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.
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
3.
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
4.
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
5.
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
6.
Unicode
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Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the worlds writing systems. As of June 2016, the most recent version is Unicode 9.0, the standard is maintained by the Unicode Consortium. Unicodes success at unifying character sets has led to its widespread, the standard has been implemented in many recent technologies, including modern operating systems, XML, Java, and the. NET Framework. Unicode can be implemented by different character encodings, the most commonly used encodings are UTF-8, UTF-16 and the now-obsolete UCS-2. UTF-8 uses one byte for any ASCII character, all of which have the same values in both UTF-8 and ASCII encoding, and up to four bytes for other characters. UCS-2 uses a 16-bit code unit for each character but cannot encode every character in the current Unicode standard, UTF-16 extends UCS-2, using one 16-bit unit for the characters that were representable in UCS-2 and two 16-bit units to handle each of the additional characters. Many traditional character encodings share a common problem in that they allow bilingual computer processing, Unicode, in intent, encodes the underlying characters—graphemes and grapheme-like units—rather than the variant glyphs for such characters. In the case of Chinese characters, this leads to controversies over distinguishing the underlying character from its variant glyphs. In text processing, Unicode takes the role of providing a unique code point—a number, in other words, Unicode represents a character in an abstract way and leaves the visual rendering to other software, such as a web browser or word processor. This simple aim becomes complicated, however, because of concessions made by Unicodes designers in the hope of encouraging a more rapid adoption of Unicode, the first 256 code points were made identical to the content of ISO-8859-1 so as to make it trivial to convert existing western text. For other examples, see duplicate characters in Unicode and he explained that he name Unicode is intended to suggest a unique, unified, universal encoding. In this document, entitled Unicode 88, Becker outlined a 16-bit character model, Unicode could be roughly described as wide-body ASCII that has been stretched to 16 bits to encompass the characters of all the worlds living languages. In a properly engineered design,16 bits per character are more than sufficient for this purpose, Unicode aims in the first instance at the characters published in modern text, whose number is undoubtedly far below 214 =16,384. By the end of 1990, most of the work on mapping existing character encoding standards had been completed, the Unicode Consortium was incorporated in California on January 3,1991, and in October 1991, the first volume of the Unicode standard was published. The second volume, covering Han ideographs, was published in June 1992, in 1996, a surrogate character mechanism was implemented in Unicode 2.0, so that Unicode was no longer restricted to 16 bits. The Microsoft TrueType specification version 1.0 from 1992 used the name Apple Unicode instead of Unicode for the Platform ID in the naming table, Unicode defines a codespace of 1,114,112 code points in the range 0hex to 10FFFFhex. Normally a Unicode code point is referred to by writing U+ followed by its hexadecimal number, for code points in the Basic Multilingual Plane, four digits are used, for code points outside the BMP, five or six digits are used, as required. Code points in Planes 1 through 16 are accessed as surrogate pairs in UTF-16, within each plane, characters are allocated within named blocks of related characters
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
Scientific notation
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Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations, on scientific calculators it is known as SCI display mode. In scientific notation all numbers are written in the form m × 10n, where the exponent n is an integer, however, the term mantissa may cause confusion because it is the name of the fractional part of the common logarithm. If the number is then a minus sign precedes m. In normalized notation, the exponent is chosen so that the value of the coefficient is at least one. Decimal floating point is an arithmetic system closely related to scientific notation. Any given integer can be written in the form m×10^n in many ways, in normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 3. 5×102 and this form allows easy comparison of numbers, as the exponent n gives the numbers order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1, the 10 and exponent are often omitted when the exponent is 0. Normalized scientific form is the form of expression of large numbers in many fields, unless an unnormalized form. Normalized scientific notation is often called exponential notation—although the latter term is general and also applies when m is not restricted to the range 1 to 10. Engineering notation differs from normalized scientific notation in that the exponent n is restricted to multiples of 3, consequently, the absolute value of m is in the range 1 ≤ |m| <1000, rather than 1 ≤ |m| <10. Though similar in concept, engineering notation is rarely called scientific notation, engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. A significant figure is a digit in a number that adds to its precision and this includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore,1,230,400 usually has five significant figures,1,2,3,0, and 4, when a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the place holding zeroes are no longer required, thus 1,230,400 would become 1.2304 ×106. However, there is also the possibility that the number may be known to six or more significant figures, thus, an additional advantage of scientific notation is that the number of significant figures is clearer
19.
India
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India, officially the Republic of India, is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and it is bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast. It shares land borders with Pakistan to the west, China, Nepal, and Bhutan to the northeast, in the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives. Indias Andaman and Nicobar Islands share a border with Thailand. The Indian subcontinent was home to the urban Indus Valley Civilisation of the 3rd millennium BCE, in the following millennium, the oldest scriptures associated with Hinduism began to be composed. Social stratification, based on caste, emerged in the first millennium BCE, early political consolidations took place under the Maurya and Gupta empires, the later peninsular Middle Kingdoms influenced cultures as far as southeast Asia. In the medieval era, Judaism, Zoroastrianism, Christianity, and Islam arrived, much of the north fell to the Delhi sultanate, the south was united under the Vijayanagara Empire. The economy expanded in the 17th century in the Mughal empire, in the mid-18th century, the subcontinent came under British East India Company rule, and in the mid-19th under British crown rule. A nationalist movement emerged in the late 19th century, which later, under Mahatma Gandhi, was noted for nonviolent resistance, in 2015, the Indian economy was the worlds seventh largest by nominal GDP and third largest by purchasing power parity. Following market-based economic reforms in 1991, India became one of the major economies and is considered a newly industrialised country. However, it continues to face the challenges of poverty, corruption, malnutrition, a nuclear weapons state and regional power, it has the third largest standing army in the world and ranks sixth in military expenditure among nations. India is a constitutional republic governed under a parliamentary system. It is a pluralistic, multilingual and multi-ethnic society and is home to a diversity of wildlife in a variety of protected habitats. The name India is derived from Indus, which originates from the Old Persian word Hindu, the latter term stems from the Sanskrit word Sindhu, which was the historical local appellation for the Indus River. The ancient Greeks referred to the Indians as Indoi, which translates as The people of the Indus, the geographical term Bharat, which is recognised by the Constitution of India as an official name for the country, is used by many Indian languages in its variations. Scholars believe it to be named after the Vedic tribe of Bharatas in the second millennium B. C. E and it is also traditionally associated with the rule of the legendary emperor Bharata. Gaṇarājya is the Sanskrit/Hindi term for republic dating back to the ancient times, hindustan is a Persian name for India dating back to the 3rd century B. C. E. It was introduced into India by the Mughals and widely used since then and its meaning varied, referring to a region that encompassed northern India and Pakistan or India in its entirety
20.
South Asia
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Topographically, it is dominated by the Indian Plate, which rises above sea level as Nepal and northern parts of India situated south of the Himalayas and the Hindu Kush. South Asia is bounded on the south by the Indian Ocean and on land by West Asia, Central Asia, East Asia, the current territories of Afghanistan, Bangladesh, Bhutan, Maldives, Nepal, India, Pakistan, and Sri Lanka form the countries of South Asia. The South Asian Association for Regional Cooperation is an economic cooperation organisation in the region which was established in 1985, South Asia covers about 5.1 million km², which is 11. 51% of the Asian continent or 3. 4% of the worlds land surface area. The population of South Asia is about 1.749 billion or about one fourth of the worlds population, overall, it accounts for about 39. 49% of Asias population and is home to a vast array of peoples. The area of South Asia and its extent is not clear cut as systemic. Aside from the region of South Asia, formerly part of the British Empire, there is a high degree of variation as to which other countries are included in South Asia. Modern definitions of South Asia are consistent in including Afghanistan, India, Pakistan, Bangladesh, Sri Lanka, Nepal, Bhutan, Myanmar is included by some scholars in South Asia, but in Southeast Asia by others. Some do not include Afghanistan, others question whether Afghanistan should be considered a part of South Asia or the Middle East, the mountain countries of Nepal and Bhutan, and the island countries of Sri Lanka and Maldives are generally included as well. Myanmar is often added, and by various deviating definitions based on often substantially different reasons, the British Indian Ocean Territory, the common concept of South Asia is largely inherited from the administrative boundaries of the British Raj, with several exceptions. The Aden Colony, British Somaliland and Singapore, though administered at various times under the Raj, have not been proposed as any part of South Asia. Additionally Burma was administered as part of the Raj until 1937, the 562 princely states that were protected by but not directly ruled by the Raj became administrative parts of South Asia upon joining Union of India or Dominion of Pakistan. China and Myanmar have also applied for the status of members of SAARC. This bloc of countries include two independent countries that were not part of the British Raj – Nepal, and Bhutan, Afghanistan was a British protectorate from 1878 until 1919, after the Afghans lost to the British in the Second Anglo-Afghan war. The United Nations Statistics Divisions scheme of sub-regions include all eight members of the SAARC as part of Southern Asia, population Information Network includes Afghanistan, Bangladesh, Burma, India, Nepal, Pakistan and Sri Lanka as part of South Asia. Maldives, in view of its characteristics, was admitted as a member Pacific POPIN subregional network only in principle, the Hirschman–Herfindahl index of the United Nations Economic and Social Commission for Asia and the Pacific for the region includes only the original seven signatories of SAARC. The British Indian Ocean Territory is connected to the region by a publication of Janes for security considerations, the inclusion of Myanmar in South Asia is without consensus, with many considering it a part of southeast Asia and others including it within South Asia. Afghanistan was of importance to the British colonial empire, especially after the Second Anglo-Afghan War over 1878–1880, Afghanistan remained a British protectorate until 1919, when a treaty with Vladimir Lenin included the granting of independence to Afghanistan. Following Indias partition, Afghanistan has generally included in South Asia
21.
Thai language
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Thai, also known as Siamese or Central Thai, is the national and official language of Thailand and the native language of the Thai people and the vast majority of Thai Chinese. Thai is a member of the Tai group of the Tai–Kadai language family, over half of the words in Thai are borrowed from Pali, Sanskrit and Old Khmer. It is a tonal and analytic language, Thai also has a complex orthography and relational markers. Spoken Thai is mutually intelligible with Laotian, Thai is the official language of Thailand, natively spoken by over 20 million people. Standard Thai is based on the register of the classes of Bangkok. In addition to Central Thai, Thailand is home to other related Tai languages, Isan, the language of the Isan region of Thailand, a collective term for the various Lao dialects spoken in Thailand that show some Siamese Thai influences, which is written with the Thai script. It is spoken by about 20 million people, Thais from both inside and outside the Isan region often simply call this variant Lao when speaking informally. Northern Thai, spoken by about 6 million in the independent kingdom of Lanna. Shares strong similarities with Lao to the point that in the past the Siamese Thais referred to it as Lao. Southern Thai, spoken by about 4.5 million Phu Thai, spoken by half a million around Nakhon Phanom Province. Phuan, spoken by 200,000 in central Thailand and Isan, Shan, spoken by about 100,000 in north-west Thailand along the border with the Shan States of Burma, and by 3.2 million in Burma. Lü, spoken by about 1,000,000 in northern Thailand, and 600,000 more in Sipsong Panna, Burma, nyaw language, spoken by 50,000 in Nakhon Phanom Province, Sakhon Nakhon Province, Udon Thani Province of Northeast Thailand. Song, spoken by about 30,000 in central and northern Thailand, Elegant or Formal Thai, official and written version, includes respectful terms of address, used in simplified form in newspapers. Rhetorical Thai, used for public speaking, religious Thai, used when discussing Buddhism or addressing monks. Royal Thai, influenced by Khmer, this is used when addressing members of the family or describing their activities. Most Thais can speak and understand all of these contexts, street and Elegant Thai are the basis of all conversations. Rhetorical, religious, and royal Thai are taught in schools as the national curriculum, many scholars believe that the Thai script is derived from the Khmer script, which is modeled after the Brahmic script from the Indic family. However, in appearance, Thai is closer to Thai Dam script, the language and its script are closely related to the Lao language and script
22.
Lao language
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Lao, also referred to as Laotian, Thai Noi, is a tonal language of the Tai–Kadai language family. It is the language of Laos, and also spoken in the northeast of Thailand. The Lao language serves as an important lingua franca as the country of Laos consists of ethnic groups. Spoken Lao is mutually intelligible with the Thai language, Lao, like many languages in Laos, is written in the Lao script, an abugida. Although there is no standard, the Vientiane dialect has become the de facto standard. Oral history of the migrations is preserved in the legends of Khun Borom, Tai speakers in what is now Laos pushed out or absorbed earlier groups of Mon–Khmer and Austronesian languages. These Tai peoples are classified by the Lao government as Lao Loum or lowland Lao, Lao and Thai are also very similar and share most of their basic vocabulary, but differences in many basic words limit inter-comprehension. The Lao language consists primarily of native Lao words, because of Buddhism, however, Pali has contributed numerous terms, especially relating to religion and in conversation with members of the Sangha. Due to their proximity, Lao has influenced the Khmer and Thai languages. Formal writing has an amount of foreign loanwords, especially Pali and Sanskrit terms, much as Latin. For politeness, pronouns are used, plus ending statements with ແດ່ or ເດີ້, negative statements are made more polite by ending with ດອກ. The following are formal register examples, many consonants in Lao make a phonemic contrast between labialized and plain versions. The complete inventory of Lao consonants is as shown in the table below, hence, final /p/, /t/, and /k/ sounds are pronounced as, and respectively. * The glottal stop appears at the end when no final follows a short vowel, All vowels make a phonemic length distinction. The vowels are as shown in the table, Diphthongs are all centering diphthongs with falling sonority. There are six tones in unchecked syllables, that is. The number of tones is reduced to four in checked syllables. The only consonant clusters allowed are syllable initial clusters /kw/ or /kʰw/, any consonant may appear in the onset, but the labialized consonants do not occur before rounded vowels
23.
Khmer language
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Khmer /kmɛər/ or Cambodian is the language of the Khmer people and the official language of Cambodia. With approximately 16 million speakers, it is the second most widely spoken Austroasiatic language, Khmer has been influenced considerably by Sanskrit and Pali, especially in the royal and religious registers, through Hinduism and Buddhism. The vast majority of Khmer speakers speak Central Khmer, the dialect of the plain where the Khmer are most heavily concentrated. Within Cambodia, regional accents exist in remote areas but these are regarded as varieties of Central Khmer, outside of Cambodia, three distinct dialects are spoken by ethnic Khmers native to areas that were historically part of the Khmer Empire. The Northern Khmer dialect is spoken by over a million Khmers in the regions of Northeast Thailand and is treated by some linguists as a separate language. Khmer is primarily an analytic, isolating language, there are no inflections, conjugations or case endings. Instead, particles and auxiliary words are used to indicate grammatical relationships, general word order is subject–verb–object, and modifiers follow the word they modify. Classifiers appear after numbers when used to count nouns, though not always so consistently as in languages like Chinese, in spoken Khmer, topic-comment structure is common and the perceived social relation between participants determines which sets of vocabulary, such as pronouns and honorifics, are proper. Khmer differs from neighboring languages such as Thai, Burmese, Lao, words are stressed on the final syllable, hence many words conform to the typical Mon–Khmer pattern of a stressed syllable preceded by a minor syllable. The language has been written in the Khmer script, an abugida descended from the Brahmi script via the southern Indian Pallava script, approximately 79% of Cambodians are able to read Khmer. Khmer is a member of the Austroasiatic language family, the family in an area that stretches from the Malay Peninsula through Southeast Asia to East India. Austroasiatic, which also includes Mon, Vietnamese and Munda, has been studied since 1856 and was first proposed as a family in 1907. Despite the amount of research, there is doubt about the internal relationship of the languages of Austroasiatic. Diffloth places Khmer in a branch of the Mon-Khmer languages. In these classification schemes Khmers closest genetic relatives are the Bahnaric and Pearic languages, more recent classifications doubt the validity of the Mon-Khmer sub-grouping and place the Khmer language as its own branch of Austroasiatic equidistant from the other 12 branches of the family. Khmer is spoken by some 13 million people in Cambodia, where it is the official language and it is also a second language for most of the minority groups and indigenous hill tribes there. Additionally there are a million speakers of Khmer native to southern Vietnam and 1.4 million in northeast Thailand, Khmer dialects, although mutually intelligible, are sometimes quite marked. The dialects form a continuum running roughly north to south, the following is a classification scheme showing the development of the modern Khmer dialects
24.
Vietnamese language
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Vietnamese /ˌviɛtnəˈmiːz/ is an Austroasiatic language that originated in the north of modern-day Vietnam, where it is the national and official language. It is the language of the Vietnamese people, as well as a first or second language for the many ethnic minorities of Vietnam. As the result of Vietnamese emigration and cultural influence, Vietnamese speakers are found throughout the world, notably in East and Southeast Asia, North America, Australia, Vietnamese has also been officially recognized as a minority language in the Czech Republic. It is part of the Austroasiatic language family of which it has by far the most speakers, Vietnamese vocabulary has borrowings from Chinese, and it formerly used a modified set of Chinese characters called chữ nôm given vernacular pronunciation. The Vietnamese alphabet in use today is a Latin alphabet with diacritics for tones. As the national language, Vietnamese is spoken throughout Vietnam by ethnic Vietnamese, Vietnamese is also the native language of the Gin minority group in southern Guangxi Province in China. A significant number of speakers also reside in neighboring Cambodia. In the United States, Vietnamese is the sixth most spoken language, with over 1.5 million speakers and it is the third most spoken language in Texas, fourth in Arkansas and Louisiana, and fifth in California. Vietnamese is the seventh most spoken language in Australia, in France, it is the most spoken Asian language and the eighth most spoken immigrant language at home. Vietnamese is the official and national language of Vietnam. It is the first language of the majority of the Vietnamese population, in the Czech Republic, Vietnamese has been recognized as one of 14 minority languages, on the basis of communities that have either traditionally or on a long-term basis resided in the country. This status grants Czech citizens from the Vietnamese community the right to use Vietnamese with public authorities, Vietnamese is increasingly being taught in schools and institutions outside of Vietnam. Since the 1980s, Vietnamese language schools have been established for youth in many Vietnamese-speaking communities around the world, furthermore, there has also been a number of Germans studying Vietnamese due to increased economic investment in Vietnam. Vietnamese is taught in schools in the form of immersion to a varying degree in Cambodia, Laos. Classes teach students subjects in Vietnamese and another language, furthermore, in Thailand, Vietnamese is one of the most popular foreign languages in schools and colleges. Vietnamese was identified more than 150 years ago as part of the Mon–Khmer branch of the Austroasiatic language family. Later, Muong was found to be closely related to Vietnamese than other Mon–Khmer languages. The term Vietic was proposed by Hayes, who proposed to redefine Viet–Muong as referring to a subbranch of Vietic containing only Vietnamese and Muong
25.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
26.
Altitude
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Altitude or height is defined based on the context in which it is used. As a general definition, altitude is a measurement, usually in the vertical or up direction. The reference datum also often varies according to the context, although the term altitude is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage. Vertical distance measurements in the direction are commonly referred to as depth. In aviation, the altitude can have several meanings, and is always qualified by explicitly adding a modifier. Parties exchanging altitude information must be clear which definition is being used, aviation altitude is measured using either mean sea level or local ground level as the reference datum. When flying at a level, the altimeter is always set to standard pressure. On the flight deck, the instrument for measuring altitude is the pressure altimeter. There are several types of altitude, Indicated altitude is the reading on the altimeter when it is set to the local barometric pressure at mean sea level. In UK aviation radiotelephony usage, the distance of a level, a point or an object considered as a point, measured from mean sea level. Absolute altitude is the height of the aircraft above the terrain over which it is flying and it can be measured using a radar altimeter. Also referred to as radar height or feet/metres above ground level, true altitude is the actual elevation above mean sea level. It is indicated altitude corrected for temperature and pressure. Height is the elevation above a reference point, commonly the terrain elevation. Pressure altitude is used to indicate flight level which is the standard for reporting in the U. S. in Class A airspace. Pressure altitude and indicated altitude are the same when the setting is 29.92 Hg or 1013.25 millibars. Density altitude is the altitude corrected for non-ISA International Standard Atmosphere atmospheric conditions, aircraft performance depends on density altitude, which is affected by barometric pressure, humidity and temperature. On a very hot day, density altitude at an airport may be so high as to preclude takeoff and these types of altitude can be explained more simply as various ways of measuring the altitude, Indicated altitude – the altitude shown on the altimeter
27.
Spaceflight
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Spaceflight is ballistic flight into or through outer space. Spaceflight can occur with spacecraft with or without humans on board, examples of human spaceflight include the U. S. Apollo Moon landing and Space Shuttle programs and the Russian Soyuz program, as well as the ongoing International Space Station. Examples of unmanned spaceflight include space probes that leave Earth orbit, as well as satellites in orbit around Earth and these operate either by telerobotic control or are fully autonomous. Spaceflight is used in exploration, and also in commercial activities like space tourism. Additional non-commercial uses of spaceflight include space observatories, reconnaissance satellites, a spaceflight typically begins with a rocket launch, which provides the initial thrust to overcome the force of gravity and propels the spacecraft from the surface of the Earth. Once in space, the motion of a spacecraft—both when unpropelled, some spacecraft remain in space indefinitely, some disintegrate during atmospheric reentry, and others reach a planetary or lunar surface for landing or impact. The first theoretical proposal of space using rockets was published by Scottish astronomer and mathematician William Leitch. More well-known is Konstantin Tsiolkovskys work, Исследование мировых пространств реактивными приборами, spaceflight became an engineering possibility with the work of Robert H. Goddards publication in 1919 of his paper A Method of Reaching Extreme Altitudes. His application of the de Laval nozzle to liquid fuel rockets improved efficiency enough for travel to become possible. He also proved in the laboratory that rockets would work in the vacuum of space, nonetheless and his attempt to secure an Army contract for a rocket-propelled weapon in the first World War was defeated by the November 11,1918 armistice with Germany. Nonetheless, Goddards paper was influential on Hermann Oberth, who in turn influenced Wernher von Braun. Von Braun became the first to produce modern rockets as guided weapons, von Brauns V-2 was the first rocket to reach space, at an altitude of 189 kilometers on a June 1944 test flight. At the end of World War II, von Braun and most of his rocket team surrendered to the United States, over the same period, the Soviet Union secretly tried but failed to develop the N1 rocket to give them the capability to land one person on the Moon. Rockets are the only means currently capable of reaching orbit or beyond, other non-rocket spacelaunch technologies have yet to be built, or remain short of orbital speeds. Spaceports are situated away from human habitation for noise and safety reasons. ICBMs have various special launching facilities, a launch is often restricted to certain launch windows. These windows depend upon the position of bodies and orbits relative to the launch site. The biggest influence is often the rotation of the Earth itself, once launched, orbits are normally located within relatively constant flat planes at a fixed angle to the axis of the Earth, and the Earth rotates within this orbit
28.
Irish language
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Irish, also referred to as Gaelic or Irish Gaelic, is a Goidelic language of the Indo-European language family originating in Ireland and historically spoken by the Irish people. Irish is spoken as a first language by a minority of Irish people. Irish enjoys constitutional status as the national and first official language of the Republic of Ireland and it is also among the official languages of the European Union. The public body Foras na Gaeilge is responsible for the promotion of the language throughout the island of Ireland and it has the oldest vernacular literature in Western Europe. The fate of the language was influenced by the power of the English state in Ireland. Elizabethan officials viewed the use of Irish unfavourably, as being a threat to all things English in Ireland and its decline began under English rule in the 17th century. In the latter part of the 19th century, there was a decrease in the number of speakers. Irish-speaking areas were hit especially hard, by the end of British rule, the language was spoken by less than 15% of the national population. Since then, Irish speakers have been in the minority, efforts have been made by the state, individuals and organisations to preserve, promote and revive the language, but with mixed results. Around the turn of the 21st century, estimates of native speakers ranged from 20,000 to 80,000 people. In the 2011 Census, these numbers had increased to 94,000 and 1.3 million, there are several thousand Irish speakers in Northern Ireland. It has been estimated that the active Irish-language scene probably comprises 5 to 10 per cent of Irelands population, there has been a significant increase in the number of urban Irish speakers, particularly in Dublin. In Gaeltacht areas, however, there has been a decline of the use of Irish. Údarás na Gaeltachta predicted that, by 2025, Irish will no longer be the language in any of the designated Gaeltacht areas. Survey data suggest that most Irish people think highly of Irish as a marker of identity. It has also argued that newer urban groups of Irish speakers are a disruptive force in this respect. In An Caighdeán Oifigiúil the name of the language is Gaeilge, before the spelling reform of 1948, this form was spelled Gaedhilge, originally this was the genitive of Gaedhealg, the form used in Classical Irish. Older spellings of this include Gaoidhealg in Classical Irish and Goídelc in Old Irish, the modern spelling results from the deletion of the silent dh in the middle of Gaedhilge, whereas Goidelic languages, used to refer to the language family including Irish, comes from Old Irish
29.
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
30.
Zonohedron
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A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of segments in three-dimensional space. Zonohedra were originally defined and studied by E. S. Fedorov, more generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope. The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron, each primary parallelohedron is combinatorially equivalent to one of five types, the rhombohedron, hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron. Let be a collection of three-dimensional vectors, with each vector vi we may associate a line segment. The Minkowski sum forms a zonohedron, and all zonohedra that contain the origin have this form, the vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, by choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. Generators parallel to the edges of an octahedron form a truncated octahedron, the Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Both of these zonohedra are simple, as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles. Any simple zonohedron corresponds in this way to a simplicial arrangement, simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane, which were studied by Grünbaum. There are also many examples that do not fit into these three families. Any prism over a polygon with an even number of sides forms a zonohedron. These prisms can be formed so that all faces are regular, two faces are equal to the regular polygon from which the prism was formed. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, the truncated cuboctahedron, with 12 squares,8 hexagons, and 6 octagons. The truncated icosidodecahedron, with 30 squares,20 hexagons and 12 decagons, in addition, certain Catalan solids are again zonohedra, The rhombic dodecahedron is the dual of the cuboctahedron. The rhombic triacontahedron is the dual of the icosidodecahedron, zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron
31.
Harmonic divisor number
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In mathematics, a harmonic divisor number, or Ore number, is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are 1,6,28,140,270,496,672,1638,2970,6200,8128,8190, for example, the harmonic divisor number 6 has the four divisors 1,2,3, and 6. Their harmonic mean is an integer,411 +12 +13 +16 =2, the number 140 has divisors 1,2,4,5,7,10,14,20,28,35,70, and 140. All of the terms in this formula are multiplicative, but not completely multiplicative, therefore, the harmonic mean H is also multiplicative. This means that, for any integer n, the harmonic mean H can be expressed as the product of the harmonic means for the prime powers in the factorization of n. For any integer M, as Ore observed, the product of the mean and arithmetic mean of its divisors equals M itself. Therefore, M is harmonic, with mean of divisors k, if. Ore showed that every number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M, therefore, the average of the divisors is M, where τ denotes the number of divisors of M. For any M, τ is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d. But, no perfect number can be a square, this follows from the form of even perfect numbers. Therefore, for a perfect number M, τ is even, Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers. W. H. Mills showed that any odd harmonic divisor number above 1 must have a power factor greater than 107. Cohen & Sorli showed that there are no odd harmonic divisor numbers smaller than 1024, Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small harmonic divisor numbers. From these results, lists are known of all harmonic divisor numbers up to 2×109, an Identity Concerning Averages of Divisors of a Given Integer. Numbers Whose Positive Divisors Have Small Integral Harmonic Mean, Cohen, Graeme L. Sorli, Ronald M. Odd harmonic numbers exceed 1024. On Divisors of Odd Perfect Numbers, on the averages of the divisors of a number
32.
Highly composite number
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A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan, the related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The initial or smallest 38 highly composite numbers are listed in the table below, the number of divisors is given in the column labeled d. The table below shows all the divisors of one of these numbers, the 15, 000th highly composite number can be found on Achim Flammenkamps website. Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, also, except in two special cases n =4 and n =36, the last exponent ck must equal 1. It means that 1,4, and 36 are the only square highly composite numbers, saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. Note, that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example,96 =25 ×3 satisfies the conditions and has 12 divisors but is not highly composite since there is a smaller number 60 which has the same number of divisors. If Q denotes the number of composite numbers less than or equal to x. The first part of the inequality was proved by Paul Erdős in 1944 and we have 1.13862 < lim inf log Q log log x ≤1.44 and lim sup log Q log log x ≤1.71. Highly composite numbers higher than 6 are also abundant numbers, one need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. It is false that all composite numbers are also Harshad numbers in base 10. The first HCN that is not a Harshad number is 245,044,800, which has a sum of 27. 10 of the first 38 highly composite numbers are highly composite numbers. The sequence of composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors. A positive integer n is a composite number if d ≥ d for all m ≤ n. The counting function QL of largely composite numbers satisfies c ≤ log Q L ≤ d for positive c, d with 0.2 ≤ c ≤ d ≤0.5. Because the prime factorization of a composite number uses all of the first k primes
33.
Motzkin number
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In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords between n points on a circle. The Motzkin numbers are named after Theodore Motzkin, and have diverse applications in geometry, combinatorics. The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle, the following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle. Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers, a Motzkin prime is a Motzkin number that is prime. Guibert, Pergola & Pinzani showed that vexillary involutions are enumerated by Motzkin numbers
34.
Prime factor
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In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of an integer is a list of the integers prime factors, together with their multiplicities. The fundamental theorem of arithmetic says that every integer has a single unique prime factorization. To shorten prime factorizations, factors are expressed in powers. For example,360 =2 ×2 ×2 ×3 ×3 ×5 =23 ×32 ×5, in which the factors 2,3 and 5 have multiplicities of 3,2 and 1, respectively. For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa divides n exactly. For a positive n, the number of prime factors of n. Perfect square numbers can be recognized by the fact all of their prime factors have even multiplicities. For example, the number 144 has the prime factors 144 =2 ×2 ×2 ×2 ×3 ×3 =24 ×32. These can be rearranged to make the more visible,144 =2 ×2 ×2 ×2 ×3 ×3 = × =2 =2. Because every prime factor appears a number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on, positive integers with no prime factors in common are said to be coprime. Two integers a and b can also be defined as if their greatest common divisor gcd =1. Euclids algorithm can be used to determine whether two integers are coprime without knowing their prime factors, the runs in a time that is polynomial in the number of digits involved. The integer 1 is coprime to every integer, including itself. This is because it has no prime factors, it is the empty product and this implies that gcd =1 for any b ≥1. The function, ω, represents the number of prime factors of n, while the function, Ω. If n = ∏ i =1 ω p i α i, for example,24 =23 ×31, so ω =2 and Ω =3 +1 =4
35.
Bell number
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In combinatorial mathematics, the Bell numbers count the number of partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. The nth of these numbers, Bn, counts the number of different ways to partition a set that has n elements, or equivalently. Outside of mathematics, the number also counts the number of different rhyme schemes for n-line poems. As well as appearing in counting problems, these numbers have a different interpretation, in particular, Bn is the nth moment of a Poisson distribution with mean 1. In general, Bn is the number of partitions of a set of size n, a partition of a set S is defined as a set of nonempty, pairwise disjoint subsets of S whose union is S. For example, B3 =5 because the 3-element set can be partitioned in 5 distinct ways, b0 is 1 because there is exactly one partition of the empty set. Every member of the empty set is a nonempty set, therefore, the empty set is the only partition of itself. As suggested by the set notation above, we consider neither the order of the partitions nor the order of elements within each partition and this means that the following partitionings are all considered identical. If, instead, different orderings of the sets are considered to be different partitions, If a number N is a squarefree positive integer, then Bn gives the number of different multiplicative partitions of N. These are factorizations of N into numbers greater than one, treating two factorizations as the same if they have the same factors in a different order. A rhyme scheme describes which lines rhyme with other. Thus, the 15 possible four-line rhyme schemes are AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, and ABCD. The Bell numbers come up in a card shuffling problem mentioned in the addendum to Gardner, of these, the number that return the deck to its original sorted order is exactly Bn. Thus, the probability that the deck is in its original order after shuffling it in this way is Bn/nn, probability that would describe a uniformly random permutation of the deck. Related to card shuffling are several problems of counting special kinds of permutations that are also answered by the Bell numbers. For instance, the nth Bell number equals number of permutations on n items in which no three values that are in sorted order have the last two of three consecutive. The permutations that avoid the generalized patterns 12-3, 32-1, 3-21, 1-32, 3-12, 21-3, the permutations in which every 321 pattern can be extended to a 3241 pattern are also counted by the Bell numbers
36.
Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
37.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
38.
Islam
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Islam is an Abrahamic monotheistic religion which professes that there is only one and incomparable God and that Muhammad is the last messenger of God. It is the worlds second-largest religion and the major religion in the world, with over 1.7 billion followers or 23% of the global population. Islam teaches that God is merciful, all-powerful, and unique, and He has guided mankind through revealed scriptures, natural signs, and a line of prophets sealed by Muhammad. The primary scriptures of Islam are the Quran, viewed by Muslims as the word of God. Muslims believe that Islam is the original, complete and universal version of a faith that was revealed many times before through prophets including Adam, Noah, Abraham, Moses. As for the Quran, Muslims consider it to be the unaltered, certain religious rites and customs are observed by the Muslims in their family and social life, while social responsibilities to parents, relatives, and neighbors have also been defined. Besides, the Quran and the sunnah of Muhammad prescribe a comprehensive body of moral guidelines for Muslims to be followed in their personal, social, political, Islam began in the early 7th century. Originating in Mecca, it spread in the Arabian Peninsula. The expansion of the Muslim world involved various caliphates and empires, traders, most Muslims are of one of two denominations, Sunni or Shia. Islam is the dominant religion in the Middle East, North Africa, sizable Muslim communities are also found in Horn of Africa, Europe, China, Russia, Mainland Southeast Asia, Philippines, Northern Borneo, Caucasus and the Americas. Converts and immigrant communities are found in almost every part of the world, Islam is a verbal noun originating from the triliteral root s-l-m which forms a large class of words mostly relating to concepts of wholeness, submission, safeness and peace. In a religious context it means voluntary submission to God, Islām is the verbal noun of Form IV of the root, and means submission or surrender. Muslim, the word for an adherent of Islam, is the active participle of the verb form. The word sometimes has connotations in its various occurrences in the Quran. In some verses, there is stress on the quality of Islam as a state, Whomsoever God desires to guide. Other verses connect Islām and dīn, Today, I have perfected your religion for you, I have completed My blessing upon you, still others describe Islam as an action of returning to God—more than just a verbal affirmation of faith. In the Hadith of Gabriel, islām is presented as one part of a triad that also includes imān, Islam was historically called Muhammadanism in Anglophone societies. This term has fallen out of use and is said to be offensive because it suggests that a human being rather than God is central to Muslims religion
39.
Prime gap
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A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g is the difference between the -th and the prime numbers, i. e. g n = p n +1 − p n. We have g1 =1, g2 = g3 =2, the sequence of prime gaps has been extensively studied, however many questions and conjectures remain unanswered. By the definition of gn every prime can be written as p n +1 =2 + ∑ i =1 n g i. The first, smallest, and only odd prime gap is 1 between the only prime number,2, and the first odd prime,3. All other prime gaps are even, there is only one pair of gaps between three consecutive odd natural numbers for which all are prime. These gaps are g2 and g3 between the primes 3,5, and 7, for any prime number P, we write P# for P primorial, that is, the product of all prime numbers up to and including P. Therefore, there exist gaps between primes that are large, i. e. for any prime number P. Another way to see that arbitrarily large prime gaps must exist is the fact that the density of primes approaches zero, in fact, by this theorem, P# is very roughly a number the size of exp, and near exp the average distance between consecutive primes is P. In reality, prime gaps of P numbers can occur at much smaller than P#. Although the average gap between primes increases as the logarithm of the integer, the ratio of the prime gap to the integers involved decreases. This is a consequence of the prime number theorem, see below, on the other hand, the ratio of the gap to the number of digits of the integers involved does increase without bound. This is a consequence of a result by Westzynthius, see below, in the opposite direction, the twin prime conjecture asserts that gn =2 for infinitely many integers n. As of March 2017 the largest known prime gap with identified probable prime gap ends has length 5103138, with 216849-digit probable primes found by Robert W. Smith. The largest known prime gap with identified proven primes as gap ends has length 1113106, with 18662-digit primes found by P. Cami, M. Jansen and we say that gn is a maximal gap, if gm < gn for all m < n. As of August 2016 the largest known maximal gap has length 1476 and it is the 75th maximal gap, and it occurs after the prime 1425172824437699411. Other record maximal gap terms can be found at A002386, usually the ratio of gn / ln is called the merit of the gap gn. In 1931, E. Westzynthius proved that maximal prime gaps grow more than logarithmically and that is, lim sup n → ∞ g n log p n = ∞
40.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
41.
Pluto
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Pluto is a dwarf planet in the Kuiper belt, a ring of bodies beyond Neptune. It was the first Kuiper belt object to be discovered, Pluto was discovered by Clyde Tombaugh in 1930 and was originally considered to be the ninth planet from the Sun. After 1992, its planethood was questioned following the discovery of objects of similar size in the Kuiper belt. In 2005, Eris, which is 27% more massive than Pluto, was discovered and this led the International Astronomical Union to define the term planet formally in 2006, during their 26th General Assembly. That definition excluded Pluto and reclassified it as a dwarf planet, Pluto is the largest and second-most-massive known dwarf planet in the Solar System and the ninth-largest and tenth-most-massive known object directly orbiting the Sun. It is the largest known trans-Neptunian object by volume but is less massive than Eris, like other Kuiper belt objects, Pluto is primarily made of ice and rock and is relatively small—about one-sixth the mass of the Moon and one-third its volume. It has an eccentric and inclined orbit during which it ranges from 30 to 49 astronomical units or AU from the Sun. This means that Pluto periodically comes closer to the Sun than Neptune, light from the Sun takes about 5.5 hours to reach Pluto at its average distance. Pluto has five moons, Charon, Styx, Nix, Kerberos. Pluto and Charon are sometimes considered a system because the barycenter of their orbits does not lie within either body. The IAU has not formalized a definition for binary dwarf planets, on July 14,2015, the New Horizons spacecraft became the first spacecraft to fly by Pluto. During its brief flyby, New Horizons made detailed measurements and observations of Pluto, on October 25,2016, at 05,48 pm ET, the last bit of data was received from New Horizons from its close encounter with Pluto on July 14,2015. In the 1840s, Urbain Le Verrier used Newtonian mechanics to predict the position of the then-undiscovered planet Neptune after analysing perturbations in the orbit of Uranus. Subsequent observations of Neptune in the late 19th century led astronomers to speculate that Uranuss orbit was being disturbed by another planet besides Neptune, by 1909, Lowell and William H. Pickering had suggested several possible celestial coordinates for such a planet. Lowell and his observatory conducted his search until his death in 1916, unknown to Lowell, his surveys had captured two faint images of Pluto on March 19 and April 7,1915, but they were not recognized for what they were. There are fourteen other known prediscovery observations, with the oldest made by the Yerkes Observatory on August 20,1909. Percivals widow, Constance Lowell, entered into a legal battle with the Lowell Observatory over her late husbands legacy. Tombaughs task was to image the night sky in pairs of photographs, then examine each pair
42.
Markov number
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The first few Markov numbers are 1,2,5,13,29,34,89,169,194,233,433,610,985,1325. Appearing as coordinates of the Markov triples, etc, there are infinitely many Markov numbers and Markov triples. There are two ways to obtain a new Markov triple from an old one. First, one may permute the 3 numbers x, y, z, second, if is a Markov triple then by Vieta jumping so is. Applying this operation twice returns the same triple one started with, joining each normalized Markov triple to the 1,2, or 3 normalized triples one can obtain from this gives a graph starting from as in the diagram. This graph is connected, in other words every Markov triple can be connected to by a sequence of these operations. If we start, as an example, with we get its three neighbors, and in the Markov tree if x is set to 1,5 and 13, respectively. For instance, starting with and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers, starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers. All the Markov numbers on the adjacent to 2s region are odd-indexed Pell numbers. Thus, there are infinitely many Markov triples of the form, likewise, there are infinitely many Markov triples of the form, where Px is the xth Pell number. Aside from the two smallest singular triples and, every Markov triple consists of three distinct integers, odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32. In his 1982 paper, Don Zagier conjectured that the nth Markov number is given by m n =13 e C n + o with C =2.3523414972 …. Moreover, he pointed out that x 2 + y 2 + z 2 =3 x y z +4 /9, the conjecture was proved by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry. The nth Lagrange number can be calculated from the nth Markov number with the formula L n =9 −4 m n 2, the Markov numbers are sums of pairs of squares. If X⋅Y⋅Z =1 then Tr = Tr, so more symmetrically if X, Y, and Z are in SL2 with X⋅Y⋅Z =1, cambridge Tracts in Mathematics and Mathematical Physics. Markov spectrum problem, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Markoff, A. Sur les formes quadratiques binaires indéfinies
43.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
44.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers
45.
Representation theory
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The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system. Representation theory is pervasive across fields of mathematics, for two reasons, secondly, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics, the success of representation theory has led to numerous generalizations. One of the most general is in category theory, let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard space of column vectors over the real or complex numbers respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers, there are three main sorts of algebraic objects for which this can be done, groups, associative algebras and Lie algebras. The set of all invertible n × n matrices is a group under matrix multiplication, matrix addition and multiplication make the set of all n × n matrices into an associative algebra and hence there is a corresponding representation theory of associative algebras. If we replace matrix multiplication MN by the matrix commutator MN − NM, then the n × n matrices become instead a Lie algebra, there are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group G or algebra A on a vector space V is a map Φ, G × V → V or Φ, A × V → V with two properties. First, for any g in G, the map φ, V → V v ↦ Φ is linear, the requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation is ignored. Equation is an expression of the associativity of matrix multiplication. This doesnt hold for the commutator and also there is no identity element for the commutator. This approach is more concise and more abstract. The vector space V is called the space of φ. It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context, otherwise the notation can be used to denote a representation. When V is of dimension n, one can choose a basis for V to identify V with Fn
46.
Monster group
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The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, the Monster group contains all but six of the other sporadic groups as subquotients. Robert Griess has called these 6 exceptions pariahs, and refers to the other 20 as the happy family and it is difficult to make a good constructive definition of the Monster because of its complexity. Martin Gardner wrote an account of the monster group in his June 1980 Mathematical Games column in Scientific American. The Monster was predicted by Bernd Fischer and Robert Griess as a group containing a double cover of Fischers Baby Monster group as a centralizer of an involution. The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and it was not clear in the 1970s that the Monster actually existed. Griess constructed M as the group of the Griess algebra. In his 1982 paper he referred to the Monster as the Friendly Giant, John Conway and Jacques Tits subsequently simplified this construction. Griesss construction showed that the Monster existed, Thompson showed that its uniqueness would follow from the existence of a 196, 883-dimensional faithful representation. A proof of the existence of such a representation was announced by Norton, Griess, Meierfrankenfeld & Segev gave the first complete published proof of the uniqueness of the Monster. The Monster was a culmination of a development of simple groups and can be built from any 2 of 3 subquotients, the Fischer group Fi24, the Baby Monster. The Schur multiplier and the automorphism group of the Monster are both trivial. The minimal degree of a complex representation is 196,883. The smallest faithful linear representation over any field has dimension 196,882 over the field with 2 elements, the smallest faithful permutation representation of the Monster is on 24 ·37 ·53 ·74 ·11 ·132 ·29 ·41 ·59 ·71 points. The Monster can be realized as a Galois group over the rational numbers, the Monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of small representations, for example, the simple groups A100 and SL20 are far larger, but easy to calculate with as they have small permutation or linear representations. All sporadic groups other than the Monster also have linear representations small enough that they are easy to work with on a computer. Performing calculations with these matrices is possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes
47.
Fourier series
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In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a set of simple oscillating functions, the discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1, Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem. The study of Fourier series is a branch of Fourier analysis, the Mémoire introduced Fourier analysis, specifically Fourier series. Through Fouriers research the fact was established that a function can be represented by a trigonometric series. The first announcement of this discovery was made by Fourier in 1807. The heat equation is a differential equation. These simple solutions are now sometimes called eigensolutions, Fouriers idea was to model a complicated heat source as a superposition of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series, from a modern point of view, Fouriers results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fouriers results with greater precision, in this section, s denotes a function of the real variable x, and s is integrable on an interval, for real numbers x0 and P. We will attempt to represent s in that interval as a sum, or series. Outside the interval, the series is periodic with period P and it follows that if s also has that property, the approximation is valid on the entire real line. We can begin with a summation, s N = A02 + ∑ n =1 N A n ⋅ sin . S N is a function with period P. The inverse relationships between the coefficients are, A n = a n 2 + b n 2 ϕ n = atan2 , when the coefficients are computed as follows, s N approximates s on, and the approximation improves as N → ∞. The infinite sum, s ∞, is called the Fourier series representation of s, both components of a complex-valued function are real-valued functions that can be represented by a Fourier series. This is the formula as before except cn and c−n are no longer complex conjugates. In particular, the Fourier series converges absolutely and uniformly to s whenever the derivative of s is square integrable, if a function is square-integrable on the interval, then the Fourier series converges to the function at almost every point
48.
J-invariant
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In mathematics, Kleins j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL defined on the upper half-plane of complex numbers. It is the unique such function which is away from a simple pole at the cusp such that j =0, j =1728. Rational functions of j are modular, and in fact give all modular functions, classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it also has surprising connections to the symmetries of the Monster group. While the j-invariant can be defined purely in terms of infinite sums. Every elliptic curve E over C is a torus, and thus can be identified with a rank 2 lattice. This is done by identifying opposite edges of each parallelogram in the lattice, thus their quotient, and therefore j, is a modular function of weight zero, in particular a meromorphic function H → C invariant under the action of SL. As explained below, j is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over C and the complex numbers. The two transformations τ → τ +1 and τ → -τ−1 together generate a group called the modular group, in other words, for every c in C, there is a unique τ in the fundamental region such that c = j. Thus, j has the property of mapping the region to the entire complex plane. As a Riemann surface, the region has genus 0, and every modular function is a rational function in j. In other words, the field of functions is C. The j-invariant has many properties, If τ is any CM point. These special values are called singular moduli, the field extension Q/Q is abelian, that is, it has an abelian Galois group. Let Λ be the lattice in C generated by and it is easy to see that all of the elements of Q which fix Λ under multiplication form a ring with units, called an order. The other lattices with generators, associated in like manner to the same order define the algebraic conjugates j of j over Q. Ordered by inclusion, the maximal order in Q is the ring of algebraic integers of Q. These classical results are the point for the theory of complex multiplication. In 1937 Theodor Schneider proved the result that if τ is a quadratic irrational number in the upper half plane then j is an algebraic integer
49.
Monstrous moonshine
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In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979 and this vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. Let r n =1,196883,21296876,842609326,18538750076,19360062527,293553734298 and he won the Fields Medal in 1998 in part for his solution of the conjecture. The Frenkel–Lepowsky–Meurman construction starts with two tools, The construction of a lattice vertex operator algebra VL for an even lattice L of rank n. In physical terms, this is the chiral algebra for a bosonic string compactified on a torus Rn/L and it can be described roughly as the tensor product of the group ring of L with the oscillator representation in n dimensions. For the case in question, one sets L to be the Leech lattice, in physical terms, this describes a bosonic string propagating on a quotient orbifold. The construction of Frenkel–Lepowsky–Meurman was the first time appeared in conformal field theory. Attached to the –1 involution of the Leech lattice, there is an involution h of VL, and an irreducible h-twisted VLmodule, to get the Moonshine Module, one takes the fixed point subspace of h in the direct sum of VL and its twisted module. This was provided by Frenkel–Lepowsky–Meurmans construction and analysis of the Moonshine Module, a Lie algebra m, called the monster Lie algebra, is constructed from V using a quantization functor. It is a generalized Kac–Moody Lie algebra with an action by automorphisms. Using the Goddard–Thorn no-ghost theorem from string theory, the root multiplicities are found to be coefficients of J, one uses the Koike–Norton–Zagier infinite product identity to construct a generalized Kac–Moody Lie algebra by generators and relations. The identity is proved using the fact that Hecke operators applied to J yield polynomials in J, by comparing root multiplicities, one finds that the two Lie algebras are isomorphic, and in particular, the Weyl denominator formula for m is precisely the Koike–Norton–Zagier identity. Using Lie algebra homology and Adams operations, a twisted denominator identity is given for each element and these identities are related to the McKay–Thompson series Tg in much the same way that the Koike–Norton–Zagier identity is related to J. These relations are strong enough that one needs to check that the first seven terms agree with the functions given by Conway. The lowest terms are given by the decomposition of the seven lowest degree homogeneous spaces given in the first step. Borcherds was later quoted as saying I was over the moon when I proved the moonshine conjecture, I dont actually know, as I have not tested this theory of mine. More recent work has simplified and clarified the last steps of the proof, Conway and Norton suggested in their 1979 paper that perhaps moonshine is not limited to the monster, but that similar phenomena may be found for other groups. In 1987, Norton combined Queens results with his own computations to formulate the Generalized Moonshine conjecture, each f is either a constant function, or a Hauptmodul
50.
Minute and second of arc
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A minute of arc, arcminute, arc minute, or minute arc is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn, one minute of arc is 1/21600 of a turn, a second of arc, arcsecond, or arc second is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, and π/648000 of a radian. To express even smaller angles, standard SI prefixes can be employed, the number of square arcminutes in a complete sphere is 4 π2 =466560000 π ≈148510660 square arcminutes. The standard symbol for marking the arcminute is the prime, though a single quote is used where only ASCII characters are permitted. One arcminute is thus written 1′ and it is also abbreviated as arcmin or amin or, less commonly, the prime with a circumflex over it. The standard symbol for the arcsecond is the prime, though a double quote is commonly used where only ASCII characters are permitted. One arcsecond is thus written 1″ and it is also abbreviated as arcsec or asec. In celestial navigation, seconds of arc are used in calculations. This notation has been carried over into marine GPS receivers, which normally display latitude and longitude in the format by default. An arcsecond is approximately the angle subtended by a U. S. dime coin at a distance of 4 kilometres, a milliarcsecond is about the size of a dime atop the Eiffel Tower as seen from New York City. A microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth, since antiquity the arcminute and arcsecond have been used in astronomy. The principal exception is Right ascension in equatorial coordinates, which is measured in units of hours, minutes. These small angles may also be written in milliarcseconds, or thousandths of an arcsecond, the unit of distance, the parsec, named from the parallax of one arcsecond, was developed for such parallax measurements. It is the distance at which the radius of the Earths orbit would subtend an angle of one arcsecond. The ESA astrometric space probe Gaia is hoped to measure star positions to 20 microarcseconds when it begins producing catalog positions sometime after 2016, there are about 1.3 trillion µas in a turn. Currently the best catalog positions of stars actually measured are in terms of milliarcseconds, apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red supergiant with a diameter of 0.05 arcsecond. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1 arcsecond, space telescopes are not affected by the Earths atmosphere but are diffraction limited. For example, the Hubble space telescope can reach a size of stars down to about 0. 1″
51.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
52.
Parsec
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The parsec is a unit of length used to measure large distances to objects outside the Solar System. One parsec is the distance at which one astronomical unit subtends an angle of one arcsecond, a parsec is equal to about 3.26 light-years in length. The nearest star, Proxima Centauri, is about 1.3 parsecs from the Sun, most of the stars visible to the unaided eye in the nighttime sky are within 500 parsecs of the Sun. The parsec unit was likely first suggested in 1913 by the British astronomer Herbert Hall Turner, named from an abbreviation of the parallax of one arcsecond, it was defined so as to make calculations of astronomical distances quick and easy for astronomers from only their raw observational data. Partly for this reason, it is still the unit preferred in astronomy and astrophysics, though the light-year remains prominent in science texts. This corresponds to the definition of the parsec found in many contemporary astronomical references. Derivation, create a triangle with one leg being from the Earth to the Sun. As that point in space away, the angle between the Sun and Earth decreases. A parsec is the length of that leg when the angle between the Sun and Earth is one arc-second. One of the oldest methods used by astronomers to calculate the distance to a star is to record the difference in angle between two measurements of the position of the star in the sky. The first measurement is taken from the Earth on one side of the Sun, and the second is approximately half a year later. The distance between the two positions of the Earth when the two measurements were taken is twice the distance between the Earth and the Sun. The difference in angle between the two measurements is twice the angle, which is formed by lines from the Sun. Then the distance to the star could be calculated using trigonometry. 5-parsec distance of 61 Cygni, the parallax of a star is defined as half of the angular distance that a star appears to move relative to the celestial sphere as Earth orbits the Sun. Equivalently, it is the angle, from that stars perspective. The star, the Sun and the Earth form the corners of a right triangle in space, the right angle is the corner at the Sun. Therefore, given a measurement of the angle, along with the rules of trigonometry. A parsec is defined as the length of the adjacent to the vertex occupied by a star whose parallax angle is one arcsecond
53.
Catalan number
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In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. They are named after the Belgian mathematician Eugène Charles Catalan, using zero-based numbering, the nth Catalan number is given directly in terms of binomial coefficients by C n =1 n +1 =. = ∏ k =2 n n + k k for n ≥0, an alternative expression for Cn is C n = − =1 n +1 for n ≥0, which is equivalent to the expression given above because = n n +1. This shows that Cn is an integer, which is not immediately obvious from the first formula given and this expression forms the basis for a proof of the correctness of the formula. They also satisfy, C0 =1 and C n +1 =2 n +2 C n, which can be a more efficient way to calculate them. Asymptotically, the Catalan numbers grow as C n ∼4 n n 3 /2 π in the sense that the quotient of the nth Catalan number, some sources use just C n ≈4 n n 3 /2. The only Catalan numbers Cn that are odd are those for which n = 2k −1, the only prime Catalan numbers are C2 =2 and C3 =5. The Catalan numbers have an integral representation C n = ∫04 x n ρ d x where ρ =12 π4 − x x and this means that the Catalan numbers are a solution of the Hausdorff moment problem on the interval instead of. The orthogonal polynomials having the weight function ρ on are H n = ∑ k =0 n k, there are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics, Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers, following are some examples, with illustrations of the cases C3 =5 and C4 =14. Cn is the number of Dyck words of length 2n, a Dyck word is a string consisting of n Xs and n Ys such that no initial segment of the string has more Ys than Xs. For n =3, for example, we have the five different parenthesizations of four factors. It follows that Cn is the number of binary trees with n +1 leaves. Cn is the number of lattice paths along the edges of a grid with n × n square cells. A monotonic path is one which starts in the left corner, finishes in the upper right corner. Counting such paths is equivalent to counting Dyck words, X stands for move right, the following hexagons illustrate the case n =4, Cn is the number of stack-sortable permutations of. These are the permutations that avoid the pattern 231, Cn is the number of permutations of that avoid the pattern 123, that is, the number of permutations with no three-term increasing subsequence. For n =3, these permutations are 132,213,231,312 and 321
54.
Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime
55.
Square pyramidal number
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In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid. The first few square pyramidal numbers are,1,5,14,30,55,91,140,204,285,385,506,650,819 and this is a special case of Faulhabers formula, and may be proved by a mathematical induction. An equivalent formula is given in Fibonaccis Liber Abaci, in modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L of a polyhedron P is a polynomial that counts the number of points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid base is a unit square with integer coordinates. The square pyramidal numbers can also be expressed as sums of binomial coefficients, the smaller tetrahedral number represents 1 +3 +6 + ⋯ + T and the larger 1 +3 +6 + ⋯ + T. Offsetting the larger and adding, we arrive at 1,1 +3,3 +6 …, Square pyramidal numbers are also related to tetrahedral numbers in a different way, P n =14. The sum of two square pyramidal numbers is an octahedral number. Augmenting a pyramid whose base edge has n balls by adding to one of its faces a tetrahedron whose base edge has n −1 balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism and this geometric dissection leads to another relation, P n = n −. Besides 1, there is one other number that is both a square and a pyramid number,4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918, in the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes. Also, P n = − which is the difference of two pentatope numbers and this can be seen by expanding, n − n = n = n and dividing through by 24. A common mathematical puzzle involves finding the number of squares in a n by n square grid. This number can be derived as follows, The number of 1 ×1 boxes found in the grid is n2, the number of 2 ×2 boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of 2 ×2 boxes, the number of k × k boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of k × k boxes and it follows that the number of squares in an n × n square grid is, n 2 +2 +2 +2 + … +12 = n 6