1.
15 (number)
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15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name, in spoken English, the numbers 15 and 50 are often confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed,15 /fɪfˈtiːn/ vs 50 /ˈfɪfti/, however, in dates such as 1500 or when contrasting numbers in the teens, the stress generally shifts to the first syllable,15 /ˈfɪftiːn/. In a 24-hour clock, the hour is in conventional language called three or three oclock. A composite number, its divisors being 1,3 and 5. A repdigit in binary and quaternary, in hexadecimal, as well as all higher bases,15 is represented as F. the 4th discrete semiprime and the first member of the discrete semiprime family. It is thus the first odd discrete semiprime, the number proceeding 15,14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21, the smallest number that can be factorized using Shors quantum algorithm. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet, the aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members. 15 is the composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree, fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33. 15 and 16 form a Ruth-Aaron pair under the definition in which repeated prime factors are counted as often as they occur. There are 15 solutions to Známs problem of length 7, if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems. Group 15 of the table are sometimes known as the pnictogens. 15 Madadgar is designated as a number in Pakistan, for mobile phones, similar to the international GSM emergency number 112, if 112 is used in Pakistan. 112 can be used in an emergency if the phone is locked. The Hanbali Sunni madhab states that the age of fifteen of a solar or lunar calendar is when ones taklif begins and is the stage whereby one has his deeds recorded. In the Hebrew numbering system, the number 15 is not written according to the method, with the letters that represent 10 and 5
2.
9 (number)
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9 is the natural number following 8 and preceding 10. In the NATO phonetic alphabet, the digit 9 is called Niner, five-digit produce PLU codes that begin with 9 are organic. Common terminal digit in psychological pricing, Nine is a number that appears often in Indian Culture and mythology. Nine influencers are attested in Indian astrology, in the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements, Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. Navaratri is a festival dedicated to the nine forms of Durga. Navaratna, meaning 9 jewels may also refer to Navaratnas - accomplished courtiers, Navratan - a kind of dish, according to Yoga, the human body has nine doors - two eyes, two ears, the mouth, two nostrils, and the openings for defecation and procreation. In Indian aesthetics, there are nine kinds of Rasa, Nine is considered a good number in Chinese culture because it sounds the same as the word long-lasting. Nine is strongly associated with the Chinese dragon, a symbol of magic, there are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales –81 yang and 36 yin, all three numbers are multiples of 9 as well as having the same digital root of 9. The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City, the name of the area called Kowloon in Hong Kong literally means, nine dragons. The nine-dotted line delimits certain island claims by China in the South China Sea, the nine-rank system was a civil service nomination system used during certain Chinese dynasties. 9 Points of the Heart / Heart Master Channels in Traditional Chinese Medicine, the nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. The Ennead is a group of nine Egyptian deities, who, in versions of the Osiris myth. The Nine Worthies are nine historical, or semi-legendary figures who, in Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil. The nine Muses in Greek mythology are Calliope, Clio, Erato, Euterpe, Melpomene, Polyhymnia, Terpsichore, Thalia and it takes nine days to fall from heaven to earth, and nine more to fall from earth to Tartarus—a place of torment in the underworld. Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo, according to Georges Ifrah, the origin of the 9 integers can be attributed to ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0. In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot, the Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, as time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller
3.
10 (number)
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10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十
4.
12 (number)
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12 is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a highly composite number, divisible by 2,3,4. It is central to systems of counting, including the Western calendar and units of time. The word twelve is the largest number with a name in English. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance and it derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels Book of John. It has cognates in every Germanic language, whose Proto-Germanic ancestor has been reconstructed as *twaliƀi, from *twa and suffix *-lif- or *-liƀ- of uncertain meaning. It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19, every other Indo-European language instead uses a form of two+ten, such as the Latin duōdecim. The usual ordinal form is twelfth but dozenth or duodecimal is also used in some contexts, similarly, a group of twelve things is usually a dozen but may also be referred to as a duodecad. The adjective referring to a group of twelve is duodecuple, as with eleven, the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan, with the implicit meaning that two is left after having already counted to ten. The Lithuanian suffix is also considered to share a similar development, the suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten. While, as mentioned above,12 has its own name in Germanic languages such as English and German, it is a number in many other languages, e. g. Italian dodici. In Germany, according to an old tradition, the numbers 0 through 12 were spelt out, the Duden now calls this tradition outdated and no longer valid, but many writers still follow it. Another system spells out all numbers written in one or two words, Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1,2,3,4,6 and 12. Twelve is also a composite number, the next one being twenty-four. Twelve is also a highly composite number, the next one being sixty. It is the first composite number of the form p2q, a square-prime,12 has an aliquot sum of 16. Accordingly,12 is the first abundant number and demonstrates an 8-member aliquot sequence,12 is the 3rd composite number in the 3-aliquot tree, the only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant, Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number
5.
17 (number)
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17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution
6.
19 (number)
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19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
7.
20 (number)
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20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants
8.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
9.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
10.
30 (number)
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30 is the natural number following 29 and preceding 31. Thirty is the sum of the first four squares, which makes it a square pyramidal number and it is a primorial and is the smallest Giuga number. 30 is the smallest sphenic number, and the smallest of the form 2 ×3 × r,30 has an aliquot sum of 42, the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises Thirty has but one number for which it is the aliquot sum, adding up some subsets of its divisors gives 30, hence 30 is a semiperfect number. 30 is the largest number such that all smaller than itself. A polygon with thirty sides is called a triacontagon, the icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a graph with 30 vertices. The atomic number of zinc is 30 Messier object M30, a magnitude 8, the duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series began on June 19,1803 BC. The duration of Saros series 30 was 1316.2 years, Thirty is, Used to indicate the end of a newspaper story, a copy editors typographical notation. S. Judas Iscariot betrayed Jesus for 30 pieces of silver, one of the rallying-cries of the 1960s student/youth protest movement was the slogan, Dont trust anyone over thirty. In Franz Kafkas novel The Trial Joseph wakes up on the morning of his birthday to find himself under arrest for an unspecified crime. After making many attempts to find the nature of the crime or the name of his accuser. The number of uprights that formed the Sarsen Circle at Stonehenge, western Christianitys most prolific 20th century essayist, F. W. Also in that essay Boreham writes It was said of Keats, in tennis, the number 30 represents the second point gained in a game. Under NCAA rules for basketball, the offensive team has 30 seconds to attempt a shot. As of 2012, three of the four major leagues in the United States and Canada have 30 teams each. The California Angels baseball team retired the number in honor of its most notable wearer, Nolan Ryan, the San Francisco Giants extended the same honor to Orlando Cepeda
11.
40 (number)
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Despite being related to the word four, the modern spelling of 40 is forty. The archaic form fourty is now considered a misspelling, the modern spelling possibly reflects a pronunciation change due to the horse–hoarse merger. Forty is a number, an octagonal number, and as the sum of the first four pentagonal numbers. Adding up some subsets of its divisors gives 40, hence 40 is a semiperfect number, given 40, the Mertens function returns 0. 40 is the smallest number n with exactly 9 solutions to the equation φ = n, Forty is the number of n-queens problem solutions for n =7. Since 402 +1 =1601 is prime,40 is a Størmer number,40 is a repdigit in base 3 and a Harshad number in base 10. Negative forty is the temperature at which the Fahrenheit and Celsius scales correspond. It is referred to as either minus forty or forty below, the planet Venus forms a pentagram in the night sky every eight years with it returning to its original point every 40 years with a 40-day regression. The duration of Saros series 40 was 1280.1 years, lunar eclipse series which began on -1387 February 12 and ended on -71 April 12. The duration of Saros series 40 was 1316.2 years, the number 40 is used in Jewish, Christian, Islamic, and other Middle Eastern traditions to represent a large, approximate number, similar to umpteen. In the Hebrew Bible, forty is often used for periods, forty days or forty years. Rain fell for forty days and forty nights during the Flood, spies explored the land of Israel for forty days. The Hebrew people lived in the Sinai desert for forty years and this period of years represents the time it takes for a new generation to arise. Moses life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, and his return to lead his people out, several Jewish leaders and kings are said to have ruled for forty years, that is, a generation. Examples include Eli, Saul, David, and Solomon, goliath challenged the Israelites twice a day for forty days before David defeated him. He went up on the day of Tammuz to beg forgiveness for the peoples sin. He went up on the first day of Elul and came down on the day of Tishrei. A mikvah consists of 40 seah of water 40 lashes is one of the punishments meted out by the Sanhedrin, One of the prerequisites for a man to study Kabbalah is that he is forty years old
12.
60 (number)
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60 is the natural number following 59 and preceding 61. Being three times 20, it is called three score in older literature. It is a number, with divisors 1,2,3,4,5,6,10,12,15,20,30. Because it is the sum of its divisors, it is a unitary perfect number. Being ten times a number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6 and it is the smallest number with exactly 12 divisors. It is the sum of a pair of twin primes and the sum of four consecutive primes and it is adjacent to two primes. It is the smallest number that is the sum of two odd primes in six ways, the smallest non-solvable group has order 60. There are four Archimedean solids with 60 vertices, the icosahedron, the rhombicosidodecahedron, the snub dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs, there are also two Archimedean solids with 60 edges, the snub cube and the icosidodecahedron. The skeleton of the forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares, in geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, a number system with base 60 is called sexagesimal. It is the smallest positive integer that is written only the smallest. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule and this ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 is an isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz
13.
80 (number)
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80 is the natural number following 79 and preceding 81. 80 is, the sum of Eulers totient function φ over the first sixteen integers, a semiperfect number, since adding up some subsets of its divisors gives 80. Palindromic in bases 3,6,9,15,19 and 39, a repdigit in bases 3,9,15,19 and 39. A Harshad number in bases 2,3,4,5,6,7,9,10,11,13,15 and 16 The Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes. Every solvable configuration of the Fifteen puzzle can be solved in no more than 80 single-tile moves, the atomic number of mercury According to Exodus 7,7, Moses was 80 years old when he initially spoke to Pharaoh on behalf of his people. Today,80 years of age is the age limit for cardinals to vote in papal elections. Jerry Rice wore the number 80 for the majority of his NFL career
14.
90 (number)
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90 is the natural number preceded by 89 and followed by 91. In English speech, the numbers 90 and 19 are often confused, when carefully enunciated, they differ in which syllable is stressed,19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in such as 1999, and when contrasting numbers in the teens and when counting, such as 17,18,19. 90 is, a perfect number because it is the sum of its unitary divisors. A semiperfect number because it is equal to the sum of a subset of its divisors, a Perrin number, preceded in the sequence by 39,51,68. Palindromic and a repdigit in bases 14,17,29, a Harshad number since 90 is divisible by the sum of its base 10 digits. In normal space, the angles of a rectangle measure 90 degrees each. Also, in a triangle, the angle opposing the hypotenuse measures 90 degrees. Thus, an angle measuring 90 degrees is called a right angle, ninety is, the atomic number of thorium, an actinide. As an atomic weight,90 identifies an isotope of strontium, the latitude in degrees of the North and the South geographical poles. NFL, New York Jets Dennis Byrds #90 is retired +90 is the code for international direct dial phone calls to Turkey,90 is the code for the French département Belfort
15.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
16.
Numeral system
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A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
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Tetradecimal
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This is a list of numeral systems, that is, writing systems for expressing numbers. Numeral systems are classified here as to whether they use positional notation, the common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system and this turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
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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
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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
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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
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Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
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Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
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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
32.
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
33.
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
34.
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
35.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
36.
Euler's totient function
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In number theory, Eulers totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ or ϕ and it can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd is equal to 1. The integers k of this form are referred to as totatives of n. For example, the totatives of n =9 are the six numbers 1,2,4,5,7 and 8. They are all relatively prime to 9, but the three numbers in this range,3,6, and 9 are not, because gcd = gcd =3. As another example, φ =1 since for n =1 the only integer in the range from 1 to n is 1 itself, Eulers totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ = φφ. This function gives the order of the group of integers modulo n. It also plays a key role in the definition of the RSA encryption system, leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it, he wrote πD for the multitude of less than D. This definition varies from the current definition for the totient function at D =1 but is otherwise the same, the now-standard notation φ comes from Gausss 1801 treatise Disquisitiones Arithmeticae. Although Gauss didnt use parentheses around the argument and wrote φA, thus, it is often called Eulers phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, so it is referred to as Eulers totient function. Jordans totient is a generalization of Eulers, the cototient of n is defined as n − φ. It counts the number of positive integers less than or equal to n that have at least one factor in common with n. There are several formulas for computing φ and it states φ = n ∏ p ∣ n, where the product is over the distinct prime numbers dividing n. The proof of Eulers product formula depends on two important facts and this means that if gcd =1, then φ = φ φ. If p is prime and k ≥1, then φ = p k − p k −1 = p k −1 = p k, proof, since p is a prime number the only possible values of gcd are 1, p, p2
37.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
38.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
39.
Complement (set theory)
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In set theory, the complement of a set A refers to elements not in A. The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A. If A and B are sets, then the complement of A in B, also termed the set-theoretic difference of B and A, is the set of elements in B. The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard, if R is the set of real numbers and Q is the set of rational numbers, then R ∖ Q is the set of irrational numbers. Let A, B, and C be three sets, the following identities capture notable properties of relative complements, C ∖ = ∪. C ∖ = ∪, with the important special case C ∖ = demonstrating that intersection can be expressed using only the relative complement operation. If A is a set, then the complement of A is the set of elements not in A. Formally. The absolute complement of A is usually denoted by A ∁, other notations include A c, A ¯, A ′, ∁ U A, and ∁ A. Assume that the universe is the set of integers, if A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of multiples of 3, then the complement of B is the set of numbers congruent to 1 or 2 modulo 3, assume that the universe is the standard 52-card deck. If the set A is the suit of spades, then the complement of A is the union of the suits of clubs, diamonds, and hearts. If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts, let A and B be two sets in a universe U. The following identities capture important properties of complements, De Morgans laws. Complement laws, A ∪ A ∁ = U, if A ⊂ B, then B ∁ ⊂ A ∁. Involution or double complement law, ∁ = A, relationships between relative and absolute complements, A ∖ B = A ∩ B ∁. Relationship with set difference, A ∁ ∖ B ∁ = B ∖ A, the first two complement laws above show that if A is a non-empty, proper subset of U, then is a partition of U. In the LaTeX typesetting language, the command \setminus is usually used for rendering a set difference symbol, when rendered, the \setminus command looks identical to \backslash except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin
40.
Topological space
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Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion, the branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. The utility of the notion of a topology is shown by the fact there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application, the most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so this is given first. Note, A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy and this axiomatization is due to Felix Hausdorff. Let X be a set, the elements of X are usually called points, let N be a function assigning to each x in X a non-empty collection N of subsets of X. The elements of N will be called neighbourhoods of x with respect to N, the function N is called a neighbourhood topology if the axioms below are satisfied, and then X with N is called a topological space. If N is a neighbourhood of x, then x ∈ N, in other words, each point belongs to every one of its neighbourhoods. If N is a subset of X and includes a neighbourhood of x, I. e. every superset of a neighbourhood of a point x in X is again a neighbourhood of x. The intersection of two neighbourhoods of x is a neighbourhood of x, any neighbourhood N of x includes a neighbourhood M of x such that N is a neighbourhood of each point of M. The first three axioms for neighbourhoods have a clear meaning, the fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X. A standard example of such a system of neighbourhoods is for the real line R, given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. A topological space is a pair, where X is a set and τ is a collection of subsets of X, satisfying the following axioms, The empty set. Any union of members of τ still belongs to τ, the intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X, given X =, the collection τ = of only the two subsets of X required by the axioms forms a topology of X, the trivial topology. Given X =, the collection τ = of six subsets of X forms another topology of X, given X = and the collection τ = P, is a topological space. τ is called the discrete topology, using de Morgans laws, the above axioms defining open sets become axioms defining closed sets, The empty set and X are closed. The intersection of any collection of closed sets is also closed, the union of any finite number of closed sets is also closed
41.
Catalan number
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In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. They are named after the Belgian mathematician Eugène Charles Catalan, using zero-based numbering, the nth Catalan number is given directly in terms of binomial coefficients by C n =1 n +1 =. = ∏ k =2 n n + k k for n ≥0, an alternative expression for Cn is C n = − =1 n +1 for n ≥0, which is equivalent to the expression given above because = n n +1. This shows that Cn is an integer, which is not immediately obvious from the first formula given and this expression forms the basis for a proof of the correctness of the formula. They also satisfy, C0 =1 and C n +1 =2 n +2 C n, which can be a more efficient way to calculate them. Asymptotically, the Catalan numbers grow as C n ∼4 n n 3 /2 π in the sense that the quotient of the nth Catalan number, some sources use just C n ≈4 n n 3 /2. The only Catalan numbers Cn that are odd are those for which n = 2k −1, the only prime Catalan numbers are C2 =2 and C3 =5. The Catalan numbers have an integral representation C n = ∫04 x n ρ d x where ρ =12 π4 − x x and this means that the Catalan numbers are a solution of the Hausdorff moment problem on the interval instead of. The orthogonal polynomials having the weight function ρ on are H n = ∑ k =0 n k, there are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics, Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers, following are some examples, with illustrations of the cases C3 =5 and C4 =14. Cn is the number of Dyck words of length 2n, a Dyck word is a string consisting of n Xs and n Ys such that no initial segment of the string has more Ys than Xs. For n =3, for example, we have the five different parenthesizations of four factors. It follows that Cn is the number of binary trees with n +1 leaves. Cn is the number of lattice paths along the edges of a grid with n × n square cells. A monotonic path is one which starts in the left corner, finishes in the upper right corner. Counting such paths is equivalent to counting Dyck words, X stands for move right, the following hexagons illustrate the case n =4, Cn is the number of stack-sortable permutations of. These are the permutations that avoid the pattern 231, Cn is the number of permutations of that avoid the pattern 123, that is, the number of permutations with no three-term increasing subsequence. For n =3, these permutations are 132,213,231,312 and 321
42.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
43.
Truncated cube
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In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces,36 edges, and 24 vertices, if the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + √2. The area A and the volume V of a cube of edge length a are. The truncated cube has five special orthogonal projections, centered, on a vertex, the last two correspond to the B2 and A2 Coxeter planes. The truncated cube can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, the following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ, where ξ = √2 −1. The parameter ξ can be varied between ±1, a value of 1 produces a cube,0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces. The truncated cube can be dissected into a cube, with six square cupola around each of the cubes faces. This dissection can also be seen within the cubic honeycomb, with cube, tetrahedron. This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupola and the central cube and this excavated cube has 16 triangles,12 squares, and 4 octagons. It shares the vertex arrangement with three nonconvex uniform polyhedra, The truncated cube is related to polyhedra and tlings in symmetry. The truncated cube is one of a family of uniform polyhedra related to the cube and this polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations, and Coxeter group symmetry, and a series of polyhedra and tilings n.8.8. A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces and it is one of a sequence of alternate truncations of polyhedra and tiling. It has 24 vertices and 36 edges, and is a cubic Archimedean graph, spinning truncated cube Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, cromwell, P. Polyhedra, CUP hbk, pbk. Ch.2 p. 79-86 Archimedean solids Eric W. Weisstein, Weisstein, Eric W. Truncated cubical graph. 3D convex uniform polyhedra o3x4x - tic
44.
Truncated octahedron
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In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces,36 edges, and 24 vertices, since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV, containing square and hexagonal faces, like the cube, it can tessellate 3-dimensional space, as a permutohedron. Its dual polyhedron is the tetrakis hexahedron, if the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths 9/8√2 and 3/2√2. A truncated octahedron is constructed from an octahedron with side length 3a by the removal of six right square pyramids. These pyramids have both base side length and lateral side length of a, to form equilateral triangles, the base area is then a2. Note that this shape is similar to half an octahedron or Johnson solid J1. The truncated octahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the B2 and A2 Coxeter planes. The truncated octahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, all permutations of are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √2 centered at the origin. The vertices are also the corners of 12 rectangles whose long edges are parallel to the coordinate axes. The edge vectors have Cartesian coordinates and permutations of these, the face normals of the 6 square faces are, and. The face normals of the 8 hexagonal faces are, the dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either −1/3 or −1/√3. The dihedral angle is approximately 1.910633 radians at edges shared by two hexagons or 2.186276 radians at edges shared by a hexagon and a square. The truncated octahedron can be dissected into an octahedron, surrounded by 8 triangular cupola on each face. Therefore, the octahedron is the permutohedron of order 4, each vertex corresponds to a permutation of. The area A and the volume V of an octahedron of edge length a are. There are two uniform colorings, with symmetry and octahedral symmetry, and two 2-uniform coloring with dihedral symmetry as a truncated triangular antiprism
45.
Rhombic dodecahedron
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In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types and it is a Catalan solid, and the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron and its polyhedral dual is the cuboctahedron. The long diagonal of each face is exactly √2 times the length of the diagonal, so that the acute angles on each face measure arccos. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the region of space while moving face A to face B. The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron can be used to tessellate three-dimensional space. It can be stacked to fill a space much like hexagons fill a plane and this polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. It is the Brillouin zone of body centered cubic crystals, some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron, the rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent, but the chemical bonds lie on the remaining edges, the graph of the rhombic dodecahedron is nonhamiltonian. The last two correspond to the B2 and A2 Coxeter planes, the rhombic dodecahedron is a parallelohedron, a space-filling polyhedron. Other symmetry constructions of the dodecahedron are also space-filling. For example, with 4 square faces, and 60-degree rhombic faces and it be seen as a cuboctahedron with square pyramids augmented on the top and bottom. In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces and it has the same topology but different geometry. The rhombic faces in this form have the golden ratio, another topologically equivalent variation, sometimes called a trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites. It has 8 vertices adjusted in or out in sets of 4. Variations can be parametrized by, where b is determined from a for planar faces and this polyhedron is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry
46.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
47.
Duality (mathematics)
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Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues theorem is self-dual in this sense under the standard duality in projective geometry, many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. From a category theory viewpoint, duality can also be seen as a functor and this functor assigns to each space its dual space, and the pullback construction assigns to each arrow f, V → W its dual f∗, W∗ → V∗. In the words of Michael Atiyah, Duality in mathematics is not a theorem, the following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. A simple, maybe the most simple, duality arises from considering subsets of a fixed set S, to any subset A ⊆ S, the complement Ac consists of all those elements in S which are not contained in A. It is again a subset of S, taking the complement has the following properties, Applying it twice gives back the original set, i. e. c = A. This is referred to by saying that the operation of taking the complement is an involution, an inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction Bc ⊆ Ac. Given two subsets A and B of S, A is contained in Bc if and only if B is contained in Ac. This duality appears in topology as a duality between open and closed subsets of some fixed topological space X, a subset U of X is closed if, because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of sets is open, so dually. The interior of a set is the largest open set contained in it, because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U. A duality in geometry is provided by the cone construction. Given a set C of points in the plane R2, unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set C. Instead, C ∗ ∗ is the smallest cone containing C which may be bigger than C. Therefore this duality is weaker than the one above, in that Applying the operation twice gives back a possibly bigger set, the other two properties carry over without change, It is still true that an inclusion C ⊆ D is turned into an inclusion in the opposite direction. Given two subsets C and D of the plane, C is contained in D ∗ if, a very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V*. Its elements are the k-linear maps φ, V → k, the three properties of the dual cone carry over to this type of duality by replacing subsets of R2 by vector space and inclusions of such subsets by linear maps. That is, Applying the operation of taking the dual vector space twice gives another vector space V**, there is always a map V → V**
48.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers
49.
Bravais lattices
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This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same, when the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its frontiers. A crystal is made up of an arrangement of one or more atoms repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space, the 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In two-dimensional space, there are 5 Bravais lattices, grouped into four crystal families, the unit cells are specified according to the relative lengths of the cell edges and the angle between them. The area of the cell can be calculated by evaluating the norm || a × b ||. The properties of the families are given below, In three-dimensional space. These are obtained by combining one of the six families with one of the centering types. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes, similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, the unit cells are specified according to the relative lengths of the cell edges and the angles between them. The volume of the cell can be calculated by evaluating the triple product a ·, where a, b. The properties of the families are given below, In four dimensions. Of these,23 are primitive and 41 are centered, ten Bravais lattices split into enantiomorphic pairs. Bravais, A. Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans lespace, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry. Catalogue of Lattices Smith, Walter Fox
50.
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
51.
Silicon
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Silicon is a chemical element with symbol Si and atomic number 14. A hard and brittle crystalline solid with a metallic luster. It is a member of group 14 in the table, along with carbon above it and germanium, tin, lead. It is not very reactive, although more reactive than germanium, Silicon is the eighth most common element in the universe by mass, but very rarely occurs as the pure element in the Earths crust. It is most widely distributed in dusts, sands, planetoids, over 90% of the Earths crust is composed of silicate minerals, making silicon the second most abundant element in the Earths crust after oxygen. Most silicon is used commercially without being separated, and often with little processing of the natural minerals, such use includes industrial construction with clays, silica sand, and stone. Silicate is used in Portland cement for mortar and stucco, and mixed with sand and gravel to make concrete for walkways, foundations. Silicates are used in whiteware ceramics such as porcelain, and in traditional quartz-based soda-lime glass, Silicon compounds such as silicon carbide are used as abrasives and components of high-strength ceramics. Elemental silicon also has an impact on the modern world economy. Most free silicon is used in the refining, aluminium-casting. Silicon is the basis of the widely used synthetic polymers called silicones, Silicon is an essential element in biology, although only tiny traces are required by animals. However, various sea sponges and microorganisms, such as diatoms and radiolaria, silica is deposited in many plant tissues, such as in the bark and wood of Chrysobalanaceae and the silica cells and silicified trichomes of Cannabis sativa, horsetails and many grasses. Silicon is a solid at room temperature, with a point of 1,414 °C. Like water, it has a density in a liquid state than in a solid state and it expands when it freezes. With a relatively high conductivity of 149 W·m−1·K−1, silicon conducts heat well. In its crystalline form, pure silicon has a gray color, like germanium, silicon is rather strong, very brittle, and prone to chipping. Silicon, like carbon and germanium, crystallizes in a cubic crystal structure with a lattice spacing of 0.5430710 nm. The outer electron orbital of silicon, like that of carbon, has four valence electrons, the 1s, 2s, 2p and 3s subshells are completely filled while the 3p subshell contains two electrons out of a possible six
52.
Atomic weight
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Relative atomic mass is a dimensionless physical quantity. In its modern definition, it is the ratio of the mass of atoms of an element in a given sample to one unified atomic mass unit. The unified atomic mass unit, symbol u, is defined being 1⁄12 of the mass of a carbon-12 atom, the mass of atoms can vary, due to the presence of various isotopes of that element. Since both values in the ratio are expressed in the unit, the resulting value is dimensionless. Within one source, it is a average over the individual atom weights present. Between sources, the weight can vary when the sources origin resulted in different isotopic concentrations. These differences are real and measurable, and can be used to identify a sample to its origin, for example, a sample of elemental carbon from volcanic methane will have a different relative atomic mass than one collected from plant or animal tissues. The well-known standard atomic weight, or atomic weight, is a usage of relative atomic mass, it is the relative atomic mass. For these sources, research reports are used by the CIAAW of the International Union of Pure, the standard atomic weights are reprinted in a wide variety of textbooks, commercial catalogues, and periodic table wall charts. They are what chemists loosely call atomic weights and it is the most published form of the relative atomic mass. Both terms are officially sanctioned by IUPAC, the term relative atomic mass now seems to be gaining as the preferred term over atomic weight, although in the case of standard atomic weight, this shorter term continues to be used. This comparison is the quotient of the two weights, which makes the value dimensionless and this quotient also explains the word relative, the sample mass value is made relative to carbon-12. It is uses as a synonym for atomic weight, in some circumstances may even be synonymous with standard atomic weight. Here the unified atomic mass unit refers to 1⁄12 of the mass of an atom of 12C in its ground state. The IUPAC definition of relative atomic mass is, An atomic weight of an element from a source is the ratio of the average mass per atom of the element to 1/12 of the mass of an atom of 12C. The definition deliberately specifies An atomic weight…, as an element will have different relative atomic masses depending on the source, for example, boron from Turkey has a lower relative atomic mass than boron from California, because of its different isotopic composition. Nevertheless, given the cost and difficulty of isotope analysis, it is usual to use the values of standard atomic weights which are ubiquitous in chemical laboratories. Older historical relative scales used either the relative isotopic mass for reference
53.
Nitrogen
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Nitrogen is a chemical element with symbol N and atomic number 7. It was first discovered and isolated by Scottish physician Daniel Rutherford in 1772, although Carl Wilhelm Scheele and Henry Cavendish had independently done so at about the same time, Rutherford is generally accorded the credit because his work was published first. Nitrogen is the lightest member of group 15 of the periodic table, the name comes from the Greek πνίγειν to choke, directly referencing nitrogens asphyxiating properties. It is an element in the universe, estimated at about seventh in total abundance in the Milky Way. At standard temperature and pressure, two atoms of the element bind to form dinitrogen, a colourless and odorless diatomic gas with the formula N2, dinitrogen forms about 78% of Earths atmosphere, making it the most abundant uncombined element. Nitrogen occurs in all organisms, primarily in amino acids, in the nucleic acids, the human body contains about 3% nitrogen by mass, the fourth most abundant element in the body after oxygen, carbon, and hydrogen. The nitrogen cycle describes movement of the element from the air, into the biosphere and organic compounds, many industrially important compounds, such as ammonia, nitric acid, organic nitrates, and cyanides, contain nitrogen. The extremely strong bond in elemental nitrogen, the second strongest bond in any diatomic molecule. Synthetically produced ammonia and nitrates are key industrial fertilisers, and fertiliser nitrates are key pollutants in the eutrophication of water systems. Apart from its use in fertilisers and energy-stores, nitrogen is a constituent of organic compounds as diverse as Kevlar used in high-strength fabric, Nitrogen is a constituent of every major pharmacological drug class, including antibiotics. Many notable nitrogen-containing drugs, such as the caffeine and morphine or the synthetic amphetamines. Nitrogen compounds have a long history, ammonium chloride having been known to Herodotus. They were well known by the Middle Ages, alchemists knew nitric acid as aqua fortis, as well as other nitrogen compounds such as ammonium salts and nitrate salts. The mixture of nitric and hydrochloric acids was known as aqua regia, celebrated for its ability to dissolve gold, the discovery of nitrogen is attributed to the Scottish physician Daniel Rutherford in 1772, who called it noxious air. Though he did not recognise it as a different chemical substance, he clearly distinguished it from Joseph Blacks fixed air. The fact that there was a component of air that does not support combustion was clear to Rutherford, Nitrogen was also studied at about the same time by Carl Wilhelm Scheele, Henry Cavendish, and Joseph Priestley, who referred to it as burnt air or phlogisticated air. Nitrogen gas was inert enough that Antoine Lavoisier referred to it as air or azote, from the Greek word άζωτικός. In an atmosphere of nitrogen, animals died and flames were extinguished
54.
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
55.
Messier 14
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Messier 14 is a globular cluster in the constellation Ophiuchus. It was discovered by Charles Messier in 1764, at a distance of about 30,000 light-years, M14 contains several hundred thousand stars. At an apparent magnitude of +7.6 it can be observed with binoculars. Medium-sized telescopes will show some hint of the stars of which the brightest is of magnitude +14. The total luminosity of M14 is in the order of 400,000 times that of the Sun corresponding to a magnitude of -9.12. The shape of the cluster is decidedly elongated, M14 is about 100 light-years across. A total of 70 variable stars are known in M14, many of the W Virginis variety common in globular clusters, in 1938, a nova appeared, although this was not discovered until photographic plates from that time were studied in 1964. It is estimated that the nova reached a brightness of magnitude +9.2. Slightly over 3° southwest of M14 lies the faint globular cluster NGC6366. SEDS Messier pages on M14 M14, Galactic Globular Clusters Database page Messier 14 on WikiSky, DSS2, SDSS, GALEX, IRAS, Hydrogen α, X-Ray, Astrophoto, Sky Map, Articles and images
56.
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
57.
Globular cluster
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A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes, the name of this category of star cluster is derived from the Latin globulus—a small sphere. A globular cluster is known more simply as a globular. Globular clusters are found in the halo of a galaxy and contain considerably more stars and are older than the less dense open clusters. Globular clusters are common, there are about 150 to 158 currently known globular clusters in the Milky Way. These globular clusters orbit the Galaxy at radii of 40 kiloparsecs or more, larger galaxies can have more, Andromeda Galaxy, for instance, may have as many as 500. Some giant elliptical galaxies such as M87, have as many as 13,000 globular clusters, every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters. The Sagittarius Dwarf galaxy and the disputed Canis Major Dwarf galaxy appear to be in the process of donating their associated globular clusters to the Milky Way and this demonstrates how many of this galaxys globular clusters might have been acquired in the past. Although it appears that globular clusters contain some of the first stars to be produced in the galaxy, their origins, the first known globular cluster, now called M22, was discovered in 1665 by Abraham Ihle, a German amateur astronomer. However, given the small aperture of early telescopes, individual stars within a cluster were not resolved until Charles Messier observed M4 in 1764. The first eight globular clusters discovered are shown in the table, subsequently, Abbé Lacaille would list NGC104, NGC4833, M55, M69, and NGC6397 in his 1751–52 catalogue. The M before a number refers to Charles Messiers catalogue, while NGC is from the New General Catalogue by John Dreyer, when William Herschel began his comprehensive survey of the sky using large telescopes in 1782 there were 34 known globular clusters. Herschel discovered another 36 himself and was the first to virtually all of them into stars. He coined the term globular cluster in his Catalogue of a Second Thousand New Nebulae, the number of globular clusters discovered continued to increase, reaching 83 in 1915,93 in 1930 and 97 by 1947. A total of 152 globular clusters have now discovered in the Milky Way galaxy. These additional, undiscovered globular clusters are believed to be hidden behind the gas, beginning in 1914, Harlow Shapley began a series of studies of globular clusters, published in about 40 scientific papers. He examined the RR Lyrae variables in the clusters and would use their period–luminosity relationship for distance estimates, later, it was found that RR Lyrae variables are fainter than Cepheid variables, which caused Shapley to overestimate the distance to the clusters. Of the globular clusters within the Milky Way, the majority are found in a halo around the core
58.
Ophiuchus
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Ophiuchus /ɒfiˈjuːkəs/ is a large constellation located around the celestial equator. Its name is from the Greek Ὀφιοῦχος Ophioukhos, serpent-bearer, Ophiuchus was one of the 48 constellations listed by the 2nd-century astronomer Ptolemy, and it remains one of the 88 modern constellations. It was formerly referred to as Serpentarius /sɜːrpənˈtɛəriəs/ and Anguitenens, Ophiuchus is located between Aquila, Serpens and Hercules, northwest of the center of the Milky Way. The southern part lies between Scorpius to the west and Sagittarius to the east, in the northern hemisphere, it is best visible in summer. It is located opposite Orion in the sky, Ophiuchus straddles the equator but lies predominately to its south. However, Rasalhague, a conspicuous star in its north, is circumpolar north of 78° north latitude. The constellation extends southward to −30° declination, segments of the ecliptic that lie within Ophiuchus lie south of −20° declination. A determination of exactly where these stars are visible on Earth would depend on atmospheric refraction, in contrast to Orion, it is in the period November–January when Ophiuchus is in the daytime sky and thus not visible at most latitudes. However, for much of the Arctic Circle in the Northern Hemispheres winter months, stars are then visible at twilight for a few hours around local noon, low in the South. In countries close to the equator Ophiuchus appears overhead in June around midnight, the brightest stars in Ophiuchus include α Ophiuchi, called Rasalhague, at magnitude 2.07, and η Ophiuchi, known as Sabik, at magnitude 2.43. Other bright stars in the constellation include β Ophiuchi, Cebalrai and λ Ophiuchi, RS Ophiuchi is part of a class called recurrent novae, whose brightness increase at irregular intervals by hundreds of times in a period of just a few days. It is thought to be at the brink of becoming a type-1a supernova, barnards Star, one of the nearest stars to the Solar System, lies in Ophiuchus. It is located to the left of β and just north of the V-shaped group of stars in an area that was occupied by the now-obsolete constellation of Taurus Poniatovii. In 2005, astronomers using data from the Green Bank Telescope discovered a superbubble so large that it extends beyond the plane of the galaxy and it is called the Ophiuchus Superbubble. In April 2007, astronomers announced that the Swedish-built Odin satellite had made the first detection of clouds of molecular oxygen in space, the supernova of 1604 was first observed on 9 October 1604, near θ Ophiuchi. Johannes Kepler saw it first on 16 October and studied it so extensively that the supernova was subsequently called Keplers Supernova and he published his findings in a book titled De stella nova in pede Serpentarii. Galileo used its brief appearance to counter the Aristotelian dogma that the heavens are changeless. In 2009 it was announced that GJ1214, a star in Ophiuchus, undergoes repeated, the planets low density suggests that the planet may have a substantial component of low-density gas—possibly hydrogen or steam
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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
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Irregular galaxy
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An irregular galaxy is a galaxy that does not have a distinct regular shape, unlike a spiral or an elliptical galaxy. Irregular galaxies do not fall into any of the classes of the Hubble sequence. Collectively they are thought to make up about a quarter of all galaxies, some irregular galaxies were once spiral or elliptical galaxies but were deformed by an uneven external gravitational force. Irregular galaxies may contain abundant amounts of gas and dust and this is not necessarily true for dwarf irregulars. There are three types of irregular galaxies, An Irr-I galaxy is an irregular galaxy that features some structure. Subtypes with some spiral structure are called Sm galaxies Subtypes without spiral structure are called Im galaxies, an Irr-II galaxy is an irregular galaxy that does not appear to feature any structure that can place it into the Hubble sequence. They may represent a version of the faint blue galaxies known to exist in deep field galaxy surveys. Some of the galaxies, especially of the Magellanic type, are small spiral galaxies that are being distorted by the gravity of a larger neighbor. The Magellanic Cloud galaxies were once classified as irregular galaxies, the Large Magellanic Cloud has since been re-classified as type SBm a type of barred spiral galaxy, the barred Magellanic spiral type. The Small Magellanic Cloud remains classified as a galaxy of type Im under current Galaxy morphological classification. Dwarf galaxy Dwarf elliptical galaxy Peculiar galaxy Galaxy morphological classification Irregular galaxy at Encyclopædia Britannica
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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
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Pegasus (constellation)
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Pegasus is a constellation in the northern sky, named after the winged horse Pegasus in Greek mythology. It was one of the 48 constellations listed by the 2nd-century astronomer Ptolemy, with an apparent magnitude varying between 2.37 and 2.45, the brightest star in Pegasus is the orange supergiant Epsilon Pegasi, also known as Enif, which marks the horses muzzle. Alpha, Beta, and Gamma, together with Alpha Andromedae form the asterism known as the Square of Pegasus. Twelve star systems have found to have exoplanets. 51 Pegasi was the first Sun-like star discovered to have an exoplanet companion, the Babylonian constellation IKU had four stars of which three were later part of the Greek constellation Hippos. Pegasus, in Greek mythology, was a horse with magical powers. One myth regarding his powers says that his hooves dug out a spring, Hippocrene, Pegasus was the one who delivered Medusas head to Polydectes, after which he travelled to Mount Olympus in order to be the bearer of thunder and lightning for Zeus. Eventually, he became the horse to Bellerophon, who was asked to kill the Chimera and succeeded with the help of Athena, despite this success, after the death of his children, Bellerophon asked Pegasus to take him to Mount Olympus. Though Pegasus agreed, he plummeted back to Earth after Zeus either threw a thunderbolt at him or made Pegasus buck him off. In ancient Persia, Pegasus was depicted by al-Sufi as a horse facing east, unlike most other uranographers. In al-Sufis depiction, Pegasuss head is made up of the stars of Lacerta the lizard and its right foreleg is represented by β Peg and its left foreleg is represented by η Peg, μ Peg, and λ Peg, its hind legs are marked by 9 Peg. The back is represented by π Peg and μ Cyg, in Chinese astronomy, the modern constellation of Pegasus lies in The Black Tortoise of the north, where the stars were classified in several separate asterisms of stars. Epsilon and Theta Pegasi are joined with Alpha Aquarii to form Wei 危 rooftop, in Hindu astronomy, the Great Square of Pegasus contained the 26th and 27th lunar mansions. More specifically, it represented a bedstead that was a place for the Moon. Covering 1121 square degrees, Pegasus is the seventh-largest of the 88 constellations, the three-letter abbreviation for the constellation, as adopted by the IAU in 1922, is Peg. The official constellation boundaries, as set by Eugène Delporte in 1930, are defined as a polygon of 35 segments. In the equatorial system the right ascension coordinates of these borders lie between 21h 12. 6m and 00h 14. 6m, while the declination coordinates are between 2. 33° and 36. 61°. Its position in the Northern Celestial Hemisphere means that the constellation is visible to observers north of 53°S
63.
Gospel of Matthew
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The Gospel According to Matthew is the first book of the New Testament. The narrative tells how the Messiah, Jesus, rejected by Israel, most scholars believe the Gospel of Matthew was composed between AD80 and 90, with a range of possibility between AD70 to 110. The title Son of David identifies Jesus as the healing and miracle-working Messiah of Israel, as Son of Man he will return to judge the world, an expectation which his disciples recognise but of which his enemies are unaware. As Son of God he is God revealing himself through his son, the gospel reflects the struggles and conflicts between the evangelists community and the other Jews, particularly with its sharp criticism of the scribes and Pharisees. The original versions of the Gospel of Matthew and the gospels are lost. The oldest relatively complete extant manuscripts of the Bible are the Codex Vaticanus and the Codex Sinaiticus, besides these, there exist manuscript fragments ranging from a few verses to whole chapters. P104 and P67 are notable fragments of Matthew, in the process of recopying, variations slipped in, different regional manuscript traditions emerged, and corrections and adjustments were made. The Gospel of Matthew is anonymous, the author is not named within the text, the consensus is that Papias does not describe the Gospel of Matthew as we know it, and it is generally accepted that Matthew was written in Greek, not in Aramaic or Hebrew. The majority view of scholars is that Mark was the first gospel to be composed. The author of Matthew did not, however, simply copy Mark, an additional 220 verses, shared by Matthew and Luke but not found in Mark, from a second source, a hypothetical collection of sayings to which scholars give the name Quelle, or the Q source. The author also had at his disposal the Greek scriptures, both as book-scrolls and in the form of collections, and, if Papias is correct. The majority view among scholars is that Matthew was a product of the last quarter of the 1st century, the Christian community to which Matthew belonged, like many 1st-century Christians, were still part of the larger Jewish community, hence the designation Jewish Christian to describe them. The author of Matthew wrote for a community of Greek-speaking Jewish Christians located probably in Syria, Matthew, alone among the gospels, alternates five blocks of narrative with five of discourse, marking each off with the phrase When Jesus had finished. The Gospel of Matthew begins with the words The Book of Genealogy of Jesus Christ, John baptizes Jesus, and the Holy Spirit descends upon him. Jesus prays and meditates in the wilderness for forty days, and is tempted by Satan and his early ministry by word and deed in Galilee meets with much success, and leads to the Sermon on the Mount, the first of the discourses. The sermon presents the ethics of the kingdom of God, introduced by the Beatitudes and it concludes with a reminder that the response to the kingdom will have eternal consequences, and the crowds amazed response leads into the next narrative block. From the authoritative words of Jesus the gospel turns to three sets of three miracles interwoven with two sets of two stories, followed by a discourse on mission and suffering. Opposition to Jesus comes to a head with accusations that his deeds are done through the power of Satan, Jesus in turn accuses his opponents of blaspheming the Holy Spirit
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Abraham
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Abraham, originally Abram, is the first of the three patriarchs of Judaism. His story features in the texts of all the Abrahamic religions and Abraham plays a prominent role as an example of faith in Judaism, Christianity. The biblical narrative revolves around the themes of posterity and land, Abraham is called by God to leave the house of his father Terah and settle in the land originally given to Canaan, but which God now promises to Abraham and his progeny. Various candidates are put forward who might inherit the land after Abraham, Abraham later marries Keturah and has six more sons, but on his death, when he is buried beside Sarah, it is Isaac who receives all Abrahams goods, while the other sons receive only gifts. Terah, the ninth in descent from Noah, was the father of three sons, Abram, Nahor, and Haran, Haran was the father of Lot, and died in his native city, Ur of the Chaldees. Abram married Sarah, who was barren, Terah, with Abram, Sarai, and Lot, then departed for Canaan, but settled in a place named Haran, where Terah died at the age of 205. Abram was 75 years old when he left Haran with his wife Sarai, his nephew Lot, and the substance and souls that they had acquired, and traveled to Shechem in Canaan. There was a famine in the land of Canaan, so that Abram and Lot and their households. On the way Abram told his wife Sarai to say that she was his sister, however, God afflicted Pharaoh and his household with great plagues, for which he tried to find the reason. Upon discovering that Sarai was a woman, Pharaoh demanded that they and their household leave immediately. When they came back to the Bethel and Hai area, Abrams and this became a problem for the herdsmen who were assigned to each familys cattle. But Lot chose to go east to the plain of Jordan where the land was well watered everywhere as far as Zoar, Abram went south to Hebron and settled in the plain of Mamre, where he built another altar to worship God. During the rebellion of the Jordan River cities against Elam, Abrams nephew, the Elamite army came to collect the spoils of war, after having just defeated the king of Sodoms armies. Lot and his family, at the time, were settled on the outskirts of the Kingdom of Sodom which made them a visible target, one person who escaped capture came and told Abram what happened. Once Abram received this news, he immediately assembled 318 trained servants, Abrams force headed north in pursuit of the Elamite army, who were already worn down from the Battle of Siddim. When they caught up with them at Dan, Abram devised a plan by splitting his group into more than one unit. Not only were able to free the captives, Abrams unit chased and slaughtered the Elamite King Chedorlaomer at Hobah. They freed Lot, as well as his household and possessions, upon Abrams return, Sodoms king came out to meet with him in the Valley of Shaveh, the kings dale
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David
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David was, according to the Hebrew Bible, the second king of the United Kingdom of Israel and Judah, reigning in c. He is described as a man after Gods own heart in 1 Samuel 13,14 and Acts 13,22. The Hebrew prophets regarded him as the ancestor of the future messiah, the New Testament says he was an ancestor of Jesus. God is angered when Saul, Israels king, unlawfully offers a sacrifice and later disobeys a divine instruction to not only all of the Amalekites. Consequently, he sends the prophet Samuel to anoint David, the youngest son of Jesse of Bethlehem, God sends an evil spirit to torment Saul. Sauls courtiers recommend that he send for David, a man skillful on the lyre, wise in speech, and brave in battle. So David enters Sauls service as one of the royal armour-bearers, and plays the lyre to soothe the king, war comes between Israel and the Philistines, and the giant Goliath challenges the Israelites to send out a champion to face him in single combat. David, sent by his father to bring provisions to his brothers serving in Sauls army, refusing the kings offer of the royal armour, he kills Goliath with his sling. Saul inquires the name of the heros father. Saul sets David over his army, all Israel loves David, but his popularity causes Saul to fear him. Saul plots his death, but Sauls son Jonathan, one of those who loves David, warns him of his fathers schemes and David flees. He becomes a vassal of the Philistine king Achish of Gath, but Achishs nobles question his loyalty, Jonathan and Saul are killed, and David is anointed king over Judah. In the north, Sauls son Ish-Bosheth is anointed king of Israel, with the death of Sauls son, the elders of Israel come to Hebron and David is anointed king over all Israel. He conquers Jerusalem, previously a Jebusite stronghold, and makes it his capital. He brings the Ark of the Covenant to the city, intending to build a temple for God, Nathan also prophesies that God has made a covenant with the house of David, Your throne shall be established forever. David wins more victories over the Philistines, while the Moabites, Edomites, Amalekites, Ammonites, during a battle to conquer the Ammonite capital of Rabbah, David seduces Bathsheba and causes the death of her husband Uriah the Hittite. In response, Nathan prophesies the punishment that shall fall upon him, in fulfillment of these words Davids son Absalom rebels. The rebellion ends at the battle of the Wood of Ephraim, Absaloms forces are routed, and Absalom is caught by his long hair in the branches of a tree, and killed by Joab, contrary to Davids order. Joab was the commander of Davids army, David laments the death of his favourite son, O my son Absalom, my son, my son Absalom
66.
Babylonian captivity
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The Babylonian captivity or Babylonian exile is the period in Jewish history during which a number of Judahites of the ancient Kingdom of Judah were captives in Babylonia. After the Battle of Carchemish in 605 BCE, Nebuchadnezzar, the king of Babylon, besieged Jerusalem, the dates, numbers of deportations, and numbers of deportees given in the biblical accounts vary. These deportations are dated to 597 BCE for the first, with others dated at 587/586 BCE, after the fall of Babylon to the Persian king Cyrus the Great in 539 BCE, exiled Judeans were permitted to return to Judah. According to the book of Ezra, construction of the second temple in Jerusalem began around 537 BCE. All these events are considered significant in Jewish history and culture, the return of the exiles was a gradual process rather than a single event, and many of the deportees or their descendants did not return. In the late 7th century BCE, the kingdom of Judah was a client state of the Assyrian empire, in the last decades of the century, Assyria was overthrown by Babylon, an Assyrian province. Egypt, fearing the rise of the Neo-Babylonian empire, seized control of Assyrian territory up to the Euphrates river in Syria. In the process Josiah, the king of Judah, was killed in a battle with the Egyptians at the Battle of Megiddo, after the defeat of Pharaoh Nechos army by the Babylonians at Carchemish in 605 BCE, Jehoiakim began paying tribute to Nebuchadnezzar II of Babylon. Some of the nobility of Judah were taken to Babylon. In the following years, the court of Jerusalem was divided into two parties, in support of Egypt and Babylon, after Nebuchadnezzar was defeated in battle in 601 BCE by Egypt, Judah revolted against Babylon, culminating in a three-month siege of Jerusalem beginning in late 598 BCE. Jehoiakim, the king of Judah, died during the siege and was succeeded by his son Jehoiachin at the age of eighteen. The city fell on 2 Adar 597 BCE, and Nebuchadnezzar pillaged Jerusalem and its Temple and took Jeconiah, his court, jehoiakims uncle Zedekiah was appointed king in his place, but the exiles in Babylon continued to consider Jeconiah as their Exilarch, or rightful ruler. Despite warnings by Jeremiah and others of the party, Zedekiah revolted against Babylon. Nebuchadnezzar returned, defeated the Egyptians, and again besieged Jerusalem, Nebuchadnezzar destroyed the city wall and the Temple, together with the houses of the most important citizens. Zedekiah and his sons were captured, the sons were executed in front of Zedekiah, Judah became a Babylonian province, called Yehud, putting an end to the independent Kingdom of Judah. Some time later, a member of the royal family assassinated Gedaliah and his Babylonian advisors. According to the book of Ezra, the Persian Cyrus the Great ended the exile in 538 BCE, the exile ended with the return under Zerubbabel the Prince and Joshua the Priest and their construction of the Second Temple in the period 521–516 BCE. He appointed there a king of his own choice and taking heavy tribute brought it back to Babylon, one of the tablets refers to food rations for Ya’u-kīnu, king of the land of Yahudu and five royal princes, his sons
67.
Stations of the Cross
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Stations of the Cross or the Way of the Cross, also known as Way of Sorrows or Via Crucis, refers to a series of images depicting Jesus Christ on the day of his crucifixion and accompanying prayers. The stations grew out of imitations of Via Dolorosa in Jerusalem which is believed to be the actual path Jesus walked to Mount Calvary, the object of the stations is to help the Christian faithful to make a spiritual pilgrimage through contemplation of the Passion of Christ. This will be individually or in a procession most commonly during Lent, especially on Good Friday, in a spirit of reparation for the sufferings. The style, form, and placement of the stations vary widely, the typical stations are small plaques with reliefs or paintings placed around a church nave. Modern minimalist stations can be simple crosses with a numeral in the centre, occasionally the faithful might say the stations of the cross without there being any image, such as when the pope leads the stations of the cross around the Colosseum in Rome on Good Friday. The older stations can be a series of chapels in a landscape, known as a Calvary. Examples include Sacred Mount Calvary in Italy, Kalwaria Zebrzydowska in Poland, Via Crucis of Valinhos, Fátima, in Portugal, the Stations of the Cross originated in pilgrimages to Jerusalem and a desire to reproduce Via Dolorosa. Imitating holy places was not a new concept, for example, the religious complex of Santo Stefano in Bologna, Italy, replicated the Church of the Holy Sepulchre and other religious sites, including Mount of Olives and Valley of Josaphat. After the siege of 1187, Jerusalem fell to Muslims, forty years later Franciscans were allowed back into the Holy Land. Their founder, Saint Francis of Assisi, held the Passion of Christ in special veneration and was the first person to receive stigmata, in 1217, St. Francis also founded the Custody of the Holy Land to guard and promote the devotion to holy places. Their efforts were recognized when Franciscans were officially proclaimed custodians of holy places by Pope Clement VI in 1342, in 1521, a book called Geystlich Strass was printed with illustrations of the stations in the Holy Land. During the 15th and 16th centuries the Franciscans began to build a series of shrines in Europe to duplicate their counterparts in the Holy Land. The number of stations varied between seven and thirty, seven was common and these were usually placed, often in small buildings, along the approach to a church, as in a set of 1490 by Adam Kraft, leading to the Johanneskirche in Nuremberg. A number of examples were established as attractions in their own right. In these the sculptures are often approaching life-size and very elaborate, remnants of these are often referred to as calvary hills. In 1686, in answer to their petition, Pope Innocent XI granted to the Franciscans the right to erect stations within their churches. In 1731, Pope Clement XII extended to all churches the right to have the stations, provided that a Franciscan father erected them, at the same time the number was fixed at fourteen. The early set of seven scenes was usually numbers 2,3,4,6,7,11, out of the fourteen traditional Stations of the Cross, only eight have clear scriptural foundation
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Fourteen Holy Helpers
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The Fourteen Holy Helpers are a group of saints venerated together in Roman Catholicism because their intercession is believed to be particularly effective, especially against various diseases. This group of Nothelfer originated in the 14th century at first in the Rhineland, devotion to the fourteen Auxiliary Saints began in Rhineland, now part of Germany, in the time of the Black Death. At the heart of the fourteen were three virgin martyrs, As the other began to be invoked along with these three virgin martyrs, they were represented together in works of art. Popular veneration of these saints often began in a monastery that held their relics, all of the saints except Giles were accounted martyrs. Saint Christopher and Saint Giles were invoked against the plague itself, Saint Denis was prayed to for relief from headache, Saint Blaise for ills of the throat, Saint Elmo, for abdominal maladies, Saint Barbara for fever, and Saint Vitus against epilepsy. Saint Giles was prayed to for a confession, and Saint Eustace as healer of family troubles. Domestic animals were also attacked by the plague, and so Saints George, Elmo, Pantaleon, Saint Margaret of Antioch is the patron of safe childbirth. As the saints joint cultus spread in the century, Pope Nicholas V attached indulgences to devotion of the Fourteen Holy Helpers. The individual celebrations of all fourteen are included in the General Roman Calendar as in 1954, the General Roman Calendar of Pope Pius XII and the General Roman Calendar of 1960. Comparable to the cult of the Fourteen Holy Helpers was that of the Four Holy Marshals and these were Quirinus of Neuss, Saint Anthony the Great, Pope Cornelius, and Saint Hubert. The fourteen saints are, Half the saints are regarded as historical figures while the other may only be legends, in France an extra helper is added, the Virgin Mary. The Fourteen Holy Helpers are honored in Bavaria as the vierzehn Heiligen, the Rococo pilgrimage church in the town of Bad Staffelstein was designed by Balthasar Neumann and built between 1743 and 1772. As he bent down to pick up the child, it abruptly disappeared, a short time later, the child reappeared in the same spot. This time, two candles were burning next to it, in June 1446, Leicht saw the child a third time. This time, the bore a red cross on its chest and was accompanied by thirteen other children. The child said, We are the fourteen helpers and wish to erect a chapel here, if you will be our servant, we will be yours. Shortly after, Leicht saw two burning candles descending to this spot and it is alleged that miraculous healings soon began, through the intervention of the fourteen saints. The Cistercian brothers to whom the land belonged erected a chapel, an altar was consecrated as early as 1448
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Saint
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A saint, also historically known as a hallow, is a term used for a person who is recognized as having an exceptional degree of holiness or likeness to God. Depending on the context and denomination, the term also retains its original Christian meaning, as any believer who is in Christ and in whom Christ dwells, whether in Heaven or on Earth. Depending on the religion, saints are recognized either by official ecclesiastical declaration, the English word saint comes from the Latin sanctus. The word translated the Greek ἅγιος, which derives from the verb ἁγιάζω, the word ἅγιος appears 229 times in the Greek New Testament, and its English translation 60 times in the corresponding text of the King James Version of the Bible. In the New Testament, saint did not denote the deceased who had recognized as especially holy or emulable. Many religions also use similar concepts to venerate persons worthy of some honor, the anthropologist Lawrence Babb in an article about Sathya Sai Baba asks the question Who is a saint. These saintly figures, he asserts, are the points of spiritual force-fields. They exert powerful attractive influence on followers but touch the lives of others in transforming ways as well. In the Bible, only one person is called a saint, They envied Moses also in the camp. The apostle Paul declared himself to be less than the least of all saints in Ephesians 3,8, in the Catholic Church, a saint is anyone in Heaven, whether recognized on Earth or not. There are many persons that the Church believes to be in Heaven who have not been formally canonized, sometimes the word saint also denotes living Christians. They remind us that the Church is holy, can never stop being holy and is called to show the holiness of God by living the life of Christ, the Catholic Church teaches that it does not make or create saints, but rather recognizes them. Proofs of heroicity required in the process of beatification will serve to illustrate in detail the general principles exposed above upon proof of their holiness or likeness to God. On 3 January 993, Pope John XV became the first pope to proclaim a person a saint, on the petition of the German ruler, before that time, the popular cults, or venerations, of saints had been local and spontaneous. Pope John XVIII subsequently permitted a cult of five Polish martyrs, walter of Pontoise was the last person in Western Europe to be canonized by an authority other than the Pope, Hugh de Boves, the Archbishop of Rouen, canonized him in 1153. Thenceforth a decree of Pope Alexander III in 1170 reserved the prerogative of canonization to the Pope, one source claims that there are over 10,000 named saints and beatified people from history, the Roman Martyrology and Orthodox sources, but no definitive head count. Alban Butler published Lives of the Saints in 1756, including a total of 1,486 saints, the latest revision of this book, edited by Rev. Herbert Thurston, SJ and British author Donald Attwater, contains the lives of 2,565 saints. Monsignor Robert Sarno, an official of the Congregation for the Causes of Saints of the Holy See, expressed that it is impossible to give an exact number of saints