1.
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

2.
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

3.
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

4.
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

5.
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

6.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set

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.
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

10.
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

11.
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

12.
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

13.
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

14.
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

15.
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

16.
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

17.
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

18.
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

19.
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

20.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N

21.
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

22.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n

23.
Perfect number
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In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Equivalently, a number is a number that is half the sum of all of its positive divisors i. e. σ1 = 2n. This definition is ancient, appearing as early as Euclids Elements where it is called τέλειος ἀριθμός. Euclid also proved a formation rule whereby q /2 is a perfect number whenever q is a prime of the form 2 p −1 for prime p —what is now called a Mersenne prime. Much later, Euler proved that all even numbers are of this form. This is known as the Euclid–Euler theorem and it is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first perfect number is 6 and its proper divisors are 1,2, and 3, and 1 +2 +3 =6. Equivalently, the number 6 is equal to half the sum of all its positive divisors, the next perfect number is 28 =1 +2 +4 +7 +14. This is followed by the perfect numbers 496 and 8128, in about 300 BC Euclid showed that if 2p−1 is prime then 2p−1 is perfect. The first four numbers were the only ones known to early Greek mathematics. Philo of Alexandria in his first-century book On the creation mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, and by Didymus the Blind, st Augustine defines perfect numbers in City of God in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs mentioned the next three numbers and listed a few more which are now known to be incorrect. Euclid proved that 2p−1 is a perfect number whenever 2p −1 is prime. Prime numbers of the form 2p −1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, for 2p −1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p −1 with a prime p are prime, in fact, Mersenne primes are very rare—of the 9,592 prime numbers p less than 100,000, 2p −1 is prime for only 28 of them. Nicomachus conjectured that every number is of the form 2p−1 where 2p −1 is prime. Ibn al-Haytham circa 1000 AD conjectured that every perfect number is of that form

24.
214 (song)
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Rivermaya is a Filipino rock band. Formed in 1994, it is one of bands who spearheaded the 1990s Philippine alternative rock explosion. Rivermaya is currently composed of original members Mark Escueta and Nathan Azarcon with Mike Elgar, former members include Rico Blanco, who had been the original songwriter of the band and vocalist Bamboo Mañalac, who later formed the band Bamboo and later went on his solo career. Rivermaya is the fourteenth biggest-selling artists/act in the Philippines, the bands predecessor consisted of Jesse Gonzales on vocals, Kenneth Ilagan on guitars, Nathan Azarcon on bass guitar, Rome Velayo on drums, and Rico Blanco on keyboards and backing vocals. They were managed by Lizza Nakpil and director Chito S. Roño who had the intention of molding the group into a show band. The group was then called Xaga, with Bamboo as frontman, the band members disbanded Xaga and formed the band Rivermaya. They started putting together original songs like Ulan,214, by November 1994, the band had released its first album, the self-titled Rivermaya, and its first single, Ulan, followed by 214. The band continued as a quartet and Rico Blanco became the full-time guitarist while Nathan Azarcons friend J-John Valencia filled in as session secondary guitar player on live shows. The bands second album Trip, followed with the singles Kisapmata, Himala, Flower, Princess of Disguise, whilce Portacio, creator of the X-Mens Bishop and co-founder of Image Comics, designed the album art for Trip. In 1997, the band released its album, Atomic Bomb. The album gained positive responses from listeners and received airplay from radio stations. During this period, Rivermaya also released the Rivermaya Remixed album, the band then embarked on a US and Canada tour in 1998. The band released its fifth album, aptly called Free. Free went on to be named Best Album of 2000 at the NU Rock Awards, in 2001, Nathan Azarcon departed from the group. This signaled a new line up change, Rico Blanco and Mark Escueta remained as members. They recruited three different musicians to fill in the left by the former members. Completing the lineup were familiar local guitar heroes, Victor Kakoy Legaspi and this new line-up allowed Rico Blanco to handle vocal duties full-time during live shows. Together, this lineup released Tuloy ang Ligaya, preceded by the EP Alab ng Puso, Legaspi resigned in 2004 but continues to work as session player with other musicians such as Julianne Tarroja, and Peryodiko, among others

25.
Rivermaya
–
Rivermaya is a Filipino rock band. Formed in 1994, it is one of bands who spearheaded the 1990s Philippine alternative rock explosion. Rivermaya is currently composed of original members Mark Escueta and Nathan Azarcon with Mike Elgar, former members include Rico Blanco, who had been the original songwriter of the band and vocalist Bamboo Mañalac, who later formed the band Bamboo and later went on his solo career. Rivermaya is the fourteenth biggest-selling artists/act in the Philippines, the bands predecessor consisted of Jesse Gonzales on vocals, Kenneth Ilagan on guitars, Nathan Azarcon on bass guitar, Rome Velayo on drums, and Rico Blanco on keyboards and backing vocals. They were managed by Lizza Nakpil and director Chito S. Roño who had the intention of molding the group into a show band. The group was then called Xaga, with Bamboo as frontman, the band members disbanded Xaga and formed the band Rivermaya. They started putting together original songs like Ulan,214, by November 1994, the band had released its first album, the self-titled Rivermaya, and its first single, Ulan, followed by 214. The band continued as a quartet and Rico Blanco became the full-time guitarist while Nathan Azarcons friend J-John Valencia filled in as session secondary guitar player on live shows. The bands second album Trip, followed with the singles Kisapmata, Himala, Flower, Princess of Disguise, whilce Portacio, creator of the X-Mens Bishop and co-founder of Image Comics, designed the album art for Trip. In 1997, the band released its album, Atomic Bomb. The album gained positive responses from listeners and received airplay from radio stations. During this period, Rivermaya also released the Rivermaya Remixed album, the band then embarked on a US and Canada tour in 1998. The band released its fifth album, aptly called Free. Free went on to be named Best Album of 2000 at the NU Rock Awards, in 2001, Nathan Azarcon departed from the group. This signaled a new line up change, Rico Blanco and Mark Escueta remained as members. They recruited three different musicians to fill in the left by the former members. Completing the lineup were familiar local guitar heroes, Victor Kakoy Legaspi and this new line-up allowed Rico Blanco to handle vocal duties full-time during live shows. Together, this lineup released Tuloy ang Ligaya, preceded by the EP Alab ng Puso, Legaspi resigned in 2004 but continues to work as session player with other musicians such as Julianne Tarroja, and Peryodiko, among others

26.
Asteroid belt
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The asteroid belt is the circumstellar disc in the Solar System located roughly between the orbits of the planets Mars and Jupiter. It is occupied by numerous irregularly shaped bodies called asteroids or minor planets, the asteroid belt is also termed the main asteroid belt or main belt to distinguish it from other asteroid populations in the Solar System such as near-Earth asteroids and trojan asteroids. About half the mass of the belt is contained in the four largest asteroids, Ceres, Vesta, Pallas, the total mass of the asteroid belt is approximately 4% that of the Moon, or 22% that of Pluto, and roughly twice that of Plutos moon Charon. Ceres, the belts only dwarf planet, is about 950 km in diameter, whereas Vesta, Pallas. The remaining bodies range down to the size of a dust particle, the asteroid material is so thinly distributed that numerous unmanned spacecraft have traversed it without incident. Nonetheless, collisions between large asteroids do occur, and these can form a family whose members have similar orbital characteristics. Individual asteroids within the belt are categorized by their spectra. The asteroid belt formed from the solar nebula as a group of planetesimals. Planetesimals are the precursors of the protoplanets. Between Mars and Jupiter, however, gravitational perturbations from Jupiter imbued the protoplanets with too much energy for them to accrete into a planet. Collisions became too violent, and instead of fusing together, the planetesimals, as a result,99. 9% of the asteroid belts original mass was lost in the first 100 million years of the Solar Systems history. Some fragments eventually found their way into the inner Solar System, Asteroid orbits continue to be appreciably perturbed whenever their period of revolution about the Sun forms an orbital resonance with Jupiter. At these orbital distances, a Kirkwood gap occurs as they are swept into other orbits. Classes of small Solar System bodies in other regions are the objects, the centaurs, the Kuiper belt objects, the scattered disc objects, the sednoids. On 22 January 2014, ESA scientists reported the detection, for the first definitive time, of water vapor on Ceres, the detection was made by using the far-infrared abilities of the Herschel Space Observatory. The finding was unexpected because comets, not asteroids, are considered to sprout jets. According to one of the scientists, The lines are becoming more and more blurred between comets and asteroids. This pattern, now known as the Titius–Bode law, predicted the semi-major axes of the six planets of the provided one allowed for a gap between the orbits of Mars and Jupiter

27.
Asteroid
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Asteroids are minor planets, especially those of the inner Solar System. The larger ones have also been called planetoids and these terms have historically been applied to any astronomical object orbiting the Sun that did not show the disc of a planet and was not observed to have the characteristics of an active comet. As minor planets in the outer Solar System were discovered and found to have volatile-based surfaces that resemble those of comets, in this article, the term asteroid refers to the minor planets of the inner Solar System including those co-orbital with Jupiter. There are millions of asteroids, many thought to be the remnants of planetesimals. The large majority of known asteroids orbit in the belt between the orbits of Mars and Jupiter, or are co-orbital with Jupiter. However, other orbital families exist with significant populations, including the near-Earth objects, individual asteroids are classified by their characteristic spectra, with the majority falling into three main groups, C-type, M-type, and S-type. These were named after and are identified with carbon-rich, metallic. The size of asteroids varies greatly, some reaching as much as 1000 km across, asteroids are differentiated from comets and meteoroids. In the case of comets, the difference is one of composition, while asteroids are composed of mineral and rock, comets are composed of dust. In addition, asteroids formed closer to the sun, preventing the development of the aforementioned cometary ice, the difference between asteroids and meteoroids is mainly one of size, meteoroids have a diameter of less than one meter, whereas asteroids have a diameter of greater than one meter. Finally, meteoroids can be composed of either cometary or asteroidal materials, only one asteroid,4 Vesta, which has a relatively reflective surface, is normally visible to the naked eye, and this only in very dark skies when it is favorably positioned. Rarely, small asteroids passing close to Earth may be visible to the eye for a short time. As of March 2016, the Minor Planet Center had data on more than 1.3 million objects in the inner and outer Solar System, the United Nations declared June 30 as International Asteroid Day to educate the public about asteroids. The date of International Asteroid Day commemorates the anniversary of the Tunguska asteroid impact over Siberia, the first asteroid to be discovered, Ceres, was found in 1801 by Giuseppe Piazzi, and was originally considered to be a new planet. In the early half of the nineteenth century, the terms asteroid. Asteroid discovery methods have improved over the past two centuries. This task required that hand-drawn sky charts be prepared for all stars in the band down to an agreed-upon limit of faintness. On subsequent nights, the sky would be charted again and any moving object would, hopefully, the expected motion of the missing planet was about 30 seconds of arc per hour, readily discernible by observers

28.
E number
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E numbers are codes for substances that are permitted to be used as food additives for use within the European Union and Switzerland. Commonly found on labels, their safety assessment and approval are the responsibility of the European Food Safety Authority. Having a single unified list for food additives was first agreed upon in 1962 with food colouring, in 1964, the directives for preservatives were added,1970 for antioxidants and 1974 for the emulsifiers, stabilisers, thickeners and gelling agents. They are increasingly, though rarely, found on North American packaging. In some European countries, E number is used informally as a pejorative term for artificial food additives. This is incorrect, because many components of foods have E numbers, e. g. vitamin C. NB, Not all examples of a fall into the given numeric range. Moreover, many chemicals, particularly in the E400–499 range, have a variety of purposes, the list shows all components that have or had an E-number assigned. Not all additives listed are still allowed in the EU, but are listed as they used to have an E-number, for an overview of currently allowed additives see here. Includes Lists of authorised food additives Food additives database

29.
Bell 214
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The Bell 214 is a medium-lift helicopter derived from Bell Helicopters ubiquitous UH-1 Huey series. The Bell 214ST shares the same number, but is a larger. The original development of the Model 214 was announced by Bell in 1970 under the name Huey Plus, the first prototype was based on a Bell 205 airframe equipped with a Lycoming T53-L-702 engine of 1,900 shp. The first Bell 214A demonstration prototype followed and was evaluated in Iran during field exercises with the Imperial Iranian Armed Forces, the trial was judged successful and an order for 287 214A helicopters followed. The intention was that aircraft would be constructed by Bell in their Dallas-Fort Worth facility. In the event,296 214A and 39 214C variants were delivered and it can be identified by the single large exhaust duct and wide chord rotor blades without stabilizer bars. Bell offered the Bell 214B BigLifter for civil use, the 214B was produced until 1981. Powered by a 2,930 shp Lycoming T5508D turboshaft, it has the rotor drive. The transmission is rated at 2,050 shp for take-off, the BigLifter features advanced rotor hub with elastomeric bearings, an automatic flight control system with stability augmentation, and commercial avionics. As of January 2012,29 Bell 214s were in service, including 25 Bell 214As with Iran. Approximately 41 Bell 214Bs are in commercial service, user countries are Australia, Canada, Norway, Singapore and United States. Bell 214 Huey Plus - The prototype 214 flew in 1970, powered by one Lycoming T53-L-702 turboshaft. Bell 214B BigLifter - Civil variant of the 214A, Bell 214B-1 - This variant of the Bell Model 214B is limited to a maximum 12,500 lb gross weight with an internal load due to different certification standards. The external load is the same as the 214B, the only difference between the 214B and 214B-1 is the dataplate, and flight manual

30.
Tupolev Tu-204
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The Tupolev Tu-204 is a twin-engined medium-range jet airliner capable of carrying 210 passengers, designed by Tupolev and produced by Aviastar SP and Kazan Aircraft Production Association. First introduced in 1989, it is intended to be equivalent to the Boeing 757, with slightly lower range and payload. It was developed for Aeroflot as a replacement for the medium-range Tupolev Tu-154 trijet, the latest version, with significant upgrades and improvements, is the Tu-204SM, which made its maiden flight on 29 December 2010. The Tu-204 was designed as a family of aircraft incorporating passenger, cargo, combi and it is powered by either two Aviadvigatel PS-90 or Rolls-Royce RB211 engines. The Tu-204 is produced at two of the largest Russian aircraft manufacturing plants in Ulyanovsk and Kazan, the Tu-204 cabin is available in several layouts, including the baseline single-class layout seating for 210 passengers and a two- or three-class layout designed for 164–193 passengers. A cargo version of the Tu-204 is being operated by several airlines in Europe. Seating configuration is 3-3 in economy and 2-2 in Business class, the business class cabin has a seat pitch of 810 millimetres. The passenger cabin can be divided into compartments according to class with removable bulkheads, compartments are illuminated by reflected light. Hidden lights located over and under the overhead bins create uniform, overhead bins for passenger baggage and coats are of the closed type. The volume of baggage per passenger is 0.052 cubic metres, in 1994, the first certificate for Tu-204 aircraft was issued. Subsequently issued certificates have extended estimated operational conditions and improved overall aircraft type design and it is currently undergoing the certification process with JAA. The Tu-204-100 variant, certified with PS-90A engines, complies with noise regulations described in Chapter 4 of Supplement 16 to ICAO which means it is quieter, the aircraft was certified to Russian standards AP-25. The Tu-204 is part of a new generation of Russian aircraft, the Tu-204 features many technological innovations such as fly-by-wire control systems, a glass cockpit, supercritical wings with winglets, and is available with Russian or foreign avionics. The wings and tails are relatively resistant to ice build-up, among todays airliners the Tu-204 is the only one which does not require wing anti-icing systems. During the test flight safety has been confirmed without the system on the bearing surfaces. The Tu-204 is the passenger airline model, and the Tu-204C is the basic freight or cargo model. The most-used models are the -100C and the -120C, certified in January 1995, this initial version is powered by Soloviev PS90 turbofans with 157 kN of thrust, and uses Russian avionics in addition to its Russian engines. The Tu-204-200 is a version with extra fuel for more range

31.
Type 214 submarine
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The Type 214 is a diesel-electric submarine developed by Howaldtswerke-Deutsche Werft GmbH. It features diesel propulsion with an air-independent propulsion system using Siemens polymer electrolyte membrane fuel cells. A contract to three boats for the Hellenic Navy was signed 15 February 2000 and a fourth unit was ordered in June 2002. The first boat was built at HDW in Kiel, Germany, the Hellenic Navy named them the Papanikolis-class. Due to improvements in the pressure hull materials, the Type 214 can dive nearly 400 meters and it can also carry food, fresh water and fuel for 84 days of operation. The Greece Papanikolis U214 class is equipped with a radar mast which does not penetrate the pressure hull of the submarine. In the top of the radar mast the radar transmitter is installed and this transmitter is part of the SPHINX Radar System supplied by Thales Defence Deutschland GmbH in Kiel. The radar sensor is a FMCW transceiver which cant be detected by ESM systems in medium terms and this technology is so called LPI radar, which means Low probability of intercept. The transmitting power is lower than the power of a mobile phone, Thales SPHINX radar is a tactical radar, designed for submarines. The South Korean Son Won-Il U214 Class Submarine is equipped with a SPHINX-D Radar System supplied by Thales Defence Deutschland GmbH and it uses an additional pulse transmitter in the top of the mast. The combination of high power pulse radar and a low power LPI transmitter is very effective for submarines. During surface operations, the sails with an open pulse fingerprint for ESM systems. The boat remains invisible to others, total of 9 are planned and 4 are in active duty. South Korea ordered its first three KSS-II/ Type 214 boats in 2000, which were assembled by Hyundai Heavy Industries, the Batch 2 order will add six more submarines to the Navy, to be built by Daewoo Shipbuilding & Marine Engineering. The Hong Beom-do, a guided missile submarine was launched on 7 April,2016. In 2005 Portugal awarded a contract to Howaldtswerke-Deutsche Werft for two type 214 submarines, which were delivered in 2010, the Pakistan Navy negotiated for the purchase of three Type 214 submarines to be built in Pakistan in 2008. During the IDEAS2008 exhibition, the HDW chief Walter Freitag told “The commercial contract has been finalised up to 95 per cent, ” he said. The first submarine would be delivered to the Pakistan Navy in 64 months after signing of the contract while the rest would be completed successively in 12 months

32.
U.S. Armed Forces
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The United States Armed Forces are the federal armed forces of the United States. They consist of the Army, Marine Corps, Navy, Air Force, from the time of its inception, the military played a decisive role in the history of the United States. A sense of unity and identity was forged as a result of victory in the First Barbary War. Even so, the Founders were suspicious of a permanent military force and it played an important role in the American Civil War, where leading generals on both sides were picked from members of the United States military. Not until the outbreak of World War II did a standing army become officially established. The National Security Act of 1947, adopted following World War II and during the Cold Wars onset, the U. S. military is one of the largest militaries in terms of number of personnel. It draws its personnel from a pool of paid volunteers. As of 2016, the United States spends about $580.3 billion annually to fund its military forces, put together, the United States constitutes roughly 40 percent of the worlds military expenditures. For the period 2010–14, the Stockholm International Peace Research Institute found that the United States was the worlds largest exporter of major arms, the United States was also the worlds eighth largest importer of major weapons for the same period. The history of the U. S. military dates to 1775 and these forces demobilized in 1784 after the Treaty of Paris ended the War for Independence. All three services trace their origins to the founding of the Continental Army, the Continental Navy, the United States President is the U. S. militarys commander-in-chief. Rising tensions at various times with Britain and France and the ensuing Quasi-War and War of 1812 quickened the development of the U. S. Navy, the reserve branches formed a military strategic reserve during the Cold War, to be called into service in case of war. Time magazines Mark Thompson has suggested that with the War on Terror, Command over the armed forces is established in the United States Constitution. The sole power of command is vested in the President by Article II as Commander-in-Chief, the Constitution also allows for the creation of executive Departments headed principal officers whose opinion the President can require. This allowance in the Constitution formed the basis for creation of the Department of Defense in 1947 by the National Security Act, the Defense Department is headed by the Secretary of Defense, who is a civilian and member of the Cabinet. The Defense Secretary is second in the chain of command, just below the President. Together, the President and the Secretary of Defense comprise the National Command Authority, to coordinate military strategy with political affairs, the President has a National Security Council headed by the National Security Advisor. The collective body has only power to the President

33.
Dallas, Texas
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Dallas is a major city in the U. S. state of Texas. It is the most populous city in the Dallas–Fort Worth metroplex, the citys population ranks ninth in the U. S. and third in Texas after Houston and San Antonio. The citys prominence arose from its importance as a center for the oil and cotton industries. The bulk of the city is in Dallas County, of which it is the county seat, however, sections of the city are located in Collin, Denton, Kaufman, and Rockwall counties. According to the 2010 United States Census, the city had a population of 1,197,816, the United States Census Bureaus estimate for the citys population increased to 1,300,092 as of July 1,2015. In 2016 DFW ascended to the one spot in the nation in year-over-year population growth. In 2014, the metropolitan economy surpassed Washington, D. C. to become the fifth largest in the U. S. with a 2014 real GDP over $504 billion, as such, the metropolitan areas economy is the 10th largest in the world. As of January 2017, the job count has increased to 3,558,200 jobs. The citys economy is based on banking, commerce, telecommunications, technology, energy, healthcare and medical research. The city is home to the third-largest concentration of Fortune 500 companies in the nation. Located in North Texas, Dallas is the core of the largest metropolitan area in the South. Dallas and nearby Fort Worth were developed due to the construction of railroad lines through the area allowing access to cotton, cattle. Later, France also claimed the area but never established much settlement, the area remained under Spanish rule until 1821, when Mexico declared independence from Spain, and the area was considered part of the Mexican state of Coahuila y Tejas. In 1836, the Republic of Texas, with majority Anglo-American settlers, in 1839, Warren Angus Ferris surveyed the area around present-day Dallas. John Neely Bryan established a permanent settlement near the Trinity River named Dallas in 1841, the origin of the name is uncertain. The Republic of Texas was annexed by the United States in 1845, Dallas was formally incorporated as a city on February 2,1856. With construction of railroads, Dallas became a business and trading center and it became an industrial city, attracting workers from Texas, the South and the Midwest. The Praetorian Building of 15 stories, built in 1909, was the first skyscraper west of the Mississippi and it marked the prominence of Dallas as a city

34.
Terminator 2: Judgment Day
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Terminator 2, Judgment Day is a 1991 American science-fiction action film co-written, produced and directed by James Cameron. The film stars Arnold Schwarzenegger, Linda Hamilton, Robert Patrick and it is the sequel to the 1984 film The Terminator, and the second installment in the Terminator franchise. A second, less advanced Terminator is also sent back in time to protect John, after a troubled pre-production characterized by legal disputes, Mario Kassar of Carolco Pictures emerged with the franchises property rights in early 1990. This paved the way for the completion of the screenplay by a Cameron-led production team, the production of Terminator 2 required a $102 million budget making it the most expensive film made up to that point. Much of the massive budget was spent on filming and special effects. The film was released on July 3,1991, in time for the U. S, Terminator 2 was a critical and commercial success and influenced popular culture, especially the use of visual effects in films. It received many accolades, including four Academy Awards for Best Sound Editing, Best Sound Mixing, Best Makeup, in 1995, John Connor is living in Los Angeles with foster parents. Skynet sends a new Terminator, designated as T-1000, back in time to kill John, the T-1000 arrives under a freeway, kills a policeman and assumes his identity. Meanwhile, the future John Connor has sent back a reprogrammed T-800 Terminator to protect his young counterpart, the Terminator and the T-1000 converge on John in a shopping mall, and a chase ensues after which John and the Terminator escape together on a motorcycle. Fearing that the T-1000 will kill Sarah in order to get to him and they encounter Sarah as she is escaping from the hospital, although she is initially reluctant to trust the T-800. After the trio escapes from the T-1000 in a police car, in addition, it would create machines that will hunt and kill the remnants of humanity. Finding him at his home, she wounds him but finds herself unable to kill him in front of his family, John and the Terminator arrive and inform Dyson of the future consequences of his work. They learn that much of his research has been engineered from the damaged CPU. Convincing him that these items and his designs must be destroyed, they break into the Cyberdyne building and retrieve the CPU, the police arrive and Dyson is shot, but he manages to trigger several explosives, destroying the lab and his research while sacrificing himself. The T-1000 relentlessly pursues the trio, eventually cornering them in a steel mill. The T-1000 and the Terminator engage in combat, with the advanced model severely damaging its adversary. The T-800 is seemingly shut down until its emergency back-up system brings it back online, the T-1000 nearly kills John and Sarah until the T-800 appears and shoots it into a vat of molten steel with an M79 grenade launcher, destroying it. John tosses the arm and CPU of the original Terminator into the vat as well, as Sarah expresses relief that the ordeal is over, the Terminator explains that to ensure that he is not used for reverse engineering he must also be destroyed

35.
List of Kangxi radicals
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The 214 Kangxi radicals form a system of radicals of Chinese characters. The radicals are numbered in stroke count order, originally introduced in the 1615 Zihui, they are named in relation to the Kangxi Dictionary of 1716. The system of 214 Kangxi radicals is based on the system of 540 radicals used in the Han-era Shuowen Jiezi. The radicals have between one and seventeen strokes, the number of strokes being 5 while the average number of strokes is slightly below 5.7. The ten radicals with the largest number of account for 10,665 characters. The 214 Kangxi radicals act as a de facto standard, which may not be duplicated exactly in every Chinese dictionary and they also serve as the basis for many computer encoding systems, including Unihan. The number of radicals may be reduced in modern practical dictionaries, thus, the Oxford Concise English–Chinese Dictionary, for example, has 188 radicals. The Xinhua Zidian, a pocket-sized character dictionary containing about 13,000 characters, a few dictionaries also introduce new radicals, treating groups of radicals that are used together in many different characters as a kind of radical. For example, Hanyu Da Cidian, the most inclusive available Chinese dictionary has 23,000 head character entries organised by a system of 200 radicals. The Unicode standard encoded 20,992 characters in version 1.0.1 in the CJK Unified Ideographs block and this standard followed the Kangxi order of radicals but did not encode all characters found in the Kangxi dictionary. More characters were added in later versions, adding CJK Unified Ideographs Extensions A, B, C and D as of Unicode 7.0 with further additions planned for Unicode 8.0, within each Extension, characters are also ordered by Kangxi radical and additional strokes. The Unicode Consortium maintains the Unihan Database, with a Radical-Stroke-Index, in Unicode version 3.0, a separate Kangxi Radicals block was introduced which encodes the 214 radicals in sequence, at U+2F00–2FD5. In addition, the CJK Radicals Supplement block was introduced, encoding alternative forms taken by Kangxi radicals as they appear within specific characters. For example, ⺁ CJK RADICAL CLIFF is a variant of ⼚ radical 27, itself identical in shape to the character consisting of unaugmented radical 27, Chinese etymology search radicals and receive the meaning as well as illustrations of radicals in history

36.
Dewey Decimal Classification
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The Dewey Decimal Classification, or Dewey Decimal System, is a proprietary library classification system first published in the United States by Melvil Dewey in 1876. It has been revised and expanded through 23 major editions, the latest issued in 2011 and it is also available in an abridged version suitable for smaller libraries. It is currently maintained by the Online Computer Library Center, a cooperative that serves libraries. OCLC licenses access to a version for catalogers called WebDewey. The Decimal Classification introduced the concepts of relative location and relative index which allow new books to be added to a library in their location based on subject. Libraries previously had given books permanent shelf locations that were related to the order of acquisition rather than topic, the classifications notation makes use of three-digit Arabic numerals for main classes, with fractional decimals allowing expansion for further detail. Using Arabic numerals for symbols, it is flexible to the degree that numbers can be expanded in linear fashion to cover aspects of general subjects. A library assigns a number that unambiguously locates a particular volume in a position relative to other books in the library. The number makes it possible to find any book and to return it to its place on the library shelves. The classification system is used in 200,000 libraries in at least 135 countries, the major competing classification system to the Dewey Decimal system is the Library of Congress Classification system created by the U. S. Melvil Dewey was an American librarian and self-declared reformer and he was a founding member of the American Library Association and can be credited with the promotion of card systems in libraries and business. He developed the ideas for his classification system in 1873 while working at Amherst College library. He applied the classification to the books in library, until in 1876 he had a first version of the classification. In 1876, he published the classification in pamphlet form with the title A Classification and Subject Index for Cataloguing and Arranging the Books and he used the pamphlet, published in more than one version during the year, to solicit comments from other librarians. It is not known who received copies or how many commented as only one copy with comments has survived, in March 1876, he applied for, and received copyright on the first edition of the index. The edition was 44 pages in length, with 2,000 index entries, comprised 314 pages, with 10,000 index entries. Editions 3–14, published between 1888 and 1942, used a variant of this same title, Dewey modified and expanded his system considerably for the second edition. In an introduction to that edition Dewey states that nearly 100 persons hav contributed criticisms, one of the innovations of the Dewey Decimal system was that of positioning books on the shelves in relation to other books on similar topics

37.
Theodicy
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Theodicy, in its most common form, is an attempt to answer the question of why a good God permits the manifestation of evil. Unlike a defense, which tries to demonstrate that Gods existence is possible in the light of evil. The German mathematician and philosopher Gottfried Leibniz coined the term theodicy in 1710 in his work Théodicée, Irenaeus German philosopher Max Weber saw theodicy as a social problem, based on the human need to explain puzzling aspects of the world. Sociologist Peter L. Berger argued that religion arose out of a need for order. Following the Holocaust, a number of Jewish theologians developed a new response to the problem of evil, sometimes called anti-theodicy, which maintains that God cannot be meaningfully justified. As an alternative to theodicy, a defense has been proposed by the American philosopher Alvin Plantinga, similar to a theodicy, a cosmodicy attempts to justify the fundamental goodness of the universe, and an anthropodicy attempts to justify the goodness of humanity. As defined by Alvin Plantinga, theodicy is the answer to the question of why God permits evil, the word theodicy derives from the Greek words Θεός Τheos and δίκη dikē. Theos is translated God and dikē can be translated as either trial or judgement, thus, theodicy literally means justifying God. In the Internet Encyclopedia of Philosophy, Nick Trakakis proposed an additional three requirements which must be contained within a theodicy, Common sense views of the world, widely held historical and scientific opinion. As a response to the problem of evil, a theodicy is distinct from a defence, a defence attempts to demonstrate that the occurrence of evil does not contradict Gods existence, but it does not propose that rational beings are able to understand why God permits evil. A theodicy seeks to show that it is reasonable to believe in God despite evidence of evil in the world, defenses propose solutions to the logical problem of evil, while theodicies attempt to answer the evident problem. German philosopher Max Weber interpreted theodicy as a problem. Weber framed the problem of evil as the dilemma that the good can suffer and the evil can prosper and he identified two purposes of theodicy, to explain why good people suffer, and why people prosper. Sociologist Peter L. Berger characterised religion as the attempt to build order out of a chaotic world. He believed that humans could not accept that anything in the world was meaningless and saw theodicy as an assertion that the cosmos has meaning and order, despite evidence to the contrary. Berger presented an argument similar to that of Weber, but suggested that the need for theodicy arose primarily out of the situation of human society and he believed that theodicies existed to allow individuals to transcend themselves, denying the individual in favour of the social order. The term theodicy was coined by German philosopher Gottfried Leibniz in his 1710 work, written in French, Essais de Théodicée sur la bonté de Dieu, bayle argued that, because the Bible asserts the coexistence of God and evil, this state of affairs must simply be accepted. Voltaire also includes the theme in his novel, Candide

38.
12 (number)
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12 is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a highly composite number, divisible by 2,3,4. It is central to systems of counting, including the Western calendar and units of time. The word twelve is the largest number with a name in English. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance and it derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels Book of John. It has cognates in every Germanic language, whose Proto-Germanic ancestor has been reconstructed as *twaliƀi, from *twa and suffix *-lif- or *-liƀ- of uncertain meaning. It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19, every other Indo-European language instead uses a form of two+ten, such as the Latin duōdecim. The usual ordinal form is twelfth but dozenth or duodecimal is also used in some contexts, similarly, a group of twelve things is usually a dozen but may also be referred to as a duodecad. The adjective referring to a group of twelve is duodecuple, as with eleven, the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan, with the implicit meaning that two is left after having already counted to ten. The Lithuanian suffix is also considered to share a similar development, the suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten. While, as mentioned above,12 has its own name in Germanic languages such as English and German, it is a number in many other languages, e. g. Italian dodici. In Germany, according to an old tradition, the numbers 0 through 12 were spelt out, the Duden now calls this tradition outdated and no longer valid, but many writers still follow it. Another system spells out all numbers written in one or two words, Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1,2,3,4,6 and 12. Twelve is also a composite number, the next one being twenty-four. Twelve is also a highly composite number, the next one being sixty. It is the first composite number of the form p2q, a square-prime,12 has an aliquot sum of 16. Accordingly,12 is the first abundant number and demonstrates an 8-member aliquot sequence,12 is the 3rd composite number in the 3-aliquot tree, the only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant, Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number

39.
15 (number)
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15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name, in spoken English, the numbers 15 and 50 are often confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed,15 /fɪfˈtiːn/ vs 50 /ˈfɪfti/, however, in dates such as 1500 or when contrasting numbers in the teens, the stress generally shifts to the first syllable,15 /ˈfɪftiːn/. In a 24-hour clock, the hour is in conventional language called three or three oclock. A composite number, its divisors being 1,3 and 5. A repdigit in binary and quaternary, in hexadecimal, as well as all higher bases,15 is represented as F. the 4th discrete semiprime and the first member of the discrete semiprime family. It is thus the first odd discrete semiprime, the number proceeding 15,14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21, the smallest number that can be factorized using Shors quantum algorithm. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet, the aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members. 15 is the composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree, fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33. 15 and 16 form a Ruth-Aaron pair under the definition in which repeated prime factors are counted as often as they occur. There are 15 solutions to Známs problem of length 7, if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems. Group 15 of the table are sometimes known as the pnictogens. 15 Madadgar is designated as a number in Pakistan, for mobile phones, similar to the international GSM emergency number 112, if 112 is used in Pakistan. 112 can be used in an emergency if the phone is locked. The Hanbali Sunni madhab states that the age of fifteen of a solar or lunar calendar is when ones taklif begins and is the stage whereby one has his deeds recorded. In the Hebrew numbering system, the number 15 is not written according to the method, with the letters that represent 10 and 5

40.
17 (number)
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17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution

41.
19 (number)
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19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key

42.
20 (number)
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20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants