1.
120 (number)
–
120, read as one hundred twenty, is the natural number following 119 and preceding 121. In English and other Germanic languages, it was formerly known as one hundred. This hundred of six score is now obsolete, but is described as the hundred or great hundred in historical contexts. 120 is the factorial of 5, and the sum of a twin prime pair,120 is the sum of four consecutive prime numbers, four consecutive powers of 2, and four consecutive powers of 3. It is also a sparsely totient number,120 is the smallest number to appear six times in Pascals triangle. 120 is also the smallest multiple of 6 with no adjacent prime number and it is the eighth hexagonal number and the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers,120 is the first multiply perfect number of order three. The sum of its factors sum to 360, exactly three times 120, note that perfect numbers are order two by the same definition. 120 is divisible by the number of primes below it,30 in this case, however, there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number. The sum of Eulers totient function φ over the first nineteen integers is 120,120 figures in Pierre de Fermats modified Diophantine problem as the largest known integer of the sequence 1,3,8,120. Fermat wanted to another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square. Leonhard Euler also searched for this number, but failed to find it, the internal angles of a regular hexagon are all 120 degrees. 120 is a Harshad number in base 10,120 is the atomic number of Unbinilium, an element yet to be discovered. The cubits of the height of the Temple building The age at which Moses died, in astrology, when two planets in a persons chart are 120 degrees apart from each other, this is called a trine. This is supposed to bring luck in the persons life. The height in inches of a hoop in the National Basketball Association. 120 is also, The medical telephone number in China In Austria, in the US Army, a common diameter for a mortar in mm. TT scale, a scale for model trains, is 1,120. 120 film is a medium format film developed by Kodak,120, a 2008 Turkish film The Israeli national legislature, the Knesset, has 120 seats
120 (number)
–
The
120-cell (or hecatonicosachoron) is a
convex regular 4-polytope consisting of 120
dodecahedral cells
2.
121 (number)
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121 is the natural number following 120 and preceding 122. One hundred twenty-one is a square and is the sum of three consecutive primes, there are no squares besides 121 known to be of the form 1 + p + p 2 + p 3 + p 4, where p is prime. Other such squares must have at least 35 digits, there are only two other squares known to be of the form n. Another example of 121 being of the few examples supporting a conjecture is that Fermat conjectured that 4 and 121 are the perfect squares of the form x3 -4. It is also a number and a centered octagonal number. In base 10, it is a Smith number since its digits add up to the value as its factorization. But it can not be expressed as the sum of any other number plus that numbers digits, making 121 a self number
121 (number)
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A
Chinese checkers board has 121 holes
3.
125 (number)
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125 is the natural number following 124 and preceding 126. It can be expressed as a sum of two squares in two different ways,125 = 10² + 5² = 11² + 2²,125 and 126 form a Ruth-Aaron pair under the second definition in which repeated prime factors are counted as often as they occur. Like many other powers of 5, it is a Friedman number in base 10 since 125 =51 +2
125 (number)
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This article may contain excessive, poor, or irrelevant examples. Please improve the article by adding more descriptive text and removing
less pertinent examples. See Wikipedia's
guide to writing better articles for further suggestions. (March 2010)
4.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
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Algebraic structure → Group theory
Group theory
5.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
6.
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.
7.
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
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A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.
8.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime number
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The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
9.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
Divisor
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The divisors of 10 illustrated with
Cuisenaire rods: 1, 2, 5, and 10
10.
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
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Numeral systems
Greek numerals
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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.
11.
Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
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
12.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
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
19.
Hexadecimal
–
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
–
Numeral systems
Hexadecimal
–
Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
–
Hexadecimal finger-counting scheme.
20.
Vigesimal
–
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
–
Numeral systems
Vigesimal
–
The
Maya numerals are a base-20 system.
21.
Base 36
–
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
–
Numeral systems
Base 36
–
34 senary = 22 decimal, in senary finger counting
Base 36
22.
Natural number
–
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
–
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
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
23.
Mersenne prime
–
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
Mersenne prime
–
Graph of number of digits in largest known Mersenne prime by year – electronic era. Note that the vertical scale, the number of digits, is a double logarithmic scale of the value of the prime.
24.
Perfect number
–
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Equivalently, a number is a number that is half the sum of all of its positive divisors i. e. σ1 = 2n. This definition is ancient, appearing as early as Euclids Elements where it is called τέλειος ἀριθμός. Euclid also proved a formation rule whereby q /2 is a perfect number whenever q is a prime of the form 2 p −1 for prime p —what is now called a Mersenne prime. Much later, Euler proved that all even numbers are of this form. This is known as the Euclid–Euler theorem and it is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first perfect number is 6 and its proper divisors are 1,2, and 3, and 1 +2 +3 =6. Equivalently, the number 6 is equal to half the sum of all its positive divisors, the next perfect number is 28 =1 +2 +4 +7 +14. This is followed by the perfect numbers 496 and 8128, in about 300 BC Euclid showed that if 2p−1 is prime then 2p−1 is perfect. The first four numbers were the only ones known to early Greek mathematics. Philo of Alexandria in his first-century book On the creation mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, and by Didymus the Blind, st Augustine defines perfect numbers in City of God in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs mentioned the next three numbers and listed a few more which are now known to be incorrect. Euclid proved that 2p−1 is a perfect number whenever 2p −1 is prime. Prime numbers of the form 2p −1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, for 2p −1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p −1 with a prime p are prime, in fact, Mersenne primes are very rare—of the 9,592 prime numbers p less than 100,000, 2p −1 is prime for only 28 of them. Nicomachus conjectured that every number is of the form 2p−1 where 2p −1 is prime. Ibn al-Haytham circa 1000 AD conjectured that every perfect number is of that form
Perfect number
–
Overview
25.
Largest known prime number
–
As of January 2017, the largest known prime number is 274,207,281 −1, a number with 22,338,618 digits. It was found in 2016 by the Great Internet Mersenne Prime Search, euclid proved that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers. Many of the largest known primes are Mersenne primes, as of January 2017, the six largest known primes are Mersenne primes, while the seventh is the largest known non-Mersenne prime. The last 16 record primes were Mersenne primes, the fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers. The record is held by 274,207,281 −1 with 22,338,618 digits, found by GIMPS in 2015. 717774014762912462113646879425801445107393100212927181629335931494239018213879217671164956287190498687010073391086436351 The first and last 120 digits are shown above, there are several prizes offered by the Electronic Frontier Foundation for record primes. GIMPS is also coordinating its long-range search efforts for primes of 100 million digits and larger, the record passed one million digits in 1999, earning a $50,000 prize. In 2008 the record passed ten million digits, earning a $100,000 prize, time called it the 29th top invention of 2008. Additional prizes are being offered for the first prime number found with at least one hundred million digits, both the $50,000 and the $100,000 prizes were won by participation in GIMPS. The following table lists the progression of the largest known prime number in ascending order, here Mn= 2n −1 is the Mersenne number with exponent n. The longest record-holder known was M19 =524,287, which was the largest known prime for 144 years, almost no records are known before 1456. GIMPS found the thirteen latest records on ordinary computers operated by participants around the world
Largest known prime number
–
Plot of the number of digits in largest known prime by year, since the electronic computer. Note that the vertical scale is
logarithmic. The red line is the exponential curve of
best fit: y = exp(0.188439 t - 362.591), where t is in years.
26.
Odd number
–
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
Odd number
–
Rubik's Revenge in solved state
27.
Semiprime
–
In mathematics, a semiprime is a natural number that is the product of two prime numbers. The semiprimes less than 100 are 4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94, and 95. Semiprimes that are not perfect squares are called discrete, or distinct, by definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its factors are 1,2,13. The total number of prime factors Ω for a n is two, by definition. A semiprime is either a square of a prime or square-free, the square of any prime number is a semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a number is a semiprime without knowing the two factors. A composite n non-divisible by primes ≤ n 3 is semiprime, various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits. These are considered novelties, since their construction method might prove vulnerable to factorization, for a semiprime n = pq the value of Eulers totient function is particularly simple when p and q are distinct, φ = = p q − +1 = n − +1. If otherwise p and q are the same, φ = φ = p = p2 − p = n − p and these methods rely on the fact that finding two large primes and multiplying them together is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of specific large semiprimes, the most recent such challenge closed in 2007. In practical cryptography, it is not sufficient to choose just any semiprime, the factors p and q of n should both be very large, around the same order of magnitude as the square root of n, this makes trial division and Pollards rho algorithm impractical. At the same time they should not be too close together, or else the number can be quickly factored by Fermats factorization method. The number may also be chosen so that none of p −1, p +1, q −1, or q +1 are smooth numbers, protecting against Pollards p −1 algorithm or Williams p +1 algorithm. However, these checks cannot take future algorithms or secret algorithms into account, in 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. It consisted of 1679 binary digits intended to be interpreted as a 23×73 bitmap image, the number 1679 = 23×73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns. Chens theorem Weisstein, Eric W. Semiprime
Semiprime
–
Overview
28.
Centered hexagonal number
–
The nth centered hexagonal number is given by the formula n 3 −3 =3 n +1. Expressing the formula as 1 +6 shows that the centered hexagonal number for n is 1 more than 6 times the th triangular number. The first few centered hexagonal numbers are,1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919. In base 10 one can notice that the hexagonal numbers rightmost digits follow the pattern 1–7–9–7–1, the sum of the first n centered hexagonal numbers is n3. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. In particular, prime centered hexagonal numbers are cuban primes, the difference between 2 and the nth centered hexagonal number is a number of the form 3n2 + 3n −1, while the difference between 2 and the nth centered hexagonal number is a pronic number. Hexagonal number Magic hexagon Star number
Centered hexagonal number
29.
Motzkin number
–
In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords between n points on a circle. The Motzkin numbers are named after Theodore Motzkin, and have diverse applications in geometry, combinatorics. The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle, the following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle. Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers, a Motzkin prime is a Motzkin number that is prime. Guibert, Pergola & Pinzani showed that vexillary involutions are enumerated by Motzkin numbers
Motzkin number
–
Contents
30.
Nonary
–
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
Nonary
–
Numeral systems
31.
Positive integers
–
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
Positive integers
–
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.
Positive integers
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
32.
Composite number
–
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
–
Overview
33.
USNS Mission San Luis Obispo (T-AO-127)
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SS Mission San Luis Obispo was a Type T2-SE-A2 tanker built for the United States Maritime Commission during World War II. After the war she was acquired by the United States Navy as USS Mission San Luis Obispo, later the tanker transferred to the Military Sea Transportation Service as USNS Mission San Luis Obispo. A Mission Buenaventura-class oiler, she was named for Mission San Luís Obispo de Tolosa in San Luis Obispo, chartered to Pacific Tankers, Inc. for operations, she spent the remainder of the War carrying fuel to our forces in the Pacific. She remained in this capacity until 27 March 1946 when she was returned to the Maritime Commission and laid up in the Maritime Reserve Fleet at James River, Virginia. Acquired by the Navy on 24 October 1947 she was placed in service with the Naval Transportation Service as Mission San Luis Obispo and she was transferred to the operational control of the newly created Military Sea Transportation Service on 1 October 1949 as USNS Mission San Luis Obispo. She served with MSTS until 22 June 1955 when she was returned to the U. S. Maritime Administration and laid up in the Maritime Reserve Fleet at Olympia, Washington. Reacquired by the Navy on 27 June 1956, she was placed in service with MSTS and served until 24 June 1957 and she was struck from the Naval Vessel Register on 24 September 1957. The ship was sold to the Hudson Waterways Corporation on 25 March 1966 and was renamed Seatrain Puerto Rico on 1 April 1966. The ship was transferred to the National Defense Reserve Fleet on December 19,1974, the ship was retired and broken up in 1986. This article incorporates text from the public domain Dictionary of American Naval Fighting Ships, the entry can be found here. Hendrickson, David, From Boxcars to Boxships, The Ships of Seatrain Lines, Steamboat Bill No.254,2005, pp. 89–102, photo gallery of Mission San Luis Obispo at NavSource Naval History
USNS Mission San Luis Obispo (T-AO-127)
–
History
34.
USNS Mission Buenaventura (T-AO-111)
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SS Mission Buenaventura was a Type T2-SE-A2 tanker built for the United States Maritime Commission during World War II. After the war she was acquired by the United States Navy as USS Mission Buenaventura, later the tanker transferred to the Military Sea Transportation Service as USNS Mission Buenaventura. The lead ship in her class of oilers, she was named for Mission San Buenaventura located in Ventura. Chartered to Deconhill Shipping Company, for operations, she spent the remainder of the War supporting the victorious Allied forces in the Pacific. She was returned to the Maritime Commission in March 1946 and on 30 March was laid up in the Maritime Commission Reserve Fleet at Mobile, acquired by the Navy on 18 November 1947 she was activated and transferred to the Naval Transportation Service for service as Mission Buenaventura. When the Naval Transportation Service was absorbed by the new Military Sea Transportation Service and she continued her worldwide service until 4 April 1960 when she was transferred to the Maritime Commission for layup at Mobile. She was taken out of service and struck from the Naval Vessel Register on 31 March 1972, final disposition, disposed of for scrap by MARAD sale 26 June 1978
USNS Mission Buenaventura (T-AO-111)
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USNS Mission Buenaventura
35.
Oiler (ship)
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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
Oiler (ship)
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The replenishment oiler
HMAS Sirius (right) providing fuel to the amphibious warfare ship
USS Juneau while both are underway
Oiler (ship)
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A replenishment oiler at work
36.
World War II
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World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945, although related conflicts began earlier. It involved the vast majority of the worlds countries—including all of the great powers—eventually forming two opposing alliances, the Allies and the Axis. It was the most widespread war in history, and directly involved more than 100 million people from over 30 countries. Marked by mass deaths of civilians, including the Holocaust and the bombing of industrial and population centres. These made World War II the deadliest conflict in human history, from late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, and formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Poland, Finland, Romania and the Baltic states. In December 1941, Japan attacked the United States and European colonies in the Pacific Ocean, and quickly conquered much of the Western Pacific. The Axis advance halted in 1942 when Japan lost the critical Battle of Midway, near Hawaii, in 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained all of its territorial losses and invaded Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in South Central China and Burma, while the Allies crippled the Japanese Navy, thus ended the war in Asia, cementing the total victory of the Allies. World War II altered the political alignment and social structure of the world, the United Nations was established to foster international co-operation and prevent future conflicts. The victorious great powers—the United States, the Soviet Union, China, the United Kingdom, the Soviet Union and the United States emerged as rival superpowers, setting the stage for the Cold War, which lasted for the next 46 years. Meanwhile, the influence of European great powers waned, while the decolonisation of Asia, most countries whose industries had been damaged moved towards economic recovery. Political integration, especially in Europe, emerged as an effort to end pre-war enmities, the start of the war in Europe is generally held to be 1 September 1939, beginning with the German invasion of Poland, Britain and France declared war on Germany two days later. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or even the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred simultaneously and this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935. The British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the forces of Mongolia and the Soviet Union from May to September 1939, the exact date of the wars end is also not universally agreed upon. It was generally accepted at the time that the war ended with the armistice of 14 August 1945, rather than the formal surrender of Japan
World War II
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Clockwise from top left: Chinese forces in the
Battle of Wanjialing, Australian
25-pounder guns during the
First Battle of El Alamein, German
Stuka dive bombers on the
Eastern Front in December 1943, a U.S. naval force in the
Lingayen Gulf,
Wilhelm Keitel signing the
German Instrument of Surrender, Soviet troops in the
Battle of Stalingrad
World War II
–
The
League of Nations assembly, held in
Geneva,
Switzerland, 1930
World War II
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Adolf Hitler at a German
National Socialist political rally in
Weimar, October 1930
World War II
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Italian soldiers recruited in 1935, on their way to fight the
Second Italo-Abyssinian War
37.
USS Admiral W. S. Sims (AP-127)
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USS Admiral W. S. Sims was a transport in the United States Navy. She was later renamed USNS General William O. Darby, later her name was struck and she was known simply by her hull number. In 1981, she was reclassified as IX-510, Admiral W. S. Anne Hitchcock Sims, widow of Admiral William Sims, delivered to the Navy on 27 September 1945 and commissioned the same day, Captain Edward C. She arrived at Manila on 6 November, and departed the Philippine port with 4,980 troops and passengers and she commenced her second round-trip voyage to the Philippines with her departure from San Francisco on 7 December. Arriving at Manila on the 22d, the sailed for home two days after Christmas. Re-routed on her voyage, Admiral W. S. Sims reached San Pedro with 4,973 passengers on board on 11 January 1946. She subsequently conducted one troop lift from Okinawa, sailing from San Pedro on 3 February 1946, Admiral W. S. Sims made one more voyage to the Far East as a Navy transport. After shifting down the west coast from Seattle to San Francisco, she sailed from the port on 27 March for Korean waters. The transport arrived at Jinsen, on 11 April 1946, and, after embarking 106 passengers at Jinsen sailed for Okinawa, arriving there on 15 April and embarking 910 additional passengers. Clearing Buckner Bay for the Philippines on 16 April, the ship disembarked 26 people at Manila, Admiral W. S. Sims reached San Francisco on 7 May. Decommissioned at San Francisco on 21 June 1946, she was transferred to the War Shipping Administration. Admiral W. S. Sims was struck from the Naval Vessel Register on 3 July 1946 and he was killed in action on the Italian front on 30 April 1945, while serving as Assistant Commander of the 10th Mountain Division. Operating out of New York under the Military Sea Transportation Service, between 1950 and 1953, she made more than 20 round trip voyages to Bremerhaven, Germany, and back. In November 1951, the ship veered 100 miles off course to respond to an SOS from a German freighter in the Bay of Biscay. Thirteen of General William O. Darbys sailors volunteered to man a lifeboat, departing New York on 20 June 1953, General William O. Darby proceeded to Yokosuka, Japan, via the Panama Canal, arriving at the Japanese port on 17 July to embark Korean War veterans. Returning to Seattle on 29 July, the transport made five more voyages between the west coast of the United States and Japan in the next five months. After returning to San Francisco on 23 January 1954, she sailed for the east coast on the 25th to resume operations with MSTS, ranging from North Africa to Turkey in that tour, she eventually returned to New York on 6 August 1956. Between 1956 and 1965, the ship conducted some 135 runs to Bremerhaven and back, in February 1963, General William O. Darby brought back from Bremerhaven two paintings loaned temporarily to the United States from the French Louvre, Whistlers Mother and La Madeleine
USS Admiral W. S. Sims (AP-127)
–
USS Admiral W. S. Sims (AP-127) at anchor.
38.
United States Navy
–
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
–
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
39.
Transport
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Transport or transportation is the movement of people, animals and goods from one location to another. Modes of transport include air, rail, road, water, cable, pipeline, the field can be divided into infrastructure, vehicles and operations. Transport is important because it enables trade between people, which is essential for the development of civilizations, terminals may be used both for interchange of passengers and cargo and for maintenance. Vehicles traveling on these networks may include automobiles, bicycles, buses, trains, trucks, people, helicopters, operations deal with the way the vehicles are operated, and the procedures set for this purpose including financing, legalities and policies. In the transport industry, operations and ownership of infrastructure can be public or private, depending on the country. Passenger transport may be public, where operators provide scheduled services, freight transport has become focused on containerization, although bulk transport is used for large volumes of durable items. Transport plays an important part in growth and globalization, but most types cause air pollution. While it is subsidized by governments, good planning of transport is essential to make traffic flow. A mode of transport is a solution that makes use of a type of vehicle, infrastructure. The transport of a person or of cargo may involve one mode or several of the modes, each mode has its own advantages and disadvantages, and will be chosen for a trip on the basis of cost, capability, and route. Human powered transport, a form of transportation, is the transport of people and/or goods using human muscle-power. Modern technology has allowed machines to enhance human power, human-powered vehicles have also been developed for difficult environments, such as snow and water, by watercraft rowing and skiing, even the air can be entered with human-powered aircraft. Animal-powered transport is the use of working animals for the movement of people, humans may ride some of the animals directly, use them as pack animals for carrying goods, or harness them, alone or in teams, to pull sleds or wheeled vehicles. A fixed-wing aircraft, commonly called airplane, is a craft where movement of the air in relation to the wings is used to generate lift. The term is used to distinguish this from rotary-wing aircraft, where the movement of the lift surfaces relative to the air generates lift, a gyroplane is both fixed-wing and rotary-wing. Fixed-wing aircraft range from small trainers and recreational aircraft to large airliners, two things necessary for aircraft are air flow over the wings for lift and an area for landing. The majority of aircraft also need an airport with the infrastructure to receive maintenance, restocking, refueling and for the loading and unloading of crew, cargo and passengers. While the vast majority of land and take off on land, some are capable of take off and landing on ice, snow
Transport
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People walking in front of the
bulk carrier BW Fjord
Transport
–
French
National Police use several modes of transport, each with their distinct advantages.
Transport
–
Human-powered transport remains common in developing countries.
Transport
–
An
Air France Airbus A318 lands at
London Heathrow Airport.
40.
USS Allendale (APA-127)
–
USS Allendale was a Haskell-class attack transport of the US Navy. She was built and used during World War II and she was of the VC2-S-AP5 Victory ship design type. Allendale was named for Allendale County, South Carolina, the new attack transport was assigned to Transport Division 57, Transport Squadron 19, Pacific Fleet. She held a series of training exercises along the California coast before sailing for Hawaii early in January 1945, the vessel reached Pearl Harbor on the 10th and commenced a series of training exercises in the local operating area which lasted into late March. After taking on troops and cargo, she set sail on 22 March for the Philippine Islands, stopped en route at Eniwetok, Ulithi, and Kossol Roads, Palau Islands, there, Allendale began preparations for the upcoming Ryūkyū invasion. She sortied on the 27th with Task Unit 51.13.24, arrived in the transport area on 1 April. In spite of frequent Japanese air harassment, all her cargo had been discharged by the 9th. That day, Allendale shaped a course for Saipan and reached that island on 13 April, two days later, she got underway for Pearl Harbor and ultimately sailed on to San Francisco, California. Reaching the west coast on 5 May, the transport began loading equipment, troops and she got underway on 17 May and made port calls at Pearl Harbor, Eniwetok, and Ulithi before reaching San Pedro Bay on 10 June. The ship discharged her passengers and cargo and, on the 19th, upon arriving in Oro Bay on 24 June, the transport embarked troops and supplies for transportation to the Philippines. After making a stop at Hollandia, New Guinea, on 1 July, Allendale pushed on to Manila. When her passengers had disembarked, the vessel shaped a course back toward the United States and she spent one week at Eniwetok in mid-July and then sailed directly to San Diego, California, arriving there on 4 August. Allendale was at San Diego when word of the Japanese capitualtion was flashed around the world, on 21 August, she sailed with units of the occupation forces destined for Honshū Island, Japan. The ship paid visits to Pearl Harbor and Saipan before touching at Wakayama on 27 September and she sent Army troops ashore to serve in the occupation forces and then sailed on 1 October for Leyte. There, she embarked personnel of the X Army Corps and headed back to Japan and she arrived in Hiro Wan, Honshū, on 21 October and discharged her passengers. Allendale left Japan on 27 October and commenced the voyage back to the United States and she touched at Samar, Philippines, on 1 November and embarked returning servicemen for passage to the west coast. The vessel arrived in San Francisco Bay on 22 November and soon thereafter began unloading her passengers and she entered drydock at the Hunters Point Naval Shipyard on the 27th for repairs. In early January 1946, Allendale got underway for the east coast, after transiting the Panama Canal, the transport arrived in Norfolk, Virginia, on 30 January
USS Allendale (APA-127)
–
USS Allendale (APA-127) underway, date and place unknown.
41.
Haskell-class attack transport
–
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
–
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.
42.
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)
43.
Cargo ship
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A cargo ship or freighter is any sort of ship or vessel that carries cargo, goods, and materials from one port to another. Thousands of cargo carriers ply the worlds seas and oceans each year, cargo ships are usually specially designed for the task, often being equipped with cranes and other mechanisms to load and unload, and come in all sizes. Today, they are almost always built by welded steel, cargo ships/freighters can be divided into five groups, according to the type of cargo they carry. Tankers carry petroleum products or other liquid cargo, dry bulk carriers carry coal, grain, ore and other similar products in loose form. Multi-purpose vessels, as the name suggests, carry different classes of cargo – e. g. liquid, a Reefer ship is specifically designed and used for shipping perishable commodities which require temperature-controlled, mostly fruits, meat, fish, vegetables, dairy products and other foodstuffs. Specialized types of cargo vessels include ships and bulk carriers. Cargo ships fall into two categories that reflect the services they offer to industry, liner and tramp services. Those on a published schedule and fixed tariff rates are cargo liners. Tramp ships do not have fixed schedules, users charter them to haul loads. Generally, the shipping companies and private individuals operate tramp ships. Cargo liners run on fixed schedules published by the shipping companies, each trip a liner takes is called a voyage. However, some cargo liners may carry passengers also, a cargo liner that carries 12 or more passengers is called a combination or passenger-cum-cargo line. The desire to trade routes over longer distances, and throughout more seasons of the year. Before the middle of the 19th century, the incidence of piracy resulted in most cargo ships being armed, sometimes heavily, as in the case of the Manila galleons. They were also escorted by warships. Piracy is still common in some waters, particularly in the Malacca Straits. In 2004, the governments of three nations agreed to provide better protection for the ships passing through the Straits. The waters off Somalia and Nigeria are also prone to piracy, while smaller vessels are also in danger along parts of the South American, Southeast Asian coasts, the words cargo and freight have become interchangeable in casual usage
Cargo ship
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The
Colombo Express, one of the largest container ships in the world (when she was built in 2005), owned and operated by
Hapag-Lloyd of
Germany
Cargo ship
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Loading of a general cargo vessel in 1959
Cargo ship
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A Delmas container ship unloading at the Zanzibar port in Tanzania
44.
Armadillo-class tanker
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The Armadillo class of tankers were those Type Z-ET1-S-C3 Liberty tankers that were commissioned into the United States Navy. They were given the hull classification symbols of unclassified miscellaneous vessels and this group of Liberty based tankers all served in the United States Navy during the Second World War. Each ship was commissioned in late 1943, and decommissioned in the summer of 1946 and these ships primarily served in the Asian-Pacific theater of the war. They brought aviation gasoline to remote islands in the south Pacific, required for the many reconnaissance missions
Armadillo-class tanker
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USS Porcupine, an Armadillo -class tanker
45.
Auk-class minesweeper
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The Auk class were Allied minesweepers serving with the United States Navy and the British Royal Navy during the Second World War. In total, there were 95 Auks built, the Auk class displaced 890 tons on average, and had an approximate length of 220-225 feet. They could reach a speed of about 18.1 knots. Auks were equipped with a single 3 inch gun, two 40 mm Bofors guns, and eight 20 mm Oerlikon guns. Thirty-two minesweepers were ordered by the US, intending them to be supplied to the Royal Navy under Lend-lease,12 were retained for USN use and given names and those transferred to the RN were named as the Catherine class receiving J pennant number prefixes. Eleven minesweepers of the Auk class were lost in World War II, six to direct action including USS Skill. HMS Catherine - transferred to Turkey in March 1947 and renamed Erdemli, HMS Cato Sunk 6 July 1944 by German Neger human torpedo. HMS Chamois Damaged by mine,21 July 1944, HMS Chance To Turkey, March 1947. HMS Combatant Returned to US,1946, HMS Cynthia Returned to US,1947. HMS Elfreda To Turkey, March 1947, HMS Fairy Returned to US,1946. HMS Florizel Returned to US,1946, HMS Foam Returned to US,1946. HMS Frolic To Turkey, March 1947, HMS Gazelle Returned to US,1946. HMS Gorgon Returned to US,1946, HMS Grecian To Turkey March 1947. HMS Jasper Returned to US,1946, HMS Magic Sunk 6 July 1944 by German Neger human torpedo. HMS Pique To Turkey March 1947, stricken 1973 HMS Pylades Sunk 8 July 1944 by German Biber human torpedo HMS Steadfast Returned to United States in 1946. HMS Strenuous Returned to United States 1946, but laid up until 1956 in the UK, HMS Tattoo To Turkey, March 1947. HMS Tourmaline To Turkey, March 1947
Auk-class minesweeper
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USS Chief (AM-315)
46.
Minesweeper (ship)
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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 (ship)
–
US Navy
Admirable-class minesweeper
USS Pivot in the Gulf of Mexico for sea trials on 12 July 1944
Minesweeper (ship)
–
A minesweeper cutting loose moored mines.
Minesweeper (ship)
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Siegburg, a modern
Ensdorf-class minesweeper of the
German Navy
47.
Sarah
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Sarah or Sara was the wife and also the half–sister of Abraham and the mother of Isaac as described in the Hebrew Bible and the Quran. According to Genesis 17,15, God changed her name to Sarah as part of a covenant after Hagar bore Abraham his first son, the Hebrew name Sarah indicates a woman of high rank and is translated as princess or noblewoman. Sarah was the wife of Abraham, Sarah was approximately ten years younger than her husband. She was considered beautiful to the point that Abraham feared that when they were more powerful rulers she would be taken away. Twice he purposely identified her as being his sister so that he would be treated well for her sake, no reason is given why Sarah remained barren for a long period of time. She was originally called Sarai, which is translated my princess, later she was called Sarah, i. e. princess. Terah, with Abram, Sarai and Lot, departed for Canaan, but stopped in a place named Haran, following Gods command Abram took his wife Sarai, his nephew Lot, and the wealth and persons that they had acquired, and traveled to Shechem in Canaan. Abram was 75 at this time, there was a severe famine in the land of Canaan, so that Abram and Lot and their households, travelled south to Egypt. When the Egyptians see you, they say, this is his wife. Then they will kill me but will let you live, say you are my sister, so that I will be treated well for your sake and my life will be spared because of you. When brought before Pharaoh, Sarai said that Abram was her brother, and it is possible that Sarai acquired her Egyptian handmaid Hagar during this stay. However, God afflicted Pharaohs household with great plagues, Pharaoh then realized that Sarai was Abrams wife and demanded that they leave Egypt immediately. After having lived in Canaan for ten years and still childless, Sarai suggested that Abram have a child with her Egyptian handmaid Hagar and this resulted in tension between Sarai and Hagar, and Sarai complained to her husband that the handmaid no longer respected her. At one point, Hagar fled from her mistress but returned after angels met her and she gave birth to Abrams son Ishmael when Abram was eighty-six years old. In Genesis 17 when Abram was ninety-nine years old, God declared his new name, Abraham – a father of many nations, God gave Sarai the new name Sarah, and blessed her. Abraham was given assurance that Sarah would have a son, not long afterwards, Abraham and Sarah were visited by three men. One of the visitors told Abraham that upon his next year. While at the tent entrance, Sarah overheard what was said, the visitor inquired of Abraham why Sarah laughed at the idea of bearing a child, for her age was as nothing to God
Sarah
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Sarah (right) and
Abraham hosting three angels (a Children's Bible illustration)
Sarah
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Abram’s Counsel to Sarai (watercolor circa 1896–1902 by
James Tissot)
Sarah
–
Sarai Is Taken to Pharaoh's Palace by
James Tissot.
Sarah
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Banishment of Hagar, Etching. À Paris chez Fr. Fanet, Éditeur, Rue des Saints Pères n° 10. 18th century. Sarah is seen at the left, looking on.
48.
Book of Esther
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The Book of Esther, also known in Hebrew as the Scroll, is a book in the third section of the Jewish Tanakh and in the Christian Old Testament. It relates the story of a Hebrew woman in Persia, born as Hadassah but known as Esther, the story forms the core of the Jewish festival of Purim, during which it is read aloud twice, once in the evening and again the following morning. Esther is the book in the Bible that does not explicitly mention God. The biblical Book of Esther is set in the Persian capital of Susa in the year of the reign of the Persian king Ahasuerus. Assuming that Ahasuerus is indeed Xerxes I, the events described in Esther began around the years 483–482 BCE, and concluded in March 473 BCE. The Book of Esther consists of an introduction in chapters 1 and 2, the action in chapters 3 to 9,19. The plot is structured around banquets, a word that occurs twenty times in Esther and only 24 times in the rest of the Hebrew bible. This is appropriate given that Esther describes the origin of a Jewish feast, the feast of Purim, the books theme, rather, is the reversal of destiny through a sudden and unexpected turn of events, the Jews seem destined to be destroyed, but instead are saved. The story begins with Ahasuerus, ruler of the Persian Empire, holding a banquet, initially for his court and dignitaries and afterwards for all inhabitants of the capital city. On the seventh day, Ahasuerus orders the queen, Vashti, to come, furious, Ahasuerus has her removed from her position and makes arrangements to choose a new queen from a selection of beautiful young women from throughout the empire. One of these is the Jewish orphan, Esther, after the death of her parents, she was fostered by her cousin, Mordecai. She finds favour in the Kings eyes, and is crowned his new queen, shortly afterwards, Mordecai discovers a plot by two courtiers, Bigthan and Teresh, to assassinate Ahasuerus. The conspirators are apprehended and hanged, and Mordecais service to the King is duly recorded, Ahasuerus appoints Haman as his viceroy. Mordecai, who sits at the gates, falls into Hamans disfavour. Having discovered that Mordecai is Jewish, Haman plans to not just Mordecai. She invites him to a feast in the company of Haman, during the feast, she asks them to attend a further feast the next evening. Meanwhile, Haman is again offended by Mordecai and, at his wifes suggestion, has a built to hang him. Ahasuerus is informed that Mordecai never received any recognition for this, to his surprise and horror, the King instructs Haman to do so to Mordecai
Book of Esther
–
A 13th/14th-century
scroll of the Book of Esther from
Fez,
Morocco, held at the
Musée du quai Branly in
Paris. Traditionally, a scroll of Esther is given only one roller, fixed to its lefthand side, rather than the customary two (see below).
Book of Esther
–
Tanakh (Judaism)
Book of Esther
–
The opening chapter of a hand-written scroll of the Book of Esther, with reader's pointer
Book of Esther
–
Scroll of Esther (Megillah)
49.
Persian Empire
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Persian Empire refers to any of a series of imperial dynasties centered in Persia. The first of these was the Achaemenid Empire established by Cyrus the Great in 550 BC with the conquest of Median, Lydian and Babylonian empires and it covered much of the Ancient world when it was conquered by Alexander the Great. Several later dynasties claimed to be heirs of the Achaemenids, Persia was then ruled by the Parthian Empire which supplanted the Hellenistic Seleucid Empire, and then by the Sassanian Empire which ruled up until mid 7th century. It is important to note that many of these empires referred to themselves as Persian, they were often ethnically ruled by Medes, Babylonians. Iranian dynastic history was interrupted by the Arab Muslim conquest of Persia in 651 AD, establishing the even larger Islamic Caliphate, the main religion of ancient Persia was the native Zoroastrianism, but after the seventh century, it was replaced by Islam. Since 1979 and the downfall of the Pahlavi dynasty during Iranian Revolution, Persia has had a Shiah theocratic government
Persian Empire
–
Extent of the first Persian Empire, the
Achaemenid Empire.