1.
900 AD
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Year 900 was a leap year starting on Tuesday of the Julian calendar. Spring – Forces under the Transoxianian emir Ismail ibn Ahmad are victorious at Balkh over Amr ibn al-Layth, the Samanid Dynasty rules over Khorasan, as well as Transoxiana. A few months later, the Samanids conquer the Zaydid emirate of Tabaristan and this victory marks the beginning of the dispersion of the local Shiites, by the new Sunni power. Arab–Byzantine War, Emperor Leo VI begins an offensive against the Abbasid army in Cilicia, Mesopotamia and Armenia and he also continues the war against the Muslims in Sicily and southern Italy. The Fatmids break away from the Abbasid Caliphate and migrate to North Africa and they claim to be descendants from Fatima bint Muhammad, the daughter of the Islamic prophet Muhammad. The Qarmatians of Al-Bahrayn, under Abū-Saʿīd Jannābī, score a victory over the Abbasid army led by Al-Abbas ibn Amr al-Ghanawi. Spring – Atenulf I, Lombard prince of Capua, conquers the Duchy of Benevento and he deposes Duke Radelchis II and unites the two southern Lombard duchies in Mezzogiorno. The Byzantines offer an alliance to Atenulf who directs an campaign against the Saracens. They have establish themselves on the banks of the Garigliano River, february 4 – The 7-year-old Louis IV is at an assembly at Forchheim proclaimed king of the East Frankish Kingdom. Because of his age, the reins of government is entirely in the hands of others – the Frankish nobles. The most influential of Louiss councillors are Hatto I, archbishop of Mainz, june 8 – Edward the Elder is crowned king of England at Kingston upon Thames. June 17 – Baldwin II, Count of Flanders has Fulk the Venerable, bishop of Reims, june 29 – The Venetians repel the Magyar raiders at Rialto. Summer – After the death of his wife Zoe Zaoutzaina, the Byzantine emperor Leo VI marries Eudokia Baïana. August – Abdallah, son of the Aghlabid emir Ibrahim II, represses a revolt of his Muslim subjects, and then initiates a campaign against the last Byzantine strongholds in Sicily. August 13 – Zwentibold, king of Lotharingia, is killed in battle on the Meuse River, while fighting against his rebellious subjects, october 12 – Following Magyars raids in Lombardy, king Louis III is called to Italy by the grandees. He takes Pavia, forcing king Berengar I to flee, King Donald II is killed after a 11-year reign. He is succeeded by his cousin Constantine II as king of Scotland, docibilis I of Gaeta and his Saracen mercenaries attack Capua, in vain. After the rejection of their proposal by the Bavarians, the Hungarians attack this country, occupying Pannonia and parts of Ostmark
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
Unicode
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Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the worlds writing systems. As of June 2016, the most recent version is Unicode 9.0, the standard is maintained by the Unicode Consortium. Unicodes success at unifying character sets has led to its widespread, the standard has been implemented in many recent technologies, including modern operating systems, XML, Java, and the. NET Framework. Unicode can be implemented by different character encodings, the most commonly used encodings are UTF-8, UTF-16 and the now-obsolete UCS-2. UTF-8 uses one byte for any ASCII character, all of which have the same values in both UTF-8 and ASCII encoding, and up to four bytes for other characters. UCS-2 uses a 16-bit code unit for each character but cannot encode every character in the current Unicode standard, UTF-16 extends UCS-2, using one 16-bit unit for the characters that were representable in UCS-2 and two 16-bit units to handle each of the additional characters. Many traditional character encodings share a common problem in that they allow bilingual computer processing, Unicode, in intent, encodes the underlying characters—graphemes and grapheme-like units—rather than the variant glyphs for such characters. In the case of Chinese characters, this leads to controversies over distinguishing the underlying character from its variant glyphs. In text processing, Unicode takes the role of providing a unique code point—a number, in other words, Unicode represents a character in an abstract way and leaves the visual rendering to other software, such as a web browser or word processor. This simple aim becomes complicated, however, because of concessions made by Unicodes designers in the hope of encouraging a more rapid adoption of Unicode, the first 256 code points were made identical to the content of ISO-8859-1 so as to make it trivial to convert existing western text. For other examples, see duplicate characters in Unicode and he explained that he name Unicode is intended to suggest a unique, unified, universal encoding. In this document, entitled Unicode 88, Becker outlined a 16-bit character model, Unicode could be roughly described as wide-body ASCII that has been stretched to 16 bits to encompass the characters of all the worlds living languages. In a properly engineered design,16 bits per character are more than sufficient for this purpose, Unicode aims in the first instance at the characters published in modern text, whose number is undoubtedly far below 214 =16,384. By the end of 1990, most of the work on mapping existing character encoding standards had been completed, the Unicode Consortium was incorporated in California on January 3,1991, and in October 1991, the first volume of the Unicode standard was published. The second volume, covering Han ideographs, was published in June 1992, in 1996, a surrogate character mechanism was implemented in Unicode 2.0, so that Unicode was no longer restricted to 16 bits. The Microsoft TrueType specification version 1.0 from 1992 used the name Apple Unicode instead of Unicode for the Platform ID in the naming table, Unicode defines a codespace of 1,114,112 code points in the range 0hex to 10FFFFhex. Normally a Unicode code point is referred to by writing U+ followed by its hexadecimal number, for code points in the Basic Multilingual Plane, four digits are used, for code points outside the BMP, five or six digits are used, as required. Code points in Planes 1 through 16 are accessed as surrogate pairs in UTF-16, within each plane, characters are allocated within named blocks of related characters
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
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
19.
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
20.
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
21.
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
22.
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
23.
North American Numbering Plan
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The North American Numbering Plan is a telephone numbering plan that encompasses 25 distinct regions in twenty countries primarily in North America, including the Caribbean and the U. S. territories. Not all North American countries participate in the NANP, each participating country forms a regulatory authority that has plenary control over local numbering resources. The FCC also serves as the U. S. regulator, Canadian numbering decisions are made by the Canadian Numbering Administration Consortium. The NANP divides the territories of its members into numbering plan areas which are encoded numerically with a telephone number prefix. Each telephone is assigned a telephone number unique only within its respective plan area. The telephone number consists of a central office code and a four-digit station number. The combination of a code and the telephone number serves as a destination routing address in the public switched telephone network. For international call routing, the NANP has been assigned the calling code 1 by the International Telecommunications Union. The North American Numbering Plan conforms with ITU Recommendation E.164, from its beginnings in 1876 and throughout the first part of the 20th century, the Bell System grew from essentially local or regional telephone systems. These systems expanded by growing their subscriber bases, as well as increasing their service areas by implementing additional local exchanges that were interconnected with tie trunks and it was the responsibility of each local administration to design telephone numbering plans that accommodated the local requirements and growth. As a result, the Bell System as a developed into an unorganized system of many differing local numbering systems. The diversity impeded the efficient operation and interconnection of exchanges into a system for long-distance telephone communication. The new numbering plan was accepted in October 1947, dividing most of North America into 86 Numbering Plan Areas. Each NPA was assigned a Numbering Plan Area code, often abbreviated as area code and these codes were first used by long-distance operators to establish long-distance calls between toll offices. The first customer-dialed direct call using area codes was made on November 10,1951, from Englewood, New Jersey, to Alameda, California. Direct distance dialing was introduced across the country and by the early 1960s most areas of the Bell System had been converted and it was commonplace in cities. In the following decades, the system expanded to all of the United States and its territories, Canada, Bermuda. By 1967,129 area codes had been assigned, mexican participation was planned, but implementation stopped after two area codes had been assigned and Mexico opted for an international numbering format, using country code 52
24.
Sampi
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Sampi is an archaic letter of the Greek alphabet. It later remained in use as a symbol for 900 in the alphabetic system of Greek numerals. Its modern shape, which resembles a π inclining to the right with a longish curved cross-stroke and its current name, sampi, originally probably meant san pi, i. e. like a pi, and is also of medieval origin. The letters original name in antiquity is not known and it has been proposed that sampi was a continuation of the archaic letter san, which was originally shaped like an M and denoted the sound in some other dialects. Besides san, names that have proposed for sampi include parakyisma and angma, while other historically attested terms for it are enacosis, sincope. It has been attested in the cities of Miletus, Ephesos, Halikarnassos, Erythrae, Teos, in the island of Samos, in the Ionian colony of Massilia, and in Kyzikos. In addition, in the city of Pontic Mesembria, on the Black Sea coast of Thrace, it was used on coins, Sampi occurs in positions where other dialects, including written Ionic, normally have double sigma, i. e. a long /ss/ sound. Some other dialects, particularly Attic Greek, have ττ in the same words, the sounds in question are all reflexes of the proto-Greek consonant clusters *, *, *, *, or *. Among the earliest known uses of sampi in this function is an abecedarium from Samos dated to the mid-7th century BC and this early attestation already bears witness to its alphabetic position behind omega, and it shows that its invention cannot have been much later than that of omega itself. The first known use of sampi in writing native Greek words is an inscription found on a silver plate in Ephesus. It can be dated between the late 7th century and mid 6th century BC, an inscription from Halicarnassus has the names Ἁλικαρναͳέν and the personal names Ὀαͳαͳιος and Πνυάͳιος. All of these appear to be of non-Greek, local origin. On a late 6th century bronze plate from Miletus dedicated to the sanctuary of Athena at Assesos and this is currently the first known instance of alphabetic sampi in Miletus itself, commonly assumed to be the birthplace of the numeral system and thus of the later numeric use of sampi. It has been suggested there may be an isolated example of the use of alphabetic sampi in Athens. In a famous painted black figure amphora from c.615 BC, known as the Nessos amphora, the expected regular form of the name would have been either Attic Νέττος – with a double τ – or Ionic Νέσσος. Traces of corrections that are still visible underneath the painted Τ have led to the conjecture that the painter originally wrote Νέͳος, a letter similar to Ionian sampi, but of unknown historical relation with it, existed in the highly deviant local dialect of Pamphylia in southern Asia Minor. According to Brixhe it probably stood for the sounds /s/, /ss/ and it is found in a few inscriptions in the cities of Aspendos and Perge as well as on local coins. For instance, an inscription from Perge dated to around 400 BC reads, anax, it is believed that the letter stood for some type of sibilant reflecting Proto-Greek */ktj/
25.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
26.
Sphenic number
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In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers. A sphenic number is a product pqr where p, q and this definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance,60 =22 ×3 ×5 has exactly 3 prime factors, the smallest sphenic number is 30 =2 ×3 ×5, the product of the smallest three primes. The first few numbers are 30,42,66,70,78,102,105,110,114,130,138,154,165. As of January 2016 the largest known number is × ×. It is the product of the three largest known primes, all sphenic numbers have exactly eight divisors. If we express the number as n = p ⋅ q ⋅ r, where p, q. For example,24 is not a number, but it has exactly eight divisors. All sphenic numbers are by definition squarefree, because the factors must be distinct. The Möbius function of any number is −1. The cyclotomic polynomials Φ n, taken over all sphenic numbers n, the first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17,1310 = 2×5×131, and 1311 = 3×19×23, there is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. The numbers 2013,2014, and 2015 are all sphenic, the next three consecutive sphenic years will be 2665,2666 and 2667. Semiprimes, products of two prime numbers
27.
Mertens function
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In number theory, the Mertens function is defined for all positive integers n as M = ∑ k =1 n μ where μ is the Möbius function. The function is named in honour of Franz Mertens and this definition can be extended to positive real numbers as follows, M = M. Less formally, M is the count of square-free integers up to x that have a number of prime factors. Because the Möbius function only takes the values −1,0, and +1, the Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture was proven false in 1985 by Andrew Odlyzko, however, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M, namely M = O. Since high values for M grow at least as fast as the root of x. Here, O refers to Big O notation, the true rate of growth of M is not known. An unpublished conjecture of Steve Gonek states that 0 < lim sup x → ∞ | M | x 5 /4 < ∞, probabilistic evidence towards this conjecture is given by Nathan Ng. Using the Euler product one finds that 1 ζ = ∏ p = ∑ n =1 ∞ μ n s where ζ is the Riemann zeta function and the product is taken over primes. Then, using this Dirichlet series with Perrons formula, one obtains,12 π i ∫ c − i ∞ c + i ∞ x s s ζ d s = M where c >1. Conversely, one has the Mellin transform 1 ζ = s ∫1 ∞ M x s +1 d x which holds for R e >1. A curious relation given by Mertens himself involving the second Chebyshev function is ψ = M log + M log + M log + ⋯. Assuming that there are not multiple non-trivial roots of ζ we have the formula by the residue theorem. Weyl conjectured that the Mertens function satisfied the approximate functional-differential equation y 2 − ∑ r =1 N B2 r. Another formula for the Mertens function is M = ∑ a ∈ F n e 2 π i a where F n is the Farey sequence of order n and this formula is used in the proof of the Franel–Landau theorem. M is the determinant of the n × n Redheffer matrix, using sieve methods similar to those used in prime counting, the Mertens function has been computed for all integers up to an increasing range of x. The Mertens function for all values up to x may be computed in O time. Combinatorial based algorithms can compute isolated values of M in O time, see A084237 for values of M at powers of 10
28.
Greater Toronto Area
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The Greater Toronto Area is the most populous metropolitan area in Canada. As of the 2016 census, it has a population of 6,417,516, the Greater Toronto Area is defined as the central city of Toronto, and the four regional municipalities that surround it, Durham, Halton, Peel, and York. The regional span of the Greater Toronto Area is sometimes combined with the city of Hamilton, Ontario and its region, to form the Greater Toronto. The Greater Toronto Area anchors a much larger urban agglomeration known as the Golden Horseshoe. However, it did not come into everyday usage until the mid- to late 1990s, the latter includes communities like Barrie, Guelph and the Niagara Region. The GTA continues, however, to be in use elsewhere in the Government of Ontario. For example, Oshawa, which is the centre of its own CMA, other municipalities, such as New Tecumseth in southern Simcoe County and Mono Township in Dufferin County are included in the Toronto CMA but not in the GTA. Other nearby urban areas, such as Hamilton, Barrie or St. Catharines-Niagara and Kitchener-Waterloo, are not part of the GTA or the Toronto CMA, but form their own CMAs near the GTA. Ultimately, all the places are part of the Golden Horseshoe metropolitan region, an urban agglomeration. When the Hamilton, Oshawa and Toronto CMAs are agglomerated with Brock and Scugog and it is part of the Great Lakes Megalopolis, containing an estimated 59 million people in 2011. The term Greater Toronto and Hamilton Area refers to the usual GTA plus the former Regional Municipality of Hamilton–Wentworth, the Greater Toronto Area was home to a number of First Nations groups who lived on the shore of Lake Ontario long before the first Europeans arrived in the region. At various times the Neutral, Seneca, Mohawk and Huron nations were living in the vicinity, the Mississaugas arrived in the late seventeenth or early eighteenth century, driving out the occupying Iroquois. While it is unclear as to who was the first European to reach the Toronto area, the area would later become very crucial for its series of trails and water routes that led from northern and western Canada to the Gulf of Mexico. Known as the Toronto Passage, it followed the Humber River, for this reason area became a hot spot for French fur traders. In 1787, the British negotiated the purchase of more than a million acres of land in the area of Toronto with the Mississaugas of New Credit. The Town of York would later be attacked by American forces in the War of 1812 in what is now known as the Battle of York, in 1816, Wentworth County and Halton County were created from York County. In 1851, Ontario County and Peel County were separated from York, the idea for a single government municipality would not be seriously explored until the late 1940s when planners decided the city needed to incorporate its immediate suburbs. However, due to opposition from suburban politicians, a compromise was struck which resulted in the creation of Metropolitan Toronto
29.
Homer the Great
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Homer the Great is the twelfth episode of The Simpsons sixth season. It originally aired on the Fox network in the United States on January 8,1995, in the episode, Homer joins an ancient secret society known as the Stonecutters. The episode was written by John Swartzwelder and directed by Jim Reardon, Patrick Stewart guest stars as Number One, the leader of the Springfield chapter of the Stonecutters. It features cultural references to Freemasonry and films such as Raiders of the Lost Ark, the song We Do was nominated for a Primetime Emmy Award for Outstanding Music And Lyrics. Homer notices that his colleagues Lenny and Carl are enjoying inexplicable privileges at the Springfield Nuclear Power Plant and he discovers they are part of an ancient secret society known as the Stonecutters. To join, one must either be the son of a Stonecutter or save the life of a Stonecutter, while extolling the Stonecutters at the dinner table, Homer discovers that his father is a member and is admitted. After the initiations, Homer takes great pleasure in the societys secret privileges, however, during a celebratory dinner with his fellow Stonecutters, he unwittingly destroys their Hallowed Sacred Parchment. He is stripped of his Stonecutter robes and is sentenced to walk home naked, before he leaves, however, it is discovered that Homer has a birthmark in the shape of the Stonecutter emblem, identifying him as the Chosen One who would lead the Stonecutters to greatness. Homer is crowned the new leader of the Stonecutters, initially enjoying himself, Homer soon feels isolated by his power when the other members treat him differently due to his new position, and asks Lisa for advice. She suggests that he ask the Stonecutters to do work to help the community. Unfortunately, the other Stonecutters take this the way and form a new society. Homer becomes despondent about losing his secret club, marge consoles him by telling him he is a member of a very selective club, the Simpson family. Although Homer the Great was written by John Swartzwelder, the story was suggested by executive producer David Mirkin, Mirkin did not have enough time to write the episode and asked Swartzwelder to do it. Mirkin came up with the idea while driving home from an early in the morning. Mirkin decided it would make an episode, where everyone in Springfield was a member of a Masonic society and Homer was left on the outside. The song We Do was not included in the script and was suggested by Matt Groening. It was written by the room, who threw in as many things that annoyed them as they possibly could. It was described as one of the series’ best musical numbers by Colin Jacobson at DVD Movie Guide, the episode guest stars Patrick Stewart as Number One
30.
The Simpsons
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The Simpsons is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a depiction of working-class life epitomized by the Simpson family, which consists of Homer, Marge, Bart, Lisa. The show is set in the town of Springfield and parodies American culture, society, television. The family was conceived by Groening shortly before a solicitation for a series of animated shorts with producer James L. Brooks, Groening created a dysfunctional family and named the characters after members of his own family, substituting Bart for his own name. The shorts became a part of The Tracey Ullman Show on April 19,1987. After a three-season run, the sketch was developed into a prime time show and became an early hit for Fox. Since its debut on December 17,1989,615 episodes of The Simpsons have been broadcast and its 28th season began on September 25,2016. It is the longest-running American sitcom and the longest-running American animated program, the Simpsons Movie, a feature-length film, was released in theaters worldwide on July 27,2007, and grossed over $527 million. On May 4,2015, the series was renewed for seasons 27 and 28, on November 4,2016, the series was renewed for seasons 29 and 30, consisting of 22 episodes each. The Simpsons received widespread critical acclaim throughout its first nine or ten seasons, Time named it the 20th centurys best television series, and Erik Adams of The A. V. Club named it televisions crowning achievement regardless of format, on January 14,2000, the Simpson family was awarded a star on the Hollywood Walk of Fame. It has won dozens of awards since it debuted as a series, including 31 Primetime Emmy Awards,30 Annie Awards, Homers exclamatory catchphrase Doh. has been adopted into the English language, while The Simpsons has influenced many adult-oriented animated sitcoms. Despite this, the show has also criticized for what many perceive as a decline in quality over the years. The Simpsons are a family who live in a fictional Middle American town of Springfield, Homer, the father, works as a safety inspector at the Springfield Nuclear Power Plant, a position at odds with his careless, buffoonish personality. He is married to Marge Simpson, a stereotypical American housewife, although the family is dysfunctional, many episodes examine their relationships and bonds with each other and they are often shown to care about one another. The family owns a dog, Santas Little Helper, and a cat, Snowball V, renamed Snowball II in I, both pets have had starring roles in several episodes. The show includes an array of supporting characters, co-workers, teachers, family friends, extended relatives, townspeople. The creators originally intended many of these characters as jokes or for fulfilling needed functions in the town
31.
Obi-Wan Kenobi
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Obi-Wan Ben Kenobi is a fictional character in the Star Wars franchise, portrayed by Alec Guinness and Ewan McGregor in the films. In the original trilogy, he is a mentor to Luke Skywalker, in the prequel trilogy, he is a master and friend to Anakin Skywalker. He is frequently featured as a character in various other Star Wars media. Obi-Wan Kenobi is introduced in the original Star Wars living as the hermit Ben Kenobi on the planet Tatooine, when Luke Skywalker and the droid C-3PO wander off in search of the lost droid R2-D2, Ben rescues them from a band of native Tusken Raiders. At his home, R2-D2 plays Ben a recording of Princess Leia which explains that R2-D2 contains the plans for the Death Star, Leia asks him to deliver the droid and the plans safely to the planet Alderaan in order to help the Rebel Alliance. He gives Luke his fathers lightsaber and asks Luke to accompany him to Alderaan, Luke declines, but promises to take Obi-Wan as far as Anchorhead Station. After Luke finds his uncle and aunt killed by Imperial troops, he agrees to go with Obi-Wan to Alderaan, in the spaceport city Mos Eisley, Obi-Wan uses the Force to trick Imperial troops into letting them through a military checkpoint. They enter a cantina and make a deal with two smugglers, Han Solo and Chewbacca, to fly them to Alderaan in their ship. During the journey, Obi-Wan begins instructing Luke in lightsaber training and he suddenly becomes weak and tells Luke of a great disturbance in the Force. Emerging from hyperspace, the party finds that Alderaan has been destroyed, the trio chase the TIE fighter to the Death Star, and subsequently get caught in the Death Stars tractor beam. On board the Death Star, Obi-Wan shuts down the tractor beam, Obi-Wan uses the duel to distract Vader as Luke, Leia, Han and Chewbacca escape to the Falcon. Although Vader strikes Obi-Wan down, his body mysteriously vanishes the moment he dies, at the climax of the film during the Rebel attack on the Death Star, Obi-Wan speaks to Luke through the Force to help him destroy the Imperial station. In The Empire Strikes Back, Obi-Wan Kenobi appears several times as a spirit through the Force, on the planet Hoth, he appears to instruct Luke to go to the planet Dagobah to find the exiled Jedi Master Yoda. Despite Yodas skepticism, Obi-Wan convinces his old master to continue Lukes training and he appears later to beseech Luke not to leave Dagobah to try to rescue his friends on Cloud City, although Luke ignores this advice. In Return of the Jedi, Obi-Wan again appears to Luke after Yodas death on Dagobah, Obi-Wan acknowledges that Darth Vader is indeed Lukes father, revealed by Vader in the previous film and confirmed by Yoda on his deathbed, and also reveals that Leia is Lukes twin sister. After the second Death Star is destroyed and the Empire defeated, Obi-Wan appears at the celebration in the Ewok village, alongside the spirits of Yoda, in Star Wars, Episode I – The Phantom Menace, Obi-Wan Kenobi appears as the Jedi Padawan of Jedi Master Qui-Gon Jinn. He accompanies his master in negotiations with the Trade Federation, which is blockading the planet Naboo with a fleet of spaceships and their ship is damaged in the escape, however, and they are forced to land on Tatooine, where they discover a young Anakin Skywalker. Qui-Gon senses Anakins extraordinarily strong link to the Force and brings the boy to Coruscant to begin Jedi training, when Qui-Gon and Obi-Wan return to Naboo to defeat the Trade Federation, they are met by Sith Lord Darth Maul
32.
Centered hexagonal number
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The nth centered hexagonal number is given by the formula n 3 −3 =3 n +1. Expressing the formula as 1 +6 shows that the centered hexagonal number for n is 1 more than 6 times the th triangular number. The first few centered hexagonal numbers are,1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919. In base 10 one can notice that the hexagonal numbers rightmost digits follow the pattern 1–7–9–7–1, the sum of the first n centered hexagonal numbers is n3. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. In particular, prime centered hexagonal numbers are cuban primes, the difference between 2 and the nth centered hexagonal number is a number of the form 3n2 + 3n −1, while the difference between 2 and the nth centered hexagonal number is a pronic number. Hexagonal number Magic hexagon Star number
33.
Sterling silver
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Sterling silver is an alloy of silver containing 92.5 by weight of silver and 7.5 by weight of other metals, usually copper. The sterling silver standard has a minimum millesimal fineness of 925 and these replacement metals include germanium, zinc and platinum, as well as a variety of other additives, including silicon and boron. Alloys such as argentium silver have red in recent decades, one of the earliest attestations of the term is in Old French form esterlin, in a charter of the abbey of Les Préaux, dating to either 1085 or 1104. The English chronicler Orderic Vitalis uses the Latin forms libræ sterilensium, the word in origin refers to the newly introduced Norman silver penny. According to the Oxford English Dictionary, the most plausible etymology is derivation from a late Old English steorling, there are a number of obsolete hypotheses. One suggests a connection with starling, because four birds were depicted on a penny of Edward I, in 1260, Henry III granted them a charter of protection. Because the Leagues money was not frequently debased like that of England, English traders stipulated to be paid in pounds of the Easterlings, and land for their Kontor, the Steelyard of London, which by the 1340s was also called Easterlings Hall, or Esterlingeshalle. The Hanseatic League was officially active in the London trade from 1266 to 1597 and this etymology may have been first suggested by Walter de Pinchebek with the explanation that the coin was originally made by moneyers from that region. The claim has also made in Henry Spelmans glossary as referenced in Commentaries on the Laws of England. Yet another claim on this hypothesis is from Camden, as quoted in Chambers Journal of Popular Literature, Science and Arts. By 1854, the tie between Easterling and Sterling was well-established, as Ronald Zupko quotes in his dictionary of weights, the British numismatist Philip Grierson disagrees with the star etymology, as the stars appeared on Norman pennies only for the single three-year issue from 1077–1080. In support of this he cites the fact one of the first acts of the Normans was to restore the coinage to the consistent weight and purity it had in the days of Offa. This would have perceived as a contrast to the progressive debasement of the intervening 200 years. The sterling alloy originated in continental Europe and was being used for commerce as early as the 12th century in the area that is now northern Germany. A piece of sterling silver dating from Henry IIs reign was used as a standard in the Trial of the Pyx until it was deposited at the Royal Mint in 1843 and it bears the royal stamp ENRI. REX but this was added later, in the reign of Henry III and this is equivalent to a millesimal fineness of 926. In Colonial America, sterling silver was used for currency and general goods as well, between 1634 and 1776, some 500 silversmiths created items in the “New World” ranging from simple buckles to ornate Rococo coffee pots. Although silversmiths of this era were typically familiar with all precious metals, stamping each of their pieces with their personal makers mark, colonial silversmiths relied upon their own status to guarantee the quality and composition of their products
34.
Eisenstein prime
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In mathematics, an Eisenstein prime is an Eisenstein integer z = a + b ω that is irreducible in the ring-theoretic sense, its only Eisenstein divisors are the units, a + bω itself and its associates. The associates and the conjugate of any Eisenstein prime are also prime. It follows that the absolute value squared of every Eisenstein prime is a prime or the square of a natural prime. The first few Eisenstein primes that equal a natural prime 3n −1 are,2,5,11,17,23,29,41,47,53,59,71,83,89,101. Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes, some non-real Eisenstein primes are 2 + ω,3 + ω,4 + ω,5 + 2ω,6 + ω,7 + ω,7 + 3ω. Up to conjugacy and unit multiples, the primes listed above, as of March 2017, the largest known Eisenstein prime is the seventh largest known prime 10223 ×231172165 +1, discovered by Péter Szabolcs and PrimeGrid. All larger known primes are Mersenne primes, discovered by GIMPS, real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes are congruent to 1 mod 3, thus no Mersenne prime is an Eisenstein prime
35.
Area code 929
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North American area codes 718,347, and 929 are New York City telephone area codes in the boroughs of The Bronx, Brooklyn, Queens, and Staten Island, as well as the Marble Hill section of Manhattan. They are overlaid by area code 917, which covers the entirety of New York City, in 1947, area code 212 was created as one of the original 86 North American area codes assigned by AT&T. It served the five boroughs of New York City. On September 1,1984, area code 718 was created as a split from 212 and was assigned to the boroughs of Brooklyn, Queens, in 1992, the borough of The Bronx and the Manhattan neighborhood of Marble Hill were moved from 212 to 718. On October 1,1999, area code 347 was added as an overlay to area code 718, on December 16,2009, the New York Public Service Commission approved an additional overlay of the 718/347 area code region. On January 22,2010, NeuStar-NANPA issued a release that 929 is to be the new area code to further overlay the New York City 718 and 347 area codes of boroughs outside Manhattan. Area code 929 went into effect on April 16,2011, Area code 917 overlays area codes 718,347, and 929, as well as area codes 212 and 646 in Manhattan. One Manhattan neighborhood, Marble Hill, is not in the 212/646 area code, Marble Hill used to be attached to Manhattan Island. After the Harlem River Ship Canal was built in 1895, Marble Hill was physically separated from Manhattan Island, soon after, the Spuyten Duyvil Creek was filled in with landfill, physically connecting Marble Hill to the Bronx. The Greater New York Charter of 1897 officially stated that Marble Hill is part of the borough of Manhattan, when the Bronxs area code was about to be changed from 212 to 718 in 1992, Marble Hill residents fought to stay in 212. This area code was celebrated in the 19982 Skinnee Js song 718, hip-hop group Theodore Unit released an album entitled 718, an homage to their home of Staten Island, which is contained within the 718 area code. Rapper/actor Mos Def refers to the 718 area code in his song Sunshine,718 is a track produced by DJ Premier from Jaz-O & Immobilarie album Kingz Kounty. The area code was mentioned in the song Dont Be One by American Metal band Emmure. The mixtape Return of the PLK contains the song titled 718 Nigga by Rapper Lloyd Banks, the FannyPack album See You Next Tuesday contains the song Seven One Eight. This song was featured in the Dont Trust the B---- in Apartment 23 episode Paris, the area code was mentioned as part of Barney Stinsons citation of many New York area codes in the How I Met Your Mother episode No Tomorrow. On the episode of Seinfeld entitled The Maid, Elaine says that she used to be a 718 which made her cry every night,718 was referred to in the Salon of the Dead episode of Gossip Girl as the slums. Included in the 2001 song Area Codes as one of the locations where rapper Ludacris has hoes. A number from the area code appears as Come forth and call 489-4608, and Ill be here in the lyrics of Diary
36.
Pronic number
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A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers, however, the rectangular number name has also been applied to the composite numbers. The first few numbers are,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462 …. The nth pronic number is also the difference between the odd square 2 and the st centered hexagonal number. The sum of the reciprocals of the numbers is a telescoping series that sums to 1,1 =12 +16 +112 ⋯ = ∑ i =1 ∞1 i. The partial sum of the first n terms in this series is ∑ i =1 n 1 i = n n +1, the nth pronic number is the sum of the first n even integers. It follows that all numbers are even, and that 2 is the only prime pronic number. It is also the only number in the Fibonacci sequence. The number of entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a number is present in only one of the factors n or n+1. Thus a pronic number is squarefree if and only if n and n +1 are also squarefree, the number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n +1. If 25 is appended to the representation of any pronic number. This is because 2 =100 n 2 +100 n +25 =100 n +25
37.
Star number
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A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on. The nth star number is given by the formula Sn = 6n +1, the digital root of a star number is always 1 or 4, and progresses in the sequence 1,4,1. The last two digits of a number in base 10 are always 01,13,21,33,37,41,53,61,73,81. Unique among the numbers is 35113, since its prime factors are also consecutive star numbers. Infinitely many star numbers are triangular numbers, the first four being S1 =1 = T1, S7 =253 = T22, S91 =49141 = T313. Infinitely many star numbers are also numbers, the first four being S1 =12, S5 =121 =112, S45 =11881 =1092. A star prime is a number that is prime. The first few star primes are 13,37,73,181,337,433,541,661,937, the term star number or stellate number is occasionally used to refer to octagonal numbers
38.
Double factorial
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In mathematics, the product of all the integers from 1 up to some non-negative integer n that have the same parity as n is called the double factorial or semifactorial of n and is denoted by n. = ∏ k =0 ⌈ n 2 ⌉ −1 = n ⋯ Therefore, = ∏ k =1 n 2 = n ⋯4 ⋅2, and for odd n it is n. = ∏ k =1 n +12 = n ⋯3 ⋅1, =9 ×7 ×5 ×3 ×1 =945. The double factorial should not be confused with the factorial function iterated twice, the sequence of double factorials for even n =0,2,4,6,8. Starts as 1,2,8,48,384,3840,46080,645120, the sequence of double factorials for odd n =1,3,5,7,9. Starts as 1,3,15,105,945,10395,135135, merserve states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals arising in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, the term odd factorial is sometimes used for the double factorial of an odd number. For an even integer n = 2k, k ≥0. For odd n = 2k −1, k ≥1, in this expression, the first denominator equals. and cancels the unwanted even factors from the numerator. For an odd positive integer n = 2k −1, k ≥1, double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings. For instance, n. for odd values of n counts Perfect matchings of the complete graph Kn +1 for odd n. For instance, a graph with four vertices a, b, c. Perfect matchings may be described in several equivalent ways, including involutions without fixed points on a set of n +1 items or chord diagrams. Stirling permutations, permutations of the multiset of numbers 1,1,2,2, K, k in which each pair of equal numbers is separated only by larger numbers, where k = n + 1/2. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction, heap-ordered trees, trees with k +1 nodes labeled 0,1,2. K, such that the root of the tree has label 0, each node has a larger label than its parent. An Euler tour of the tree gives a Stirling permutation, unrooted binary trees with n + 5/2 labeled leaves. Each such tree may be formed from a tree with one leaf, by subdividing one of the n tree edges
39.
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
40.
Abundant number
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In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number and its proper divisors are 1,2,3,4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance, the number 12 has an abundance of 4, for example. A number n for which the sum of divisors σ>2n, or, equivalently, the sum of proper divisors s>n. The first 28 abundant numbers are,12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,100,102,104,108,112,114,120, …. For example, the divisors of 24 are 1,2,3,4,6,8. Because 36 is more than 24, the number 24 is abundant and its abundance is 36 −24 =12. The smallest odd abundant number is 945, the smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5,7,11,13,17,19,23, and 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If A represents the smallest abundant number not divisible by the first k primes then for all ϵ >0 we have,2 − ϵ < ln A <2 + ϵ for sufficiently large k, infinitely many even and odd abundant numbers exist. The set of abundant numbers has a natural density, marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480. Every multiple of a number is abundant. For example, every multiple of 6 is abundant because the divisors include 1, n/2, n/3, every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10, every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number which is not a number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The abundancy index of n is the ratio σ/n, distinct numbers n1, n2. with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 =12 corresponds to the first abundant number, if p = is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant
41.
Primitive abundant number
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In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers. For example,20 is an abundant number because, The sum of its proper divisors is 1 +2 +4 +5 +10 =22. The sums of the divisors of 1,2,4,5 and 10 are 0,1,3,1 and 8 respectively. The first few primitive abundant numbers are,20,70,88,104,272,304,368,464,550,572, the smallest odd primitive abundant number is 945. A variant definition is abundant numbers having no abundant proper divisor and it starts,12,18,20,30,42,56,66,70,78,88,102,104,114 Every multiple of a primitive abundant number is an abundant number. Every abundant number is a multiple of an abundant number or a multiple of a perfect number. Every primitive abundant number is either a primitive semiperfect number or a weird number, there are an infinite number of primitive abundant numbers. The number of primitive abundant numbers less than or equal to n is o
42.
Primitive semiperfect number
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In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its divisors is a perfect number. The first few numbers are 6,12,18,20,24,28,30,36,40. Every multiple of a number is semiperfect. A semiperfect number that is not divisible by any smaller number is primitive. Every number of the form 2mp for a number m. In particular, every number of the form 2m is semiperfect, the smallest odd semiperfect number is 945. A semiperfect number is necessarily either perfect or abundant, an abundant number that is not semiperfect is called a weird number. With the exception of 2, all primary pseudoperfect numbers are semiperfect, every practical number that is not a power of two is semiperfect. The natural density of the set of semiperfect numbers exists, a primitive semiperfect number is a semiperfect number that has no semiperfect proper divisor. The first few semiperfect numbers are 6,20,28,88,104,272,304,350. There are infinitely many such numbers, all numbers of the form 2mp, with p a prime between 2m and 2m+1, are primitive semiperfect, but this is not the only form, for example,770. Hemiperfect number Erdős–Nicolas number Friedman, Charles N, sums of divisors and Egyptian fractions. Weisstein, Eric W. Primitive semiperfect number
43.
Hexagonal number
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A hexagonal number is a figurate number. The formula for the nth hexagonal number h n =2 n 2 − n = n =2 n ×2. The first few numbers are,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,780,861,946. Every hexagonal number is a number, but only every other triangular number is a hexagonal number. Like a triangular number, the root in base 10 of a hexagonal number can only be 1,3,6. The digital root pattern, repeating every nine terms, is 166193139. Every even perfect number is hexagonal, given by the formula M p 2 p −1 = M p /2 = h /2 = h 2 p −1 where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal, for example, the 2nd hexagonal number is 2×3 =6, the 4th is 4×7 =28, the 16th is 16×31 =496, and the 64th is 64×127 =8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130, adrien-Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. Hexagonal numbers can be rearranged into rectangular numbers of n by. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages, to avoid ambiguity, hexagonal numbers are sometimes called cornered hexagonal numbers. One can efficiently test whether a positive x is an hexagonal number by computing n =8 x +1 +14. If n is an integer, then x is the nth hexagonal number, if n is not an integer, then x is not hexagonal. The nth number of the sequence can also be expressed by using Sigma notation as h n = ∑ i =0 n −1 where the empty sum is taken to be 0. Centered hexagonal number Mathworld entry on Hexagonal Number
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ISBN
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
45.
Argentina
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Argentina, officially the Argentine Republic, is a federal republic in the southern half of South America. With a mainland area of 2,780,400 km2, Argentina is the eighth-largest country in the world, the second largest in Latin America, and the largest Spanish-speaking one. The country is subdivided into provinces and one autonomous city, Buenos Aires. The provinces and the capital have their own constitutions, but exist under a federal system, Argentina claims sovereignty over part of Antarctica, the Falkland Islands, and South Georgia and the South Sandwich Islands. The earliest recorded presence in the area of modern-day Argentina dates back to the Paleolithic period. The country has its roots in Spanish colonization of the region during the 16th century, Argentina rose as the successor state of the Viceroyalty of the Río de la Plata, a Spanish overseas viceroyalty founded in 1776. The country thereafter enjoyed relative peace and stability, with waves of European immigration radically reshaping its cultural. The almost-unparalleled increase in prosperity led to Argentina becoming the seventh wealthiest developed nation in the world by the early 20th century, Argentina retains its historic status as a middle power in international affairs, and is a prominent regional power in the Southern Cone and Latin America. Argentina has the second largest economy in South America, the third-largest in Latin America and is a member of the G-15 and it is the country with the second highest Human Development Index in Latin America with a rating of very high. Because of its stability, market size and growing high-tech sector, the description of the country by the word Argentina has to be found on a Venice map in 1536. In English the name Argentina probably comes from the Spanish language, however the naming itself is not Spanish, Argentina means in Italian of silver, silver coloured, probably borrowed from the Old French adjective argentine of silver > silver coloured already mentioned in the 12th century. The French word argentine is the form of argentin and derives of argent silver with the suffix -in. The Italian naming Argentina for the country implies Argentina Terra land of silver or Argentina costa coast of silver, in Italian, the adjective or the proper noun is often used in an autonomous way as a substantive and replaces it and it is said lArgentina. The name Argentina was probably first given by the Venitian and Genoese navigators, in Spanish and Portuguese, the words for silver are respectively plata and prata and of silver is said plateado and prateado. Argentina was first associated with the silver mountains legend, widespread among the first European explorers of the La Plata Basin. The first written use of the name in Spanish can be traced to La Argentina, a 1602 poem by Martín del Barco Centenera describing the region, the 1826 constitution included the first use of the name Argentine Republic in legal documents. The name Argentine Confederation was also used and was formalized in the Argentine Constitution of 1853. In 1860 a presidential decree settled the name as Argentine Republic
46.
Finland
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Finland, officially the Republic of Finland, is a sovereign state in Northern Europe. A peninsula with the Gulf of Finland to the south and the Gulf of Bothnia to the west, the country has borders with Sweden to the northwest, Norway to the north. Estonia is south of the country across the Gulf of Finland, Finland is a Nordic country situated in the geographical region of Fennoscandia, which also includes Scandinavia. Finlands population is 5.5 million, and the majority of the population is concentrated in the southern region,88. 7% of the population is Finnish people who speak Finnish, a Uralic language unrelated to the Scandinavian languages, the second major group are the Finland-Swedes. In terms of area, it is the eighth largest country in Europe, Finland is a parliamentary republic with a central government based in the capital Helsinki, local governments in 311 municipalities, and an autonomous region, the Åland Islands. Over 1.4 million people live in the Greater Helsinki metropolitan area, from the late 12th century, Finland was an integral part of Sweden, a legacy reflected in the prevalence of the Swedish language and its official status. In the spirit of the notion of Adolf Ivar Arwidsson, we are not Swedes, we do not want to become Russians, let us therefore be Finns, nevertheless, in 1809, Finland was incorporated into the Russian Empire as the autonomous Grand Duchy of Finland. In 1906, Finland became the nation in the world to give the right to vote to all adult citizens. Following the 1917 Russian Revolution, Finland declared itself independent, in 1918, the fledgling state was divided by civil war, with the Bolshevik-leaning Reds supported by the equally new Soviet Russia, fighting the Whites, supported by the German Empire. After a brief attempt to establish a kingdom, the became a republic. During World War II, the Soviet Union sought repeatedly to occupy Finland, with Finland losing parts of Karelia, Salla and Kuusamo, Petsamo and some islands, Finland joined the United Nations in 1955 and established an official policy of neutrality. The Finno-Soviet Treaty of 1948 gave the Soviet Union some leverage in Finnish domestic politics during the Cold War era, Finland was a relative latecomer to industrialization, remaining a largely agrarian country until the 1950s. It rapidly developed an advanced economy while building an extensive Nordic-style welfare state, resulting in widespread prosperity, however, Finnish GDP growth has been negative in 2012–2014, with a preceding nadir of −8% in 2009. Finland is a top performer in numerous metrics of national performance, including education, economic competitiveness, civil liberties, quality of life, a large majority of Finns are members of the Evangelical Lutheran Church, though freedom of religion is guaranteed under the Finnish Constitution. The first known appearance of the name Finland is thought to be on three rune-stones. Two were found in the Swedish province of Uppland and have the inscription finlonti, the third was found in Gotland, in the Baltic Sea. It has the inscription finlandi and dates from the 13th century, the name can be assumed to be related to the tribe name Finns, which is mentioned first known time AD98. The name Suomi has uncertain origins, but a candidate for a source is the Proto-Baltic word *źemē, in addition to the close relatives of Finnish, this name is also used in the Baltic languages Latvian and Lithuanian
47.
Card game
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A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games, a small number of card games played with traditional decks have formally standardized rules, but most are folk games whose rules vary by region, culture, and person. Many games that are not generally placed in the family of games do in fact use cards for some aspect of their gameplay. Similarly, some games that are placed in the game genre involve a board. Given the association of card games and gambling, the pope, Benedict XIV, a card game is played with a deck or pack of playing cards which are identical in size and shape. Each card has two sides, the face and the back, normally the backs of the cards are indistinguishable. The faces of the cards may all be unique, or there can be duplicates, the composition of a deck is known to each player. In some cases several decks are shuffled together to form a pack or shoe. The first playing cards appeared in the century during Tang dynasty China. The Song dynasty statesman and historian Ouyang Xiu has noted that playing cards arose in connection to an earlier development in the book format from scrolls to pages. During the Ming dynasty, characters from novels such as the Water Margin were widely featured on the faces of playing cards. A precise description of Chinese money playing cards survived from the 15th century, mahjong tiles are a 19th-century invention based on three-suited money playing card decks, similar to the way in which Rummikub tiles were derived recently from modern Western playing cards. The same kind of games can also be played with tiles made of wood, plastic, bone, the most notable examples of such tile sets are dominoes, mahjong tiles and Rummikub tiles. Chinese dominoes are also available as playing cards and it is not clear whether Emperor Muzong of Liao really played with domino cards as early as 969, though. Legend dates the invention of dominoes in the year 1112,500 years later domino cards were reported as a new invention. Playing cards first appeared in Europe in the last quarter of the 14th century, the 1430s in Italy saw the invention of the tarot deck, a full Latin-suited deck augmented by suitless cards with painted motifs that played a special role as trumps. Tarot card games are played with these decks in parts of Central Europe. A full tarot deck contains 14 cards in suit, low cards labeled 1–10, and court cards Valet, Chevalier, Dame, and Roi, plus the Fool or Excuse card
48.
Contract bridge
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Contract bridge, or simply bridge, is a trick-taking game using a standard 52-card deck. It is played by four players in two competing partnerships, with partners sitting opposite each other around a table. Millions of people play bridge worldwide in clubs, tournaments, online and with friends at home, making it one of the worlds most popular card games, particularly among seniors. The World Bridge Federation is the body for international competitive bridge. The game consists of deals, each progressing through four phases. During the auction, partners communicate information about their hand, including its overall strength, the cards are then played, the declaring side trying to fulfill the contract, and the defenders trying to stop the declaring side achieving its goal. The deal is scored based on the number of tricks taken, the contract, one theory is that the name bridge has its origins in the name of an earlier game. Bridge departed from whist with the creation of Biritch in the 19th century, the word biritch itself is a spelling of the Russian word Бирюч, an occupation of a diplomatic clerk or an announcer. However some experts think that the Russian origin of the game is a fallacy, another theory is that British soldiers invented the game bridge while serving in the Crimean War. Bridge is a four-player partnership trick-taking game with thirteen tricks per deal, the dominant variations of the game are rubber bridge, more common in social play, and duplicate bridge, which enables comparative scoring in tournament play. Each player is dealt thirteen cards from a standard 52-card deck, a trick starts when a player leads, i. e. plays the first card. The leader to the first trick is determined by the auction, each player, in a clockwise order, plays one card on the trick. Players must play a card of the suit as the original card led, unless they have none. The player who played the card wins the trick. Within a suit, the ace is ranked highest followed by the king, queen and jack, in a deal where the auction has determined that there is no trump suit, the trick must be won by a card of the suit led. However, in a deal there is a trump suit. If one or more plays a trump to a trick when void in the suit led. For example, if the suit is spades and a player is void in the suit led and plays a spade card
49.
Croatia
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Croatia, officially the Republic of Croatia, is a sovereign state between Central Europe, Southeast Europe, and the Mediterranean. Its capital city is Zagreb, which one of the countrys primary subdivisions. Croatia covers 56,594 square kilometres and has diverse, mostly continental, Croatias Adriatic Sea coast contains more than a thousand islands. The countrys population is 4.28 million, most of whom are Croats, the Croats arrived in the area of present-day Croatia during the early part of the 7th century AD. They organised the state into two duchies by the 9th century, tomislav became the first king by 925, elevating Croatia to the status of a kingdom. The Kingdom of Croatia retained its sovereignty for nearly two centuries, reaching its peak during the rule of Kings Petar Krešimir IV and Dmitar Zvonimir, Croatia entered a personal union with Hungary in 1102. In 1527, faced with Ottoman conquest, the Croatian Parliament elected Ferdinand I of the House of Habsburg to the Croatian throne. In 1918, after World War I, Croatia was included in the unrecognized State of Slovenes, Croats and Serbs which seceded from Austria-Hungary, a fascist Croatian puppet state backed by Fascist Italy and Nazi Germany existed during World War II. After the war, Croatia became a member and a federal constituent of the Socialist Federal Republic of Yugoslavia. On 25 June 1991 Croatia declared independence, which came wholly into effect on 8 October of the same year, the Croatian War of Independence was fought successfully during the four years following the declaration. A unitary state, Croatia is a republic governed under a parliamentary system, the International Monetary Fund classified Croatia as an emerging and developing economy, and the World Bank identified it as a high-income economy. Croatia is a member of the European Union, United Nations, the Council of Europe, NATO, the World Trade Organization, the service sector dominates Croatias economy, followed by the industrial sector and agriculture. Tourism is a significant source of revenue during the summer, with Croatia ranked the 18th most popular tourist destination in the world, the state controls a part of the economy, with substantial government expenditure. The European Union is Croatias most important trading partner, since 2000, the Croatian government constantly invests in infrastructure, especially transport routes and facilities along the Pan-European corridors. Internal sources produce a significant portion of energy in Croatia, the rest is imported, the origin of the name is uncertain, but is thought to be a Gothic or Indo-Aryan term assigned to a Slavic tribe. The oldest preserved record of the Croatian ethnonym *xъrvatъ is of variable stem, the first attestation of the Latin term is attributed to a charter of Duke Trpimir from the year 852. The original is lost, and just a 1568 copy is preserved—leading to doubts over the authenticity of the claim, the oldest preserved stone inscription is the 9th-century Branimir Inscription, where Duke Branimir is styled as Dux Cruatorvm. The inscription is not believed to be dated accurately, but is likely to be from during the period of 879–892, the area known as Croatia today was inhabited throughout the prehistoric period