1.
Integer
–
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
Integer
–
Algebraic structure → Group theory
Group theory
2.
Negative number
–
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
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
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
100 (number)
–
The
U.S. hundred-dollar bill, Series 2009.
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
Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.
5.
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
Greek numerals
–
Numeral systems
Greek numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
6.
Roman numerals
–
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
Roman numerals
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Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
–
Numeral systems
Roman numerals
–
A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
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An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
7.
Binary number
–
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
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
8.
Ternary numeral system
–
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
Ternary numeral system
–
Numeral systems
9.
Quaternary numeral system
–
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
Quaternary numeral system
–
Numeral systems
10.
Quinary
–
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
Quinary
–
Numeral systems
11.
Senary
–
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
Senary
–
Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting
Senary
12.
Octal
–
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
Octal
–
Numeral systems
13.
Duodecimal
–
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
Duodecimal
–
Numeral systems
Duodecimal
–
A duodecimal multiplication table
14.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
Hexadecimal
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Numeral systems
Hexadecimal
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Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
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Hexadecimal finger-counting scheme.
15.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
Vigesimal
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Numeral systems
Vigesimal
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The
Maya numerals are a base-20 system.
16.
Base 36
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Base 36
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Numeral systems
Base 36
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34 senary = 22 decimal, in senary finger counting
Base 36
17.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Natural number
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The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
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Natural numbers can be used for counting (one
apple, two apples, three apples, …)
18.
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
Parity (mathematics)
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Rubik's Revenge in solved state
Parity (mathematics)
19.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
On-Line Encyclopedia of Integer Sequences
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On-Line Encyclopedia of Integer Sequences
20.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
Composite number
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Overview
21.
Deficient number
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In number theory, a deficient or deficient number is a number n for which the sum of divisors σ<2n, or, equivalently, the sum of proper divisors s<n. The value 2n − σ is called the numbers deficiency, as an example, consider the number 21. Its proper divisors are 1,3 and 7, and their sum is 11, because 11 is less than 21, the number 21 is deficient. Its deficiency is 2 ×21 −32 =10, since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. An infinite number of even and odd deficient numbers exist. All odd numbers with one or two prime factors are deficient. All proper divisors of deficient or perfect numbers are deficient, there exists at least one deficient number in the interval for all sufficiently large n. Closely related to deficient numbers are perfect numbers with σ = 2n, the natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica. Almost perfect number Amicable number Sociable number Sándor, József, Mitrinović, Dragoslav S. Crstici, Borislav, the Prime Glossary, Deficient number Weisstein, Eric W. Deficient Number
Deficient number
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Overview
22.
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
Iteration
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A pentagon iteration. Connecting alternate corners of a regular pentagon produces a
pentagram which encloses a smaller inverted pentagon. Iterating the process produces a sequence of nested pentagons and pentagrams and also demonstrates
recursion.
23.
Semiprime
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In mathematics, a semiprime is a natural number that is the product of two prime numbers. The semiprimes less than 100 are 4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94, and 95. Semiprimes that are not perfect squares are called discrete, or distinct, by definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its factors are 1,2,13. The total number of prime factors Ω for a n is two, by definition. A semiprime is either a square of a prime or square-free, the square of any prime number is a semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a number is a semiprime without knowing the two factors. A composite n non-divisible by primes ≤ n 3 is semiprime, various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits. These are considered novelties, since their construction method might prove vulnerable to factorization, for a semiprime n = pq the value of Eulers totient function is particularly simple when p and q are distinct, φ = = p q − +1 = n − +1. If otherwise p and q are the same, φ = φ = p = p2 − p = n − p and these methods rely on the fact that finding two large primes and multiplying them together is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of specific large semiprimes, the most recent such challenge closed in 2007. In practical cryptography, it is not sufficient to choose just any semiprime, the factors p and q of n should both be very large, around the same order of magnitude as the square root of n, this makes trial division and Pollards rho algorithm impractical. At the same time they should not be too close together, or else the number can be quickly factored by Fermats factorization method. The number may also be chosen so that none of p −1, p +1, q −1, or q +1 are smooth numbers, protecting against Pollards p −1 algorithm or Williams p +1 algorithm. However, these checks cannot take future algorithms or secret algorithms into account, in 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. It consisted of 1679 binary digits intended to be interpreted as a 23×73 bitmap image, the number 1679 = 23×73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns. Chens theorem Weisstein, Eric W. Semiprime
Semiprime
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Overview
24.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime number
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The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
25.
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
Square number
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m = 1 2 = 1
26.
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
Divisor
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The divisors of 10 illustrated with
Cuisenaire rods: 1, 2, 5, and 10
27.
Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
Point (geometry)
–
Projecting a
sphere to a
plane.
28.
Projective plane
–
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as a plane equipped with additional points at infinity where parallel lines intersect. Thus any two lines in a projective plane intersect in one and only one point. Renaissance artists, in developing the techniques of drawing in perspective, the archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG, RP2. There are many projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces, such embeddability is a consequence of a property known as Desargues theorem, not shared by all projective planes. The last condition excludes the so-called degenerate cases, the term incidence is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression point P is incident with l is used instead of either P is on l or l passes through P. To turn the ordinary Euclidean plane into a projective plane proceed as follows and that point is considered incident with each line of the class. Different parallel classes get different points and these points are called points at infinity. Add a new line which is considered incident with all the points at infinity and this line is called the line at infinity. The extended structure is a plane and is called the Extended Euclidean Plane or the real projective plane. The process outlined above, used to obtain it, is called projective completion or projectivization and this plane can also be constructed by starting from R3 viewed as a vector space, see below. The points of the Moulton plane are the points of the Euclidean plane, to create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, the Moulton plane has parallel classes of lines and is an affine plane. It can be projectivized, as in the example, to obtain the projective Moulton plane
Projective plane
–
These parallel lines appear to intersect in the
vanishing point "at infinity". In a projective plane this is actually true.
Projective plane
–
The Moulton plane. Lines sloping down and to the right are bent where they cross the y -axis.
29.
183rd Airlift Squadron
–
The 183d Airlift Squadron is a unit of the 172d Airlift Wing of the Mississippi Air National Guard stationed at Allen C. Thompson Field Air National Guard Base, Mississippi. The squadron is equipped with the C-17 Globemaster III, the 183d was organized in 1953 as a reconnaissance unit, but converted to the airlift role in 1957. It was called to service during the first Gulf War. The squadron was constituted as the 183d Tactical Reconnaissance Squadron in 1953 and it was organized at Hawkins Field, Mississippi and extended federal recognition on 1 July 1953. The squadron was assigned to the 117th Tactical Reconnaissance Group, of the Alabama Air National Guard, the 183d was initially equipped with World War II-era Douglas RB-26C Invader night photographic reconnaissance aircraft. The black RB-26s were light bombers that were modified for reconnaissance in the late 1940s. Most of the aircraft received were unarmed Korean War veterans, which carried cameras, upon mobilization, the squadron would be gained by Tactical Air Command. In 1957, the B-26 was reaching the end of its operational service, the 183d and its support elements expanded into a group level later that year, when the 172d Air Transport Group was activated. The 183d became the flying squadron. Support elements assigned into the group were the 183d Material Squadron, 183d Air Base Squadron, the squadron moved from Hawkins to Allen C. Thompson Field, another field near Jackson, Mississippi in 1963. The group received the Douglas C-124 Globemaster II heavy intercontinental transport was received in 1966 which meant supplies,1966 was also marked by a change of mobilization command to Military Airlift Command and the name to 183d Military Airlift Squadron. Once more, TAC gained the unit if it was called to federal service and it was upgraded to new 1979 production C-130H aircraft in 1980 and continued to fly tactical airlift missions until the mid-1980s. On 12 July 1986 the first Lockheed C-141B Starlifter to be released from the active duty Air Force was assigned to the Mississippi Air National Guard, with a total of eight aircraft, the unit resumed a global airlift mission and was gained by MAC when mobilized. In March 1988 the squadron took part in the airlift of approximately 3200 troops and almost 1000 tons of cargo on an exercise to Palmerola Air Base, the 183d was the only Air National Guard unit to participate in this airlift of troops to Honduras. On 6 December 1988 the Soviet Republic of Armenia suffered a powerful earthquake, the first Air Guard aircraft to fly to Armenia was a C-141B from the 183d. From 20 December 1989 to 12 January 1990 the 183d flew 21 sorties in support of Operation Just Cause, the unit transported 403.6 tons of cargo and 1,274 passengers during the operation. On 7 August 1990 the 172ds support of Operation Desert Shield, on 24 August 1990 the 183d Airlift Squadron was one of the first two units to be called into active federal service and moved to Charleston Air Force Base, South Carolina. Until May 1991, shen the squadron was returned to control the 148 members of the 183d flew 2,880 sorties which transported 15,837 passengers and 25,949.2 tons of cargo
183rd Airlift Squadron
–
C-17 Globemaster III from the 183d Airlift Squadron
183rd Airlift Squadron
–
183rd Airlift Squadron emblem
183rd Airlift Squadron
–
C-121, C-119 ans C-46 at Hawkins Field
183rd Airlift Squadron
–
183d Military Airlift Squadron C-124 Globemaster II
30.
Mississippi Air National Guard
–
The Mississippi Air National Guard is the air force militia of the State of Mississippi, United States of America. It is, along with the Mississippi Army National Guard, an element of the Mississippi National Guard, as state militia units, the units in the Mississippi Air National Guard are not in the normal United States Air Force chain of command. They are under the jurisdiction of the Governor of Mississippi though the office of the Mississippi Adjutant General unless they are federalized by order of the President of the United States. The Mississippi Air National Guard is headquartered in Jackson, and its commander is currently Major General William O. Hill, under the Total Force concept, Mississippi Air National Guard units are considered to be Air Reserve Components of the United States Air Force. Mississippi 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. 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, the Mississippi Air National Guard origins date to 18 August 1939 with the establishment of the 153d Observation Squadron and is oldest unit of the Mississippi Air National Guard. It is one of the 29 original National Guard Observation Squadrons of the United States Army National Guard formed before World War II. The 153d Observation Squadron was ordered into service on 15 October 1940 as part of the buildup of the Army Air Corps prior to the United States entry into World War II. 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 Mississippi ANG received federal recognition on 12 September 1946 as the 153d Fighter Squadron at Key Field, Meridian. It was equipped with F-47 Thunderbolts and its mission was the air defense of the state, a second Mississippi ANG unit, the 183d Tactical Reconnaissance Squadron, was organized and federally recognized at Hawkins Field, Jackson and extended recognition as a new unit on 1 July 1953. On 1 July 1965 the 183d Air Transport Squadron was authorized to expand to a level. Today, the 172d Airlift Wing flies weekly missions into harms way to return wounded patients of the Total Force safely back to the U. S. The 186th Air Refueling Wing provides worldwide air refueling support to major commands of the United States Air Force, as well as other U. S. military forces and the military forces of allied nations. After the September 11th,2001 terrorist attacks on the United States, also, Mississippi 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. afhra. af. mil/. 172d Airlift Wing 186th Air Refueling Wing 209 Squadron on the Air National Guard website
Mississippi Air National Guard
–
153d Air Refueling Squadron KC-135E Stratotanker, Key Field AGB, Meridian. The 153d is the oldest unit in the Mississippi Air National Guard, having over 70 years of service to the state and nation
31.
172d Airlift Wing
–
The 172d Airlift Wing is a unit of the Mississippi Air National Guard, stationed at Allen C. Thompson Field Air National Guard Base, Mississippi. If activated to service, the Wing is gained by the United States Air Force Air Mobility Command. Also, the 172d Airlift Wing provides the State of Mississippi support in the event of emergency, maintains peace and order. The 183d became the new groups flying squadron, the support units assigned into the group were the 183d Material Squadron, 183d Air Base Squadron, and the 183d USAF Dispensary. The C-124 was being retired in the early 1970s and the 172d airlift mission was changed to theater support and it upgraded to new C-130H aircraft in 1980 and continued to fly tactical airlift missions until the mid-1980s. On 12 July 1986 the group returned to the airlift role when the first Lockheed C-141B Starlifter to be released from active-duty Air Force control was assigned. With a total of eight aircraft, the unit began its new mission, in March 1988 the 172d took part in the airlift of approximately 3200 troops and almost 1000 tons of cargo on an exercise to Palmerola Air Base, Honduras. The 172d was the only Air National Guard unit that participated in this airlift to Honduras, on 6 December 1988 the Soviet Republic of Armenia suffered a powerful earthquake. The first Air Guard aircraft to fly to Armenia was a C-141B from the 172d, in September 1989 a devastating hurricane struck the tiny island of St. Croix, leaving the island crippled. The 172d flew eleven emergency relief missions, hauling 465 tons of cargo and 472 passengers, from 20 December 1989 to 12 January 1990 the 172d flew 21 sorties in support of Operation Just Cause, the incursion into Panama to replace Manuel Noriega as ruler. Cargo transported during the operation amounted to 403.6 tons and 1,274 passengers were airlifted, on 7 August 1990 the 172ds support of Operations Desert Shield and Desert Storm began when aircrew members started flying voluntary missions. Approximately 98 aircrew members flew these missions before 24 August 1990, the 183d was one of the first two reserve and guard units to be activated. By May 1991 the 148 members of the 183d had flown 2,880 sorties which transported 15,837 passengers and 25,949.2 tons of cargo, in 2000, the C-141C with electronic glass cockpit was phased into service with the group. In October 2000 after the USS Cole bombing in Aden, seventeen members of the 172d deployed to Ramstein Air Base Germany, members of the 183d Aeromedical Evacuation and 183d Airlift Squadrons picked up four sailors from Ramstein and flew them home to Norfolk Naval Station. In February 2003 the 172d retired its last C-141C Starlifter in preparation for the arrival of the wings first Boeing C-17 Globemaster III, the plane was the first Globemaster III assigned to the Air National Guard and was named the Spirit of the Minutemen. afhra. af. mil/. 172d Airlift Wing history page 172d Airlift Wing@globalsecurity. org Rogers, United States Air Force Unit Designations Since 1978. ISBN 1-85780-197-0 Official website 172d Airlift Wing at globalsecurity. org
172d Airlift Wing
–
172d Wing
C-17 Globemaster III at
Camp Shelby Auxiliary Field
172d Airlift Wing
–
C-121, C-119 and C-46 at Hawkins Field
172d Airlift Wing
–
183d Military Airlift Squadron C-124 Globemaster II in South Vietnam
32.
Canadian Expeditionary Force
–
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
Canadian Expeditionary Force
–
26th Battalion of the Second Canadian Expeditionary Force, 1915
Canadian Expeditionary Force
Canadian Expeditionary Force
–
Private Joseph Pappin, 130 Battalion, Canadian Expeditionary Force.
33.
Winnipeg, Manitoba
–
Winnipeg is the capital and largest city of the province of Manitoba in Canada. It is located near the centre of North America and is 110 kilometres from the U. S. border. It is also the place of the confluence of the Red, the city is named after the nearby Lake Winnipeg, the name comes from the Western Cree words for muddy water. The region was a centre for aboriginal peoples long before the arrival of Europeans. French traders built the first fort on the site in 1738, a settlement was later founded by the Selkirk settlers of the Red River Colony in 1812, the nucleus of which was incorporated as the City of Winnipeg in 1873. As of 2011, Winnipeg is the seventh most populated municipality in Canada, being located very far inland, the local climate is extremely seasonal even by Canadian standards with average January lows of around −21 °C and average July highs of 26 °C. Known as the Gateway to the West, Winnipeg is a railway, Winnipeg was the first Canadian host of the Pan American Games. It is home to professional sports franchises, including the Winnipeg Blue Bombers, the Winnipeg Jets, Manitoba Moose. Winnipeg lies at the confluence of the Assiniboine and the Red River of the North and this point was at the crossroads of canoe routes travelled by First Nations before European contact. Winnipeg is named after nearby Lake Winnipeg, the name is a transcription of the Western Cree words for muddy or brackish water. Estimates of the date of first settlement in this area are varied, in 1805, Canadian colonists observed First Nations peoples engaged in farming activity along the Red River. The practice quickly expanded, driven by the demand by traders for provisions, the rivers provided an extensive transportation network linking northern First Peoples with those to the south along the Missouri and Mississippi rivers. The Ojibwe made some of the first maps on birch bark, sieur de La Vérendrye built the first fur trading post on the site in 1738, called Fort Rouge. French trading continued at this site for decades before the arrival of the British Hudsons Bay Company after France ceded the territory following its defeat in the Seven Years War. Many French and later British men who were trappers married First Nations women, their mixed-race children hunted, traded and they gradually developed as an ethnicity known as the Métis because of sharing a traditional culture. Lord Selkirk was involved with the first permanent settlement, the purchase of land from the Hudsons Bay Company, the North West Company built Fort Gibraltar in 1809, and the Hudsons Bay Company built Fort Douglas in 1812, both in the area of present-day Winnipeg. The two companies competed fiercely over trade, the Métis and Lord Selkirks settlers fought at the Battle of Seven Oaks in 1816. In 1821, the Hudsons Bay and North West Companies merged, Fort Gibraltar was renamed Fort Garry in 1822 and became the leading post in the region for the Hudsons Bay Company
Winnipeg, Manitoba
–
Clockwise from top:
Downtown featuring the
Canadian Museum for Human Rights,
Investors Group Field,
Saint Boniface and the
Esplanade Riel bridge, Wesley Hall at the
University of Winnipeg,
Manitoba Legislative Building.
Winnipeg, Manitoba
–
Winnipeg's old City Hall in 1887
Winnipeg, Manitoba
–
The Winnipeg General Strike, 21 June 1919
Winnipeg, Manitoba
–
River walkway near
The Forks, with
St. Boniface Cathedral in the background
34.
183rd Fighter Wing
–
The 183d Wing is a unit of the Illinois Air National Guard, stationed at Capital Airport Air National Guard Station, Springfield, Illinois. If activated to service, the wing is gained by the United States Air Force Air Combat Command. A non-flying wing, the 183d Wing is tasked with augmenting Component Numbered Air Force in both the Air and Space Operations Centers and the Air Force Forces staff, A1 to A9, the primary unit of the 183d Wing is the 183d Air Operations Group. The 183 AOG is capable of augmenting C-NAF staffs worldwide, therefore affording AOG personnel the opportunity to train at various locations - CONUS and OCONUS. The role of the AOG is to assist the AOC in organizing, planning, the 183rd AOG is aligned with Twelfth Air Force / AFSOUTH at Davis-Monthan AFB, Arizona. The 170th TFS becoming the flying squadron. Other squadrons assigned into the group were the 183d Material Squadron, 183d Combat Support Squadron, the 170th Tactical Fighter Squadron was equipped with Republic F-84F Thunderstreaks. It continued to fly the aircraft throughout the 1960s, the squadron was not activated for service during the Vietnam War, although from 1968 to 1971 many of its personnel were activated and some saw service in Southeast Asia. All F-84Fs were grounded in November 1971, after a 170th pilot was killed when his plane lost a wing during exercises at the Hardwood Gunnery Range in Findley, Wisconsin. The accident was caused by the milkbone joining bolt in the wing which had been weakened by years of flying. All RF-84F aircraft were grounded after inspections of other F-84Fs in the Air National Guard found the same issue affected many other aircraft. The problem was deemed too widespread to justify the repair of the aircraft. Along with the F-4C, a flight of RF-4C Phantom II reconnaissance aircraft were received, in 1981, the F-4Cs were exchanged for the F-4D Phantom II. The 170th saw its first General Dynamics F-16A Fighting Falcon on 7 June 1989 when two landed at Capital Airport to replace the aging F-4D Phantom II, by May 5,1990, the 170th TFS was operational with the F-16A. Its mission at the time was fighter attack and was provided the Block 15 for this job, during early 1994 the 170th FS started to exchange their Block 15 F-16A/B for Block 30 F-16C/D Fighting Falcon with the big inlet design. Most of the Block 15s were retired straight to AMARC, during the 1990s, the unit conducted numerous overseas deployments, including six to Southwest Asia, two to Denmark, one to Panama, one to Curaçao, and one to Thailand. After the September 11,2001, attack, the 170th FS increased its capability by obtaining AN/AAQ-28 LITENING targeting pods in October 2001. Training with the new pod started immediately and included some internal personnel as some were trained on use of the pod as well as training from the Wisconsin ANG, the purpose of the training was to get ready for deployment in March 2002 for Operation Enduring Freedom
183rd Fighter Wing
–
170th Fighter Squadron - 60th Anniversary F-16. USAF F-16C block 30 #87-0296 from the 170th FS is leaving Springfield IAP for its final training flight in September 2008 before the unit's inactivation
183rd Fighter Wing
–
An 183rd F-84F with other ANG fighters in the early 1970s.
183rd Fighter Wing
–
170th Expeditionary Fighter Squadron F-16C 87-0294 at Balad AB, Iraq, takes off on an Operation Iraqi Freedom mission on July 21st, 2006
35.
United States Air Force
–
The United States Air Force is the aerial warfare service branch of the United States Armed Forces and one of the seven American uniformed services. Initially part of the United States Army, the USAF was formed as a branch of the military on 18 September 1947 under the National Security Act of 1947. It is the most recent branch of the U. S. military to be formed, the U. S. Air Force is a military service organized within the Department of the Air Force, one of the three military departments of the Department of Defense. The Air Force is headed by the civilian Secretary of the Air Force, who reports to the Secretary of Defense, the U. S. Air Force provides air support for surface forces and aids in the recovery of troops in the field. As of 2015, the service more than 5,137 military aircraft,406 ICBMs and 63 military satellites. It has a $161 billion budget with 313,242 active duty personnel,141,197 civilian employees,69,200 Air Force Reserve personnel, and 105,500 Air National Guard personnel. According to the National Security Act of 1947, which created the USAF and it shall be organized, trained, and equipped primarily for prompt and sustained offensive and defensive air operations. The stated mission of the USAF today is to fly, fight, and win in air, space and we will provide compelling air, space, and cyber capabilities for use by the combatant commanders. We will excel as stewards of all Air Force resources in service to the American people, while providing precise and reliable Global Vigilance, Reach and it should be emphasized that the core functions, by themselves, are not doctrinal constructs. The purpose of Nuclear Deterrence Operations is to operate, maintain, in the event deterrence fails, the US should be able to appropriately respond with nuclear options. Dissuading others from acquiring or proliferating WMD, and the means to deliver them, moreover, different deterrence strategies are required to deter various adversaries, whether they are a nation state, or non-state/transnational actor. Nuclear strike is the ability of forces to rapidly and accurately strike targets which the enemy holds dear in a devastating manner. Should deterrence fail, the President may authorize a precise, tailored response to terminate the conflict at the lowest possible level, post-conflict, regeneration of a credible nuclear deterrent capability will deter further aggression. Finally, the Air Force regularly exercises and evaluates all aspects of operations to ensure high levels of performance. Nuclear surety ensures the safety, security and effectiveness of nuclear operations, the Air Force, in conjunction with other entities within the Departments of Defense or Energy, achieves a high standard of protection through a stringent nuclear surety program. The Air Force continues to pursue safe, secure and effective nuclear weapons consistent with operational requirements, adversaries, allies, and the American people must be highly confident of the Air Forces ability to secure nuclear weapons from accidents, theft, loss, and accidental or unauthorized use. This day-to-day commitment to precise and reliable nuclear operations is the cornerstone of the credibility of the NDO mission, positive nuclear command, control, communications, effective nuclear weapons security, and robust combat support are essential to the overall NDO function. OCA is the method of countering air and missile threats, since it attempts to defeat the enemy closer to its source
United States Air Force
–
First
F-35 Lightning II of the
33rd Fighter Wing arrives at
Eglin AFB
United States Air Force
–
Seal of the Department of the Air Force (
United States Air Force portal)
United States Air Force
–
U.S. Air Force airmen from the 720th STG jumping out of a
C-130J Hercules aircraft during water rescue training in the
Florida panhandle
United States Air Force
–
Combat Controllers participating in
Operation Enduring Freedom provide air traffic control to a
C-130 taking off from a remote airfield.
36.
Abraham Lincoln Capital Airport
–
Abraham Lincoln Capital Airport is a civil-military public airport in Sangamon County, Illinois. It is three miles northwest of downtown Springfield, the capital of Illinois, the airport is owned by the Springfield Airport Authority. The airport is home to Capital Airport Air National Guard Station and it is home to the 183d Fighter Wing, an Illinois Air National Guard unit operationally gained by the Air Combat Command and State Headquarters, Illinois Air National Guard. Historically a fighter unit, the 183 FW consists of 321 full-time and 800 part-time military personnel, the airport covers 2,300 acres at an elevation of 598 feet. It has three runways, 4/22 is 8,001 by 150 feet concrete, 13/31 is 7,400 by 150 feet asphalt, 18/36 is 5,300 by 150 feet asphalt/concrete. In 2009 the airport had 33,903 aircraft operations, average 92 per day, 71% general aviation, 19% air taxi, 10% military,169 aircraft were then based at the airport, 78% single-engine, 17% multi-engine, 4% jet and 1% helicopter. Allegiant Air is the airline with mainline jets, McDonnell Douglas MD-80s. American Eagle and United Express flights from Springfield are regional jets, the airport was served by Ozark Airlines with Douglas DC-9s and Fairchild Hiller FH-227s to St. Louis and Chicago OHare. Air Illinois flew BAC One-Elevens and HS 748s, Handley Page Jetstreams, Air Illinois HS 748s flew nonstop to now closed Meigs Field on the lakefront of Lake Michigan next to downtown Chicago. The airport has a Subway, a shop, an automated teller machine, TV
Abraham Lincoln Capital Airport
–
Abraham Lincoln Capital Airport
Abraham Lincoln Capital Airport
–
FAA airport diagram
37.
Springfield, Illinois
–
Springfield is the capital of the U. S. state of Illinois and the county seat of Sangamon County. The citys population of 116,250 as of the 2010 U. S. Census makes it the sixth most populous city. It is the largest city in central Illinois, present-day Springfield was settled by European Americans in the late 1810s, around the time Illinois became a state. The most famous resident was Abraham Lincoln, who lived in Springfield from 1837 until 1861. The city lies on a flat plain that encompasses much of the surrounding countryside. Hilly terrain lies near the Sangamon River, lake Springfield, a large artificial lake owned by the City Water, Light & Power company, supplies the city with recreation and drinking water. Weather is fairly typical for middle latitude locations, with hot summers, spring and summer weather is like that of most midwestern cities, severe thunderstorms are common. Tornadoes hit the Springfield area in 1957 and 2006, the city is governed by a mayor–council form of government. The city proper is also the Capital Township governmental entity, in addition, the government of the state of Illinois is also based in Springfield. State government entities include the Illinois General Assembly, the Illinois Supreme Court, there are three public and three private high schools in Springfield. Public schools in Springfield are operated by District No.186, Springfields economy is marked by government jobs, and the medical field, which account for a large percentage of the citys workforce. Springfields original name was Calhoun, after Senator John C. Calhoun of South Carolina, the land that Springfield now occupies was originally settled by trappers and traders who came to the Sangamon River in 1818. The settlements first cabin was built in 1820, by John Kelly and it was located at what is now the northwest corner of Second Street and Jefferson Street. In 1821, Calhoun became the county seat of Sangamon County due to fertile soil, settlers from Kentucky, Virginia, and as far as North Carolina came to the city. By 1832, Senator Calhoun had fallen out of the favor with the public, at that time, Springfield, Massachusetts was comparable to modern-day Silicon Valley—known for industrial innovation, concentrated prosperity, and the celebrated Springfield Armory. Most importantly, it was a city that had built itself up from frontier outpost to national power through ingenuity – an example that the newly named Springfield, Illinois, sought to emulate. Kaskaskia was the first capital of the Illinois Territory from its organization in 1809, continuing through statehood in 1818, vandalia was the second state capital of Illinois from 1819 to 1839. Springfield became the third and current capital of Illinois in 1839, the designation was largely due to the efforts of Abraham Lincoln and his associates, nicknamed the Long Nine for their combined height of 54 feet
Springfield, Illinois
–
The
Illinois State Capitol in Springfield, Illinois.
Springfield, Illinois
–
Hotel damaged by the 2006 Springfield tornadoes
Springfield, Illinois
–
Astronaut Photography of Springfield Illinois taken from the International Space Station (ISS)
Springfield, Illinois
–
An image of Downtown Springfield with a view of the State capitol
38.
Focke-Wulf Ta-183
–
The Focke-Wulf Ta 183 Huckebein was a design for a jet-powered fighter aircraft intended as the successor to the Messerschmitt Me 262 and other day fighters in Luftwaffe service during World War II. It was developed only to the extent of wind tunnel models when the war ended, the name Huckebein is a reference to a trouble-making raven from an illustrated story in 1867 by Wilhelm Busch. In early 1944, the Reich Air Ministry became aware of Allied jet developments, in response, they instituted the Emergency Fighter Program which took effect on July 3,1944, ending production of most bomber and multi-role aircraft in favour of fighters, especially jet fighters. The result was a series of advanced designs, some using swept wings for improved transonic performance, Kurt Tanks design team led by Hans Multhopp designed in 1945 a fighter known as Huckebein, also known as Project V or Design II at Focke-Wulf. The aircraft was intended to use the advanced Heinkel HeS011 turbojet, the wings were swept back at 40° and were mounted in the mid-fuselage position. The box-like structure contained six fuel cells, giving the aircraft a total load of 1,565 l. The original design used a T-tail, with a long vertical stabilizer. The vertical stabilizer was swept back at 60°, and the stabilizer was swept back. Many problems beset the project, including the chance of a Dutch roll, work therefore concentrated on the much less problematic Focke-Wulf Project VII. However, when the RLM eventually rejected that design, Huckebein was again brought to the fore, the Ta 183 had a short fuselage with the air intake passing under the cockpit and proceeding to the rear where the single engine was located. The pilot sat in a cockpit with a bubble canopy which provided excellent vision. The primary armament of the aircraft consisted of four 30 mm MK108 cannons arranged around the air intake, multhopps team also seriously explored a second version of the basic design, known as Design III, a modified Design II. The first of these had only minor modifications, with differently shaped wingtips. The second version had a reduced sweepback to 32°, allowing the wing, the tail was also redesigned, using a short horizontal boom to mount the control surfaces just above the line of the rear fuselage. This version looks considerably more conventional to the eye, although somewhat stubby due to the short overall length of the HeS011. The second of two schemes was entered in the official competition ordered by the Oberkommando der Luftwaffe at the end of 1944. On 28 February 1945, the Luftwaffe High Command examined the various Emergency Fighter proposals and selected the Junkers EF128 to be developed and produced, a total of 16 prototypes were to be built, allowing the tail unit to be interchanged between the Design II and III variations. Of the Versuchs aircraft, the Ta 183 V1-V3 were to be powered by the Jumo 004B turbojet with somewhat lengthened rear fuselages to accommodate them, the Ta 183 V4-V14 were intended to be A-0 series pre-production aircraft and V15-V16 were to be static test aircraft
Focke-Wulf Ta-183
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Ta 183 Huckebein
Focke-Wulf Ta-183
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A menacing "Hans Huckebein", the namesake raven for the Ta 183
Focke-Wulf Ta-183
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A Model of Focke-Wulf Ta 183 Design II
Focke-Wulf Ta-183
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Pulqui II on takeoff
39.
Fighter aircraft
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A fighter aircraft is a military aircraft designed primarily for air-to-air combat against other aircraft, as opposed to bombers and attack aircraft, whose main mission is to attack ground targets. The hallmarks of a fighter are its speed, maneuverability, many fighters have secondary ground-attack capabilities, and some are designed as dual-purpose fighter-bombers, often aircraft that do not fulfill the standard definition are called fighters. This may be for political or national security reasons, for advertising purposes, a fighters main purpose is to establish air superiority over a battlefield. Since World War I, achieving and maintaining air superiority has been considered essential for victory in conventional warfare, the word fighter did not become the official English-language term for such aircraft until after World War I. In the British Royal Flying Corps and Royal Air Force these aircraft were referred to as scouts into the early 1920s, the U. S. Army called their fighters pursuit aircraft from 1916 until the late 1940s. In most languages a fighter aircraft is known as a hunter, exceptions include Russian, where a fighter is an истребитель, meaning exterminator, and Hebrew where it is matose krav. As a part of nomenclature, a letter is often assigned to various types of aircraft to indicate their use. In Russia I was used, while the French continue to use C and this has always been the case, for instance the Sopwith Camel and other fighting scouts of World War I performed a great deal of ground-attack work. Several aircraft, such as the F-111 and F-117, have received fighter designations but had no fighter capability due to political or other reasons, the F-111B variant was originally intended for a fighter role with the U. S. Navy, but it was cancelled. This blurring follows the use of fighters from their earliest days for attack or strike operations against ground targets by means of strafing or dropping small bombs, versatile multirole fighter-bombers such as the F/A-18 Hornet are a less expensive option than having a range of specialized aircraft types. An interceptor is generally an aircraft intended to target bombers and so often trades maneuverability for climb rate, fighters were developed in World War I to deny enemy aircraft and dirigibles the ability to gather information by reconnaissance. Early fighters were very small and lightly armed by later standards, and most were built with a wooden frame, covered with fabric. As control of the airspace over armies became increasingly important all of the major powers developed fighters to support their military operations, between the wars, wood was largely replaced by steel tubing, then aluminium tubing, and finally aluminium stressed skin structures began to predominate. By World War II, most fighters were all-metal monoplanes armed with batteries of guns or cannons. By the end of the war, turbojet engines were replacing piston engines as the means of propulsion, further increasing aircraft speed. Since the weight of the engine was so less than on piston engined fighters. This in turn required the development of ejection seats so the pilot could escape, in the 1950s, radar was fitted to day fighters, since pilots could no longer see far enough ahead to prepare for any opposition. Since then, radar capabilities have grown enormously and are now the method of target acquisition
Fighter aircraft
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A fly by of an
F-22 Raptor, a pair of
F-86 Sabres, and a
P-38 Lightning during Heritage Flight training at
Davis–Monthan Air Force Base
Fighter aircraft
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Landing on
HMS Furious in a
Sopwith Pup scout
Fighter aircraft
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Hawker Sea Hurricanes in formation
Fighter aircraft
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MiG-17 underside
40.
USNS Henry Gibbins (T-AP-183)
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USNS Henry Gibbins was a troop transport that served with the United States Military Sea Transportation Service during the 1950s. Prior to her MSTS service, she served as US Army transport USAT Henry Gibbins during World War II and she later served with the New York Maritime Academy as TS Empire State IV and with the Massachusetts Maritime Academy as USTS Bay State. Laid down on 23 August 1941, she was launched on 11 November 1942 as Biloxi and she was delivered to the Army Transportation Service 27 February 1943 as Henry Gibbins and served the Army as a troop transport during World War II. Following the war, the Henry Gibbins was refitted as a war brides ship, Henry Gibbins was acquired by the U. S. Navy from the U. S. Army on 1 March 1950, and assigned to the Military Sea Transportation Service. During the Korean War she transported men and equipment from New York City to the Caribbean and Canal Zone ports, prior to their assignment in the Pacific. In 1953, Henry Gibbins operated on the New York to Bremerhaven, Germany, from 1954 until late 1959 the veteran transport steamed from New York to the Caribbean over 75 times, sailed to the Mediterranean on 3 occasions and crossed the Atlantic to Northern Europe 8 times. During this time Henry Gibbins shuttled thousands of troops and tons of supplies between the United States and her foreign bases. Henry Gibbins was transferred from MSTS to the United States Maritime Administration 2 December 1959, at Fort Schuyler, New York, the college named her TS Empire State IV and she retained that name until being transferred to the Massachusetts Maritime Academy in 1973. At that time she was renamed USTS Bay State, during the winter of 1976-77, one of the coldest in fifty years, the Bay State suffered serious ice damage to her hull at her berth in Buzzards Bay at the southern end of the Cape Cod Canal. The hull plates were repaired and the continued to serve as a training vessel for two more years. In the summer of 1977 she carried cadets to Europe, in the summer of 1978 she made a training cruise to the Mediterranean. The vessel was returned to the Maritime Administration after her final training cruise in 1978, between the hull damage she had sustained in 1977, her age, and an increase in Massachusetts Maritime Academys enrollment, she no longer suited the Academys requirements. According to the U. S. Maritime Administration, the ship was scrapped in 1983 after suffering an engine room fire, USAT Henry Gibbins - DANFS Online. Haven, The Dramatic Story of 1,000 World War II Refugees and How They Came to America
USNS Henry Gibbins (T-AP-183)
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USNS Henry Gibbins (T-AP-183), c. 1952
41.
Troop transport
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A troopship is a ship used to carry soldiers, either in peacetime or wartime. Attack transports, a variant of ocean-going troopship adapted to transporting invasion forces ashore, landing ships beach themselves and bring their troops directly ashore. Ships to transport troops were used in Antiquity. Ancient Rome used the navis lusoria, a vessel powered by rowers and sail, to move soldiers on the Rhine. HMT Olympic even rammed and sank a U-boat during one of its wartime crossings, individual liners capable of exceptionally high speed transited without escorts, smaller or older liners with poorer performance were protected by operating in convoys. The British government aided both Cunard and the White Star Line to construct the liners RMS Mauretania, RMS Aquitania, RMS Olympic, when the vulnerability of these ships to return fire was realized most were used instead as troopship or hospital ships. RMS Queen Mary and RMS Queen Elizabeth were two of the most famous converted liners of World War II, when they were fully converted, each could carry well over 10,000 troops per trip. Queen Mary holds the record, with 15,740 troops on a single passage in late July 1943. The modified Liberties were capable of transporting up to 450,550,30 Type C4 ship-based General G. O. Squier-class, the largest carrying over 6,000 passengers. A class of Victory ship-based dedicated troopship was developed late in World War II, a total of 84 such VC2-S-AP2 hull conversions was completed. A class of Type C3 ship – comprising mainly C3-S-A2 and C3-S-A3 hulls – was also converted to dedicated troopships, at least 15 classes of Attack Transport, consisting of at least 400 ships specially equipped for landing invasion forces rather than general troop movement. The designation HMT would normally replace RMS, MV or SS for ships converted to troopship duty with the United Kingdoms Royal Navy, initially troopships adapted as attack transports were designated AP, starting in 1942 keel-up attack transports received the designation APA. In the era of the Cold War the United States designed the SS United States so that it could easily be converted from a liner to a troopship, in case of war. More recently, RMS Queen Elizabeth 2 and the SS Canberra were requisitioned by the Royal Navy to carry British soldiers to the Falklands War, by the end of the twentieth century, nearly all long-distance personnel transfer was done by airlift in military transport aircraft
Troop transport
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Soldiers climb down netting on the sides of the
attack transport USS McCawley (APA-4) on 14 June 1943, rehearsing for landings on
New Georgia
Troop transport
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A
Bayfield-class attack transport underway with its complement of landing craft, the
USS Hamblen (APA-114)
Troop transport
–
Nicknamed the "Grey Ghost",
RMS Queen Mary holds the all-time record for most troops on one passage, 15,740 on a late July 1943 run from the U.S. to Europe
Troop transport
–
A U.S.
General G. O. Squier-class troop transport
42.
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
United States Navy
United States Navy
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United States Navy portal
United States Navy
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USS Constellation vs L'Insurgente during the
Quasi-War
United States Navy
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USS Constitution vs HMS Guerriere during the
War of 1812
43.
Cargo ship
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A cargo ship or freighter is any sort of ship or vessel that carries cargo, goods, and materials from one port to another. Thousands of cargo carriers ply the worlds seas and oceans each year, cargo ships are usually specially designed for the task, often being equipped with cranes and other mechanisms to load and unload, and come in all sizes. Today, they are almost always built by welded steel, cargo ships/freighters can be divided into five groups, according to the type of cargo they carry. Tankers carry petroleum products or other liquid cargo, dry bulk carriers carry coal, grain, ore and other similar products in loose form. Multi-purpose vessels, as the name suggests, carry different classes of cargo – e. g. liquid, a Reefer ship is specifically designed and used for shipping perishable commodities which require temperature-controlled, mostly fruits, meat, fish, vegetables, dairy products and other foodstuffs. Specialized types of cargo vessels include ships and bulk carriers. Cargo ships fall into two categories that reflect the services they offer to industry, liner and tramp services. Those on a published schedule and fixed tariff rates are cargo liners. Tramp ships do not have fixed schedules, users charter them to haul loads. Generally, the shipping companies and private individuals operate tramp ships. Cargo liners run on fixed schedules published by the shipping companies, each trip a liner takes is called a voyage. However, some cargo liners may carry passengers also, a cargo liner that carries 12 or more passengers is called a combination or passenger-cum-cargo line. The desire to trade routes over longer distances, and throughout more seasons of the year. Before the middle of the 19th century, the incidence of piracy resulted in most cargo ships being armed, sometimes heavily, as in the case of the Manila galleons. They were also escorted by warships. Piracy is still common in some waters, particularly in the Malacca Straits. In 2004, the governments of three nations agreed to provide better protection for the ships passing through the Straits. The waters off Somalia and Nigeria are also prone to piracy, while smaller vessels are also in danger along parts of the South American, Southeast Asian coasts, the words cargo and freight have become interchangeable in casual usage
Cargo ship
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The
Colombo Express, one of the largest container ships in the world (when she was built in 2005), owned and operated by
Hapag-Lloyd of
Germany
Cargo ship
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Loading of a general cargo vessel in 1959
Cargo ship
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A Delmas container ship unloading at the Zanzibar port in Tanzania
44.
USS Haraden (DD-183)
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The first USS Haraden was a Wickes-class destroyer in the United States Navy following World War I. She was later transferred to the Royal Canadian Navy as HMCS Columbia, named for Jonathan Haraden, she was launched by Newport News Shipbuilding and Dry Dock Company, Newport News, Virginia on 4 July 1918, sponsored by Miss Mabel B. Stephens, great-niece of Captain Jonathan Haraden, Haraden was commissioned at Norfolk Navy Yard on 7 June 1919, to Lieutenant Commander R. H. Haraden was assigned to U. S. Naval Forces in European Waters, after calling at Newport and this duty occupied her until 23 October 1919, when she departed for Norfolk, Virginia, arriving on 18 November. The destroyer departed Norfolk on 7 April 1920 for Charleston, South Carolina, after an extensive overhaul at New York, ending on 2 May, Haraden sailed for Newport and training operations off New England. She returned to Charleston on 12 October 1921 and to Philadelphia on 10 April 1922, with the mounting tensions in 1939, Haraden was called back to active service and recommissioned at Philadelphia 4 December 1939. After shakedown training at Guantanamo Bay, Cuba, the destroyer performed neutrality patrol in Cuban waters briefly and she subsequently conducted neutrality patrol in waters off Block Island and Nantucket Shoals, and made three training cruises in Chesapeake Bay. Arriving Boston Navy Yard 7 September 1940, Haraden was one of the fifty over-age destroyers to be sent to the United Kingdom in exchange for bases and she sailed on 18 September for Halifax, Nova Scotia, and decommissioned there for transfer to the British on 24 September 1940. Her name was struck from the Navy List 8 January 1941, Columbia first underwent refit and then was assigned to convoy duties in the Atlantic. Her first major action began 15 October 1941 when she joined convoy SC-8, Columbia, and the other escorts fought valiantly, but nine merchantmen from the convoy were sunk before reaching England. After the U. S. s entry into the war Columbia was reassigned to convoy ships from New York to St. Johns, Newfoundland, the first leg of the transatlantic journey. She escorted convoys and performed anti-submarine patrol until 25 February 1944 and this article incorporates text from the public domain Dictionary of American Naval Fighting Ships. The entry can be found here
USS Haraden (DD-183)
–
History
45.
Wickes class destroyer
–
The Wickes-class destroyers were a class of 111 destroyers built by the United States Navy in 1917–19. Along with the 6 preceding Caldwell-class and 156 subsequent Clemson-class destroyers, only a few were completed in time to serve in World War I, including USS Wickes, the lead ship of the class. While some were scrapped in the 1930s, the rest served through World War II, most of these were converted to other uses, nearly all in U. S. service had half their boilers and one or more stacks removed to increase fuel and range or accommodate troops. Others were transferred to the British Royal Navy and the Royal Canadian Navy, All were scrapped within a few years after World War II. The destroyer type was at time a relatively new class of fighting ship for the U. S. Navy. The type arose in response to torpedo boats that had been developing from 1865, a series of destroyers had been built over the preceding years, designed for high smooth water speed, with indifferent results, especially poor performance in heavy seas and poor fuel economy. The lesson of these destroyers was the appreciation of the need for true seakeeping and seagoing abilities. There were few cruisers in the Navy, which was a fleet of battleships and destroyers so destroyers performed scouting missions. A report of October 1915 by Captain W. S. Sims noted that the smaller destroyers used fuel far too quickly, and that war games showed the need for fast vessels with a larger radius of action. As a result, the size of U. S. destroyer classes increased steadily, starting at 450 tons and rising to over 1,000 tons between 1905 and 1916. The increase in size has never stopped, with some US Arleigh Burke-class destroyers now up to 10,800 tons full load. With World War I then in its year and tensions between the U. S. and Germany increasing, the U. S. needed to expand its navy. The Naval Appropriation Act of 1916 called for a second to none. The Act authorized 10 battleships,6 Lexington-class battlecruisers,10 Omaha-class scout cruisers, a subsequent General Board recommendation for further destroyers to combat the submarine threat resulted in a total of 267 Wickes- and Clemson-class destroyers completed. However, the design of the ships remained optimized for operation with the battleship fleet, the requirements of the new design were high speed and mass production. The development of warfare during World War I created a requirement for destroyers in numbers that had not been contemplated before the war. A top speed of 35 knots was needed for operation with the Lexington battlecruisers, the final design had a flush deck and four smokestacks. It was a fairly straightforward evolution of the preceding Caldwell class, General dissatisfaction with the earlier 1,000 ton designs led to the fuller hull form of the flush deck type
Wickes class destroyer
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USS Wickes (DD-75)
Wickes class destroyer
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8 ships of Wickes class destroyer, New York Shipbuilding Corporation, Camden, New Jersey, 1919.
46.
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
Destroyer
–
USS Winston S. Churchill, an
Arleigh Burke-class guided missile destroyer of the
United States Navy
Destroyer
Destroyer
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The
Imperial Japanese Navy 's Kotaka (1887)
Destroyer
–
HMS Havock the first modern destroyer, commissioned in 1894
47.
USS Samuel S. Miles (DE-183)
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USS Samuel S. Miles was a Cannon-class destroyer escort built for the United States Navy during World War II. She served in the Pacific Ocean and provided escort service against submarine and air attack for Navy vessels and she returned home at wars end with eight battle stars to her credit. She was laid down on 5 July 1943 by the Federal Shipbuilding, newark, New Jersey, launched on 3 October 1943, sponsored by Mrs. Samuel S. Miles, and commissioned on 4 November 1943, Lt. Comdr. Following shakedown off Bermuda, Samuel S. Miles departed New York City, on 30 December 1943, and steamed via the Panama Canal to the Marshall Islands, arriving on 19 February 1944. Serving as a ship in the Marshall Islands area, she protected fleet oilers during fast carrier air strikes against the Caroline Islands. Next she guarded oilers during the capture of Saipan and Tinian and she supported the Leyte and Luzon, Philippine Islands, campaigns in late 1944 and early 1945. Samuel S. Miles sank Japanese submarine I-177 near the Palau Islands on 3 October, after guarding the invasion force at Iwo Jima in February, she screened the bombardment group that pounded Okinawa, where she destroyed one enemy plane on 27 March. A kamikaze near-miss killed one of her crew members on 11 April, after screening escort carriers operating north of Okinawa, she sailed to the west coast in July. After overhaul, she voyaged via the Panama Canal to Norfolk, Virginia, reaching St. Johns River, Florida, on 8 November 1945, she decommissioned and entered the Reserve Fleet on 28 March 1946. Samuel S, Miles received eight battle stars for World War II service and this article incorporates text from the public domain Dictionary of American Naval Fighting Ships. The entry can be found here
USS Samuel S. Miles (DE-183)
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USS Samuel S. Miles (DE-183)
USS Samuel S. Miles (DE-183)
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French Destroyer Escort Arabe (F717)
48.
Cannon class destroyer escort
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The Cannon class was a class of destroyer escorts were built by the United States primarily for ocean anti-submarine warfare escort service during World War II. The lead ship, USS Cannon, was commissioned on 26 September 1943 at Wilmington, of the 116 ships ordered 44 were canceled and six commissioned directly into the Free French Forces. Destroyer escorts were regular companions escorting the cargo ships. BRP Rajah Humabon of the Philippine Navy, formerly USS Atherton, the class was also known as the DET type from their Diesel Electric Tandem drive. The DETs substitution for a propulsion plant was the primary difference with the predecessor Buckley class. The DET was in turn replaced with a direct drive diesel plant to yield the design of the successor Edsall class, a total of 72 ships of the Cannon class were built. Evans as Berbère, served 1952-1960 USS Riddle as Kabyle, served 1950-1959 USS Samuel S
Cannon class destroyer escort
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USS Cannon (DE-99)
Cannon class destroyer escort
–
BRP Rajah Humabon (PF-11) of the Philippine Navy
49.
USS Seal (SS-183)
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USS Seal, a Salmon-class submarine, was the second ship of the United States Navy to be named for the seal, a sea mammal valued for its skin and oil. Her keel was laid down on 25 May 1936 by the Electric Boat Company in Groton, the boat was commissioned on 30 April 1937, Lieutenant Karl G. Hensel in command. That exercise, to test the ability to control the approaches to Central America. In March, Seal returned to the Haiti–Cuba area for exercises with Destroyer Division 4, in April, she proceeded to New London, Connecticut, for overhaul which included modification of her main engines. In June, the submarine again sailed south, transited the Panama Canal, and continued on to San Diego, California, in Hawaii from July to September, she took soundings for the Hydrographic Office and participated in various local exercises. At the end of the month, she returned to San Diego. During the next two years, she conducted exercises and provided services to ships and to United States Navy and United States Army air units along the West Coast. In the fall of 1941, Submarine Division 21 - of which she was now a part - was transferred to the Asiatic Fleet. Departing Pearl Harbor on 24 October, she reached Manila on 10 November and she headed north to intercept Japanese forces moving into northern Luzon to reinforce those already landed at Vigan and Aparri, Cagayan. From the Vigan area, the moved into the approaches to Lingayen Gulf. By 20 January, she was patrolling east of the Celebes to intercept enemy traffic into Kema. On 27 January, she was ordered to patrol off Kendari, Seal arrived at Soerabaja on 5 February. Daily air raids necessitated diving during the day and precluded repairs to her engines, which smoked excessively, on 11 February, she departed for Tjilatjap on the south coast of Java, and on 14 February, she went alongside Holland. That same day, the Japanese moved into southern Sumatra, and on 19 February, Allied forces counterattacked, and as air and surface forces hit the Japanese fleet, Seal departed Tjilatjap and transited Lombok Strait to patrol north of Java. On 24 February, she attacked two convoys, only damaging one freighter, the next day, she unsuccessfully attacked an enemy warship formation. On 1 March, as the Japanese moved against Soerabaja, she was similarly disappointed, on 21 March, she headed for Fremantle, Western Australia—the Netherlands East Indies had fallen. Arriving on 9 April, Seal departed again on 12 May and worked her way through the Malay Archipelago, the Celebes Sea, and the Sulu Sea to her patrol area off the Indochina coast. During the early morning hours of 28 May, she entered the South China Sea, on 7 June, while off Cam Ranh Bay, she attacked an eight-ship convoy and underwent a seven-hour depth charging by surface ships and aircraft
USS Seal (SS-183)
–
History
50.
Salmon class submarine
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The United States Navy Salmon-class submarines were an important developmental step in the design of the fleet submarine concept during the 1930s. Also, their 11,000 nautical miles unrefueled range would allow them to operate in Japanese home waters and these rugged and dependable boats provided yeoman service during World War II, along with their immediate successors, the similar Sargo class. In some references, the Salmons and Sargos are called the New S Class, 1st, authorized under the Fiscal Year 1936 provision of the Vinson-Trammell Act, two distinct, but very similar, designs were developed, to be built by three different constructors. The Electric Boat Company of Groton, Connecticut designed and built Salmon, Seal, the Navys lead submarine design entity, the Portsmouth Naval Shipyard of Kittery, Maine submitted a design for the Government group, which became Snapper and Stingray. Using the Portsmouth plans and acting as a yard, the Mare Island Naval Shipyard of Vallejo. This difference led to casualties in Snapper and Sturgeon, and to the loss of Squalus, larger than the design of the Porpoise-class, the conning tower installed by Electric Boat had two concave spherical ends, The Portsmouth design had a concave end aft and a convex one forward. Portsmouth and Mare Island ran into difficulties with their conning towers. The problem was fixed, but the experience caused the government yards to adopt the double concave design for the next several years. Externally, there were differences in the shape of the upper edge of the aft end of the conning tower fairwater. The Electric Boat design had a gradual taper to this bulwark. Also, as built the Electric Boat trio had two 34 foot periscopes and this resulted in a fairly small periscope shear support structure above the fairwater. The three Government boats had one 34 foot and one 40 foot periscope and this necessitated a taller shear, the Electric Boat-built Porpoises had been built to an all-welded design. Conservative engineers and shipfitters at the Government yards stuck with tried, Electric Boats method proved superior, providing a stronger and tighter boat, as well as preventing leakage of fuel oil tanks after depth charge attacks. Finally convinced of the efficacy of Electric Boats innovation, Government yards finally converted wholesale to welding for their three Salmons and the Navy was entirely happy with the results, the six boats of this class were straight forward derivations of the later boats of the preceding Porpoise class. Although considered to be successful in most respects, valuable lessons had been learned from the Porpoises, the Salmons were longer, heavier, and faster versions with a better internal arrangement and a heavier armament. Two additional torpedo tubes were added to the aft torpedo room, for a total of four forward, the development of the Torpedo Data Computer, making broadside attacks practical, had made stern tubes more desirable. In order to access the weapons in these tubes, the boat had to surface, small boats stowed there for running sailors ashore for liberty were removed and set in the water. The weapons were extracted from the one by one and winched up to the main deck
Salmon class submarine
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USS Salmon on trials in 1938
Salmon class submarine
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Periscope photo of
Japanese destroyer Yamakaze sinking.