exegetical derivation. The minimum size of a community is 10 men, one more is required to achieve a majority, but a simple majority cannot convict, and so an additional judge is required. Finally, a court should not have a number of judges to prevent deadlocks. This court dealt with religious matters. The Great Sanhedrin was made up of a Nasi, who functioned as head or representing president, but was not a member of the court, an Av Beit Din, the chief of the court, and sixty-nine general members. In the Second Temple period, the Great Sanhedrin met in the Hall of Hewn Stones in the Temple in Jerusalem, the court convened every day except festivals and Shabbat. In the late 3rd century, to persecution, its authoritative decisions were issued under the name of Beit HaMidrash. Historically, the last binding decision of the Great Sanhedrin appeared in 358 CE, the Great Sanhedrin was dissolved after continued persecution by the Eastern Roman Empire and aspiring Christendom. Over the centuries, there have been attempts to revive the institution, such as the Grand Sanhedrin convened by Napoleon Bonaparte and modern attempts in Israel. The Hasmonean court in the Land of Israel, presided over by Alexander Jannaeus, king of Judea until 76 BCE, the exact nature of this early Sanhedrin is not clear. It may have been a body of sages or priests, or a political, legislative, only after the destruction of the Second Temple was the Sanhedrin made up only of sages. In the Second Temple period, the Great Sanhedrin met in the Hall of Hewn Stones in the Temple in Jerusalem, the court convened every day except festivals and Shabbat. After the destruction of the Second Temple in 70, the Sanhedrin was re-established in Yavneh with reduced authority, the seat of the Patriarchate moved to Usha under the presidency of Gamaliel II in 80 CE. In 116 it moved back to Yavneh, and then back to Usha. Rabbinic texts indicate that following the Bar Kokhba revolt, southern Galilee became the seat of learning in the Land of Israel. This region was the location of the court of the Patriarch which was situated first at Usha, then at Bet Shearim, later at Sepphoris and finally at Tiberias
50.
Old Testament
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Its counterpart is the New Testament, the second part of the Christian Bible. The books that comprise the Old Testament canon differ between Christian Churches as well as their order and names. The most common Protestant canon comprises 39 books, the Catholic canon comprises 46 books, the 39 books in common to all the Christian canons corresponds to 24 books of the Tanakh, with some differences of order, and there are some differences in text. The additional number reflects the split of texts in the Christian Bibles into separate books, for example, Kings, Samuel and Chronicles, Ezra–Nehemiah, the books which are part of a Christian Old Testament but which are not part of the Hebrew canon are sometimes described as deuterocanonical. In general, Protestant bibles do not include books in its canon. The Old Testament consists of translations of many books by various authors produced over a period of centuries. The canon formed in stages, first the Pentateuch by around 400 BC, then the Prophets during the Hasmonean dynasty, and finally the remaining books. The Old Testament contains 39 or 46 or more books, divided, very broadly, into the Pentateuch, the books, the wisdom books. For the Orthodox canon, Septuagint titles are provided in parentheses when these differ from those editions, for the Catholic canon, the Douaic titles are provided in parentheses when these differ from those editions. Likewise, the King James Version references some of these books by the spelling when referring to them in the New Testament. The Talmud in Bava Batra 14b gives a different order for the books in Neviim and Ketuvim and this order is also cited in Mishneh Torah Hilchot Sefer Torah 7,15. The order of the books of the Torah is universal through all denominations of Judaism and they are present in a few historic Protestant versions, the German Luther Bible included such books, as did the English 1611 King James Version. Empty table cells indicate that a book is absent from that canon, several of the books in the Eastern Orthodox canon are also found in the appendix to the Latin Vulgate, formerly the official Bible of the Roman Catholic Church. The books of Joshua, Judges, Samuel and Kings follow, there is a broad consensus among scholars that these originated as a single work during the Babylonian exile of the 6th century BC. The two Books of Chronicles cover much the material as the Pentateuch and Deuteronomistic history and probably date from the 4th century BC. Chronicles, and Ezra–Nehemiah, were finished during the 3rd century BC. Catholic and Orthodox Old Testaments contain two to four Books of Maccabees, written in the 2nd and 1st centuries BC and these history books make up around half the total content of the Old Testament. God is consistently depicted as the one who created or put into order the world, the Old Testament stresses the special relationship between God and his chosen people, Israel, but includes instructions for proselytes as well
51.
Protestantism
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Protestantism is a form of Christianity which originated with the Reformation, a movement against what its followers considered to be errors in the Roman Catholic Church. It is one of the three divisions of Christendom, together with Roman Catholicism and Orthodoxy. The term derives from the letter of protestation from German Lutheran princes in 1529 against an edict of the Diet of Speyer condemning the teachings of Martin Luther as heretical. Although there were earlier breaks from or attempts to reform the Roman Catholic Church—notably by Peter Waldo, John Wycliffe, Protestants reject the notion of papal supremacy and deny the Roman Catholic doctrine of transubstantiation, but disagree among themselves regarding the real presence of Christ in the Eucharist. The Five solae summarize the reformers basic differences in theological beliefs, in the 16th century, Lutheranism spread from Germany into Denmark, Norway, Sweden, Finland, the Baltic states, and Iceland. Reformed churches were founded in Germany, Hungary, the Netherlands, Scotland, Switzerland and France by such reformers as John Calvin, Huldrych Zwingli, the political separation of the Church of England from Rome under King Henry VIII brought England and Wales into this broad Reformation movement. Protestants developed their own culture, which made major contributions in education, the humanities and sciences, the political and social order, the economy and the arts, some Protestant denominations do have a worldwide scope and distribution of membership, while others are confined to a single country. A majority of Protestants are members of a handful of families, Adventism, Anglicanism, Baptist churches, Reformed churches, Lutheranism, Methodism. Nondenominational, evangelical, charismatic, independent and other churches are on the rise, and constitute a significant part of Protestant Christianity. Six princes of the Holy Roman Empire and rulers of fourteen Imperial Free Cities, the edict reversed concessions made to the Lutherans with the approval of Holy Roman Emperor Charles V three years earlier. During the Reformation, the term was used outside of the German politics. The word evangelical, which refers to the gospel, was more widely used for those involved in the religious movement. Nowadays, this word is still preferred among some of the historical Protestant denominations in the Lutheran and Calvinist traditions in Europe, above all the term is used by Protestant bodies in the German-speaking area, such as the EKD. In continental Europe, an Evangelical is either a Lutheran or a Calvinist, the German word evangelisch means Protestant, and is different from the German evangelikal, which refers to churches shaped by Evangelicalism. The English word evangelical usually refers to Evangelical Protestant churches, and it traces its roots back to the Puritans in England, where Evangelicalism originated, and then was brought to the United States. Protestantism as a term is now used in contradistinction to the other major Christian traditions, i. e. Roman Catholicism. Initially, Protestant became a term to mean any adherent to the Reformation movement in Germany and was taken up by Lutherans. Even though Martin Luther himself insisted on Christian or Evangelical as the only acceptable names for individuals who professed Christ, French and Swiss Protestants preferred the word reformed, which became a popular, neutral and alternative name for Calvinists
52.
Anglican Church
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The Anglican Communion is an international association of autonomous churches consisting of the Church of England and national and regional Anglican churches in full communion with it. Full participation in the life of each church is available to all communicant Anglicans. The Archbishop of Canterbury, Primate of All England, has a place of honour among the bishops of the Anglican churches and he is recognised as primus inter pares. The archbishop does not exercise authority in the provinces outside England, the churches of the Anglican Communion considers themselves to be part of the nicos one, holy, catholic and apostolic church and to be both Catholic and Reformed. For some adherents, Anglicanism represents a non-papal Catholicism, for others a form of Protestantism though without a dominant guiding figure such as Luther, Knox, Calvin, for others, their self-identity represents some combination of the two. The communion encompasses a spectrum of belief and practice including evangelical, liberal. With a membership estimated at around 85 million members, the Anglican Communion is the third largest Christian communion in the world, after the Catholic Church and the Eastern Orthodox Church. Some of these churches are known as Anglican, such as the Anglican Church of Canada, some, for example the Church of Ireland, the Scottish and American Episcopal churches, and some other associated churches have a separate name. The Anglican Communion has no legal existence nor any governing structure which might exercise authority over the member churches. There is an Anglican Communion Office in London, under the aegis of the Archbishop of Canterbury, the Communion is held together by a shared history, expressed in its ecclesiology, polity and ethos and also by participation in international consultative bodies. Early in its development, Anglicanism developed a vernacular prayer book, unlike other traditions, Anglicanism has never been governed by a magisterium nor by appeal to one founding theologian, nor by an extra-credal summary of doctrine. Instead, Anglicans have typically appealed to the Book of Common Prayer and its offshoots as a guide to Anglican theology and this had the effect of inculcating the principle of Lex orandi, lex credendi as the foundation of Anglican identity and confession. These parameters were most clearly articulated in the rubrics of the successive prayer books. With the expansion of the British Empire, and hence the growth of Anglicanism outside Great Britain and Ireland, the first major expression of this were the Lambeth Conferences of the communions bishops, first convened by Archbishop of Canterbury Charles Longley in 1869. One of the influential early resolutions of the conference was the so-called Chicago-Lambeth Quadrilateral of 1888. Its intent was to provide the basis for discussions of reunion with the Roman Catholic and Orthodox Churches, the Apostles Creed, as the Baptismal Symbol, and the Nicene Creed, as the sufficient statement of the Christian faith. The two Sacraments ordained by Christ Himself - Baptism and the Supper of the Lord - ministered with unfailing use of Christs Words of Institution, and of the elements ordained by Him. The Historic Episcopate, locally adapted in the methods of its administration to the needs of the nations
53.
Thirty-Nine Articles
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The Thirty-nine Articles of Religion are the historically defining statements of doctrines and practices of the Church of England with respect to the controversies of the English Reformation. The Thirty-nine Articles form part of the Book of Common Prayer used by both the Church of England and the Episcopal Church. When Henry VIII broke with the Roman Catholic Church and was excommunicated, he formed a new Church of England, at this point, he needed to determine what its doctrines and practices would be in relation to the Roman Catholic Church and the new Protestant movements in continental Europe. These positions began with the Ten Articles in 1536, and concluded with the finalisation Thirty-nine articles in 1571, the Thirty-nine articles ultimately served to define the doctrine of the Church of England as it related to Calvinist doctrine and Roman Catholic practice. The articles went through at least five major revisions prior to their finalisation in 1571, the first attempt was the Ten Articles in 1536, which showed some slightly Protestant leanings—the result of an English desire for a political alliance with the German Lutheran princes. The next revision was the Six Articles in 1539 which swung away from all reformed positions, and then the Kings Book in 1543, during the reign of Edward VI, Henry VIIIs only son, the Forty-Two Articles were written under the direction of Archbishop Thomas Cranmer in 1552. It was in this document that Calvinist thought reached the zenith of its influence in the English Church and these articles were never put into action, due to Edward VIs death and the reversion of the English Church to Roman Catholicism under Henry VIIIs elder daughter, Mary I. The articles pulled back some of the more extreme Calvinist thinking. The Thirty-nine articles were finalised in 1571, and incorporated into the Book of Common Prayer, the Ten Articles were first published in 1536 by Thomas Cranmer. They were the first guidelines of the Church of England as it became independent of Rome, the Institution of the Christian Man, published in 1537, was written by a committee of 46 divines and bishops headed by Thomas Cranmer. The purpose of the work, along with the Ten Articles of the year, was to implement the reforms of Henry VIII in separating from the Catholic Church. It was considered reformatory in basic orientation, though it was not strongly Lutheran, the work functioned as an official formulary of the reformed Anglican faith in England. It was later superseded by other creedal and official statements during the reigns of Edward VI and Elizabeth I. It would evolve into the Kings Book, the work was a noble endeavor on the part of the bishops to promote unity, and to instruct the people in Church doctrine. The Germans presented, as a basis of agreement, a number of Articles based on the Lutheran Confession of Augsburg, bishops Tunstall, Stokesley and others were not won over by these Protestant arguments and did everything they could to avoid agreement. They were willing to separate from Rome, but their plan was to unite with the Greek Church, the bishops also refused to eliminate what the Germans called the Abuses allowed by the Anglican Church. Stokesley considered these customs to be essential because the Greek Church practised them, in opposition, Cranmer favoured a union with German Protestants. The king, unwilling to break with Catholic practices, dissolved the conference, Henry had felt uneasy about the appearance of the Lutheran doctors and their theology within his kingdom
54.
John Buchan
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John Buchan, 1st Baron Tweedsmuir GCMG GCVO CH PC was a Scottish novelist, historian and Unionist politician who served as Governor General of Canada, the 15th since Canadian Confederation. He eventually wrote propaganda for the British war effort in the First World War, in 1935 he was appointed Governor General of Canada by King George V, on the recommendation of Prime Minister of Canada R. B. Bennett, to replace the Earl of Bessborough and he occupied the post until his death in 1940. Buchan proved to be enthusiastic about literacy, as well as the evolution of Canadian culture, Buchan was born in Perth, Scotland. He was the first child of John Buchan—a Free Church of Scotland minister—and Helen Jane Buchan, Buchan was brought up in Kirkcaldy, Fife, and spent many summer holidays with his maternal grandparents in Broughton, in the Scottish Borders. The childhood he and his sister, Anna, shared was documented in her memoir, Unforgettable, after attending Hutchesons Grammar School, Buchan was awarded a scholarship to the University of Glasgow at age 17, where he studied classics, wrote poetry, and became a published author. With a junior Hulme scholarship, he moved on in 1895 to study Literae Humaniores at Brasenose College, Oxford, where his friends included Hilaire Belloc, Raymond Asquith, and Aubrey Herbert. It was at around the time of his graduation from Oxford that Buchan had his first portrait painted, together, Buchan and his wife had four children, Alice, John, William, and Alastair, two of whom would spend most of their lives in Canada. With the outbreak of the First World War, Buchan went to write for the British War Propaganda Bureau and he continued to write fiction, and in 1915 published his most famous work, The Thirty-Nine Steps, a spy-thriller set just prior to World War I. The novel featured Buchans oft used hero, Richard Hannay, whose character was based on Edmund Ironside, a sequel, Greenmantle, came the following year. Buchan then enlisted in the British Army and was commissioned as a lieutenant in the Intelligence Corps. It was difficult for him, given his close connections to many of Britains military leaders, following the close of the war, Buchan turned his attention to writing on historical subjects, along with his usual thrillers and novels. Robert Graves, who lived in nearby Islip, mentioned his being recommended by Buchan for a position at the newly founded Cairo University. In a 1927 by-election, Buchan was elected as the Unionist Party Member of Parliament for the Combined Scottish Universities, politically, he was of the Unionist-Nationalist tradition, believing in Scotlands promotion as a nation within the British Empire. Buchan remarked in a speech to parliament, I believe every Scotsman should be a Scottish nationalist, if it could be proved that a Scottish parliament were desirable. He found himself profoundly affected by John Morleys Life of Gladstone, after the United Free Church of Scotland joined in 1929 with the Church of Scotland, Buchan remained an active elder of St. Columbas Church in London, as well as of the Oxford Presbyterian parish. In 1933 and 1934 Buchan was further appointed as the King George Vs Lord High Commissioner to the General Assembly of the Church of Scotland, beginning in 1930 Buchan aligned himself with Zionism and the related Palestine All Party Parliamentary Group. In recognition of his contributions to literature and education, on 1 January 1932, in 1935 Buchans literary work was adapted for the cinema with the completion of Alfred Hitchcocks The 39 Steps, starring Robert Donat as Richard Hannay, though with Buchans story much altered
55.
Novel
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A novel is any relatively long piece of written narrative fiction, normally in prose, and typically published as a book. The genre has also described as possessing, a continuous. This view sees the novels origins in Classical Greece and Rome, medieval, early modern romance, the latter, an Italian word used to describe short stories, supplied the present generic English term in the 18th century. The romance is a closely related long prose narrative, Romance, as defined here, should not be confused with the genre fiction love romance or romance novel. Other European languages do not distinguish between romance and novel, a novel is le roman, der Roman, il romanzo, a novel is a long, fictional narrative which describes intimate human experiences. Most European languages use the word romance for extended narratives, fictionality is most commonly cited as distinguishing novels from historiography. However this can be a problematic criterion, historians would also invent and compose speeches for didactic purposes. Novels can, on the hand, depict the social, political and personal realities of a place and period with clarity. Even in the 19th century, fictional narratives in verse, such as Lord Byrons Don Juan, Alexander Pushkins Yevgeniy Onegin, vikram Seths The Golden Gate, composed of 590 Onegin stanzas, is a more recent example of the verse novel. Both in 12th-century Japan and 15th-century Europe, prose fiction created intimate reading situations, on the other hand, verse epics, including the Odyssey and Aeneid, had been recited to a select audiences, though this was a more intimate experience than the performance of plays in theaters. A new world of Individualistic fashion, personal views, intimate feelings, secret anxieties, conduct and gallantry spread with novels, the novel is today the longest genre of narrative prose fiction, followed by the novella, short story, and flash fiction. However, in the 17th century critics saw the romance as of epic length, the length of a novel can still be important because most literary awards use length as a criterion in the ranking system. Urbanization and the spread of printed books in Song Dynasty China led to the evolution of oral storytelling into consciously fictional novels by the Ming dynasty, parallel European developments did not occur for centuries, and awaited the time when the availability of paper allowed for similar opportunities. By contrast, Ibn Tufails Hayy ibn Yaqdhan and Ibn al-Nafis Theologus Autodidactus are works of didactic philosophy, in this sense, Hayy ibn Yaqdhan would be considered an early example of a philosophical novel, while Theologus Autodidactus would be considered an early theological novel. Epic poetry exhibits some similarities with the novel, and the Western tradition of the novel back into the field of verse epics. Then at the beginning of the 18th century, French prose translations brought Homers works to a wider public, longus is the author of the famous Greek novel, Daphnis and Chloe. Romance or chivalric romance is a type of narrative in prose or verse popular in the circles of High Medieval. In later romances, particularly those of French origin, there is a tendency to emphasize themes of courtly love
56.
Alfred Hitchcock
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Sir Alfred Joseph Hitchcock KBE was an English film director and producer, at times referred to as The Master of Suspense. He pioneered many elements of the suspense and psychological thriller genres and he had a successful career in British cinema with both silent films and early talkies and became renowned as Englands best director. Hitchcock moved to Hollywood in 1939, and became a US citizen in 1955 and he also fashioned for himself a recognisable directorial style. Hitchcocks stylistic trademarks include the use of movement that mimics a persons gaze. In addition, he framed shots to maximise anxiety, fear, or empathy and his work often features fugitives on the run alongside icy blonde female characters. Prior to 1980, there had long been talk of Hitchcock being knighted for his contribution to film, Hitchcock later received his knighthood from Queen Elizabeth II in the 1980 New Year Honours. Hitchcock directed more than fifty films in a career spanning six decades and is often regarded as one of the most influential directors in cinematic history. His flair was for narrative, cruelly withholding crucial information and engaging the emotions of the audience like no one else, Hitchcocks first thriller, The Lodger, A Story of the London Fog, helped shape the thriller genre in film. His 1929 film, Blackmail, is cited as the first British sound feature film, while Rear Window, Vertigo, North by Northwest. Alfred Joseph Hitchcock was born on 13 August 1899 in Leytonstone and he was the second son and the youngest of three children of William Hitchcock, a greengrocer and poulterer, and Emma Jane Hitchcock. He was named after his fathers brother, Hitchcock was raised as a Roman Catholic, and sent to Salesian College, Battersea, and the Jesuit grammar school St Ignatius College in Stamford Hill, London. His parents were both of half-English and half-Irish ancestry and he often described a lonely and sheltered childhood that was worsened by his obesity. Around age five, Hitchcock recalled that to him for behaving badly. This incident implanted a lifelong fear of policemen in Hitchcock, and such harsh treatment, sources vary on Hitchcocks performance in school. Gene Adair reports that by most accounts, Alfred was only an average, or slightly above-average, however, McGilligan writes that Hitchcock certainly excelled academically. When Hitchcock was 15, his father died, in that same year, he left St. Ignatius to study at the London County Council School of Engineering and Navigation in Poplar, London. After leaving, he became a draftsman and advertising designer with a company called Henleys. Hitchcock joined a regiment of the Royal Engineers in 1917
57.
The Thirty-Nine Steps
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The Thirty-Nine Steps is an adventure novel by the Scottish author John Buchan. It first appeared as a serial in Blackwoods Magazine in August and September 1915 before being published in form in October that year by William Blackwood and Sons. It is the first of five novels featuring Richard Hannay, a hero with a stiff upper lip. John Buchan wrote The Thirty-Nine Steps while he was ill in bed with a duodenal ulcer, the novel was his first shocker, as he called it — a story combining personal and political dramas. The novel marked a point in Buchans literary career and introduced his adventuring hero. He described a shocker as an adventure where the events in the story are unlikely, Buchans son, William, later wrote that the name of the book originated when the authors daughter was counting the stairs at a private nursing home in Broadstairs, where Buchan was convalescing. There was a staircase leading down to the beach. My sister, who was six, and who had just learnt to count properly, went down them and gleefully announced. Some time later the house was demolished and a section of the stairs, One night he is buttonholed by a stranger, a well-travelled American, who claims to be in fear for his life. The man appears to know of an anarchist plot to destabilise Europe, beginning with a plan to assassinate the Greek Premier, Constantine Karolides, during his forthcoming visit to London. The man reveals his name to be Franklin P. Scudder, a spy, and remarks that he is dead. Scudder explains that he has faked his own death in order to avert suspicion, Scudder claims to be following a ring of German spies called the Black Stone who are trying to steal British plans for the outbreak of war. Hannay lets Scudder hide in his flat, and sure enough the day another man is discovered having apparently committed suicide in the same building. Four days later Hannay returns home to find Scudder dead with a knife through his heart. Hannay fears that the murderers will come for him next, and he also feels a duty to take up Scudders cause and save Karolides from the assassination. He decides to go into hiding in Scotland and then to contact the authorities at the last minute, in order to escape from his flat unseen, he bribes the milkman into lending him his uniform and exits wearing it, escaping from the German spies watching the house. Carrying Scudders pocket-book, he catches a train leaving from London St. Pancras station. Arriving at a station somewhere in Galloway, Hannay lodges in a shepherds cottage
58.
Jack Benny
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Jack Benny was an American comedian, vaudevillian, radio, television and film actor, and violinist. Recognized as a leading American entertainer of the 20th century, Benny portrayed his character as a miser, in character, he would claim to be 39 years of age, regardless of his actual age. Benny was known for comic timing and the ability to cause laughter with a pregnant pause or a single expression and his radio and television programs, popular from the 1930s to the 1970s, were a major influence on the sitcom genre. Benny was born in Chicago, Illinois, and grew up in nearby Waukegan and he was the son of Meyer Kubelsky and Emma Sachs Kubelsky. Meyer was an owner and later a haberdasher who had immigrated to America from Poland. Benny began studying violin, an instrument that became his trademark, at the age of 6 and he loved the instrument, but hated practice. His music teacher was Otto Graham Sr. a neighbor and father of Otto Graham of NFL fame, at 14, Benny was playing in dance bands and his high school orchestra. He was a dreamer and poor at his studies, and was expelled from high school. He did poorly in business school later and at attempts to join his fathers business, at age 17, he began playing the violin in local vaudeville theaters for $7.50 a week. He was joined by Ned Miller, a composer and singer. In 1911, Benny was playing in the theater as the young Marx Brothers. Minnie, their mother, enjoyed Bennys violin playing and invited him to accompany her boys in their act, Bennys parents refused to let their son go on the road at 17, but it was the beginning of his long friendship with the Marx Brothers, especially Zeppo Marx. The next year, Benny formed a musical duo with pianist Cora Folsom Salisbury. This provoked famous violinist Jan Kubelik, who feared that the young vaudevillian with a name would damage his reputation. Under legal pressure, Benjamin Kubelsky agreed to change his name to Ben K. Benny, when Salisbury left the act, Benny found a new pianist, Lyman Woods, and renamed the act From Grand Opera to Ragtime. They worked together for five years and slowly integrated comedy elements into the show and they reached the Palace Theater, the Mecca of Vaudeville, and did not do well. Benny left show business briefly in 1917 to join the United States Navy during World War I, and often entertained the troops with his violin playing. One evening, his performance was booed by the troops, so with prompting from fellow sailor and actor Pat OBrien, he ad-libbed his way out of the jam
59.
The Cure
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The Cure are an English rock band formed in Crawley, West Sussex in 1976. The band has experienced several changes, with vocalist, guitarist. During the early 1980s, the increasingly dark and tormented music was a staple of the emerging gothic rock genre. After the release of Pornography in 1982, the future was uncertain. The band is estimated to have sold 27 million records as of 2004 and have released thirteen albums, ten EPs. The 1989 album Disintegration is regarded as the commercial and critical peak. That band consisted of Robert Smith on piano, Michael Mick Dempsey on guitar, Laurence Lol Tolhurst on percussion, Marc Ceccagno on lead guitar and Alan Hill on bass guitar. In January 1976 while at St. Wilfrids Comprehensive School Ceccagno formed a 5-piece rock band with Smith on guitar and Dempsey on bass and they called themselves Malice and rehearsed David Bowie, Jimi Hendrix and Alex Harvey songs in a local church hall. By late April 1976, Ceccagno and the two school friends had left, and Tolhurst, Martin Creasy, and Porl Thompson had joined the band. This lineup played all three of Malices only documented live shows during December 1976, both Malice and Easy Cure auditioned several vocalists before Smith assumed the role of Easy Cures frontman in September 1977. That year, Easy Cure won a talent competition with German label Hansa Records, although the band recorded tracks for the company, none were ever released. Following disagreements in March 1978 over the direction the band should take, Smith later recalled, We were very young. They just thought they could turn us into a teen group and they actually wanted us to do cover versions and we always refused. Thompson was dropped from the band in May, and the trio were soon renamed The Cure by Smith. Later that month, the recorded their first sessions as a trio at Chestnut Studios in Sussex. The demo found its way to Polydor Records scout Chris Parry, the Cure released their debut single Killing an Arab in December 1978 on the Small Wonder label as a stopgap until Fiction finalised distribution arrangements with Polydor. The band placed a label that denied the racist connotations on the singles 1979 reissue on Fiction. The Cure released their debut album Three Imaginary Boys in May 1979, because of the bands inexperience in the studio, Parry and engineer Mike Hedges took control of the recording
60.
Bloodflowers
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Bloodflowers is the eleventh studio album by British alternative rock band The Cure, released in February 2000. The album is seen as a return to form by critics. Robert Smith has expressed on several occasions that the album is the part in his trilogy, the first being the 1982 album Pornography. The album is the last so far to feature extensive use of keyboards, Bloodflowers was released on 15 February 2000 by record label Fiction. It was a success, debuting at number 16 on the US Billboard 200 albums chart. It was nominated for a Grammy Award for Best Alternative Music Album in 2001, in 2002, the band performed Pornography, Disintegration, and Bloodflowers in their entirety to a Berlin audience, and released the recording on DVD in 2003, titled The Cure, Trilogy. On the 2007–2008 4Tour, the band played Maybe Someday at various shows, out of This World, Watching Me Fall, The Last Day of Summer and Bloodflowers were last performed on the 2016 North American tour. 39 was played at the first London date of the 2016 World Tour as part of the encore, except for the 2002 Trilogy shows in Berlin, the other songs have not been played since the 2000 Dream Tour. Coming Up and Spilt Milk have not been performed live at all, Bloodflowers received a generally favourable response from critics. Entertainment Weekly called it one of the bands most affecting works, a less favourable review came from Trouser Press, which wrote Bloodflowers feels like a forced recreation of the earlier gloomy classics. The album sounds completely uninspired, as Smith and company go through the motions of Cure-ness, all tracks written by The Cure. Other tracks recorded Possession – was released in the Join the Dots box set, just Say Yes – original version released on the Greatest Hits Demos & Rarities Microsite in 2001, rerecorded version released on the Greatest Hits CD. – cover version with different music circulates P2P networks, heavy World – instrumental on Lost Flowers demo, speculated to be released on the Bloodflowers reissue. Everything Forever – instrumental on the Lost Flowers demo, speculated to be on the Bloodflowers reissue
61.
Tenacious D
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Tenacious D is an American comedy rock duo that was formed in Los Angeles, California in 1994. Composed of lead vocalist and guitarist Jack Black and lead guitarist and vocalist Kyle Gass, Tenacious Ds studio releases, and its live performances, feature a full band lineup, including such musicians as guitarist John Konesky and bassist John Spiker. Drummer Dave Grohl has played on every studio release, the band first gained popularity in 1997 when they starred in their eponymous television series and began to support large rock acts. In 2001, they released Tenacious D, their debut album featuring a full band, the first single, Tribute, was the bands most successful achieving their only Top 10 in any chart, until they released The Metal, which was shown on Saturday Night Live. In 2006, they starred in, and recorded the soundtrack for, in support of the film, the band went on a world tour, appearing for the first time with a full band. They released their newest album Rize of the Fenix on May 15,2012, Tenacious Ds music showcases Blacks theatrical vocal delivery and Gass acoustic guitar playing abilities. Critics have described their fusion of vulgar absurdist comedy with music as mock rock. Jack Black and Kyle Gass initially met in Los Angeles in 1985, Black admits the duo did not see eye to eye due to animosity between the two as Gass – who was the main musician for the Actors Gang – felt threatened by Black. This all changed in 1989 in Edinburgh, Scotland, during the Edinburgh Fringe of 1989 and they were performing Tim Robbins and Adam Simons play Carnage. They would also work on productions at the group together regularly too. The two were just friends between 1989 and 1994 and did not play any concerts or record any music and their second song came about when Black was listening to Metallicas One in 1994 and told Gass that it was the best song in the world. Gass told Black that they couldnt write the best song in the world but Black put a twist on it and said they could write a tribute. Gass played an A minor chord at his apartment and the two spent three full days crafting the song, when it was done Gass mentioned they knew they had something, the song was comedic and evolved their comedy music persona. At their first concert, at Als Bar, the only played their one song Tribute. Black and Gass gave them the choice between Pets or Meat, Balboas Biblical Theatre and The Axe Lords Featuring Gorgazons Mischief. Tenacious D—a basketball term used by commentators to describe robust defensive positioning in basketball —did not get the majority of votes, however, cross, with Mr. Show writer Bob Odenkirk, continued his involvement with Tenacious D by producing three half-hour shows based on the band. The series, entitled Tenacious D, premiered on HBO in 1997, according to Gass, the series was cancelled after HBO requested ten episodes with the stipulation that he and Black would have to relinquish their role as executive producers, and only write songs. After the series aired, the continued to perform live
62.
Rize of the Fenix
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Rize of the Fenix is the third studio album by American rock band Tenacious D. Produced by John Kimbrough, it was released in North America on May 15,2012 by Columbia Records. In addition to core members Jack Black and Kyle Gass, the album marks the return of John Konesky, John Spiker and Dave Grohl. The album was nominated for a Grammy Award for Best Comedy Album at the 2013 ceremony, in November 2006, Jack Black expressed wishes to take a year-long break from acting, though Kyle Gass hinted a desire for Tenacious D to end at their current highpoint. However, Black confirmed that an album would be recorded by announcing that a new song has been written for it entitled Deth Starr. He said that the album would likely be released in 2010, music magazine Billboard quoted Black as revealing that We just laid down a hot were calling it the bomb track. Its a very powerful recording called Deth Starr so it has nothing to do with the Star Wars, adding that Its kind of sci-fi and he performed a vocal sample of the song along to keyboards. At one point, Gass hinted that the third album may be called Tenacious D 3-D, reasoning that Its the third record. Theres going to be a 3 and a D, so you have to connect them, Dave Grohl confirmed that he would appear as the drummer on the album, after performing on both Tenacious D and The Pick of Destiny. In terms of themes for the new songs, Black noted that Were gonna be talking about love, there are gonna be some songs about sex. In a May 2011 interview at Attack of the Show, Black announced that three songs on their album would be named Rize of the Fenix, distinguishable by either letter or number. Also in the interview, he named another song called Señorita. John Konesky has estimated that the new album will come out in spring 2012, in February 2012, it was revealed that the title of the album would in fact be spelled Rize of the Fenix and was released on May 15,2012. A music video for To Be the Best was released on The A. V and it guest stars Maria Menounos, Tim Robbins, Val Kilmer, Jimmy Kimmel, Dave Grohl, Yoshiki, and Josh Groban. On April 18,2012 a video was released on the TenaciousDSME YouTube channel titled Where Have We Been, the iTunes pre-order bonus track 5 Needs was originally performed by Tenacious D in their cameo in the 1996 film Bio-Dome. The band released the album on April 28,2012, sans bonus tracks, on their album website www. rizeofthefenix. com the band posted the entire album to stream to counteract the leaking of the music video for Rize of the Fenix. This was promoted by their Facebook page, the video was released for free on May 1 as an iTunes download. The next Tenacious D music video was for Roadie, released on May 8 via Funny or Die, the video featured Danny McBride as the Roadie
63.
A Night at the Opera (Queen album)
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A Night at the Opera is the fourth studio album by the British rock band Queen, released on 21 November 1975. Co-produced by Roy Thomas Baker and Queen, it was the most expensive album ever recorded at the time of its release, the album takes its name from the Marx Brothers film A Night at the Opera, which the band watched one night at the studio complex when recording. 4 on the Billboard 200 and became the bands first Platinum selling album in the US, the worldwide sales for the album are over 6 million copies. A Night at the Opera incorporates a range of styles, from ballads and songs in a music hall style, to hard rock tracks. It also produced the bands most successful single in the UK, Bohemian Rhapsody, other than on the live album, he said it was dedicated to a motherfucker I used to know. After the song together, it was agreed that the author should have his way. As with Bohemian Rhapsody, most of the parts on this song were initially played on piano by Mercury. Death on Two Legs remained on the setlist until, and well into, The Game Tour in 1980, however, the piano introduction was played during the Hot Space and Works tours. Lazing on a Sunday Afternoon is another song by Mercury and he played piano and performed all of the vocals. The lead vocal was sung in the studio and reproduced through headphones in a tin bucket elsewhere in the studio, a microphone picked up the sound from the bucket, which gives it a hollow megaphone sound. The guitar solo is also reported to have recorded on the vocal track, as there were no more tracks to record on. The key change going into the solo is a tritone relationship, making it a jarring. Im in Love with My Car is amongst Roger Taylors most famous songs in the Queen catalogue, the song was initially taken as a joke by May, who thought that Taylor was not serious when he heard a demo recording. Taylor played the guitars in the demo, but they were later re-recorded by May on his Red Special. The lead vocals were performed by Taylor on the studio version, the revving sounds at the conclusion of the song were recorded by Taylors then current car, an Alfa Romeo. The lyrics were inspired by one of the roadies, Johnathan Harris. The song is dedicated to him, the album says, Dedicated to Johnathan Harris, the song was often played live during the 1977–81 period. Taylor sang it from the drums while Mercury played piano and provided backing vocals and it was played in the Queen + Paul Rodgers Tour in 2005 and the Rock the Cosmos Tour in 2008
64.
Queen (band)
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Queen are a British rock band that formed in London in 1970. Their classic line-up was Freddie Mercury, Brian May, Roger Taylor, before forming Queen, Brian May and Roger Taylor had played together in a band named Smile. Freddie Mercury was a fan of Smile and encouraged them to experiment with elaborate stage. Mercury joined the band in 1970, suggested Queen as a new band name, John Deacon was recruited before the band recorded their eponymous debut album in 1973. The latter featured Bohemian Rhapsody, which stayed at one in the UK for nine weeks. The bands 1977 album News of the World contained We Will Rock You and We Are the Champions, by the early 1980s, Queen were one of the biggest stadium rock bands in the world. Their performance at the 1985 Live Aid concert has been ranked among the greatest in history by various music publications. In 1991, Mercury died of bronchopneumonia, a complication of AIDS, since then, May and Taylor have performed under the name of Queen with Paul Rodgers and Adam Lambert as vocalists on several tours. The band have released a total of 18 number-one albums,18 number-one singles, estimates of their record sales generally range from 150 million to 300 million records, making them one of the worlds best-selling music artists. Queen received the Outstanding Contribution to British Music Award from the British Phonographic Industry in 1990 and they were inducted into the Rock and Roll Hall of Fame in 2001. In 1968, guitarist Brian May, a student at Londons Imperial College, may placed an advertisement on a college notice board for a Mitch Mitchell/Ginger Baker type drummer, Roger Taylor, a young dental student, auditioned and got the job. While attending Ealing Art College, Tim Staffell became friends with Farrokh Bulsara, Bulsara felt that he and the band had the same tastes and soon became a keen fan of Smile. In 1970, after Staffell left to join the band Humpy Bong, the band had a number of bass players during this period who did not fit with the bands chemistry. It was not until February 1971 that they settled on John Deacon and they recorded four of their own songs, Liar, Keep Yourself Alive, The Night Comes Down and Jesus, for a demo tape, no record companies were interested. It was also around this time Freddie changed his surname to Mercury, inspired by the line Mother Mercury, on 2 July 1971, Queen played their first show in the classic line-up of Mercury, May, Taylor and Deacon at a Surrey college outside London. Having attended art college, Mercury also designed Queens logo, called the Queen crest, the logo combines the zodiac signs of all four members, two lions for Leo, a crab for Cancer, and two fairies for Virgo. The lions embrace a stylised letter Q, the crab rests atop the letter with flames rising directly above it, There is also a crown inside the Q and the whole logo is over-shadowed by an enormous phoenix. The whole symbol bears a resemblance to the Royal coat of arms of the United Kingdom
65.
The 39 Clues
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It consists of four series, The Clue Hunt, Cahills vs. Vespers, Unstoppable, and Doublecross. They chronicle the adventures of two siblings, Amy and Dan Cahill, who discover that their family, the Cahills, has been the most influential family in history. The first story arc concerns Dan and Amys quest to find the 39 Clues and this series primary audience is age 8–12. Since the release of the first novel, The Maze of Bones, on September 9,2008, the books have gained popularity, positive reception, and commercial success. As of July 2010, the series has about 8.5 million copies in print and has been translated into 24 languages. The publisher of the books is Scholastic Press in the United States, steven Spielberg acquired film rights to the series in June 2008, and a film based on the books was set to be released in 2016. The series also originated tie-in merchandise, including collectible cards and an interactive Internet game, the first series revolves around orphans Amy and Dan Cahill, who discover upon their grandmothers death that the Cahill family has shaped most of world history. Amy and Dan pursue the Clues while evading the sabotage of other Cahills, each book chronicles one location which Amy, Dan, and their au pair Nellie travel to and focuses on one historical character associated with a Clue. The Maze of Bones is the first book in the series, written by Rick Riordan, Amy and Dans grandmother, Grace Cahill, changes her will shortly before her death. Amy and Dan enter the Clue hunt, competing against more experienced Clue hunters, the Holts, Alistair Oh, the Starlings, the Kabras, Jonah Wizard, pursuing the clue hidden in Graces library leads to the Franklin Institute. Then the Starlings get an injury made by the Holts. There, Dan and Amy discover Benjamin Franklin has hidden a clue in Paris, after convincing their au pair, Nellie, to chaperone their trip, Amy and Dan travel to Paris, where they follow a trail of ciphers into the catacombs. Then, at an old church, they find the clue in a vial, the Kabras steal the vial, but Dan solves the puzzle and discovers the clue, iron solute. Amys internet searches lead to the location of the second Clue, Vienna. One False Note, second book in the series, was written by Gordon Korman, en route to Vienna, the Holts steal from Amy, Dan, and Nellie sheet music, a code that leads to the Clue, forcing them to rely on Dans photographic memory. They go to an archive to find the diary of Mozarts sister, Maria Anna Nannerl Mozart, the diary leads them to Salzburg, Mozarts birthplace, their search there ends in an explosion. Dan, Amy, and Nellie then go to Venice, where Dan, Ian and Natalie Kabra attack them and play Mozarts first harpsichord, triggering a booby trap that knocks Ian out. Natalie gets hit with a dart from her gun from Dan
66.
Glorious 39
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Glorious 39 is a 2009 British war thriller film written and directed by Stephen Poliakoff. Starring Romola Garai, Bill Nighy, Julie Christie, Jeremy Northam, Christopher Lee, David Tennant, Jenny Agutter, the film was released on 20 November 2009. In present-day London, Michael Walton visits his cousins, Walter and Oliver Page, Michael, interested in family history, asks them about his great aunt, Anne Keyes, the sister of his grandmother, Celia. Anne, an actress, was the eldest of the three Keyes children, desperate for children, her father, Alexander, a Member of Parliament, and mother, Maud, had adopted her, however, Maud subsequently gave birth to Ralph and Celia. Michael is curious to learn what happened to Anne, which leads Walter to reminisce about the summer of 1939, on the day of Alexanders birthday, Anne has prepared a table in the garden to celebrate. Annes friend, the outspoken MP Hector and lover, the reserved Lawrence are present for the festivities, when Alexander arrives that night, he also brings a guest, the quiet government employee, Joseph Balcombe. During dinner, Hector rants about Britains lack of action against Nazi Germany, noting that while his view is unpopular and it is later revealed that he has been one of those calling out for a new prime minister. The next day, while looking for a cat, Anne finds her in one of the propertys sheds. She finds gramophone records labelled Foxtrot, which, when played, prove to contain recorded meetings, Alexander reveals that he has allowed Balcombe to store government documents in the shed. Two weeks later, Anne is notified that Hector has been found dead, Anne wonders if Balcombe had anything to do with Hectors death. Alexander brushes off the idea, but does offer to ask Balcombe to remove the records from the shed, while there, the picnic-goers take off for a walk, including Aunt Elizabeth, leaving Anne to watch over baby Oliver. Anne awakens to find Oliver and his pushchair missing and she follows his cries to no avail, and when the family returns, they search, too, until they find him in his pushchair on a lane. Anne vehemently denies moving the baby, but the incident plants roots of doubt about Annes word, Balcombe removes the records that night, but Anne has secretly kept two of them. The family then returns to London because Parliament has been recalled, while there, Anne listens to the records. One contains a recording of a distressed Hector pleading with Balcombe to cease calling him, however, the maid bursts into the room, which causes the gramophone to fall, and the record to break into pieces. On 1 September Anne gives a record to her fellow actor and friend, Gilbert. Anne travels back to Norfolk to keep Aunt Elizabeth company, where she listens to the second recording, on it, she recognizes Balcombes voice, along with another, her brother, Ralph. Ralph is heard suggesting the name thin man dancing for a covert operation, the name he suggested is a reference to a childhood toy with which the siblings played
67.
World War II
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World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945, although related conflicts began earlier. It involved the vast majority of the worlds countries—including all of the great powers—eventually forming two opposing alliances, the Allies and the Axis. It was the most widespread war in history, and directly involved more than 100 million people from over 30 countries. Marked by mass deaths of civilians, including the Holocaust and the bombing of industrial and population centres. These made World War II the deadliest conflict in human history, from late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, and formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Poland, Finland, Romania and the Baltic states. In December 1941, Japan attacked the United States and European colonies in the Pacific Ocean, and quickly conquered much of the Western Pacific. The Axis advance halted in 1942 when Japan lost the critical Battle of Midway, near Hawaii, in 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained all of its territorial losses and invaded Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in South Central China and Burma, while the Allies crippled the Japanese Navy, thus ended the war in Asia, cementing the total victory of the Allies. World War II altered the political alignment and social structure of the world, the United Nations was established to foster international co-operation and prevent future conflicts. The victorious great powers—the United States, the Soviet Union, China, the United Kingdom, the Soviet Union and the United States emerged as rival superpowers, setting the stage for the Cold War, which lasted for the next 46 years. Meanwhile, the influence of European great powers waned, while the decolonisation of Asia, most countries whose industries had been damaged moved towards economic recovery. Political integration, especially in Europe, emerged as an effort to end pre-war enmities, the start of the war in Europe is generally held to be 1 September 1939, beginning with the German invasion of Poland, Britain and France declared war on Germany two days later. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or even the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred simultaneously and this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935. The British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the forces of Mongolia and the Soviet Union from May to September 1939, the exact date of the wars end is also not universally agreed upon. It was generally accepted at the time that the war ended with the armistice of 14 August 1945, rather than the formal surrender of Japan
68.
CBS
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CBS is an American commercial broadcast television network that is a flagship property of CBS Corporation. The company is headquartered at the CBS Building in New York City with major facilities and operations in New York City. CBS is sometimes referred to as the Eye Network, in reference to the iconic logo. It has also called the Tiffany Network, alluding to the perceived high quality of CBS programming during the tenure of William S. Paley. It can also refer to some of CBSs first demonstrations of color television, the network has its origins in United Independent Broadcasters Inc. a collection of 16 radio stations that was purchased by Paley in 1928 and renamed the Columbia Broadcasting System. Under Paleys guidance, CBS would first become one of the largest radio networks in the United States, in 1974, CBS dropped its former full name and became known simply as CBS, Inc. In 2000, CBS came under the control of Viacom, which was formed as a spin-off of CBS in 1971, CBS Corporation is controlled by Sumner Redstone through National Amusements, which also controls the current Viacom. The television network has more than 240 owned-and-operated and affiliated stations throughout the United States. The origins of CBS date back to January 27,1927, Columbia Phonographic went on the air on September 18,1927, with a presentation by the Howard Barlow Orchestra from flagship station WOR in Newark, New Jersey, and fifteen affiliates. Operational costs were steep, particularly the payments to AT&T for use of its land lines, in early 1928 Judson sold the network to brothers Isaac and Leon Levy, owners of the networks Philadelphia affiliate WCAU, and their partner Jerome Louchenheim. With the record out of the picture, Paley quickly streamlined the corporate name to Columbia Broadcasting System. He believed in the power of advertising since his familys La Palina cigars had doubled their sales after young William convinced his elders to advertise on radio. By September 1928, Paley bought out the Louchenheim share of CBS, during Louchenheims brief regime, Columbia paid $410,000 to A. H. Grebes Atlantic Broadcasting Company for a small Brooklyn station, WABC, which would become the networks flagship station. WABC was quickly upgraded, and the relocated to 860 kHz. The physical plant was relocated also – to Steinway Hall on West 57th Street in Manhattan, by the turn of 1929, the network could boast to sponsors of having 47 affiliates. Paley moved right away to put his network on a financial footing. In the fall of 1928, he entered talks with Adolph Zukor of Paramount Pictures. The deal came to fruition in September 1929, Paramount acquired 49% of CBS in return for a block of its stock worth $3.8 million at the time
69.
Reality show
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It differs from documentary television in that the focus tends to be on drama, personal conflict, and entertainment rather than educating viewers. The genre has various standard tropes, including confessionals used by cast members to express their thoughts, an early example of the genre was the 1991 Dutch series Nummer 28, which was the first show to bring together strangers and record their interactions. It then exploded as a phenomenon in the late 1990s and early 2000s with the success of the series Survivor, Idols. These shows and a number of others became global franchises, spawning local versions in dozens of countries, Reality television as a whole has become a fixture of television programming. There are grey areas around what is classified as reality television, Reality television has faced significant criticism since its rise in popularity. Much of the criticism has centered on the use of the word reality, Television formats portraying ordinary people in unscripted situations are almost as old as the television medium itself. Precedents for television that portrayed people in unscripted situations began in the late 1940s, queen for a Day was an early example of reality-based television. The 1946 television game show Cash and Carry sometimes featured contestants performing stunts, debuting in 1948, Allen Funts hidden camera show Candid Camera broadcast unsuspecting ordinary people reacting to pranks. In 1948, talent search shows Ted Macks Original Amateur Hour and Arthur Godfreys Talent Scouts featured amateur competitors, in the 1950s, game shows Beat the Clock and Truth or Consequences involved contestants in wacky competitions, stunts, and practical jokes. Confession was a show which aired from June 1958 to January 1959. The radio series Nightwatch tape-recorded the daily activities of Culver City, the series You Asked for It incorporated audience involvement by basing episodes around requests sent in by postcard from viewers. First broadcast in the United Kingdom in 1964, the Granada Television documentary Seven Up, broadcast interviews with a dozen ordinary 7-year-olds from a broad cross-section of society and inquired about their reactions to everyday life. Every seven years, a film documented the life of the same individuals during the period, titled the Up Series, episodes include 7 Plus Seven,21 Up. The program was structured as a series of interviews with no element of plot, however, it did have the then-new effect of turning ordinary people into celebrities. The first reality show in the modern sense may have been the series The American Sportsman, Another precursor may be considered Mutual of Omahas Wild Kingdom which aired from 1963 through 1988. This show featured zoologist Marlin Perkins traveling across the globe and illustrating the variety of animal life on the planet. Though mostly a travelogue, it was popular in syndication and new episodes were produced through the eighties. The 12-part 1973 PBS series An American Family showed a nuclear family going through a divorce, unlike many later reality shows, it was more or less documentary in purpose and style
70.
Survivor (U.S. TV series)
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The American series premiered on May 31,2000, on CBS. It is hosted by television personality Jeff Probst, who is also an executive producer, the show maroons a group of strangers in an isolated location, where they must provide food, water, fire, and shelter for themselves. The contestants compete in challenges for rewards and immunity from elimination, the American version has been very successful. From the 2000–01 through the 2005–06 television seasons, its first eleven seasons rated amongst the top ten most watched shows, Jeff Probst won the award for Outstanding Host for a Reality or Reality-Competition Program four consecutive times after the award was introduced in 2008. In 2007, the series was included in Time magazines list of the 100 greatest TV shows of all-time, in 2013, TV Guide ranked it at #39 on its list of the 60 Best Series of All Time. The series was renewed for a 34th season, Survivor, Game Changers, the series has been renewed through the 2017–18 television season. The first U. S. season of Survivor followed the general format as the Swedish series. Sixteen or more players are split between two or more tribes, are taken to an isolated location and are forced to live off the land with meager supplies for 39 days. Once about half the players are remaining, the tribes are merged into a single tribe, most players that are voted out at this stage form the games jury. Once down to two or three people, a final Tribal Council is held where the remaining players plead their case to the jury members, the jury then votes for which player should be considered the Sole Survivor and win the shows grand prize. In all seasons for the United States version, this has included a $1 million prize in addition to the Sole Survivor title, some seasons have included additional prizes, the United States version is produced by Mark Burnett and hosted by Jeff Probst. Each competition is called a season, has a unique name, the first season was broadcast as a summer replacement show in 2000. Starting with Survivor, Africa, there have two seasons aired during each U. S. television season. In the first season, there was a 75-person crew, by season 22, the crew had grown to 325 people. There have been a total of 498 contestants that have competed on Survivors 33 seasons, the original idea of Survivor was developed by Charlie Parsons in 1994 under the name Castaway. Parsons formed Planet24 with Bob Geldof to produce the show and tried to have the BBC broadcast it, Parsons went to Swedish television and was able to find a broadcaster, ultimately producing Expedition Robinson in 1997. The show was a success, and plans for international versions were made, Mark Burnett intended to be the person to bring the show to the United States, though he recognized that the Swedish version was a bit crude and mean-spirited. Burnett retooled the concept to use better production values, based on his prior Eco-Challenge show, Burnett spent about a year trying to find a broadcaster that would take the show, retooling the concept based on feedback
71.
The Honeymooners
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The Honeymooners is an American television sitcom created by and starring Jackie Gleason, based on a recurring 1951–55 comedy sketch of the same name. The popularity of the sketches led Gleason to rework The Honeymooners as a filmed half-hour series, the final episode of The Honeymooners aired on September 22,1956, although Gleason revived the characters sporadically until 1978. The Honeymooners was one of the first U. S. television shows to portray working-class married couples in a gritty, the program is also popular internationally, particularly in Canada, Poland and Scandinavian countries Norway and Sweden. The majority of The Honeymooners focused on its four principal characters on fixed sets within a Brooklyn apartment building, played by Jackie Gleason—a bus driver for the fictional Gotham Bus Company based out of Brooklyn, NY. He is never seen driving a bus, but is shown at the bus depot. Ralph is frustrated by his lack of success, and often develops get-rich-quick schemes, Ralph is very short tempered, frequently resorting to bellowing, insults and hollow threats. Well-hidden beneath the layers of bluster, however, is a soft-hearted man who loves his wife and is devoted to his best pal. Ralph enjoys bowling, playing pool and being a member in the Loyal Order of Raccoon Lodge. Ralph was given membership in the union for real New York City bus drivers during the run of the show. Ralph Kramden is the inspiration for the animated character Fred Flintstone, Alice, played in the first seven episodes by Pert Kelton and by Audrey Meadows throughout the Classic 39, is Ralphs patient but sharp-tongued wife of roughly 15 years. She often finds herself bearing the brunt of Ralphs insults, which she returns with biting sarcasm and she has grown accustomed to his empty threats, One of these days. Right in the kisser. or BANG, ZOOM, to which she usually replies, Ahhh, shaddap. She studied to be a secretary before her marriage, and works briefly in that capacity when Ralph is laid off, wilma Flintstone is based on Alice Kramden. Another foil for Ralph is Alices mother, who is even sharper-tongued than her daughter and she despises Ralph as a bad provider. Alices father is mentioned but never seen. Alices sister, Agnes, appeared in one episode, Ralph and Alice lived with her mother for six years after getting married before they got their own apartment. Ralphs mother is mentioned, but appears in one episode. Ralphs father is mentioned in one episode as having given Ralph a cornet he learned to play as a boy
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United States Constitution
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The United States Constitution is the supreme law of the United States of America. The Constitution, originally comprising seven articles, delineates the national frame of government, Articles Four, Five and Six entrench concepts of federalism, describing the rights and responsibilities of state governments and of the states in relationship to the federal government. Article Seven establishes the procedure used by the thirteen States to ratify it. In general, the first ten amendments, known collectively as the Bill of Rights, offer specific protections of individual liberty, the majority of the seventeen later amendments expand individual civil rights protections. Others address issues related to federal authority or modify government processes and procedures, Amendments to the United States Constitution, unlike ones made to many constitutions worldwide, are appended to the document. All four pages of the original U. S, according to the United States Senate, The Constitutions first three words—We the People—affirm that the government of the United States exists to serve its citizens. From September 5,1774 to March 1,1781, the Continental Congress functioned as the government of the United States. The process of selecting the delegates for the First and Second Continental Congresses underscores the revolutionary role of the people of the colonies in establishing a governing body. The Articles of Confederation and Perpetual Union was the first constitution of the United States and it was drafted by the Second Continental Congress from mid-1776 through late-1777, and ratification by all 13 states was completed by early 1781. Under the Articles of Confederation, the governments power was quite limited. The Confederation Congress could make decisions, but lacked enforcement powers, implementation of most decisions, including modifications to the Articles, required unanimous approval of all thirteen state legislatures. The Continental Congress could print money but the currency was worthless, Congress could borrow money, but couldnt pay it back. No state paid all their U. S. taxes, some paid nothing, some few paid an amount equal to interest on the national debt owed to their citizens, but no more. No interest was paid on debt owed foreign governments, by 1786, the United States would default on outstanding debts as their dates came due. Internationally, the Articles of Confederation did little to enhance the United States ability to defend its sovereignty, most of the troops in the 625-man United States Army were deployed facing – but not threatening – British forts on American soil. They had not been paid, some were deserting and others threatening mutiny, spain closed New Orleans to American commerce, U. S. officials protested, but to no effect. Barbary pirates began seizing American ships of commerce, the Treasury had no funds to pay their ransom, if any military crisis required action, the Congress had no credit or taxing power to finance a response. Domestically, the Articles of Confederation was failing to bring unity to the sentiments and interests of the various states