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.
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
20.
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
21.
Sphenic number
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In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers. A sphenic number is a product pqr where p, q and this definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance,60 =22 ×3 ×5 has exactly 3 prime factors, the smallest sphenic number is 30 =2 ×3 ×5, the product of the smallest three primes. The first few numbers are 30,42,66,70,78,102,105,110,114,130,138,154,165. As of January 2016 the largest known number is × ×. It is the product of the three largest known primes, all sphenic numbers have exactly eight divisors. If we express the number as n = p ⋅ q ⋅ r, where p, q. For example,24 is not a number, but it has exactly eight divisors. All sphenic numbers are by definition squarefree, because the factors must be distinct. The Möbius function of any number is −1. The cyclotomic polynomials Φ n, taken over all sphenic numbers n, the first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17,1310 = 2×5×131, and 1311 = 3×19×23, there is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. The numbers 2013,2014, and 2015 are all sphenic, the next three consecutive sphenic years will be 2665,2666 and 2667. Semiprimes, products of two prime numbers
Sphenic number
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Overview
22.
3 (number)
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3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
3 (number)
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The
Shield of the Trinity is a diagram of the Christian doctrine of the Trinity
23.
5 (number)
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5 is a number, numeral, and glyph. It is the number following 4 and preceding 6. Five is the prime number. Because it can be written as 221 +1, five is classified as a Fermat prime, therefore a regular polygon with 5 sides is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also the number that is part of more than one pair of twin primes. Five is conjectured to be the only odd number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being 2 plus 3,5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation. Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers,5 is the length of the hypotenuse of the smallest integer-sided right triangle. In bases 10 and 20,5 is a 1-automorphic number,5 and 6 form a Ruth–Aaron pair under either definition. There are five solutions to Známs problem of length 6 and this is related to the fact that the symmetric group Sn is a solvable group for n ≤4 and not solvable for n ≥5. While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar, K5, Five is also the number of Platonic solids. A polygon with five sides is a pentagon, figurate numbers representing pentagons are called pentagonal numbers. Five is also a square pyramidal number, Five is the only prime number to end in the digit 5, because all other numbers written with a 5 in the ones-place under the decimal system are multiples of five. As a consequence of this,5 is in base 10 a 1-automorphic number, vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions, unlike expansions with all other prime denominators, because they are prime factors of ten, the base. When written in the system, all multiples of 5 will end in either 5 or 0
5 (number)
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The fives of all four suits in
playing cards
24.
7 (number)
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7 is the natural number following 6 and preceding 8. Seven, the prime number, is not only a Mersenne prime. It is also a Newman–Shanks–Williams prime, a Woodall prime, a prime, a lucky prime, a happy number, a safe prime. Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers, Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree. N =7 is the first natural number for which the statement does not hold, Two nilpotent endomorphisms from Cn with the same minimal polynomial. 7 is the only number D for which the equation 2n − D = x2 has more than two solutions for n and x natural, in particular, the equation 2n −7 = x2 is known as the Ramanujan–Nagell equation. 7 is the dimension, besides the familiar 3, in which a vector cross product can be defined. 7 is the lowest dimension of an exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere. 999,999 divided by 7 is exactly 142,857, for example, 1/7 =0.142857142857. and 2/7 =0.285714285714. In fact, if one sorts the digits in the number 142857 in ascending order,124578, the remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example,628 ÷7 =89 5/7, here 5 is the remainder, so in this case,628 ÷7 =89.714285. Another example,5238 ÷7 =748 2/7, hence the remainder is 2, in this case,5238 ÷7 =748.285714. A seven-sided shape is a heptagon, the regular n-gons for n ≤6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons are called heptagonal numbers, Seven is also a centered hexagonal number. Seven is the first integer reciprocal with infinitely repeating sexagesimal representation, There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers. There are seven types of catastrophes. When rolling two standard six-sided dice, seven has a 6 in 36 probability of being rolled, the greatest of any number, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved, in quaternary,7 is the smallest prime with a composite sum of digits
7 (number)
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Seven Days of Creation - 1765 book
7 (number)
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Graph of the probability distribution of the sum of 2 six-sided dice
7 (number)
–
The
Seven Lucky Gods in
Japanese mythology
25.
17 (number)
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17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution
17 (number)
–
No row 17 in
Alitalia planes.
26.
19 (number)
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19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
19 (number)
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A 19x19
Go board
19 (number)
–
19 is a
centered triangular number
27.
31 (number)
–
31 is the natural number following 30 and preceding 32. As a Mersenne prime,31 is related to the perfect number 496,31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a triangular number, the lowest prime centered pentagonal number. For the Steiner tree problem,31 is the number of possible Steiner topologies for Steiner trees with 4 terminals, at 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31,31 is a repdigit in base 5, and base 2. The numbers 31,331,3331,33331,333331,3333331, for a time it was thought that every number of the form 3w1 would be prime. Here,31 divides every fifteenth number in 3w1, the atomic number of gallium Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is visible to the naked eye in a modestly dark sky. The New General Catalogue object NGC31, a galaxy in the constellation Phoenix The Saros number of the solar eclipse series which began on -1805 January 31. The duration of Saros series 31 was 1316.2 years, the Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, the jersey number 31 has been retired by several North American sports teams in honor of past playing greats, In Major League Baseball, The San Diego Padres, for Dave Winfield. The Chicago Cubs, for Ferguson Jenkins and Greg Maddux, the Atlanta Braves, also for Maddux. The New York Mets, for Mike Piazza, in the NBA, The Boston Celtics, for Cedric Maxwell. The Indiana Pacers, for Reggie Miller, in the NHL, The Edmonton Oilers, for Grant Fuhr. The New York Islanders, for Billy Smith, in the NFL, The Atlanta Falcons, for William Andrews. The New Orleans Saints, for Jim Taylor, NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company. In ice hockey goaltenders often wear the number 31, in football the number 31 has been retired by Queens Park Rangers F. C.31 from the Prime Pages
31 (number)
–
31 is a
centered pentagonal number
28.
37 (number)
–
37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37
37 (number)
–
House number in
Baarle (in its Belgian part)
29.
Square (algebra)
–
In mathematics, a square is the result of multiplying a number by itself. The verb to square is used to denote this operation, squaring is the same as raising to the power 2, and is denoted by a superscript 2, for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the adjective which corresponds to squaring is quadratic. The square of an integer may also be called a number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, for instance, the square of the linear polynomial x +1 is the quadratic polynomial x2 + 2x +1. One of the important properties of squaring, for numbers as well as in other mathematical systems, is that. That is, the function satisfies the identity x2 =2. This can also be expressed by saying that the function is an even function. The squaring function preserves the order of numbers, larger numbers have larger squares. In other words, squaring is a function on the interval. Hence, zero is its global minimum, the only cases where the square x2 of a number is less than x occur when 0 < x <1, that is, when x belongs to an open interval. This implies that the square of an integer is never less than the original number, every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of one number, itself. For this reason, it is possible to define the square root function, no square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. There are several uses of the squaring function in geometry. The name of the squaring function shows its importance in the definition of the area, the area depends quadratically on the size, the area of a shape n times larger is n2 times greater. The squaring function is related to distance through the Pythagorean theorem and its generalization, Euclidean distance is not a smooth function, the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance, which has a paraboloid as its graph, is a smooth, the dot product of a Euclidean vector with itself is equal to the square of its length, v⋅v = v2
Square (algebra)
–
The
composition of the tiling
Image:ConformId.jpg (understood as a function on the complex plane) with the complex square function
Square (algebra)
–
5⋅5, or 5 2 (5 squared), can be shown graphically using a
square. Each block represents one unit, 1⋅1, and the entire square represents 5⋅5, or the area of the square.
30.
On-Line Encyclopedia of Integer Sequences
–
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
–
On-Line Encyclopedia of Integer Sequences
31.
Galaxy
–
A galaxy is a gravitationally bound system of stars, stellar remnants, interstellar gas, dust, and dark matter. The word galaxy is derived from the Greek galaxias, literally milky, Galaxies range in size from dwarfs with just a few billion stars to giants with one hundred trillion stars, each orbiting its galaxys center of mass. Galaxies are categorized according to their morphology as elliptical, spiral. Many galaxies are thought to have holes at their active centers. The Milky Ways central black hole, known as Sagittarius A*, has a four million times greater than the Sun. Recent estimates of the number of galaxies in the observable universe range from 200 billion to 2 trillion or more, most of the galaxies are 1,000 to 100,000 parsecs in diameter and separated by distances on the order of millions of parsecs. The space between galaxies is filled with a gas having an average density of less than one atom per cubic meter. The majority of galaxies are organized into groups, clusters. At the largest scale, these associations are generally arranged into sheets and filaments surrounded by immense voids. In Greek mythology, Zeus places his son born by a mortal woman, the infant Heracles, on Heras breast while she is asleep so that the baby will drink her divine milk and will thus become immortal. Hera wakes up while breastfeeding and then realizes she is nursing a baby, she pushes the baby away, some of her milk spills and. In the astronomical literature, the capitalized word Galaxy is often used to refer to our galaxy, the Milky Way, to distinguish it from the other galaxies in our universe. The English term Milky Way can be traced back to a story by Chaucer c. 1380, See yonder, lo, the Galaxyë Which men clepeth the Milky Wey, For hit is whyt. However, the word Universe was later understood to mean the entirety of existence, so this expression fell into disuse and the objects instead became known as galaxies. Tens of thousands of galaxies have been catalogued, but only a few have well-established names, such as the Andromeda Galaxy, the Magellanic Clouds, the Whirlpool Galaxy, and the Sombrero Galaxy. Astronomers work with numbers from certain catalogues, such as the Messier catalogue, the NGC, the IC, the CGCG, all of the well-known galaxies appear in one or more of these catalogues but each time under a different number. For example, Messier 109 is a galaxy having the number 109 in the catalogue of Messier, but also codes NGC3992, UGC6937, CGCG 269-023, MCG +09-20-044. One of the arguments to do so is that these impressive objects deserve better than uninspired codes, for instance, Bodifee and Berger propose the informal, descriptive name Callimorphus Ursae Majoris for the well-formed barred galaxy Messier 109 in Ursa Major
Galaxy
–
NGC 4414, a typical spiral galaxy in the
constellation Coma Berenices, is about 55,000
light-years in diameter and approximately 60 million light-years away from Earth
Galaxy
–
A
fish-eye mosaic of the Milky Way arching at a high inclination across the night sky, shot from a dark-sky location in Chile
Galaxy
–
Photograph of the "Great Andromeda Nebula" from 1899, later identified as the
Andromeda Galaxy
Galaxy
–
NGC 3923 Elliptical Shell Galaxy-Hubble Space Telescope photograph
32.
Asteroid belt
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The asteroid belt is the circumstellar disc in the Solar System located roughly between the orbits of the planets Mars and Jupiter. It is occupied by numerous irregularly shaped bodies called asteroids or minor planets, the asteroid belt is also termed the main asteroid belt or main belt to distinguish it from other asteroid populations in the Solar System such as near-Earth asteroids and trojan asteroids. About half the mass of the belt is contained in the four largest asteroids, Ceres, Vesta, Pallas, the total mass of the asteroid belt is approximately 4% that of the Moon, or 22% that of Pluto, and roughly twice that of Plutos moon Charon. Ceres, the belts only dwarf planet, is about 950 km in diameter, whereas Vesta, Pallas. The remaining bodies range down to the size of a dust particle, the asteroid material is so thinly distributed that numerous unmanned spacecraft have traversed it without incident. Nonetheless, collisions between large asteroids do occur, and these can form a family whose members have similar orbital characteristics. Individual asteroids within the belt are categorized by their spectra. The asteroid belt formed from the solar nebula as a group of planetesimals. Planetesimals are the precursors of the protoplanets. Between Mars and Jupiter, however, gravitational perturbations from Jupiter imbued the protoplanets with too much energy for them to accrete into a planet. Collisions became too violent, and instead of fusing together, the planetesimals, as a result,99. 9% of the asteroid belts original mass was lost in the first 100 million years of the Solar Systems history. Some fragments eventually found their way into the inner Solar System, Asteroid orbits continue to be appreciably perturbed whenever their period of revolution about the Sun forms an orbital resonance with Jupiter. At these orbital distances, a Kirkwood gap occurs as they are swept into other orbits. Classes of small Solar System bodies in other regions are the objects, the centaurs, the Kuiper belt objects, the scattered disc objects, the sednoids. On 22 January 2014, ESA scientists reported the detection, for the first definitive time, of water vapor on Ceres, the detection was made by using the far-infrared abilities of the Herschel Space Observatory. The finding was unexpected because comets, not asteroids, are considered to sprout jets. According to one of the scientists, The lines are becoming more and more blurred between comets and asteroids. This pattern, now known as the Titius–Bode law, predicted the semi-major axes of the six planets of the provided one allowed for a gap between the orbits of Mars and Jupiter
Asteroid belt
–
By far the largest object within the belt is
Ceres. The total mass of the asteroid belt is significantly less than
Pluto 's, and approximately twice that of Pluto's moon
Charon.
Asteroid belt
–
Sun Jupiter trojans Orbits of
planets
Asteroid belt
–
Giuseppe Piazzi, discoverer of
Ceres, the largest object in the asteroid belt. For several decades after its discovery Ceres was known as a planet, after which it was reclassified as asteroid number 1. In 2006 it was recognized to be a dwarf planet.
Asteroid belt
–
951 Gaspra, the first asteroid imaged by a spacecraft, as viewed during
Galileo ' s 1991 flyby; colors are exaggerated
33.
Asteroid
–
Asteroids are minor planets, especially those of the inner Solar System. The larger ones have also been called planetoids and these terms have historically been applied to any astronomical object orbiting the Sun that did not show the disc of a planet and was not observed to have the characteristics of an active comet. As minor planets in the outer Solar System were discovered and found to have volatile-based surfaces that resemble those of comets, in this article, the term asteroid refers to the minor planets of the inner Solar System including those co-orbital with Jupiter. There are millions of asteroids, many thought to be the remnants of planetesimals. The large majority of known asteroids orbit in the belt between the orbits of Mars and Jupiter, or are co-orbital with Jupiter. However, other orbital families exist with significant populations, including the near-Earth objects, individual asteroids are classified by their characteristic spectra, with the majority falling into three main groups, C-type, M-type, and S-type. These were named after and are identified with carbon-rich, metallic. The size of asteroids varies greatly, some reaching as much as 1000 km across, asteroids are differentiated from comets and meteoroids. In the case of comets, the difference is one of composition, while asteroids are composed of mineral and rock, comets are composed of dust. In addition, asteroids formed closer to the sun, preventing the development of the aforementioned cometary ice, the difference between asteroids and meteoroids is mainly one of size, meteoroids have a diameter of less than one meter, whereas asteroids have a diameter of greater than one meter. Finally, meteoroids can be composed of either cometary or asteroidal materials, only one asteroid,4 Vesta, which has a relatively reflective surface, is normally visible to the naked eye, and this only in very dark skies when it is favorably positioned. Rarely, small asteroids passing close to Earth may be visible to the eye for a short time. As of March 2016, the Minor Planet Center had data on more than 1.3 million objects in the inner and outer Solar System, the United Nations declared June 30 as International Asteroid Day to educate the public about asteroids. The date of International Asteroid Day commemorates the anniversary of the Tunguska asteroid impact over Siberia, the first asteroid to be discovered, Ceres, was found in 1801 by Giuseppe Piazzi, and was originally considered to be a new planet. In the early half of the nineteenth century, the terms asteroid. Asteroid discovery methods have improved over the past two centuries. This task required that hand-drawn sky charts be prepared for all stars in the band down to an agreed-upon limit of faintness. On subsequent nights, the sky would be charted again and any moving object would, hopefully, the expected motion of the missing planet was about 30 seconds of arc per hour, readily discernible by observers
Asteroid
–
253 Mathilde, a
C-type asteroid measuring about 50 kilometres (30 mi) across, covered in craters half that size. Photograph taken in 1997 by the
NEAR Shoemaker probe.
Asteroid
–
2013 EC, shown here in radar images, has a provisional designation
Asteroid
–
⚵
Asteroid
–
243 Ida and its moon Dactyl. Dactyl is the first satellite of an asteroid to be discovered.
34.
195 Broadway
–
195 Broadway is a 29-story building on Broadway in the Financial District of the New York City borough of Manhattan. It was the headquarters of American Telephone and Telegraph, as well as Western Union for a time. It occupies almost an entire block on one side of Broadway and it also has the address 15 Dey Street, well known as the site of one end of the first transcontinental telephone call. 195 Broadway is also known as the Telephone Building, Telegraph Building, or Western Union Building, the building is still in use. The 1987 film Wall Street used the ground floor lobby as Charlie Sheens characters office. The building includes an entrance to the Fulton Street station on the IRT Lexington Avenue Line, from 1885 to 1910, AT&T was headquartered at 125 Milk Street in Boston. In 1912, Vail developed a plan for a 29-story headquarters building that would be constructed on Broadway on the block stretching from Dey Street to Fulton Street. The plan entailed constructing one wing on the Dey Street corner, the first portion of the building, the Dey Street wing completed in 1916, was an L-shaped structure at the corner of Dey Street and Broadway with an extension reaching Fulton Street. William W. Bosworth, the architect who designed the John D. Rockefeller estate at Kykuit, was commissioned to create the Fulton Street wing of the building. Bosworths designed featured layers of granite columns in Doric and Ionic styles. The statue was cast in bronze and covered with over 40,000 pieces of gold leaf, after a court-ordered divestiture of Western Union, the statues official title was changed to Genius of Electricity. The statue was renamed Spirit of Communication in the 1930s, but has been known by its nickname. One of sculptor Paul Manships earliest public works was The Four Elements, in 1978, AT&T commissioned a new building at 550 Madison Avenue. This new AT&T Building was designed by Philip Johnson and quickly became an icon of the new Postmodern architectural style, the building was completed in 1984, the very year of the Bell System divestiture. It proved to be too large for the corporation and in 1993, AT&T leased the building to Sony
195 Broadway
–
195 Broadway
195 Broadway
–
195 Broadway on right center
195 Broadway
–
Commemorative plaque
35.
Financial District, Manhattan
–
The neighborhood roughly overlaps with the boundaries of the New Amsterdam settlement in the late 17th century. The Financial District has witnessed growth in its population to approximately 43,000 as of 2014, the Financial District encompasses roughly the area south of City Hall Park in Lower Manhattan but excludes Battery Park and Battery Park City. The former World Trade Center complex was located in the neighborhood until the September 11 attacks, the heart of the Financial District is often considered to be the corner of Wall Street and Broad Street, both of which are contained entirely within the district. Until the late 20th and early 21st century, the neighborhood was considered to be primarily a destination for daytime traders and office workers from around New York City, the Financial District is part of Manhattan Community Board 1, which also includes five other neighborhoods. The Financial District has a number of tourist attractions such as the adjacent South Street Seaport Historic District, the New York City Police Museum, bowling Green is the starting point of traditional ticker-tape parades on Broadway, where here it is also known as the Canyon of Heroes. The Museum of Jewish Heritage and the Skyscraper Museum are both in adjacent Battery Park City which is home to the World Financial Center
Financial District, Manhattan
–
The Financial District of
Lower Manhattan viewed from
New York Harbor, near the
Statue of Liberty, October 2013
Financial District, Manhattan
Financial District, Manhattan
–
One World Trade Center
Financial District, Manhattan
–
70 Pine Street
36.
195th Fighter Squadron
–
The 195th Fighter Squadron is a unit of the Arizona Air National Guard 162d Fighter Wing located at Tucson Air National Guard Base, Arizona. The 195th is equipped with the F-16 Fighting Falcon, see, 373d Fighter Group for expanded World War II history Formed at Westover Field, Massachusetts in August 1943. During World War II the 410th Fighter Squadron was assigned to the European Theater of Operations and it was equipped with P-47 Thunderbolts. The 410th flew its first combat mission on 8 May 1944 and it then took part in preinvasion activities, e. g. escorting B-26 Marauders to attack airdromes, bridges and railroads in Occupied France. The squadron bombed such targets as troops in the Falaise-Argentan area in August 1944, during the Battle of the Bulge, December 1944 - January 1945, the 410th concentrated on the destruction of bridges, marshalling yards and highways. It flew reconnaissance missions to ground operations in the Rhine Valley in March 1945, hitting airfields, motor transports. The squadron continued tactical air operations until 4 May 1945, returned to the United States and prepared for transfer to the Pacific Theater during the Summer of 1945, the Japanese Capitulation in August led to the squadrons inactivation in November 1945. The wartime 410th Fighter Squadron was re-activated and re-designated as the 195th Fighter Squadron and it was allotted to the California Air National Guard, on 24 May 1946. It was organized at Van Nuys Airport, California on 16 September 1946, the 195th Fighter Squadron was entitled to the history, honors, and colors of the 410th. The squadron was equipped with F-51D Mustangs and was assigned to the 146th Fighter Group, as part of the Continental Air Command Fourth Air Force, the squadron trained for tactical fighter missions and air-to-air combat. The unit was called to federal service on 1 March 1951 for duty in the Korean War. Its parent 146th FG was transferred to Moody AFB, Georgia, the 195th FS, however, remained at Van Nuys Airport as part of the Air Defense Command 28th Air Division. It was released from duty and returned to California state control on 11 December 1952. After the Korean War, the squadron was equipped with the long-range F-51H Mustang and remained a part of Air Defense Command, in February 1954, by July 1955 the transition from the F-51H Mustang to the F-86A Sabre was complete. The squadron was re-designated a Fighter Interceptor unit with an air defense mission for the Los Angeles area, during the 1950s, the squadron received newer F-86F Sabres in 1957 and later F-86H Sabre day interceptors in 1959. In 1960, the 146th FIG was reassigned to Military Air Transport Service, with air transportation recognized as a critical wartime need, the 146th was re-designated the 146th Air Transport Wing. During the Berlin Crisis of 1961, both the Wing and squadron were federalized on 1 October 1961, from Van Nuys, the 195th augmented MATS airlift capability worldwide in support of the Air Force’s needs. It returned again to California state control on 31 August 1962, the C-97s were retired in 1970 and the unit was transferred to Tactical Air Command
195th Fighter Squadron
–
195th Fighter Squadron General Dynamics F-16C Block 42H Fighting Falcon 90-0716 flies over Tucson, Arizona
195th Fighter Squadron
–
195th Fighter-Bomber Squadron - North American F-86A Sabre Formation over Los Angeles, 1954
195th Fighter Squadron
–
1972 photo of 195th TAS C-130A 56-0468 in Alaska
195th Fighter Squadron
–
195th TFS A-7D 71-330, about 1985
37.
Arizona Air National Guard
–
The Arizona Air National Guard is the air force militia of the State of Arizona, United States of America. It is, along with the Arizona Army National Guard, an element of the Arizona National Guard, as state militia units, the units in the Arizona Air National Guard are not typically in the normal United States Air Force chain of command unless federalized. They are under the jurisdiction of the Governor of Arizona through the office of the Arizona Adjutant General unless they are federalized by order of the President of the United States. The Arizona Air National Guard is headquartered at the Joint Force Headquarters in Phoenix, under the Total Force concept, Arizona Air National Guard units are an Air Reserve Components of the United States Air Force. Arizona 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. The wing consists of three squadrons and more than 70 F-16 aircraft, the 162d conducts international pilot training in support of foreign military sales program. It operates and maintains extensive radar and communications to train student Weapons Directors for postings throughout the Air Force and 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 Arizona ANG was founded by Barry Goldwater, and its origins date to the formation of the 197th Fighter Squadron at Luke Army Airfield, Glendale and it was equipped with F-51D Mustangs and its mission was the air defense of the state. Today, the Arizona ANG performs an air refueling mission. Its 111th Space Operations Squadron is the first space communications unit in the United States military, after the September 11th,2001 terrorist attacks on the United States, elements of every Air National Guard unit in Arizona has been activated in support of the Global War on Terrorism. Also, Arizona ANG units have deployed overseas as part of Operation Enduring Freedom in Afghanistan. Arizona Wing Civil Air Patrol This article incorporates public domain material from the Air Force Historical Research Agency website http, Arizona Air National Guard 161st Air Refueling Wing 162nd Fighter Wing
Arizona Air National Guard
–
197th Air Refueling Squadron - KC-135R Stratotanker taking off from Sky Harbor Air National Guard Base, Phoenix. The 197th ARS is the oldest unit in the Arizona Air National Guard, having over 60 years of service to the state and nation
Arizona Air National Guard
–
Arizona Air National Guard F-51H Mustang 44-64455 The Mustang was flown by the 197th Fighter Squadron from 1946 until 1954.
38.
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.
39.
USNS Leroy Grumman (T-AO-195)
–
USNS Leroy Grumman is a Henry J. Kaiser-class fleet replenishment oiler of the United States Navy. The Henry J. Kaiser-class oilers were preceded by the shorter Cimarron-class fleet replenishment oilers, Leroy Grumman has an overall length of 206.5 metres. It has a beam of 29.7 metres and a draft of 11 metres, the oiler has a displacement of 41,353 tonnes at full load. It has a capacity of 180,000 imperial barrels of fuel or fuel oil. It can carry a dry load of 690 square metres and can refrigerate 128 pallets of food, the ship is powered by two 10 PC4.2 V570 Colt-Pielstick diesel engines that drive two shafts, this gives a power of 25.6 megawatts. The Henry J. Kaiser-class oilers have maximum speeds of 20 knots and they were built without armaments but can be fitted with close-in weapon systems. The ship has a platform but not any maintenance facilities. It is fitted with five fuelling stations, these can fill two ships at the time and the ship is capable of pumping 900,000 US gallons of diesel or 540,000 US gallons of jet fuel per hour. It has a complement of eighty-nine civilians, twenty-nine spare crew and she entered non-commissioned U. S. Navy service under the control of the Military Sealift Command with a primarily civilian crew on 3 August 1989. Leroy Grumman serves in the United States Atlantic Fleet and this article includes information collected from the Naval Vessel Register, which, as a U. S. government publication, is in the public domain. The entry can be found here, navSource Online, Service Ship Photo Archive, USNS Leroy Grumman USNS Leroy Grumman Wildenberg, Thomas. Gray Steel and Black Oil, Fast Tankers and Replenishment at Sea in the U. S. Navy, 1912-1995
USNS Leroy Grumman (T-AO-195)
–
USNS Leroy Grumman (T-AO-195)
40.
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
41.
Fleet replenishment oiler
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A replenishment oiler is a naval auxiliary ship with fuel tanks and dry cargo holds, which can conduct underway replenishment on the high seas. Several countries have used replenishment oilers, the US Navy hull classification symbol for this type of ship was AOR. Replenishment oilers are slower and carry fewer dry stores than the US Navys fast combat support ships, the development of the oiler paralleled the change from coal- to oil-fired boilers in warships. Though arguments related to fuel security were made against such a change, one of the first generation of blue-water navy oiler support vessels was the British RFA Kharki, active 1911 in the run-up to the First World War. Such vessels heralded the transition from coal to oil as the fuel of warships and removed the need to rely on, modern examples of the fast combat support ship include the large British Fort-Class, displacing and 669 ft in length, and the Australian HMAS Sirius. For smaller navies, such as the Royal New Zealand Navy, such ships are designed to carry large amounts of fuel and dry stores for the support of naval operations far away from port. Replenishment oilers are also equipped with extensive medical and dental facilities than smaller ships can provide. Such ships are equipped with multiple refueling gantries to refuel and resupply ships at a time. The process of refueling and supplying ships at sea is called underway replenishment, furthermore, such ships often are designed with helicopter decks and hangars. This allows the operation of rotary-wing aircraft, which allows the resupply of ships by helicopter and this process is called vertical replenishment. They may also carry man-portable air-defense systems for air defense capability. Kaiser-class oiler (United States Navy and Chilean Navy ex-USNS Andrew J
Fleet replenishment oiler
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The replenishment oiler
HMAS Sirius (right) providing fuel to the amphibious warfare ship
USS Juneau while both are underway
Fleet replenishment oiler
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A replenishment oiler at work
42.
USS Lenawee (APA-195)
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USS Lenawee was a Haskell-class attack transport built and used by the US Navy in World War II and saw further service during the Korean War and Vietnam War. She was a Victory ship design, VC2-S-AP5 and she was named after Lenawee County, Michigan, USA. Built to transport troops to hostile shores, Lenawee picked up her complement of landing craft at San Francisco. Following amphibious training in the Hawaiian Islands, she sailed 27 January 1945 for Saipan, in the Marianas the final rehearsals for her entrance into the battle zone were held, and 1,503 troops of the 5th Marines and the 62th Naval Construction Battalion embarked. After a 3-day voyage, she arrived at Iwo Jima on D-Day,19 February, withdrawn on the 27th, she retired to Guam to discharge Marine casualties and prepare for the final large-scale amphibious operation of World War II. Sailing south to Espiritu Santo, she embarked over 1,000 troops of the Army 27th Division to reinforce the Okinawa invasion forces. Landing troops and cargo each day and retiring to open sea each night, as part of Commodore J. B. McGoverns Transport Squadron 16, she transported troops from the Philippines to Japan and was present in Tokyo Bay with 1,135 troops of the 1st Cavalry when the Japanese surrendered 2 September. The outbreak of the Korean War caused her to recommission 30 September 1950, with San Diego as her home port, she operated part of each year, except 1952 and 1956, in the Far East. Her first voyage began 22 March 1951 when she departed for Yokosuka, operating mainly among the Japanese Islands, she twice transported men and supplies to the Korean theater before returning home 27 November. Again in May 1953 Lenawee returned to duties in Korean waters and was at Inchon in July when the final truce was signed. The Chinese offshore islands and Vietnam proved to be the new areas in the Far East. S. Ambassador to China Karl L. Ranking for a first hand observation, even without such crises, the Navy never loses its alertness, continually training for any eventuality. Each year amphibious operations were held with marines either off the California coast, in the Hawaiians, in mid-August 1957, USS Wantuck got underway from San Diego for Hawaii on the first leg of a voyage to Japan. Wantuck suffered two men killed—one who drowned in the engine room and another scalded by high-pressure steam—and five injured. The submarine rescue ship USS Florikan and fleet ocean tug USS Cree came to Wantuck s aid, while Lenawee took some of Wantucks injured men aboard, Wantuck arrived in San Diego under tow on the evening of 16 August 1957. Deemed not worth repairing, Wantuck was decommissioned at San Diego on 15 November 1957 and her name was struck from the Navy List on 4 March 1958. Beginning in 1963 the South China Sea became a scene of operations for Lenawee
USS Lenawee (APA-195)
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USS Lenawee (APA-195) operating out of
San Diego, California in March, 1957.
43.
Haskell-class attack transport
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Haskell-class attack transports were amphibious assault ships of the United States Navy created in 1944. They were designed to transport 1,500 troops and their combat equipment, the Haskells were very active in the World War II Pacific Theater of Operations, landing Marines and Army troops and transporting casualties at Iwo Jima and Okinawa. Ships of the class were among the first Allied ships to enter Tokyo Bay at the end of World War II, after the end of World War II, most participated in Operation Magic Carpet, the massive sealift of US personnel back to the United States. A few of the Haskell class were reactivated for the Korean War, the Haskell class, Maritime Commission standard type VC2-S-AP5, is a sub‑type of the World War II Victory ship design. 117 were launched in 1944 and 1945, with 14 more being finished as another VC2 type or canceled, the VC2-S-AP5 design was intended for the transport and assault landing of over 1,500 troops and their heavy combat equipment. During Operation Magic Carpet, up to 1,900 personnel per ship were carried homeward, the Haskells carried 25 landing craft to deliver the troops and equipment right onto the beach. The 23 main boats were the 36 feet long, LCVP, the LCVP was designed to carry 36 equipped troops. The other 2 landing craft were the 50 foot long LCM, capable of carrying 60 troops or 30 tons of cargo, the Haskell-class ships were armed with one 5/38 caliber gun, twelve Bofors 40 mm guns, and ten Oerlikon 20 mm guns. See List of Haskell-class attack transports, Haskell-class attack transports included APA-117, USS Haskell, the lead ship, through APA-247, the never completed USS Mecklenburg. The hulls for APA-181 through APA-186 were repurposed to be hospital ships before they were named, ultimately those hospital ships were built on larger C4 plan and the six VC2 hulls were built in a merchant configuration. APA-240 through APA-247 were named, but cancelled in 1945 when the war ended, with the special exception of the USS Marvin H. McIntyre, the Haskell-class ships were all named after counties of the United States. Most of the Haskell-class ships were mothballed in 1946, with only a few remaining in service, many of the Haskell class were scrapped in 1973-75. A few were converted into Missile Range Instrumentation Ships, the USS Gage, the last remaining ship in the Haskell configuration, was scrapped in 2009 at ESCO Marine, in Brownsville, Tx. The USS Sherburne, which was converted and renamed USS Range Sentinel, lasted until she was scrapped in 2012
Haskell-class attack transport
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USS Noble, a ship of the Haskell class, in 1956
Haskell-class attack transport
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The
USS St. Mary's in
San Francisco Bay,
California, in late 1945 or early 1946. She is returning troops from the western Pacific to the United States as part of Operation Magic Carpet. Note the long homeward bound pennant trailing from her after mast, and the sign on shore (in the right distance) stating " Welcome Home, Well Done."
Haskell-class attack transport
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The
USS Rutland lowering an LCM off
Iwo Jima, 1945.
44.
Attack transport
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Attack transport is a United States Navy ship classification for a variant of ocean-going troopship adapted to transporting invasion forces ashore. Unlike standard troopships – often drafted from commercial shipping fleets – that rely on either a quay or tenders and they are not to be confused with landing ships, which beach themselves to bring their troops directly ashore, or their general British equivalent, the Landing ship, infantry. A total of 388 APA and AKA attack transports were built for service in World War II in at least fifteen classes, depending on class they were armed with one or two 5-inch guns and a variety of 40 mm and 20 mm anti-aircraft weapons. Some of these were outfitted with heavy boat davits and other arrangements to enable them to handle landing craft] for amphibious assault operations. In 1942, when the AP number series had extended beyond 100. Therefore, the new classification of attack transport was created and numbers assigned to fifty-eight APs then in commission or under construction, the actual reclassification of these ships was not implemented until February 1943, by which time two ships that had APA numbers assigned had been lost. Another two transports sunk in 1942, USS George F. Elliott and USS Leedstown, were configured as attack transports. In addition, as part of the 1950s modernization of the Navys amphibious force with faster ships, as a result, only attack transport ships were assigned for the assault, without support from any companion attack cargo ships. This created extreme logistics burdens for the force because it resulted in considerable overloading of the transports with both men and equipment. To compound problems, these forces were not able to assemble or train together before executing the Aleutian invasion on 11 May 1943, lack of equipment and training subsequently resulted in confusion during the landings on Attu. By the end of the 1950s, it was clear that boats would soon be superseded by amphibious tractors and helicopters for landing assault troops. These could not be supported by attack transports in the numbers required, by 1969, when the surviving attack transports were redesignated LPA, only a few remained in commissioned service. The last of these were decommissioned in 1980 and sold abroad, the APA/LPA designation may, therefore, now be safely considered extinct. Nearly identical ships used to transport vehicles, supplies and landing craft, Landing Ship Infantry This article incorporates text from the public domain Dictionary of American Naval Fighting Ships. APA/LPA -- Attack Transports by the US Naval Historical Center
Attack transport
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The
USS American Legion was a
Harris-class attack transport launched in 1919 that saw extensive service in World War II
Attack 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
Attack transport
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A loaded
Bayfield-class attack transport underway, the
USS Hamblen (APA-114)
45.
Minesweeper
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A minesweeper is a small naval warship designed to engage in minesweeping. Using various mechanisms intended to counter the threat posed by naval mines, although naval warfare has a long history, the earliest known usage of the naval mine dates to the Ming dynasty. Dedicated minesweepers, however, only appear in the record several centuries later, to the Crimean War. In the Crimean War, minesweepers consisted of British rowboats trailing grapnels to snag the mines, despite the use of mines in the American Civil War, there are no records of effective minesweeping being used. Officials in the Union Army attempted to create the first minesweeper but were plagued by flawed designs, minesweeping technology picked up in the Russo-Japanese War, using aging torpedo boats as minesweepers. In Britain, naval leaders recognized before the outbreak of World War I that the development of sea mines was a threat to the nations shipping, sir Arthur Wilson noted the real threat of the time was blockade aided by mines and not invasion. A Trawler Section of the Royal Navy Reserve became the predecessor of the mine sweeping forces with specially designed ships and these reserve Trawler Section fishermen and their trawlers were activated, supplied with mine gear, rifles, uniforms and pay as the first minesweepers. The dedicated, purpose-built minesweeper first appeared during World War I with the Flower-class minesweeping sloop, by the end of the War, naval mine technology had grown beyond the ability of minesweepers to detect and remove. Minesweeping made significant advancements during World War II, combatant nations quickly adapted ships to the task of minesweeping, including Australias 35 civilian ships that became Auxiliary Minesweepers. Both Allied and Axis countries made heavy use of minesweepers throughout the war, historian Gordon Williamson wrote that Germanys minesweepers alone formed a massive proportion of its total strength, and are very much the unsung heroes of the Kriegsmarine. Naval mines remained a threat even after the war ended, after the Second World War, allied countries worked on new classes of minesweepers ranging from 120-ton designs for clearing estuaries to 735-ton oceangoing vessels. The United States Navy even used specialized Mechanized Landing Craft to sweep shallow harbors in, as of June 2012, the U. S. Navy had four minesweepers deployed to the Persian Gulf to address regional instabilities. Minesweepers are equipped with mechanical or electrical devices, known as sweeps, mechanical sweeps are devices designed to cut the anchoring cables of moored mines, and preferably attach a tag to help the subsequent localization and neutralization. They are towed behind the minesweeper, and use a body to maintain the sweep at the desired depth. Influence sweeps are equipment, often towed, that emulate a particular ship signature, the most common such sweeps are magnetic and acoustic generators. There are two modes of operating an influence sweep, MSM and TSM, MSM sweeping is founded on intelligence on a given type of mine, and produces the output required for detonation of this mine. If such intelligence is unavailable, the TSM sweeping instead reproduces the influence of the ship that is about to transit through the area. TSM sweeping thus clears mines directed at this ship without knowledge of the mines, however, mines directed at other ships might remain
Minesweeper
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US Navy
Admirable-class minesweeper
USS Pivot in the Gulf of Mexico for sea trials on 12 July 1944
Minesweeper
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A minesweeper cutting loose moored mines.
Minesweeper
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Siegburg, a modern
Ensdorf-class minesweeper of the
German Navy
46.
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
47.
USS Sealion (SS-195)
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USS Sealion, a Sargo-class submarine, was the first ship of the United States Navy to be named for the sea lion, any of several large, eared seals native to the Pacific. Her keel was laid down on 20 June 1938 by the Electric Boat Company of Groton and she was launched on 25 May 1939 sponsored by Mrs. Augusta K. Bloch, wife of Admiral Claude C. Bloch, Commander-in-Chief, United States Fleet, and commissioned on 27 November 1939, following shakedown, Sealion, assigned to Submarine Division 17, prepared for overseas deployment. In the spring of 1940, she sailed, with her division for the Philippine Islands, arriving at Cavite in the fall to commence operations as a unit of the Asiatic Fleet. Into October 1941, she ranged from Luzon into the Sulu Archipelago, then, with her sister ship Seadragon, another submarine in SubDiv 202, she prepared for a regular overhaul at the Cavite Navy Yard. By 8 December, her yard period had begun, and, the first bomb struck the aft end of her conning tower and exploded outside the hull, over the control room. In addition, one crewman, Howard Firth, died while a POW, Sealion flooded immediately and settled down by the stern with 40% of her main deck underwater and a 15-degree list to starboard. The destruction of the yard made repairs impossible, and she was ordered destroyed. All salvageable equipment was taken off, depth charges were placed inside, and on 25 December, eli Thomas Reich, who was executive officer and engineer on Sealion when it was sunk, assumed command of the second Sealion in March 1944. In a graphic story in El Alamein no Shinden, the Sealion stops the German invasion of England, Operation Sealion
USS Sealion (SS-195)
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History
48.
Sargo-class submarine
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Similar to the previous Salmon class, they were built between 1937 and 1939. With a top speed of 21 knots, a range of 11,000 nautical miles, in some references, the Salmons and Sargos are called the New S Class, 1st and 2nd Groups. The Sargo-class submarine USS Swordfish had the distinction of being the first US Navy vessel to sink a Japanese ship in World War II. In most features the Sargos were a repeat of the Salmons, except for the return to full diesel-electric drive for the last four boats, the first six Sargos were driven by a composite direct-drive and diesel-electric plant in the same manner as the Salmons. In this arrangement, two engines in the forward engine room drove generators. In the after engine room, two engines were clutched to reduction gears which sat forward of the engines, with vibration-isolating hydraulic clutches. Two high-speed electric motors, driven by the engines or batteries, were also connected to each reduction gear. The Bureau of Steam Engineering and the General Board desired a full diesel-electric plant, hart, the only experienced submariner on the General Board, who pointed out that a full diesel-electric system could be disabled by flooding. Technical problems went against the use of two large direct-drive diesels in place of the four-engine composite plant, no engine of suitable power to reach the desired 21-knot speed existed in the US, and the current vibration-isolating hydraulic clutches were not capable of transmitting enough power. It was also not practical to gear two engines to each shaft, so a full diesel-electric plant was adopted for the last four Sargos, and remained standard for all subsequent conventionally-powered US submarines. Four of the class were equipped with the troublesome Hooven-Owens-Rentschler double-acting diesels, an attempt to produce more power from a smaller engine than other contemporary designs, the double-acting system proved unreliable in service. During World War II, all had their engines replaced with GM-Winton 16-278A engines, instead of a single hard rubber case, it had two concentric hard rubber cases with a layer of soft rubber between them. This was to prevent sulfuric acid leakage in the event one case cracked during depth-charging and this remained the standard battery design until replaced with Sargo II and GUPPY batteries in submarines upgraded under the Greater Underwater Propulsion Power Program after World War II. The original Mark 21 3-inch /50 caliber deck gun proved to be too light in service and it lacked sufficient punch to finish off crippled or small targets quickly enough to suit the crews. It was replaced by the Mark 9 4-inch /50 caliber gun in 1943-44, from commissioning until late 1941 the first six Sargos were based first at San Diego, later at Pearl Harbor. The last four were sent to the Philippines shortly after commissioning, the Japanese occupation of southern Indo-China and the August 1941 American-British-Dutch retaliatory oil embargo had raised international tensions. After the Japanese attack on Pearl Harbor on 7 December 1941 and he was assigned sixteen Salmons or Sargos, the entirety of both classes. Seven Porpoise-class and six S-boats rounded out the force, fortunately, the Japanese did not bomb the Philippines until 10 December 1941, so almost all of the submarines were able to get underway before an attack
Sargo-class submarine
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USS Sargo
Sargo-class submarine
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Periscope photo of a Japanese merchant ship torpedoed by Seawolf sinking.
49.
Submarine
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A submarine is a watercraft capable of independent operation underwater. It differs from a submersible, which has more limited underwater capability, the term most commonly refers to a large, crewed vessel. It is also used historically or colloquially to refer to remotely operated vehicles and robots, as well as medium-sized or smaller vessels, such as the midget submarine. The noun submarine evolved as a form of submarine boat, by naval tradition, submarines are usually referred to as boats rather than as ships. Although experimental submarines had been built before, submarine design took off during the 19th century, Submarines were first widely used during World War I, and now figure in many navies large and small. Civilian uses for submarines include marine science, salvage, exploration and facility inspection, Submarines can also be modified to perform more specialized functions such as search-and-rescue missions or undersea cable repair. Submarines are also used in tourism, and for undersea archaeology, most large submarines consist of a cylindrical body with hemispherical ends and a vertical structure, usually located amidships, which houses communications and sensing devices as well as periscopes. In modern submarines, this structure is the sail in American usage, a conning tower was a feature of earlier designs, a separate pressure hull above the main body of the boat that allowed the use of shorter periscopes. There is a propeller at the rear, and various hydrodynamic control fins, smaller, deep-diving and specialty submarines may deviate significantly from this traditional layout. Submarines use diving planes and also change the amount of water, Submarines have one of the widest ranges of types and capabilities of any vessel. Submarines can work at greater depths than are survivable or practical for human divers, modern deep-diving submarines derive from the bathyscaphe, which in turn evolved from the diving bell. In 1578, the English mathematician William Bourne recorded in his book Inventions or Devises one of the first plans for an underwater navigation vehicle and its unclear whether he ever carried out his idea. The first submersible of whose construction there exists reliable information was designed and built in 1620 by Cornelis Drebbel and it was propelled by means of oars. By the mid-18th century, over a dozen patents for submarines/submersible boats had been granted in England, in 1747, Nathaniel Symons patented and built the first known working example of the use of a ballast tank for submersion. His design used leather bags that could fill with water to submerge the craft, a mechanism was used to twist the water out of the bags and cause the boat to resurface. In 1749, the Gentlemens Magazine reported that a design had initially been proposed by Giovanni Borelli in 1680. By this point of development, further improvement in design stagnated for over a century, until new industrial technologies for propulsion. The first military submarine was the Turtle, a hand-powered acorn-shaped device designed by the American David Bushnell to accommodate a single person and it was the first verified submarine capable of independent underwater operation and movement, and the first to use screws for propulsion
Submarine
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A
Russian Navy Typhoon-class submarine underway. Also known as "Project 941".
Submarine
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Drebbel, the first navigable submarine
Submarine
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The French submarine Plongeur
Submarine
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The
Nordenfelt -designed,
Ottoman submarine
Abdül Hamid
50.
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
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BRP Rajah Humabon (PF-11) of the Philippine Navy