1.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
2.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
3.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
4.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
5.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
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
19.
Parity (mathematics)
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
20.
Abundant number
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In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number and its proper divisors are 1,2,3,4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance, the number 12 has an abundance of 4, for example. A number n for which the sum of divisors σ>2n, or, equivalently, the sum of proper divisors s>n. The first 28 abundant numbers are,12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,100,102,104,108,112,114,120, …. For example, the divisors of 24 are 1,2,3,4,6,8. Because 36 is more than 24, the number 24 is abundant and its abundance is 36 −24 =12. The smallest odd abundant number is 945, the smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5,7,11,13,17,19,23, and 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If A represents the smallest abundant number not divisible by the first k primes then for all ϵ >0 we have,2 − ϵ < ln A <2 + ϵ for sufficiently large k, infinitely many even and odd abundant numbers exist. The set of abundant numbers has a natural density, marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480. Every multiple of a number is abundant. For example, every multiple of 6 is abundant because the divisors include 1, n/2, n/3, every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10, every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number which is not a number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The abundancy index of n is the ratio σ/n, distinct numbers n1, n2. with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 =12 corresponds to the first abundant number, if p = is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant
21.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
22.
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
23.
Thue-Morse sequence
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The first few steps of this procedure yield the strings 0 then 01,0110,01101001,0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. There are several equivalent ways of defining the Thue–Morse sequence, to compute the nth element tn, write the number n in binary. If the number of ones in this expansion is odd then tn =1. For this reason John H. Conway et al. call numbers n satisfying tn =1 odious numbers and numbers for which tn =0 evil numbers. In other words, tn =0 if n is an evil number, if this bit is at an even index, tn differs from tn −1, and otherwise it is the same as tn −1. The resulting algorithm takes constant time to each sequence element. The Thue–Morse sequence is the sequence tn satisfying the relation for all non-negative integers n. So, the first element is 0, then once the first 2n elements have been specified, forming a string s, then the next 2n elements must form the bitwise negation of s. Now we have defined the first 2n+1 elements, and we recurse, spelling out the first few steps in detail, We start with 0. The bitwise negation of 0 is 1, combining these, the first 2 elements are 01. The bitwise negation of 01 is 10, combining these, the first 4 elements are 0110. The bitwise negation of 0110 is 1001, combining these, the first 8 elements are 01101001. The sequence can also be defined by, ∏ i =0 ∞ = ∑ j =0 ∞ t j x j and that is, there are many instances of XX, where X is some string. For instance, with k =0, we have A = T0 =0, however, there are no cubes, instances of XXX. There are also no overlapping squares, instances of 0X0X0 or 1X1X1, notice that T2n is palindrome for any n >1. Further, let qn be a word obtain from T2n by counting ones between consecutive zeros, for instance, q1 =2 and q2 =2102012 and so on. The words Tn do not contain overlapping squares in consequence, the words qn are palindrome squarefree words, the easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps. Then nX can be set to any multiple of m that is larger than twice the length of X, but the Morse sequence is uniformly recurrent without being periodic, not even eventually periodic
24.
Powerful number
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A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a number is the product of a square and a cube, that is, a number m of the form m = a2b3. Powerful numbers are known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful, in the other direction, suppose that m is powerful, with prime factorization m = ∏ p i α i, where each αi ≥2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi - γi. Then, all values βi are nonnegative integers, and all values γi are either zero or three, so m = =23 supplies the desired representation of m as a product of a square. Informally, given the prime factorization of m, take b to be the product of the factors of m that have an odd exponent. Because m is powerful, each prime factor with an odd exponent has an exponent that is at least 3, in addition, each prime factor of m/b3 has an even exponent, so m/b3 is a perfect square, so call this a2, then m = a2b3. The representation m = a2b3 calculated in this way has the property that b is squarefree, the sum of the reciprocals of the powerful numbers converges. More generally, the sum of the reciprocals of the sth powers of the numbers is equal to ζ ζ ζ whenever it converges. Let k denote the number of numbers in the interval. Then k is proportional to the root of x. More precisely, c x 1 /2 −3 x 1 /3 ≤ k ≤ c x 1 /2, c = ζ / ζ =2.173 …, the two smallest consecutive powerful numbers are 8 and 9. However, one of the two numbers in a pair formed in this way must be a square. According to Guy, Erdős has asked whether there are many pairs of consecutive powerful numbers such as in which neither number in the pair is a square. Jaroslaw Wroblewski showed that there are indeed infinitely many such pairs by showing that 33c2 +1 = 73d2 has infinitely many solutions and it is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers. Any odd number is a difference of two squares,2 = k2 + 2k +1, so 2 − k2 = 2k +1. Similarly, any multiple of four is a difference of the squares of two numbers that differ by two,2 − k2 = 4k +4
25.
Practical number
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In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. Practical numbers were used by Fibonacci in his Liber Abaci in connection with the problem of representing rational numbers as Egyptian fractions, Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators. The name practical number is due to Srinivasan and he noted that the subdivision of money, weights and measures involved numbers like 4,12,16,20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10. He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of numbers that was completed by Stewart. This characterization makes it possible to determine whether a number is practical by examining its prime factorization, every even perfect number and every power of two is also a practical number. Practical numbers have also shown to be analogous with prime numbers in many of their properties. If the ordered set of all divisors of the number n is d 1, d 2. D j with d 1 =1 and d j = n, in other words the ordered sequence of all divisors d 1 < d 2 <. < d j of a number has to be a complete sub-sequence. This partial characterization was extended and completed by Stewart and Sierpiński who showed that it is straightforward to determine whether a number is practical from its prime factorization, a positive integer greater than one with prime factorization n = p 1 α1. P k α k is if and only if each of its prime factors p i is small enough for p i −1 to have a representation as a sum of smaller divisors. The condition stated above is necessary and sufficient for a number to be practical, in the other direction, the condition is sufficient, as can be shown by induction. Since q ≤ σ and n / p k α k can be shown by induction to be practical, we can find a representation of q as a sum of divisors of n / p k α k. The divisors representing r, together with p k α k times each of the divisors representing q, the only odd practical number is 1, because if n >2 is an odd number, then 2 cannot be expressed as the sum of distinct divisors of n. More strongly, Srinivasan observes that other than 1 and 2, the product of two practical numbers is also a practical number. More strongly the least common multiple of any two numbers is also a practical number. Equivalently, the set of all numbers is closed under multiplication. From the above characterization by Stewart and Sierpiński it can be seen that if n is a practical number, in the set of all practical numbers there is a primitive set of practical numbers
26.
Heptomino
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A heptomino is a polyomino of order 7, that is, a polygon in the plane made of 7 equal-sized squares connected edge-to-edge. The name of type of figure is formed with the prefix hept-. When rotations and reflections are not considered to be distinct shapes, when reflections are considered distinct, there are 196 one-sided heptominoes. When rotations are also considered distinct, there are 760 fixed heptominoes, the figure shows all possible free heptominoes, coloured according to their symmetry groups,84 heptominoes have no symmetry. Their symmetry group consists only of the identity mapping,9 heptominoes have an axis of reflection symmetry aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a parallel to the sides of the squares. 7 heptominoes have an axis of symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection,4 heptominoes have point symmetry, also known as rotational symmetry of order 2. Their symmetry group has two elements, the identity and the 180° rotation,3 heptominoes have two axes of reflection symmetry, both aligned with the gridlines. Their symmetry group has four elements, the identity, two reflections and the 180° rotation and it is the dihedral group of order 2, also known as the Klein four-group. 1 heptomino has two axes of symmetry, both aligned with the diagonals. Its symmetry group also has four elements and its symmetry group is also the dihedral group of order 2 with four elements. This results in 84 ×8 + ×4 + ×2 =760 fixed heptominoes, of the 108 free heptominoes,101 satisfy the Conway criterion and 3 more can form a patch satisfying the criterion. Thus, only 4 heptominoes fail to satisfy the criterion and, in fact, although a complete set of the 108 free heptominoes has a total of 756 squares, it is not possible to tile a rectangle with them. The proof of this is trivial, since there is one heptomino which has a hole and it is also impossible to pack them into a 757-square rectangle with a one-square hole because 757 is a prime number
27.
Polyomino
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A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares and it may be regarded as a finite subset of the regular square tiling with a connected interior. Polyominoes are classified according to how many cells they have, Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in Fairy Chess Review between the years 1937 to 1957, under the name of dissection problems. The name polyomino was invented by Solomon W. Golomb in 1953, related to polyominoes are polyiamonds, formed from equilateral triangles, polyhexes, formed from regular hexagons, and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, like many puzzles in recreational mathematics, polyominoes raise many combinatorial problems. The most basic is enumerating polyominoes of a given size, no formula has been found except for special classes of polyominoes. A number of estimates are known, and there are algorithms for calculating them, Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only simply connected polyominoes, there are three common ways of distinguishing polyominoes for enumeration, free polyominoes are distinct when none is a rigid transformation of another. Translating, rotating, reflecting, or glide reflecting a free polyomino does not change its shape, one-sided polyominoes are distinct when none is a translation or rotation of another. Translating or rotating a one-sided polyomino does not change its shape, fixed polyominoes are distinct when none is a translation of another. Translating a fixed polyomino will not change its shape, the following table shows the numbers of polyominoes of various types with n cells. As of 2004, Iwan Jensen has enumerated the fixed polyominoes up to n =56, free polyominoes have been enumerated up to n =28 by Tomás Oliveira e Silva. The dihedral group D4 is the group of symmetries of a square and this group contains four rotations and four reflections. It is generated by alternating reflections about the x-axis and about a diagonal, one free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of D4. However, those images are not necessarily distinct, the more symmetry a free polyomino has, therefore, a free polyomino that is invariant under some or all non-trivial symmetries of D4 may correspond to only 4,2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group D4, the following table shows the numbers of polyominoes with n squares, sorted by symmetry groups. Each polyomino of order n+1 can be obtained by adding a square to a polyomino of order n and this leads to algorithms for generating polyominoes inductively
28.
Iteration
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Iteration is the act of repeating a process, either to generate an unbounded sequence of outcomes, or with the aim of approaching a desired goal, target or result. Each repetition of the process is called an iteration. In the context of mathematics or computer science, iteration is a building block of algorithms. Iteration in mathematics may refer to the process of iterating a function i. e. applying a function repeatedly, iteration of apparently simple functions can produce complex behaviours and difficult problems - for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate solutions to certain mathematical problems. Newtons method is an example of an iterative method, manual calculation of a numbers square root is a common use and a well-known example. Iteration in computing is the marking out of a block of statements within a computer program for a defined number of repetitions. That block of statements is said to be iterated, a computer scientist might also refer to block of statements as an iteration. In the example above, the line of code is using the value of i as it increments and this idea is found in the old adage, Practice makes perfect. Unlike computing and math, educational iterations are not predetermined, instead, in algorithmic situations, recursion and iteration can be employed to the same effect. Some types of programming languages, known as functional programming languages, are designed such that they do not set up block of statements for explicit repetition as with the for loop, instead, those programming languages exclusively use recursion. Each piece of work will be divided repeatedly until the amount of work is as small as it can possibly be, the algorithm then reverses and reassembles the pieces into a complete whole. The classic example of recursion is in list-sorting algorithms such as Merge Sort, the code below is an example of a recursive algorithm in the Scheme programming language that will output the same result as the pseudocode under the previous heading. In Object-Oriented Programming, an iterator is an object that ensures iteration is executed in the way for a range of different data structures, saving time. An iteratee is an abstraction which accepts or rejects data during an iteration, recursion Fractal Iterated function Infinite compositions of analytic functions
29.
Conjecture
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In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found. Conjectures such as the Riemann hypothesis or Fermats Last Theorem have shaped much of history as new areas of mathematics are developed in order to prove them. In number theory, Fermats Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics, two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23,1852 when Francis Guthrie, while trying to color the map of counties of England, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. The four color theorem was proven in 1976 by Kenneth Appel and it was the first major theorem to be proved using a computer. Appel and Hakens approach started by showing that there is a set of 1,936 maps. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. The Hauptvermutung of geometric topology is the conjecture that any two triangulations of a triangulable space have a refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze and this conjecture is now known to be false. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion, the manifold version is true in dimensions m ≤3
30.
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
31.
Palindrome
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A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward, such as madam or racecar. Sentence-length palindromes may be written when allowances are made for adjustments to capital letters, punctuation, and word dividers, such as A man, a plan, was it a car or a cat I saw. Composing literature in palindromes is an example of constrained writing, the word palindrome was coined by the English playwright Ben Jonson in the 17th century from the Greek roots palin and dromos. Palindromes date back at least to 79 AD, as a palindrome was found as a graffito at Herculaneum and this palindrome, called the Sator Square, consists of a sentence written in Latin, Sator Arepo Tenet Opera Rotas. It is remarkable for the fact that the first letters of each form the first word, the second letters form the second word. Hence, it can be arranged into a square that reads in four different ways. As such, they can be referred to as palindromatic, the palindromic Latin riddle In girum imus nocte et consumimur igni describes the behavior of moths. It is likely that this palindrome is from medieval rather than ancient times, byzantine Greeks often inscribed the palindrome, Wash sins, not only face ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ, on baptismal fonts. This practice was continued in many English churches, some well-known English palindromes are, Able was I ere I saw Elba, A man, a plan, a canal - Panama. Madam, Im Adam and Never odd or even, English palindromes of notable length include mathematician Peter Hiltons Doc, note, I dissent. A fast never prevents a fatness, I diet on cod and Scottish poet Alastair Reids T. Eliot, top bard, notes putrid tang emanating, is sad, Id assign it a name, gnat dirt upset on drab pot toilet. The most familiar palindromes in English are character-unit palindromes, the characters read the same backward as forward. Some examples of words are redivider, noon, civic, radar, level, rotor, kayak, reviver, racecar, redder, madam. There are also word-unit palindromes in which the unit of reversal is the word, word-unit palindromes were made popular in the recreational linguistics community by J. A. Lindon in the 1960s. Occasional examples in English were created in the 19th century, several in French and Latin date to the Middle Ages. Palindromes often consist of a sentence or phrase, e. g, mr. Owl ate my metal worm, Was it a cat I saw. Or Go hang a salami, Im a lasagna hog, punctuation, capitalization, and spaces are usually ignored. Some, such as Rats live on no evil star, Live on time, emit no evil, semordnilap is a name coined for words that spell a different word in reverse
32.
House at 196 Main Street
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The House at 196 Main Street, also known as the Hiram Eaton House, is a historic house at 196 Main Street in Wakefield, Massachusetts. The house was built in the 1840s or 1850s, probably for Hiram Eaton, the house is a well-preserved Greek Revival house, 2 1⁄2 stories in height and five bays wide, with a side-gable roof pierced by three gabled dormers. It has a plan and its front porch is supported by delicately fluted columns. The house was listed on the National Register of Historic Places in 1989, National Register of Historic Places listings in Wakefield, Massachusetts National Register of Historic Places listings in Middlesex County, Massachusetts
33.
Wakefield, Massachusetts
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Wakefield is a town in Middlesex County, Massachusetts in the Greater Boston metropolitan area, incorporated in 1812 and located about 12.5 mi north-northwest of Downtown Boston. The 73rd most populous municipality in Massachusetts, Wakefields population was 24,932 at the 2010 census, Wakefield was first settled in 1638 and was originally known as Lynn Village. It officially separated from Lynn and incorporated as Reading in 1644 when the first church and this first corn mill was built on the Mill River on Water Street, and later small saw mills were built on the Mill River and the Saugus River. The old parish became known as the Old or South Parish when in 1713 the North Parish was established. This North Parish later became the town of North Reading, in 1769 the West Parish was established. In 1812 the Old or South Parish of Reading separated from Reading and was incorporated as South Reading. At the time it was spelled South Redding, not South Reading, the railroad was chartered and built in 1844 between Wilmington and Boston. This later became the line of the Boston and Maine Railroad. The Boston and Maine Foundry was built in 1854 and was reincorporated as the Smith. The Boston Ice Company cut and shipped ice from Lake Quannapowitt starting in 1851, the Rattan Works was established in 1856 by Cyrus Wakefield. This later grew into the Wakefield Rattan Company and at one time had a thousand employees, in 1868 Cyrus Wakefield donated land and money for a new town hall, and in thanks the town voted to change its name from South Reading to Wakefield. The town hall, currently named for William J. Lee, is located at 1 Lafayette Street, in 1856 the South Reading Public Library was established, which later became the Beebe Town Library. In 1923, the Lucius Beebe Memorial Library was built and established by Junius Beebe, the first weekly newspaper in Wakefield was established in 1858. One of the oldest and largest manufacturers of flying model airplane toys in the world, Paul K. Guillow, the company is particularly notable for its extensive line of balsa wood model airplane kits. Route 128 was built along the edge of the town by 1958. American Mutual had over 1000 employees, most of them commuting to work via Route 128, by the late 1980s American Mutual was in liquidation due to the Woburn W. R. Grace litigation. The headquarters building was sold to the Beal Company and was home to Boston Technology Inc. which invented and manufactured corporate voice mail systems that operated on computer systems, in April 1971, a fire burned down much of the amusement park. The area now consists of office buildings and is called Edgewater Park
34.
Canadian Expeditionary Force
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The Canadian Expeditionary Force was the designation of the field force created by Canada for service overseas in the First World War. The force fielded several combat formations on the Western Front in France and Belgium, the Canadian Cavalry Brigade and the Canadian Independent Force, which were independent of the Canadian Corps, also fought on the Western Front. The CEF also had a reserve and training organization in England. The Germans went so far as to call them storm troopers for their combat efficiency. In August 1918, the CEFs Canadian Siberian Expeditionary Force travelled to revolution-torn Russia and it reinforced an anti-Bolshevik garrison in Vladivostok during the winter of 1918–19. At this time, another force of Canadian soldiers were placed in Archangel, the Canadian Expeditionary Force was mostly volunteers, as conscription was not enforced until the end of the war when call-ups began in January 1918. Ultimately, only 24,132 conscripts arrived in France before the end of the war, Canada was the senior Dominion in the British Empire and automatically at war with Germany upon the British declaration. According to Canadian historian Dr. Serge Durflinger at the Canadian War Museum, of the first contingent formed at Valcartier, Quebec in 1914, fully two-thirds were men born in the United Kingdom. By the end of the war in 1918, at least fifty per cent of the CEF consisted of British-born men, many British nationals from the United Kingdom or other territories who were resident in Canada also joined the CEF. As several CEF battalions were posted to the Bermuda Garrison before proceeding to France, although the Bermuda Militia Artillery and Bermuda Volunteer Rifle Corps both sent contingents to the Western Front, the first would not arrive there til June 1915. By then, many Bermudians had already been serving on the Western Front in the CEF for months, Bermudians in the CEF enlisted under the same terms as Canadians, and all male British Nationals resident in Canada became liable for conscription under the Military Service Act,1917. Two tank battalions were raised in 1918 but did not see service, most of the infantry battalions were broken up and used as reinforcements, with a total of fifty being used in the field, including the mounted rifle units, which were re-organized as infantry. The artillery and engineering units underwent significant re-organization as the war progressed, a distinct entity within the Canadian Expeditionary Force was the Canadian Machine Gun Corps. It consisted of several machine gun battalions, the Eatons, Yukon, and Borden Motor Machine Gun Batteries. During the summer of 1918, these units were consolidated into four machine gun battalions, the Canadian Corps with its four infantry divisions comprised the main fighting force of the CEF. The Canadian Cavalry Brigade also served in France, the 1915 Battle of Ypres, the first engagement of Canadian forces in the Great War, changed the Canadian perspective on war. Ypres exposed Canadian soldiers and their commanders to modern war and they had already experienced the effects of shellfire and developed a reputation for aggressive trench raiding despite their lack of formal training and generally inferior equipment. In April 1915, they were introduced to yet another facet of modern war, the Germans employed chlorine gas to create a hole in the French lines adjacent to the Canadian force and poured troops into the gap
35.
World War I
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World War I, also known as the First World War, the Great War, or the War to End All Wars, was a global war originating in Europe that lasted from 28 July 1914 to 11 November 1918. More than 70 million military personnel, including 60 million Europeans, were mobilised in one of the largest wars in history and it was one of the deadliest conflicts in history, and paved the way for major political changes, including revolutions in many of the nations involved. The war drew in all the worlds great powers, assembled in two opposing alliances, the Allies versus the Central Powers of Germany and Austria-Hungary. These alliances were reorganised and expanded as more nations entered the war, Italy, Japan, the trigger for the war was the assassination of Archduke Franz Ferdinand of Austria, heir to the throne of Austria-Hungary, by Yugoslav nationalist Gavrilo Princip in Sarajevo on 28 June 1914. This set off a crisis when Austria-Hungary delivered an ultimatum to the Kingdom of Serbia. Within weeks, the powers were at war and the conflict soon spread around the world. On 25 July Russia began mobilisation and on 28 July, the Austro-Hungarians declared war on Serbia, Germany presented an ultimatum to Russia to demobilise, and when this was refused, declared war on Russia on 1 August. Germany then invaded neutral Belgium and Luxembourg before moving towards France, after the German march on Paris was halted, what became known as the Western Front settled into a battle of attrition, with a trench line that changed little until 1917. On the Eastern Front, the Russian army was successful against the Austro-Hungarians, in November 1914, the Ottoman Empire joined the Central Powers, opening fronts in the Caucasus, Mesopotamia and the Sinai. In 1915, Italy joined the Allies and Bulgaria joined the Central Powers, Romania joined the Allies in 1916, after a stunning German offensive along the Western Front in the spring of 1918, the Allies rallied and drove back the Germans in a series of successful offensives. By the end of the war or soon after, the German Empire, Russian Empire, Austro-Hungarian Empire, national borders were redrawn, with several independent nations restored or created, and Germanys colonies were parceled out among the victors. During the Paris Peace Conference of 1919, the Big Four imposed their terms in a series of treaties, the League of Nations was formed with the aim of preventing any repetition of such a conflict. This effort failed, and economic depression, renewed nationalism, weakened successor states, and feelings of humiliation eventually contributed to World War II. From the time of its start until the approach of World War II, at the time, it was also sometimes called the war to end war or the war to end all wars due to its then-unparalleled scale and devastation. In Canada, Macleans magazine in October 1914 wrote, Some wars name themselves, during the interwar period, the war was most often called the World War and the Great War in English-speaking countries. Will become the first world war in the sense of the word. These began in 1815, with the Holy Alliance between Prussia, Russia, and Austria, when Germany was united in 1871, Prussia became part of the new German nation. Soon after, in October 1873, German Chancellor Otto von Bismarck negotiated the League of the Three Emperors between the monarchs of Austria-Hungary, Russia and Germany
36.
People's Volunteer Army
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The Peoples Volunteer Army was the armed forces deployed by the Peoples Republic of China during the Korean War. The Peoples Volunteer Army entered Korea on October 19,1950, although the United Nations forces were under United States command, this army was officially a UN police force. In order to avoid a war with the US and other UN members. Technically speaking, the PVA was the PLAs North East Frontier Force, the CPVA soldier was reasonably well clothed, in keeping with the PLAs guerrilla origin and egalitarian attitudes. All ranks wore a cotton or woolen green or khaki shirt and trousers combination with leaders uniforms being different in cut and with red piping and collar tabs. The nominal strength of a PLA division was 9,500 men, with a regiment comprising 3,000, however, many divisions sent to the Korean War were below-strength while the divisions stationed opposite Taiwan were above-strength. There was also variation in organization and equipment as well as in the quantity and quality of the military equipment, some of the PLAs equipment was from the Imperial Japanese Army or were captured from the Kuomintang military forces. Some Czechoslovak-made weapons were purchased on the open market by the PRC. The Peoples Republic of China had issued warnings that they would intervene if any non-South Korean forces crossed the 38th parallel, Truman regarded the warnings as a bold attempt to blackmail the UN. On October 8,1950, the day after American troops crossed the parallel, Chairman Mao issued the order for the NEFF to be moved to the Yalu River, ready to cross. Mao Zedong sought Soviet aid and saw intervention as essentially defensive and we must be prepared for the US to declare. War with China, he told Joseph Stalin, premier Zhou Enlai was sent to Moscow to add force to Maos cabled arguments. Mao delayed his forces while waiting for Soviet help, and the attack was thus postponed from 13 October to 19 October. Soviet assistance was limited to providing air support no closer than 60 miles from the battlefront, the MiG-15s in PRC colours would be an unpleasant surprise to the UN pilots, they would hold local air superiority against the F-80 Shooting Stars until newer F-86 Sabres were deployed. The Soviet role was known to the U. S. and it has been alleged by the Chinese that the Soviets had agreed to full scale air support, which never occurred south of Pyongyang, and helped accelerate the Sino-Soviet Split. On October 15,1950, Truman went to Wake Island to discuss the possibility of Chinese intervention, macArthur reassured Truman that if the Chinese tried to get down to Pyongyang there would be the greatest slaughter. On October 19,1950, Pyongyang, North Koreas capital, on the same day, the PVA formally crossed the Yalu River under strict secrecy. The Chinese assault began on October 25,1950, under the command of Peng Dehuai with 270,000 PVA troops, after these initial engagements, the Chinese withdrew into the mountains
37.
Korean War
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The Korean War began when North Korea invaded South Korea. The United Nations, with the United States as the principal force, China came to the aid of North Korea, and the Soviet Union gave some assistance. Korea was ruled by Japan from 1910 until the days of World War II. In August 1945, the Soviet Union declared war on Japan, as a result of an agreement with the United States, U. S. forces subsequently moved into the south. By 1948, as a product of the Cold War between the Soviet Union and the United States, Korea was split into two regions, with separate governments, both governments claimed to be the legitimate government of all of Korea, and neither side accepted the border as permanent. The conflict escalated into open warfare when North Korean forces—supported by the Soviet Union, on that day, the United Nations Security Council recognized this North Korean act as invasion and called for an immediate ceasefire. On 27 June, the Security Council adopted S/RES/83, Complaint of aggression upon the Republic of Korea and decided the formation, twenty-one countries of the United Nations eventually contributed to the UN force, with the United States providing 88% of the UNs military personnel. After the first two months of war, South Korean forces were on the point of defeat, forced back to the Pusan Perimeter, in September 1950, an amphibious UN counter-offensive was launched at Inchon, and cut off many North Korean troops. Those who escaped envelopment and capture were rapidly forced back north all the way to the border with China at the Yalu River, at this point, in October 1950, Chinese forces crossed the Yalu and entered the war. Chinese intervention triggered a retreat of UN forces which continued until mid-1951, after these reversals of fortune, which saw Seoul change hands four times, the last two years of fighting became a war of attrition, with the front line close to the 38th parallel. The war in the air, however, was never a stalemate, North Korea was subject to a massive bombing campaign. Jet fighters confronted each other in combat for the first time in history. The fighting ended on 27 July 1953, when an armistice was signed, the agreement created the Korean Demilitarized Zone to separate North and South Korea, and allowed the return of prisoners. However, no treaty has been signed, and the two Koreas are technically still at war. Periodic clashes, many of which are deadly, continue to the present, in the U. S. the war was initially described by President Harry S. Truman as a police action as it was an undeclared military action, conducted under the auspices of the United Nations. In South Korea, the war is referred to as 625 or the 6–2–5 Upheaval. In North Korea, the war is referred to as the Fatherland Liberation War or alternatively the Chosǒn War. In China, the war is called the War to Resist U. S
38.
196th Infantry Brigade (United States)
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The 196th Infantry Brigade, also known as the Charger Brigade was first formed on 24 June 1921 as part of the United States Army Reserves 98th Division with the responsibility of training soldiers. During World War II, the 98th initially defended Kauai, Hawaii and Maui, Hawaii, the Division began intensive training in May 1945 to prepare for the invasion of Japan, but the war ended before they could depart Hawaii. The 196th served in Vietnam from 15 July 1966 through 29 June 1972, the Brigade was reactivated in September 1965 at Fort Devens Massachusetts, where it was originally scheduled to be sent to the Dominican Republic. Instead the Army rushed it to Vietnam, the Brigade departing on 15 July 1966 via transport ships and it began operations almost immediately in the western area of III Corps Tactical Zone. The 196th conducted Operation Cedar Falls, Gadsden, Lancaster, Junction City, Benton, Attleboro turned into a major action after a large enemy base camp was found on 19 October 1966. In February 1967, Gen. William Westmoreland ordered the formation of a division sized Army task force to reinforce American forces in I Corps Tactical Zone, the 196th was selected to form a part of the task force. Task Force Oregon became operational on April 20,1967, when troops from the 196th landed at Chu Lai in I Corps, over the next month, it was joined by the 1st Brigade of the 101st Airborne Division and the 3rd Brigade of the 25th Infantry Division. On 25 April 1967 Task Force Oregon was redesignated the 23rd Infantry Division, later, the 1st Brigade, 101st Airborne and the 3rd Brigade, 25th Infantry Division were replaced by the 198th and 11th Light Infantry Brigades. As part of the 23rd, the 196th participated in Operations Wheeler/Wallowa, Golden Fleece, Fayette Canyon, Frederick Hill, Lamar Plain, Elk Canyon I, in early May 1968, the 2-1 Infantry of the 196th was flown in to assist at the Battle of Kham Duc. On 29 November 1971, the 196th became a temporary entity to safeguard this same area of operations. In April 1971, the 196th moved to Da Nang to assist in port security duties, the brigade suffered 1,188 KIA, and 5,591 WIA in Vietnam. Since 2001, the 196th Infantry Brigade has trained nearly 10,000 Soldiers that deployed to combat operations in Iraq, Afghanistan, Horn of Africa. The brigade also exercises training and readiness oversight for the Hawaii, Guam, annually the 196th Infantry Brigade conducts Kaimalu O Hawaii and Konfitma All Hazard CST Field Training Exercises in Hawaii and Saipan respectively. In 2007, the 196th Infantry Brigade was awarded the Army Superior Unit Award for its support to the War on Terror in preparing RC units and Soldiers for combat duty. In Season 2, episode 4 of the TV series Prison Break, the Brigade, A History, Its Organization and Employment in the US Army Summers, Harry G. Historical Atlas of the Vietnam War, http, //196th. org/History. htm http, //www. usarpac. army. mil/196th/history. htm http, //www. lzcenter. com/Operations. html Article by Pete Shotts used with permission
39.
United States Army Reserve
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The United States Army Reserve is the federal reserve force of the United States Army. Together, the Army Reserve and the Army National Guard constitute the Army element of the Reserve components of the United States Armed Forces. On 30 June 2016, Lieutenant General Charles D. Luckey became the 33rd Chief of Army Reserve, on 23 April 1908 Congress created the Medical Reserve Corps, the official predecessor of the Army Reserve. This organization provided a pool of trained Reserve officers and enlisted men for use in war. The Organized Reserve included the Officers Reserve Corps, Enlisted Reserve Corps, the Organized Reserves were redesignated 25 March 1948 as the Organized Reserve Corps. Recognizing the importance of the Organized Reserve to the World War II effort, Congress authorized retirement, a tentative troop basis for the Organized Reserve Corps, prepared in March 1946, outlined 25 divisions, three armored, five airborne, and 17 infantry. These divisions and all other Organized Reserve Corps units were to be maintained in one of three categories, labeled Class A, Class B, and Class C. The troop basis listed nine divisions as Class A, nine as Class B, eventually the War Department agreed and made the appropriate changes. Although the dispute over Class A units lasted several months, the War Department proceeded with the reorganization of the Organized Reserve Corps divisions during the summer of 1946. That all divisions were to begin as Class C units, progressing to the categories as men and equipment became available. Also, the War Department wanted to take advantage of the pool of trained reserve officers, by that time Army Ground Forces had been reorganized as an army group headquarters that commanded six geographic armies. The armies replaced the nine areas of the prewar era. The First United States Army declined to support a division. After the change, the Organized Reserve Corps had four airborne, the Second Army insisted upon the number 80 for its airborne unit because the division was to be raised in the prewar 80th Divisions area, not that of the 99th. Finally, the 103rd Infantry Division, organized in 1921 in New Mexico, Colorado, and Arizona, was moved to Iowa, Minnesota, South Dakota, a major problem in forming divisions and other units in the Organized Reserve Corps was adequate housing. While many National Guard units owned their own armories, some dating back to the nineteenth century, although the War Department requested funds for needed facilities, Congress moved slowly in response. During the summer and fall of 1951 the six army commanders in the United States, staff agencies, the army commanders urged that all divisions in the Organized Reserve Corps be infantry divisions because they believed that the reserves could not adequately support armored and airborne training. They thought thirteen, rather than twelve, reserve divisions should be maintained to provide a geographic distribution of the units
40.
98th Infantry Division (United States)
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The 98th Infantry Division was a unit of the United States Army in the closing months of World War I and during World War II. The unit is now one of the U. S. Army Reserves training divisions, the 98th Training Divisions current primary mission is to conduct Initial Entry Training for new soldiers. It is one of three training divisions subordinate to the 108th Training Command, since 1959, however, the 98th Training Division has been a unit of the U. S. Army Reserve with the primary mission of training Soldiers. S. as well as Puerto Rico. The 98th was activiated as one of the Armys infantry divisions on 15 Sep,1942 at Camp Breckinridge, Kentucky, filling its ranks primarily with soldiers from the New York and New England regions. Slated as a participant in Operation Olympic, scheduled for 1 November 1945 one of two planned invasions of Japan, the war drew to a close before the 98th was deployed to a combat zone. Instead, the 98th Division arrived in Japan on 27 Sep 1945 and served in Osaka, awards earned by 98th Division soldiers during this period include, Legion of Merit-1, Soldiers Medal-8, Bronze Star −146. Commanding Generals during the World War II era were Major General Paul L. Ransom, Major General George W. Griner, Jr. Major General Ralph C. Smith and Major General Arthur McK. On 18 April 1947, the Iroquois Division was reactivated in Rochester, New York on reserve status and it had been previously planned to be an airborne division. A note on the troop list nevertheless indicated that the unit was to be reorganized and redesignated as an airborne unit upon mobilization and was to train as such. The reorganization of 1 May 1959, redesignated the 98th Infantry Division as the 98th Division, the changes of 1968 also ushered in the designation and training of Army Reserve Drill Sergeants, a significant and enduring innovation. S. The 98th would maintain this basic organization and mission for the next 14 years, on 3 September 2004, the 98th Division received mobilization orders for Operation Iraqi Freedom. This mobilization was to be the first overseas deployment for the unit since World War II, the mission, known as the Foreign Army Training Assistance Command, consisted primarily of training the new Iraqi Army and Iraqi security forces. An expeditionary force of more than 700 Iroquois warriors were trained and equipped at four sites, Camp Atterbury, Fort Bliss, Fort Hood, the demands of Operation Iraqi Freedom required an accelerated training schedule which crammed as many warfighting skills as possible into a forty-one-day period. The 98th made full use of the 33,000 acres at Camp Atterbury and it was at Camp Atterbury that the advisory support teams, the heart of the FA-TRAC mission, transformed to cohesive units in long days. Instruction and support teams spread out across all points in Iraq from Al Kasik in the north to as far south as Umm Qasr and they established contact with Iraqi security units with the help of interpreters and helped build the six divisions of the new Iraqi Army. They also established officer and noncommissioned officer education schools at the Kirkush Military Training Base and they trained Iraqi police, the Highway Patrol, the special Police Commandos and the Iraqi Border Police. The division also fielded soldiers to other locations as Guantanamo Bay, Cuba. Five 98th Training Division soldiers were killed in action during the deployment to Iraq in 2004-05
41.
196th Reconnaissance Squadron
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The 196th Reconnaissance Squadron is a unit of the 163d Reconnaissance Wing of the California Air National Guard stationed at March Joint Air Reserve Base, California. The 196th is equipped with the MQ-1 Predator, the 196th Reconnaissance Squadrons primary mission is to support the war on terrorism by providing reconnaissance, flying the MQ-1 Predator,24 hours a day, seven days a week. The FTU trains pilots and sensor operators for ACC, and trains enlisted personnel to assemble, disassemble, maintain and repair the Predator for Air Education, during World War II the 411th was assigned to the European Theater of Operations, Ninth Air Force in Western Europe. It was equipped with P-47 Thunderbolts, the unit flew its first combat mission on 8 May 1944, a fighter sweep over Normandy. It then took part in activities, escorting B-26 Marauders to attack airdromes, bridges. The squadron bombed such targets as troops in the Falaise-Argentan area in August 1944, during the Battle of the Bulge, from December 1944 to January 1945, the 411th concentrated on the destruction of bridges, marshalling yards, and highways. It also flew missions to support ground operations in the Rhine Valley in March 1945, hitting airfields, motor transport. The squadron continued tactical air operations until 4 May 1945, the 411th returned to the United States and prepared for transfer to the Pacific Theater during the Summer of 1945. The Japanese capitulation in August led to the squadrons inactivation in November 1945, the wartime 411th Fighter Squadron was allotted to the California Air National Guard, on 24 May 1946 and redesignated as the 196th Fighter Squadron. It was organized at Norton Air Force Base, California, on 12 September 1946, the squadron was equipped with P-51D Mustangs and assigned to the 146th Fighter Group, at Van Nuys Airport by the National Guard Bureau. The squadron trained for fighter missions and air-to-air combat under the supervision of Fourth Air Force. In June 1948, the unit received 25 F-80C Shooting Star aircraft, the 196th was one of the first Air National Guard units to receive these new jets. The 196th was federalized on 10 October 1950 due to the Korean War, the groups other operational squadrons were the 128th Fighter Squadron of the Georgia Air National Guard and the 159th Fighter Squadron of the Florida Air National Guard. At George the three squadrons were equipped with Lockheed F-80Cs and began operational training. As a result, they lost their character as squadrons of the Georgia, Florida, in April 1951 the 116th Fighter-Bomber Group began receiving brand new F-84E Thunderjets directly from Republic Aviation. On 14 May the 116th Fighter-Bomber Wing received a Warning Order for an impending transfer, and they expected to be transferred to Europe. With a Readiness Date of 25 June, the 116 FBW was ready to move, however, two days later the wing received orders transferring them to Japan. Fifty-four F-84Es had to be obtained from Bergstrom AFB, Texas, the 196th FBS departed from San Diego on 10 July on the USS Sitkoh Bay
42.
California Air National Guard
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The California Air National Guard is the air force militia of the U. S. State of California. It is, along with the California Army National Guard, an element of the California National Guard, as state militia units, the units in the California Air National Guard are not in the normal United States Air Force chain of command. They are under the jurisdiction of the Governor of California through the office of the California Adjutant General unless they are federalized by order of the President of the United States. The California Air National Guard is headquartered in Sacramento, and its commander is currently Major General Jon K. Kelk, under the Total Force concept, California Air National Guard units are considered to be Air Reserve Components of the United States Air Force. California ANG units are trained and equipped by the Air Force and are gained by a Major Command of the USAF if federalized. State missions include disaster relief in times of earthquakes, hurricanes, floods and forest fires, search and rescue, protection of public services. It is under state jurisdiction and its members are employed only within the State of California and it is not subject to be called, ordered or assigned as any element of the federal armed forces. Currently in transition from a KC-135 Stratotanker air refueling mission to an MQ-1 Predator ISR wing, executing global unmanned aerial systems, combat support, and humanitarian missions. The California Air National Guard origins date to 28 August 1917 with the establishment of the 115th Aero Squadron as part of the World War I United States Army Air Service. The 115th served in France on the Western Front, constructed facilities and engaged in supply, the Militia Act of 1903 established the present National Guard system, units raised by the states but paid for by the Federal Government, liable for immediate state service. If federalized by Presidential order, they fall under the military chain of command. On 1 June 1920, the Militia Bureau issued Circular No.1 on organization of National Guard air units, initially the Unit held its meetings at Clover Field, Santa Monica, using Reserve Equipment planes for flying. Later on, the Squadron met at the National Guard Armory, in 1925, several months after its organization, the Squadron moved to permanent quarters at Griffith Park Aerodrome in Los Angeles. These unit designations were allotted and transferred to various State National Guard bureaus to provide them unit designations to re-establish them as Air National Guard units, the modern California ANG received federal recognition on 1 July 1946 as the 62d Fighter Wing at Van Nuys Airport, Van Nuys. Its 115th Bombardment Squadron was equipped with A-26 Invader light bombers, on 16 September 1946, its 146th Fighter Group was also formed at Van Nuys, with several fighter squadrons equipped with F-51 Mustangs and its mission was the air defense of the state. The 61sts mission was the air defense of Northern California, the 62d, today, units of the CA ANG perform a homeland defense mission, worldwide airlift missions, aireal firefighting, combat search and rescue, and Unmanned Aireal Reconnaissance missions. After the 11 September 2001 terrorist attacks on the United States, in December 2007, after the grounding of F-15 fighters due to potential structural problems, the California Air National Guard assumed responsibility for defense of the western United States. This was the first time that a single states fighter wing took responsibility of defense for an entire coast, also, California ANG units have been deployed overseas as part of Operation Enduring Freedom in Afghanistan and Operation Iraqi Freedom in Iraq as well as other locations as directed
43.
USNS Kanawha (T-AO-196)
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United States Naval Ship USNS Kanawha is a Henry J. Kaiser-class fleet replenishment oiler of the United States Navy in non-commissioned service in the Military Sealift Command. USNS Kanawha, the tenth Henry J. Kaiser-class ship, was laid down by the Sun Shipbuilding and Drydock Company in Chester, Pennsylvania, on 13 July 1989. She was launched on 22 September 1990 and delivered to the U. S. Navy, Kanawha is in active service today with the Military Sealift Command Naval Fleet Auxiliary Force and is assigned to the U. S. The Henry J. Kaiser-class oilers were preceded by the shorter Cimarron-class fleet replenishment oilers, Kanawha has an overall length of 206.5 metres. It has a beam of 29.7 metres and a draft of 11 metres, the oiler has a displacement of 41,353 tonnes at full load. It has a capacity of 180,000 imperial barrels of fuel or fuel oil. It can carry a dry load of 690 square metres and can refrigerate 128 pallets of food, the ship is powered by two 10 PC4.2 V570 Colt-Pielstick diesel engines that drive two shafts, this gives a power of 25.6 megawatts. The Henry J. Kaiser-class oilers have maximum speeds of 20 knots and they were built without armaments but can be fitted with close-in weapon systems. The ship has a platform but not any maintenance facilities. It is fitted with five fuelling stations, these can fill two ships at the time and the ship is capable of pumping 900,000 US gallons of diesel or 540,000 US gallons of jet fuel per hour. It has a complement of eighty-nine civilians, twenty-nine spare crew, on November 18,2010, USNS Kanawha “briefly came into contact” with HMCS Fredericton of the Royal Canadian Navy during a replenishment-at-sea manoeuvre off the coast of Florida. There were no injuries, but both ships suffered “superficial” damage consisting of scrapes and dents on both hulls and this article includes information collected from the Naval Vessel Register, which, as a U. S. government publication, is in the public domain. The entry can be found here, navSource Online, Service Ship Photo Archive T-AO-196 Kanawha USNS Kanawha Wildenberg, Thomas. Gray Steel and Black Oil, Fast Tankers and Replenishment at Sea in the U. S. Navy, 1912-1995
44.
United States Navy
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The United States Navy is the naval warfare service branch of the United States Armed Forces and one of the seven uniformed services of the United States. The U. S. Navy is the largest, most capable navy in the world, the U. S. Navy has the worlds largest aircraft carrier fleet, with ten in service, two in the reserve fleet, and three new carriers under construction. The service has 323,792 personnel on duty and 108,515 in the Navy Reserve. It has 274 deployable combat vessels and more than 3,700 operational aircraft as of October 2016, the U. S. Navy traces its origins to the Continental Navy, which was established during the American Revolutionary War and was effectively disbanded as a separate entity shortly thereafter. It played a role in the American Civil War by blockading the Confederacy. It played the role in the World War II defeat of Imperial Japan. The 21st century U. S. Navy maintains a global presence, deploying in strength in such areas as the Western Pacific, the Mediterranean. The Navy is administratively managed by the Department of the Navy, the Department of the Navy is itself a division of the Department of Defense, which is headed by the Secretary of Defense. The Chief of Naval Operations is an admiral and the senior naval officer of the Department of the Navy. The CNO may not be the highest ranking officer in the armed forces if the Chairman or the Vice Chairman of the Joint Chiefs of Staff. The mission of the Navy is to maintain, train and equip combat-ready Naval forces capable of winning wars, deterring aggression, the United States Navy is a seaborne branch of the military of the United States. The Navys three primary areas of responsibility, The preparation of naval forces necessary for the prosecution of war. The development of aircraft, weapons, tactics, technique, organization, U. S. Navy training manuals state that the mission of the U. S. Armed Forces is to prepare and conduct prompt and sustained combat operations in support of the national interest, as part of that establishment, the U. S. Navys functions comprise sea control, power projection and nuclear deterrence, in addition to sealift duties. It follows then as certain as that night succeeds the day, that without a decisive naval force we can do nothing definitive, the Navy was rooted in the colonial seafaring tradition, which produced a large community of sailors, captains, and shipbuilders. In the early stages of the American Revolutionary War, Massachusetts had its own Massachusetts Naval Militia, the establishment of a national navy was an issue of debate among the members of the Second Continental Congress. Supporters argued that a navy would protect shipping, defend the coast, detractors countered that challenging the British Royal Navy, then the worlds preeminent naval power, was a foolish undertaking. Commander in Chief George Washington resolved the debate when he commissioned the ocean-going schooner USS Hannah to interdict British merchant ships, and reported the captures to the Congress
45.
Henry J. Kaiser-class oiler
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The Henry J. Kaiser class is an American class of eighteen fleet replenishment oilers which began construction in August 1984. There are stations on both sides of each ship for underway replenishment of fuel and stores, the ships in this class have a small capacity to carry and transfer fresh and frozen foods as well as other materials, and have two dry cargo transfer rigs. Patuxent, Laramie, and Rappahannock differ from the other 15 ships in having double hulls to meet the requirements of the Oil Pollution Act of 1990. Hull separation is 6 feet at the sides and 6 feet 6 inches on the bottom, the circumstances of the construction program were convoluted and it is worthwhile to spell them out here. A second-source contract, for T-AOs 191 and 192, was awarded to Pennsylvania Shipbuilding Company, the Navy then determined that the ships were no longer needed as oilers, and undertook a study of the feasibility of converting them to ammunition ships. This study concluded that such a conversion was cost-prohibitive and the ships were placed in storage in an incomplete condition. They were sold for recycling in 2011, the class is named for its lead unit, Henry J. Kaiser, which in turn is named for the American industrialist and shipbuilder Henry J. Kaiser. The tenth through eighteenth ships were named after American rivers, which is a traditional naming convention for U. S. Navy oilers. In U. S. Navy service, the serve in a non-commissioned status in the Military Sealift Command. After joining the fleet, the 16 completed ships all saw service between 1986 and 1996, when Andrew J. Higgins became the first unit of the class to be laid up. Since then, some of the others have also spent periods out of service in reserve or in an operational status. Andrew J. Higgins never re-entered U. S. service after being laid up in 1996 and she was sold to Chile in 2009 and was commissioned into the Chilean Navy in 2010 as Almirate Montt
46.
Fleet replenishment oiler
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A replenishment oiler is a naval auxiliary ship with fuel tanks and dry cargo holds, which can conduct underway replenishment on the high seas. Several countries have used replenishment oilers, the US Navy hull classification symbol for this type of ship was AOR. Replenishment oilers are slower and carry fewer dry stores than the US Navys fast combat support ships, the development of the oiler paralleled the change from coal- to oil-fired boilers in warships. Though arguments related to fuel security were made against such a change, one of the first generation of blue-water navy oiler support vessels was the British RFA Kharki, active 1911 in the run-up to the First World War. Such vessels heralded the transition from coal to oil as the fuel of warships and removed the need to rely on, modern examples of the fast combat support ship include the large British Fort-Class, displacing and 669 ft in length, and the Australian HMAS Sirius. For smaller navies, such as the Royal New Zealand Navy, such ships are designed to carry large amounts of fuel and dry stores for the support of naval operations far away from port. Replenishment oilers are also equipped with extensive medical and dental facilities than smaller ships can provide. Such ships are equipped with multiple refueling gantries to refuel and resupply ships at a time. The process of refueling and supplying ships at sea is called underway replenishment, furthermore, such ships often are designed with helicopter decks and hangars. This allows the operation of rotary-wing aircraft, which allows the resupply of ships by helicopter and this process is called vertical replenishment. They may also carry man-portable air-defense systems for air defense capability. Kaiser-class oiler (United States Navy and Chilean Navy ex-USNS Andrew J
47.
USS Edith M. III (SP-196)
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USS Edith M. III was a United States Navy patrol vessel in commission from 1917 to 1919. Edith M. III was built by V. J. Osborn at Croton-on-Hudson, New York, as a civilian motorboat of the same name in 1909. The United States Navy purchased her for World War I service in June 1917 and commissioned her on 5 November 1917 as USS Edith M. III with Boatswain A. R. Mulkins, USNRF, in command. Edith M. III was assigned to the 3rd Naval District, decommissioned on 8 May 1919, Edith M. III was sold on 2 July 1919. List of United States Navy ships Patrol boat World War I This article incorporates text from the public domain Dictionary of American Naval Fighting Ships, the entry can be found here. Navy History and Heritage Command Online Library of Selected Images, U. S. Navy Ships, USS Edith M. III, previously the civilian motor boat Edith M. III. NavSource Online, Section Patrol Craft Photo Archive Edith M. III
48.
Motorboat
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A motorboat, speedboat, or powerboat is a boat which is powered by an engine. Some motorboats are fitted with engines, others have an outboard motor installed on the rear, containing the internal combustion engine, the gearbox. An inboard-outboard contains a hybrid of a powerplant and an outboard, where the combustion engine is installed inside the boat. There are two configurations of an inboard, V-drive and direct drive, the V-drive has become increasingly popular due to wakeboarding and wakesurfing. Motorboats vary greatly in size and configuration, from the four-meter, the earliest boat to be powered by a petrol engine was tested on the Neckar River by Gottlieb Daimler and Wilhelm Maybach in 1886, when they tested their new longcase clock engine. It had been constructed in the greenhouse in Daimlers back yard. The first public display took place on the Waldsee in Cannstatt, today a suburb of Stuttgart, the engine of this boat had a single cylinder of 1 horse power. Daimlers second launch in 1887 had a second cylinder positioned at an angle of 15 degrees to the first one, the first successful motor boat was designed by the Priestman Brothers in Hull, England, under the direction of William Dent Priestman. The company began trials of their first motorboat in 1888, the engine was powered with kerosene and used an innovative high-tension ignition system. The company was the first to large scale production of the motor boat. Working in the garden of their home in Olton, Warwickshire, they designed, in 1897, he produced a second engine similar in design to his previous one but running on benzene at 800 r. p. m. The engine drove a reversible propeller, an important part of his new engine was the revolutionary carburettor, for mixing the fuel and air correctly. His invention was known as a carburetor, because fuel was drawn into a series of wicks. He patented this invention in 1905, the Daimler Company began production of motor boats in 1897 from its manufacturing base in Coventry. The engines had two cylinders and the charge of petroleum and air was ignited by compression into a heated platinum tube. The engine gave about six horse-power, the petrol was fed by air pressure to a large surface carburettor and also an auxiliary tank which supplied the burners for heating the ignition tubes. It was not until 1901 that a safer apparatus for igniting the fuel with a spark was used in motor boats. Interest in fast motorboats grew rapidly in the years of the 20th century
49.
Clemson-class destroyer
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The Clemson class was a series of 156 destroyers which served with the United States Navy from after World War I through World War II. The Clemson class was a redesign of the Wickes class for greater fuel capacity. As finally built, the Clemson class would be a fairly straightforward expansion of the Wickes-class destroyers and these designs included a reduction in speed to between 26–28 knots by eliminating two boilers, freeing up displacement for depth charges and more fuel. This proposal foreshadowed the destroyer escorts of World War II, upgrading the gun armament from 4-inch to 5-inch guns was also considered, but only five ships were armed with 5-inch guns. In addition, the stern of the Wickes-class destroyers resulted in a large turning radius. In the end the General Board decided the 35 knots speed be retained so as to allow the Clemson class to be used as a fleet escort, the pressing need for destroyers overruled any change that would slow production compared to the proceeding Wickes class. Wing tanks for oil were installed on either side of the ships to increase the operational range. This design choice meant the oil would be stored above the waterline and create additional vulnerability. The class resulted from a General Board recommendation for further destroyers to combat the threat, culminating in a total of 267 Wickes-. However, the design of the ships remained optimized for operation with the battleship fleet, the main armament was the same as the Wickes class, four 4-inch /50 caliber guns and twelve 21-inch torpedo tubes. Although the design provided for two guns, most ships carried a single 3-inch /23 caliber AA gun, typically on the aft deckhouse. A frequent modification was replacing the aft 4-inch gun with the 3-inch gun to more room for the depth charge tracks. Anti-submarine armament was added during or after construction, typically, two depth charge tracks were provided aft, along with a Y-gun depth charge projector forward of the aft deckhouse. Despite the provision for 5-inch guns, only seven ships were built with a gun armament. USS Hovey and USS Long had twin 4-inch/50 mounts for a total of eight guns, while an increased rudder size helped, the answer would be in a redesigned stern, but this was not implemented. They were reported to be prone to rolling in light load conditions. The flush deck gave the great strength but this also made the deck very wet. Fourteen ships of the class were involved in the Honda Point Disaster in 1923, many never saw wartime service, as a significant number were decommissioned in 1930 and scrapped as part of the London Naval Treaty
50.
Destroyer
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Before World War II, destroyers were light vessels with little endurance for unattended ocean operations, typically a number of destroyers and a single destroyer tender operated together. After the war, the advent of the missile allowed destroyers to take on the surface combatant roles previously filled by battleships. This resulted in larger and more powerful guided missile destroyers more capable of independent operation, the emergence and development of the destroyer was related to the invention of the self-propelled torpedo in the 1860s. A navy now had the potential to destroy an enemy battle fleet using steam launches to launch torpedoes. Fast boats armed with torpedoes were built and called torpedo boats, the first seagoing vessel designed to fire the self-propelled Whitehead torpedo was the 33-ton HMS Lightning in 1876. She was armed with two drop collars to launch these weapons, these were replaced in 1879 by a torpedo tube in the bow. By the 1880s, the type had evolved into small ships of 50–100 tons, in response to this new threat, more heavily gunned picket boats called catchers were built which were used to escort the battle fleet at sea. The anti-torpedo boat origin of this type of ship is retained in its name in other languages, including French, Italian, Portuguese, Czech, Greek, Dutch and, up until the Second World War, Polish. At that time, and even into World War I, the function of destroyers was to protect their own battle fleet from enemy torpedo attacks. The task of escorting merchant convoys was still in the future, an important development came with the construction of HMS Swift in 1884, later redesignated TB81. This was a torpedo boat with four 47 mm quick-firing guns. At 23.75 knots, while still not fast enough to engage torpedo boats reliably. Another forerunner of the torpedo boat destroyer was the Japanese torpedo boat Kotaka, designed to Japanese specifications and ordered from the London Yarrow shipyards in 1885, she was transported in parts to Japan, where she was assembled and launched in 1887. The 165-foot long vessel was armed with four 1-pounder quick-firing guns and six torpedo tubes, reached 19 knots, in her trials in 1889, Kotaka demonstrated that she could exceed the role of coastal defense, and was capable of accompanying larger warships on the high seas. The Yarrow shipyards, builder of the parts for the Kotaka, the first vessel designed for the explicit purpose of hunting and destroying torpedo boats was the torpedo gunboat. Essentially very small cruisers, torpedo gunboats were equipped with torpedo tubes, by the end of the 1890s torpedo gunboats were made obsolete by their more successful contemporaries, the torpedo boat destroyers, which were much faster. The first example of this was HMS Rattlesnake, designed by Nathaniel Barnaby in 1885, the gunboat was armed with torpedoes and designed for hunting and destroying smaller torpedo boats. Exactly 200 feet long and 23 feet in beam, she displaced 550 tons, built of steel, Rattlesnake was un-armoured with the exception of a 3⁄4-inch protective deck
51.
World War II
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World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945, although related conflicts began earlier. It involved the vast majority of the worlds countries—including all of the great powers—eventually forming two opposing alliances, the Allies and the Axis. It was the most widespread war in history, and directly involved more than 100 million people from over 30 countries. Marked by mass deaths of civilians, including the Holocaust and the bombing of industrial and population centres. These made World War II the deadliest conflict in human history, from late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, and formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Poland, Finland, Romania and the Baltic states. In December 1941, Japan attacked the United States and European colonies in the Pacific Ocean, and quickly conquered much of the Western Pacific. The Axis advance halted in 1942 when Japan lost the critical Battle of Midway, near Hawaii, in 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained all of its territorial losses and invaded Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in South Central China and Burma, while the Allies crippled the Japanese Navy, thus ended the war in Asia, cementing the total victory of the Allies. World War II altered the political alignment and social structure of the world, the United Nations was established to foster international co-operation and prevent future conflicts. The victorious great powers—the United States, the Soviet Union, China, the United Kingdom, the Soviet Union and the United States emerged as rival superpowers, setting the stage for the Cold War, which lasted for the next 46 years. Meanwhile, the influence of European great powers waned, while the decolonisation of Asia, most countries whose industries had been damaged moved towards economic recovery. Political integration, especially in Europe, emerged as an effort to end pre-war enmities, the start of the war in Europe is generally held to be 1 September 1939, beginning with the German invasion of Poland, Britain and France declared war on Germany two days later. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or even the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred simultaneously and this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935. The British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the forces of Mongolia and the Soviet Union from May to September 1939, the exact date of the wars end is also not universally agreed upon. It was generally accepted at the time that the war ended with the armistice of 14 August 1945, rather than the formal surrender of Japan
52.
USS Logan (APA-196)
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USS Logan was a Haskell-class attack transport of the United States Navy, named for counties in Colorado, Illinois, Kansas, Kentucky, Ohio, Oklahoma, Nebraska, North Dakota, and West Virginia. The Haskell-class design, Maritime Commission standard type VC2-S-AP5, is a sub type of the World War II Victory ship design, the ship was laid down on 27 May 1944 by Kaiser Co. Vancouver, Washington, launched 19 September 1944, sponsored by Mrs. Paul E. Lattner, acquired by the Navy, departing Hawaii on 26 January 1945, the ship stopped at Saipan, to use that recently liberated rugged atoll for five more days of very realistic training. Pulling to within 1,000 yards of the beaches, Logan lowered all boats. Immediately thereafter, the beachmasters, engineers, and quartermasters were quickly dispatched ashore, for Iwo Jima the ship was assigned to the 23rd Marines of the 4th Marine Division. She landed Hq Company of the 133rd NCB on yellow beach D Day for Shore Party duty, later, on 21 February, at 0445 hours, The Logan rammed the USS NAPA along frames 98-102. The impact resulted in a 15 foot long hull breach to the NAPA, with Minimal damage done to the Logan, she continued her service in Iwo Jima until she received her departure orders. With 200 wounded soldiers resting comfortably in sick bay, the ship departed Iwo Jima on 28 February, stopping briefly at Saipan, she made Guam on 4 March and debarked the casualties. The next day she sailed back to Saipan to prepare for the assault on Okinawa, following three weeks of extensive rehearsals off Saipan and Tinian with Rear Admiral Wrights TG51. The same maneuver was successful the following day. For the next six days, Rear Admiral Wrights group laid off Okinawa, fully prepared, if needed, by 11 April the success of the campaign was assured, and the task group steamed back to Saipan. The ship was hit by a Japanese Kamikaze and Officer Percy McDonald Scarbrough, Scarbrough received the Meritorious Service Medal for his heroism and quick thinking. Logan maintained her readiness with amphibious exercises off Saipan and in the New Hebrides. She reached San Francisco on 13 August, two days before the Japanese surrender, after V-J Day the tremendous job of occupying Japan and bringing home the veteran troops still faced the Navy. Consequently, Logan departed San Francisco on 23 August to embark troops at Pearl Harbor for occupation duty in Japan and she arrived at Honshū on 27 September. On 10 October she proceeded to the Philippines, thence to the Marshall Islands, the ship arrived Seattle on 27 October 1945. In November and again in January 1946, Logan made Magic Carpet runs to the Philippines to bring the boys home and she was released from Magic Carpet on 6 March, decommissioned on 27 November 1946, and joined the Pacific Reserve Fleet at San Francisco. Logan recommissioned on 10 November 1951, during the height of the Korean War, after shakedown and refresher training off San Diego, the attack transport departed for Yokosuka, Japan, on 9 April 1952, and arrived 26 April
53.
Haskell-class attack transport
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Haskell-class attack transports were amphibious assault ships of the United States Navy created in 1944. They were designed to transport 1,500 troops and their combat equipment, the Haskells were very active in the World War II Pacific Theater of Operations, landing Marines and Army troops and transporting casualties at Iwo Jima and Okinawa. Ships of the class were among the first Allied ships to enter Tokyo Bay at the end of World War II, after the end of World War II, most participated in Operation Magic Carpet, the massive sealift of US personnel back to the United States. A few of the Haskell class were reactivated for the Korean War, the Haskell class, Maritime Commission standard type VC2-S-AP5, is a sub‑type of the World War II Victory ship design. 117 were launched in 1944 and 1945, with 14 more being finished as another VC2 type or canceled, the VC2-S-AP5 design was intended for the transport and assault landing of over 1,500 troops and their heavy combat equipment. During Operation Magic Carpet, up to 1,900 personnel per ship were carried homeward, the Haskells carried 25 landing craft to deliver the troops and equipment right onto the beach. The 23 main boats were the 36 feet long, LCVP, the LCVP was designed to carry 36 equipped troops. The other 2 landing craft were the 50 foot long LCM, capable of carrying 60 troops or 30 tons of cargo, the Haskell-class ships were armed with one 5/38 caliber gun, twelve Bofors 40 mm guns, and ten Oerlikon 20 mm guns. See List of Haskell-class attack transports, Haskell-class attack transports included APA-117, USS Haskell, the lead ship, through APA-247, the never completed USS Mecklenburg. The hulls for APA-181 through APA-186 were repurposed to be hospital ships before they were named, ultimately those hospital ships were built on larger C4 plan and the six VC2 hulls were built in a merchant configuration. APA-240 through APA-247 were named, but cancelled in 1945 when the war ended, with the special exception of the USS Marvin H. McIntyre, the Haskell-class ships were all named after counties of the United States. Most of the Haskell-class ships were mothballed in 1946, with only a few remaining in service, many of the Haskell class were scrapped in 1973-75. A few were converted into Missile Range Instrumentation Ships, the USS Gage, the last remaining ship in the Haskell configuration, was scrapped in 2009 at ESCO Marine, in Brownsville, Tx. The USS Sherburne, which was converted and renamed USS Range Sentinel, lasted until she was scrapped in 2012
54.
Attack transport
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Attack transport is a United States Navy ship classification for a variant of ocean-going troopship adapted to transporting invasion forces ashore. Unlike standard troopships – often drafted from commercial shipping fleets – that rely on either a quay or tenders and they are not to be confused with landing ships, which beach themselves to bring their troops directly ashore, or their general British equivalent, the Landing ship, infantry. A total of 388 APA and AKA attack transports were built for service in World War II in at least fifteen classes, depending on class they were armed with one or two 5-inch guns and a variety of 40 mm and 20 mm anti-aircraft weapons. Some of these were outfitted with heavy boat davits and other arrangements to enable them to handle landing craft] for amphibious assault operations. In 1942, when the AP number series had extended beyond 100. Therefore, the new classification of attack transport was created and numbers assigned to fifty-eight APs then in commission or under construction, the actual reclassification of these ships was not implemented until February 1943, by which time two ships that had APA numbers assigned had been lost. Another two transports sunk in 1942, USS George F. Elliott and USS Leedstown, were configured as attack transports. In addition, as part of the 1950s modernization of the Navys amphibious force with faster ships, as a result, only attack transport ships were assigned for the assault, without support from any companion attack cargo ships. This created extreme logistics burdens for the force because it resulted in considerable overloading of the transports with both men and equipment. To compound problems, these forces were not able to assemble or train together before executing the Aleutian invasion on 11 May 1943, lack of equipment and training subsequently resulted in confusion during the landings on Attu. By the end of the 1950s, it was clear that boats would soon be superseded by amphibious tractors and helicopters for landing assault troops. These could not be supported by attack transports in the numbers required, by 1969, when the surviving attack transports were redesignated LPA, only a few remained in commissioned service. The last of these were decommissioned in 1980 and sold abroad, the APA/LPA designation may, therefore, now be safely considered extinct. Nearly identical ships used to transport vehicles, supplies and landing craft, Landing Ship Infantry This article incorporates text from the public domain Dictionary of American Naval Fighting Ships. APA/LPA -- Attack Transports by the US Naval Historical Center