1.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
2.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
3.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
100 (number)
–
The
U.S. hundred-dollar bill, Series 2009.
4.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.
5.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek numerals
–
Numeral systems
Greek numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
6.
Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
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Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
–
Numeral systems
Roman numerals
–
A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
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An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
7.
Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
8.
Ternary numeral system
–
The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
Ternary numeral system
–
Numeral systems
9.
Quaternary numeral system
–
Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
Quaternary numeral system
–
Numeral systems
10.
Quinary
–
Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
Quinary
–
Numeral systems
11.
Senary
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Senary
–
Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting
Senary
12.
Octal
–
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
Octal
–
Numeral systems
13.
Duodecimal
–
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
Duodecimal
–
Numeral systems
Duodecimal
–
A duodecimal multiplication table
14.
Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
Hexadecimal
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Numeral systems
Hexadecimal
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Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
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Hexadecimal finger-counting scheme.
15.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
Vigesimal
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Numeral systems
Vigesimal
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The
Maya numerals are a base-20 system.
16.
Base 36
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Base 36
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Numeral systems
Base 36
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34 senary = 22 decimal, in senary finger counting
Base 36
17.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Natural number
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The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
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Natural numbers can be used for counting (one
apple, two apples, three apples, …)
18.
Parity (mathematics)
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
Parity (mathematics)
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Rubik's Revenge in solved state
Parity (mathematics)
19.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
Composite number
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Overview
20.
Deficient number
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In number theory, a deficient or deficient number is a number n for which the sum of divisors σ<2n, or, equivalently, the sum of proper divisors s<n. The value 2n − σ is called the numbers deficiency, as an example, consider the number 21. Its proper divisors are 1,3 and 7, and their sum is 11, because 11 is less than 21, the number 21 is deficient. Its deficiency is 2 ×21 −32 =10, since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. An infinite number of even and odd deficient numbers exist. All odd numbers with one or two prime factors are deficient. All proper divisors of deficient or perfect numbers are deficient, there exists at least one deficient number in the interval for all sufficiently large n. Closely related to deficient numbers are perfect numbers with σ = 2n, the natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica. Almost perfect number Amicable number Sociable number Sándor, József, Mitrinović, Dragoslav S. Crstici, Borislav, the Prime Glossary, Deficient number Weisstein, Eric W. Deficient Number
Deficient number
–
Overview
21.
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
–
The divisors of 10 illustrated with
Cuisenaire rods: 1, 2, 5, and 10
22.
8 (number)
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8 is the natural number following 7 and preceding 9. 8 is, a number, its proper divisors being 1,2. It is twice 4 or four times 2, a power of two, being 23, and is the first number of the form p3, p being an integer greater than 1. The first number which is neither prime nor semiprime, the base of the octal number system, which is mostly used with computers. In octal, one digit represents 3 bits, in modern computers, a byte is a grouping of eight bits, also called an octet. A Fibonacci number, being 3 plus 5, the next Fibonacci number is 13. 8 is the only positive Fibonacci number, aside from 1, the order of the smallest non-abelian group all of whose subgroups are normal. The dimension of the octonions and is the highest possible dimension of a division algebra. The first number to be the sum of two numbers other than itself, the discrete biprime 10, and the square number 49. It has a sum of 7 in the 4 member aliquot sequence being the first member of 7-aliquot tree. All powers of 2, have a sum of one less than themselves. A number is divisible by 8 if its last 3 digits,8 and 9 form a Ruth–Aaron pair under the second definition in which repeated prime factors are counted as often as they occur. There are a total of eight convex deltahedra, a polygon with eight sides is an octagon. Figurate numbers representing octagons are called octagonal numbers, a polyhedron with eight faces is an octahedron. A cuboctahedron has as faces six equal squares and eight regular triangles. Sphenic numbers always have exactly eight divisors, the number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example, if O is the limit of the inclusions of real orthogonal groups O ↪ O ↪ … ↪ O ↪ …. Clifford algebras also display a periodicity of 8, for example, the algebra Cl is isomorphic to the algebra of 16 by 16 matrices with entries in Cl
8 (number)
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Four
playing cards showing the "8" of all four suits
8 (number)
–
The 8-spoked
Dharmacakra represents the
Noble Eightfold Path
23.
1 (number)
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1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few
1 (number)
–
The 24-hour tower clock in
Venice, using J as a symbol for 1.
24.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set
2 (number)
–
The twos of all four suits in
playing cards
25.
4 (number)
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4 is a number, numeral, and glyph. It is the number following 3 and preceding 5. Four is the only cardinal numeral in the English language that has the number of letters as its number value. Four is the smallest composite number, its divisors being 1 and 2. Four is also a composite number. The next highly composite number is 6, Four is the second square number, the second centered triangular number. 4 is the smallest squared prime and the even number in this form. It has a sum of 3 which is itself prime. The aliquot sequence of 4 has 4 members and is accordingly the first member of the 3-aliquot tree, a number is a multiple of 4 if its last two digits are a multiple of 4. For example,1092 is a multiple of 4 because 92 =4 ×23, only one number has an aliquot sum of 4 and that is squared prime 9. Four is the smallest composite number that is equal to the sum of its prime factors, however, it is the only composite number n for which. It is also a Motzkin number, in bases 6 and 12,4 is a 1-automorphic number. In addition,2 +2 =2 ×2 =22 =4, continuing the pattern in Knuths up-arrow notation,2 ↑↑2 =2 ↑↑↑2 =4, and so on, for any number of up arrows. A four-sided plane figure is a quadrilateral which include kites, rhombi, a circle divided by 4 makes right angles and four quadrants. Because of it, four is the number of plane. Four cardinal directions, four seasons, duodecimal system, and vigesimal system are based on four, a solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces of a polyhedron. The regular tetrahedron is the simplest Platonic solid, a tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only regular polyhedron
4 (number)
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playing cards for 4
4 (number)
–
"Four" redirects here. For other uses, see
4 (disambiguation).
26.
46 (number)
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46 is the natural number following 45 and preceding 47. Forty-six is a Wedderburn-Etherington number, a number and a centered triangular number. It is the sum of the totient function for the first twelve integers,46 is the largest even integer that can not be expressed as a sum of two abundant numbers. 46 is the third semiprime with an aliquot sum. The aliquot sequence of 46 is, since it is possible to find sequences of 46 consecutive integers such that each inner member shares a factor with either the first or the last member,46 is an Erdős–Woods number. The approximate molar mass of ethanol Messier object M46, a magnitude 6.5 open cluster in the constellation Puppis, the New General Catalogue object NGC46, a star in the constellation Pisces. The Saros number of the solar eclipse series began on April 1,1371 BC. The duration of Saros series 46 was 1280.1 years, the Saros number of the lunar eclipse series which began on July 19,1358 BC and ended on October 8,12. The duration of Saros series 46 was 1370.5 years, the number of mountains in the 46 peaks of the Adirondack mountain range. People who have climbed all of them are called forty-sixers, there is also an unofficial 47th peak, the name of a defensive scheme used in American football, see 46 defense. The total of books in the Old Testament, Catholic version, the number corresponding to the word ADAM where A=1, D=4, M=40. Forty-six is also, The code for international direct dial phone calls to Sweden, the number of samurai, out of 47, who carried out the attack in the historical Ako vendetta, sometimes referred to as the 46 Ronins to discount the one samurai forced to turn back. In the title of the movie Code 46, starring Tim Robbins, several routes numbered 46 exist throughout the world. Because 46 in Japanese can be pronounced as yon roku, and yoroshiku（よろしく） means my best regards in Japanese,46 is the number of the City Chevrolet and Superflo cars driven by Cole Trickle in the movie Days of Thunder. The number of the French department Lot,46 is the number that unlocks the Destiny spaceship on the popular Sci-Fi TV show Stargate Universe. Dr. Rush discovers that the number 46 relates to the amount of human chromosomes, the number depicted in the first flag of Oklahoma, signifying the fact that Oklahoma was the 46th state to join the United States
46 (number)
–
Flag of Oklahoma (1911–1925)
27.
Square numbers
–
In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
Square numbers
–
m = 1 2 = 1
28.
Binary numeral system
–
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 numeral system
–
Numeral systems
Binary numeral system
–
Gottfried Leibniz
Binary numeral system
–
George Boole
29.
0 (number)
–
0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone
0 (number)
–
Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
0 (number)
0 (number)
–
The number 605 in Khmer numerals, from the Sambor inscription (
Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.
0 (number)
–
The back of Olmec stela C from
Tres Zapotes, the second oldest
Long Count date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of
Epi-Olmec script.
30.
Prime number
–
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
–
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.
31.
53 (number)
–
53 is the natural number following 52 and preceding 54. Fifty-three is the 16th prime number and it is also an Eisenstein prime, and a Sophie Germain prime. The sum of the first 53 primes is 5830, which is divisible by 53,53 written in hexadecimal is 35, that is, the same characters used in the decimal representation, but reversed. Four multiples of 53 share this property,371 =17316,5141 =141516,99481 =1849916, and 8520280 =82025816,53 cannot be expressed as the sum of any integer and its base-10 digits, making 53 a self number. 53 is the smallest prime number that does not divide the order of any sporadic group, the duration of Saros series 53 was 1514.5 years, and it contained 85 solar eclipses. The Saros number of the lunar eclipse series began on June 5,993 BC. The duration of Saros series 53 was 1280.1 years, fictional 53rd Precinct in the Bronx was found in the TV comedy Car 54, Where Are You. UDP and TCP port number for the Domain Name System protocol, 53-TET is a musical temperament that has a fifth that is closer to pure than our current system. 53 More Things To Do In Zero Gravity is a mentioned in The Hitchhikers Guide to the Galaxy. 53 a number used on the hand of the tulip in Infinity Train
53 (number)
–
A fan-built
Herbie
32.
184 Dejopeja
–
184 Dejopeja is a large M-type Main belt asteroid. It was discovered by J. Palisa on February 28,1878, and was named after Deiopea and this is an X-type asteroid with a diameter of 66 km and a geometric albedo of 0.190. Based upon Photometric observations taken during 2000, it has a rotation period of 6.441 ±0.001 h. The light curve is tri-modal, most likely due to an angular shape,184 Dejopeja at the JPL Small-Body Database Discovery · Orbit diagram · Orbital elements · Physical parameters
184 Dejopeja
–
A three-dimensional model of 184 Dejopeja based on its light curve.
33.
Asteroid belt
–
The asteroid belt is the circumstellar disc in the Solar System located roughly between the orbits of the planets Mars and Jupiter. It is occupied by numerous irregularly shaped bodies called asteroids or minor planets, the asteroid belt is also termed the main asteroid belt or main belt to distinguish it from other asteroid populations in the Solar System such as near-Earth asteroids and trojan asteroids. About half the mass of the belt is contained in the four largest asteroids, Ceres, Vesta, Pallas, the total mass of the asteroid belt is approximately 4% that of the Moon, or 22% that of Pluto, and roughly twice that of Plutos moon Charon. Ceres, the belts only dwarf planet, is about 950 km in diameter, whereas Vesta, Pallas. The remaining bodies range down to the size of a dust particle, the asteroid material is so thinly distributed that numerous unmanned spacecraft have traversed it without incident. Nonetheless, collisions between large asteroids do occur, and these can form a family whose members have similar orbital characteristics. Individual asteroids within the belt are categorized by their spectra. The asteroid belt formed from the solar nebula as a group of planetesimals. Planetesimals are the precursors of the protoplanets. Between Mars and Jupiter, however, gravitational perturbations from Jupiter imbued the protoplanets with too much energy for them to accrete into a planet. Collisions became too violent, and instead of fusing together, the planetesimals, as a result,99. 9% of the asteroid belts original mass was lost in the first 100 million years of the Solar Systems history. Some fragments eventually found their way into the inner Solar System, Asteroid orbits continue to be appreciably perturbed whenever their period of revolution about the Sun forms an orbital resonance with Jupiter. At these orbital distances, a Kirkwood gap occurs as they are swept into other orbits. Classes of small Solar System bodies in other regions are the objects, the centaurs, the Kuiper belt objects, the scattered disc objects, the sednoids. On 22 January 2014, ESA scientists reported the detection, for the first definitive time, of water vapor on Ceres, the detection was made by using the far-infrared abilities of the Herschel Space Observatory. The finding was unexpected because comets, not asteroids, are considered to sprout jets. According to one of the scientists, The lines are becoming more and more blurred between comets and asteroids. This pattern, now known as the Titius–Bode law, predicted the semi-major axes of the six planets of the provided one allowed for a gap between the orbits of Mars and Jupiter
Asteroid belt
–
By far the largest object within the belt is
Ceres. The total mass of the asteroid belt is significantly less than
Pluto 's, and approximately twice that of Pluto's moon
Charon.
Asteroid belt
–
Sun Jupiter trojans Orbits of
planets
Asteroid belt
–
Giuseppe Piazzi, discoverer of
Ceres, the largest object in the asteroid belt. For several decades after its discovery Ceres was known as a planet, after which it was reclassified as asteroid number 1. In 2006 it was recognized to be a dwarf planet.
Asteroid belt
–
951 Gaspra, the first asteroid imaged by a spacecraft, as viewed during
Galileo ' s 1991 flyby; colors are exaggerated
34.
Asteroid
–
Asteroids are minor planets, especially those of the inner Solar System. The larger ones have also been called planetoids and these terms have historically been applied to any astronomical object orbiting the Sun that did not show the disc of a planet and was not observed to have the characteristics of an active comet. As minor planets in the outer Solar System were discovered and found to have volatile-based surfaces that resemble those of comets, in this article, the term asteroid refers to the minor planets of the inner Solar System including those co-orbital with Jupiter. There are millions of asteroids, many thought to be the remnants of planetesimals. The large majority of known asteroids orbit in the belt between the orbits of Mars and Jupiter, or are co-orbital with Jupiter. However, other orbital families exist with significant populations, including the near-Earth objects, individual asteroids are classified by their characteristic spectra, with the majority falling into three main groups, C-type, M-type, and S-type. These were named after and are identified with carbon-rich, metallic. The size of asteroids varies greatly, some reaching as much as 1000 km across, asteroids are differentiated from comets and meteoroids. In the case of comets, the difference is one of composition, while asteroids are composed of mineral and rock, comets are composed of dust. In addition, asteroids formed closer to the sun, preventing the development of the aforementioned cometary ice, the difference between asteroids and meteoroids is mainly one of size, meteoroids have a diameter of less than one meter, whereas asteroids have a diameter of greater than one meter. Finally, meteoroids can be composed of either cometary or asteroidal materials, only one asteroid,4 Vesta, which has a relatively reflective surface, is normally visible to the naked eye, and this only in very dark skies when it is favorably positioned. Rarely, small asteroids passing close to Earth may be visible to the eye for a short time. As of March 2016, the Minor Planet Center had data on more than 1.3 million objects in the inner and outer Solar System, the United Nations declared June 30 as International Asteroid Day to educate the public about asteroids. The date of International Asteroid Day commemorates the anniversary of the Tunguska asteroid impact over Siberia, the first asteroid to be discovered, Ceres, was found in 1801 by Giuseppe Piazzi, and was originally considered to be a new planet. In the early half of the nineteenth century, the terms asteroid. Asteroid discovery methods have improved over the past two centuries. This task required that hand-drawn sky charts be prepared for all stars in the band down to an agreed-upon limit of faintness. On subsequent nights, the sky would be charted again and any moving object would, hopefully, the expected motion of the missing planet was about 30 seconds of arc per hour, readily discernible by observers
Asteroid
–
253 Mathilde, a
C-type asteroid measuring about 50 kilometres (30 mi) across, covered in craters half that size. Photograph taken in 1997 by the
NEAR Shoemaker probe.
Asteroid
–
2013 EC, shown here in radar images, has a provisional designation
Asteroid
–
⚵
Asteroid
–
243 Ida and its moon Dactyl. Dactyl is the first satellite of an asteroid to be discovered.
35.
184 Airborne Division Nembo
–
184th Paratroopers Division Nembo or 184th Divisione Paracadutisti Nembo was an airborne division of the Italian Army during World War II. The Nembo was formed in December 1942 from the 185 Parachute Regiment from 185 Paratroopers Division Folgore, the 183 Regiment was sent to North Africa, where it was destroyed. The 185 Parachute Regiment was sent to northeastern Italy to fight Yugoslav partisans, and was sent to Sicily. It also fought on the mainland when Sicily was evacuated, the rest of the division was sent to Sardinia in June 1943. After the Italian surrender, a significant part of the Nembo went over to the German side, Nembo was sent to Yugoslavia in the spring of 1943, on anti-partisan operations near the Italian border city of Gorizia. In June 1943, the 183 and 184 Parachute Regiments were sent to Sardinia to defend the island against what was expected to be the main Allied landing, the 185 Parachute Regiment was sent to Calabria. When the Allies landed in Sicily in July 1943, the 185 was sent to reinforce the island and it was selected to form the rear guard and protect the withdrawing Italian and German forces, which were evacuating to the mainland. After the allied landings in Southern Italy the division engaged the British 8th Army in the Aspromonte massif, in January 1944, the Allies launched Operation Shingle, and the division fought against them in the landings at Anzio
184 Airborne Division Nembo
–
Nembo Division collar insignia
36.
Airborne forces
–
Airborne forces are military units, usually light infantry, set up to be moved by aircraft and dropped into battle, typically by parachute. Thus, they can be placed behind enemy lines, and have the capability to deploy almost anywhere with little warning. The formations are limited only by the number and size of their aircraft, so given enough capacity a huge force can appear out of nowhere in minutes, Major Lewis H. Brereton and his superior Brigadier General Billy Mitchell suggested dropping elements of the U. S. 1st Division behind German lines near Metz, the operation was planned for February 1919 but the war ended before such an attack could be seriously planned. Mitchell conceived that US troops could be trained to utilize parachutes. Following the war, the United States Army Air Service experimented with the concept of having carried on the wings of aircraft pulled off by the opening of their parachutes. The first true paratroop drop was by Italy in November 1927, within a few years several battalions had been raised and were eventually formed into two Folgore and Nembo divisions. Although these would fight with distinction in World War II. Men drawn from the Italian parachute forces were dropped in a special operation in North Africa in 1943 in an attempt to destroy parked aircraft of the United States Army Air Forces. Subsequently, on May 10,1928, Second Lieutenant César Álvarez War Palmas Las voluntarily jumped from a height of 3,000 meters, then on May 16,1928, Major Fernando Melgar Conde and Sergeant 1st. Jose Pineda Castro, jumped from the famous Las Palmas at altitudes of 2,000 and 4,300 meters, at about the same time, the Soviet Union was also experimenting with the idea, planning to drop entire units complete with vehicles and light tanks. To help train enough experienced jumpers, parachute clubs were organized with the aim of transferring into the armed forces if needed, one of the observing parties, Germany, was particularly interested. In 1936, Major F. W. Immans was ordered to set up a school at Stendal. The military had purchased large numbers of Junkers Ju 52 aircraft which were slightly modified for use as paratroop transports in addition to their other duties. The first training class was known as Ausbildungskommando Immans and they commenced the first course on May 3,1936. Other nations, including Argentina, Peru, Japan, France, France became the first nation to organize women in an airborne unit. Recruiting 200 nurses who during peacetime would parachute into natural disasters, several groups within the German armed forces attempted to raise their own paratroop formations, resulting in confusion. As a result, Luftwaffe General Kurt Student was put in command of developing a paratrooper force to be known as the Fallschirmjäger
Airborne forces
–
Parawings worn by members of the
British Armed Forces who have undergone Parachute Training at
RAF Brize Norton.
Airborne forces
–
U.S. Air Force airmen from the 720th STG jumping out of a
C-130J Hercules aircraft during water rescue training in the
Florida panhandle
Airborne forces
–
Paratroopers from the U.S. Army's
82nd Airborne Division jump from a
C-17 Globemaster at
Ft. Bragg, N.C., during Exercise Joint Forcible Entry in April 2005.
Airborne forces
–
Eisenhower speaks with U.S. paratroops of the
502d Parachute Infantry Regiment,
101st Airborne Division on the evening of June 5, 1944.
37.
Division (military)
–
A division is a large military unit or formation, usually consisting of between 10,000 and 20,000 soldiers. Infantry divisions during the World Wars ranged between 10,000 and 30,000 in nominal strength, in most armies, a division is composed of several regiments or brigades, in turn, several divisions typically make up a corps. In the West, the first general to think of organising an army into smaller units was Maurice de Saxe, Marshal General of France. He died at the age of 54, without having implemented his idea, victor-François de Broglie put the ideas into practice. He conducted successful practical experiments of the system in the Seven Years War. The first war in which the system was used systematically was the French Revolutionary War. It made the more flexible and easy to manoeuvre. Under Napoleon, the divisions were grouped together into corps, because of their increasing size, napoleons military success spread the divisional and corps system all over Europe, by the end of the Napoleonic Wars, all armies in Europe had adopted it. In modern times, most military forces have standardized their divisional structures, the peak use of the division as the primary combat unit occurred during World War II, when the belligerents deployed over a thousand divisions. With technological advances since then, the power of each division has increased. Divisions are often formed to organize units of a particular type together with support units to allow independent operations. In more recent times, divisions have mainly been organized as combined arms units with subordinate units representing various combat arms, in this case, the division often retains the name of a more specialized division, and may still be tasked with a primary role suited to that specialization. For the most part, large cavalry units did not remain after World War II, in general, two new types of cavalry were developed, air cavalry or airmobile, relying on helicopter mobility, and armored cavalry, based on an autonomous armored formation. The former was pioneered by the 11th Air Assault Division, formed on 1 February 1963 at Fort Benning, on 29 June 1965 the division was renamed as the 1st Cavalry Division, before its departure for the Vietnam War. After the end of the Vietnam War, the 1st Cavalry Division was reorganised and re-equipped with tanks, the development of the tank during World War I prompted some nations to experiment with forming them into division-size units. Many did this the way as they did cavalry divisions, by merely replacing cavalry with AFVs. This proved unwieldy in combat, as the units had many tanks, instead, a more balanced approach was taken by adjusting the number of tank, infantry, artillery, and support units. A panzer division was a division of the Wehrmacht and the Waffen-SS of Germany during World War II
Division (military)
–
A
Priest 105mm self-propelled gun of British 3rd Infantry Division, 1944
Division (military)
–
Standard
NATO symbol for an infantry division. The Xs do not replace the division's number; instead, the two Xs represent a division (one would denote a brigade; three, a corps).
Division (military)
–
Members of the Australian 6th Division at
Tobruk, 22 January 1941
Division (military)
–
British soldiers from the 1st Armoured Division engage Iraqi Army positions with their 81mm Mortar in Iraq, 26 March 2003.
38.
World War II
–
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
–
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
–
Adolf Hitler at a German
National Socialist political rally in
Weimar, October 1930
World War II
–
Italian soldiers recruited in 1935, on their way to fight the
Second Italo-Abyssinian War
39.
Canadian Expeditionary Force
–
The Canadian Expeditionary Force was the designation of the field force created by Canada for service overseas in the First World War. The force fielded several combat formations on the Western Front in France and Belgium, the Canadian Cavalry Brigade and the Canadian Independent Force, which were independent of the Canadian Corps, also fought on the Western Front. The CEF also had a reserve and training organization in England. The Germans went so far as to call them storm troopers for their combat efficiency. In August 1918, the CEFs Canadian Siberian Expeditionary Force travelled to revolution-torn Russia and it reinforced an anti-Bolshevik garrison in Vladivostok during the winter of 1918–19. At this time, another force of Canadian soldiers were placed in Archangel, the Canadian Expeditionary Force was mostly volunteers, as conscription was not enforced until the end of the war when call-ups began in January 1918. Ultimately, only 24,132 conscripts arrived in France before the end of the war, Canada was the senior Dominion in the British Empire and automatically at war with Germany upon the British declaration. According to Canadian historian Dr. Serge Durflinger at the Canadian War Museum, of the first contingent formed at Valcartier, Quebec in 1914, fully two-thirds were men born in the United Kingdom. By the end of the war in 1918, at least fifty per cent of the CEF consisted of British-born men, many British nationals from the United Kingdom or other territories who were resident in Canada also joined the CEF. As several CEF battalions were posted to the Bermuda Garrison before proceeding to France, although the Bermuda Militia Artillery and Bermuda Volunteer Rifle Corps both sent contingents to the Western Front, the first would not arrive there til June 1915. By then, many Bermudians had already been serving on the Western Front in the CEF for months, Bermudians in the CEF enlisted under the same terms as Canadians, and all male British Nationals resident in Canada became liable for conscription under the Military Service Act,1917. Two tank battalions were raised in 1918 but did not see service, most of the infantry battalions were broken up and used as reinforcements, with a total of fifty being used in the field, including the mounted rifle units, which were re-organized as infantry. The artillery and engineering units underwent significant re-organization as the war progressed, a distinct entity within the Canadian Expeditionary Force was the Canadian Machine Gun Corps. It consisted of several machine gun battalions, the Eatons, Yukon, and Borden Motor Machine Gun Batteries. During the summer of 1918, these units were consolidated into four machine gun battalions, the Canadian Corps with its four infantry divisions comprised the main fighting force of the CEF. The Canadian Cavalry Brigade also served in France, the 1915 Battle of Ypres, the first engagement of Canadian forces in the Great War, changed the Canadian perspective on war. Ypres exposed Canadian soldiers and their commanders to modern war and they had already experienced the effects of shellfire and developed a reputation for aggressive trench raiding despite their lack of formal training and generally inferior equipment. In April 1915, they were introduced to yet another facet of modern war, the Germans employed chlorine gas to create a hole in the French lines adjacent to the Canadian force and poured troops into the gap
Canadian Expeditionary Force
–
26th Battalion of the Second Canadian Expeditionary Force, 1915
Canadian Expeditionary Force
Canadian Expeditionary Force
–
Private Joseph Pappin, 130 Battalion, Canadian Expeditionary Force.
40.
World War I
–
World War I, also known as the First World War, the Great War, or the War to End All Wars, was a global war originating in Europe that lasted from 28 July 1914 to 11 November 1918. More than 70 million military personnel, including 60 million Europeans, were mobilised in one of the largest wars in history and it was one of the deadliest conflicts in history, and paved the way for major political changes, including revolutions in many of the nations involved. The war drew in all the worlds great powers, assembled in two opposing alliances, the Allies versus the Central Powers of Germany and Austria-Hungary. These alliances were reorganised and expanded as more nations entered the war, Italy, Japan, the trigger for the war was the assassination of Archduke Franz Ferdinand of Austria, heir to the throne of Austria-Hungary, by Yugoslav nationalist Gavrilo Princip in Sarajevo on 28 June 1914. This set off a crisis when Austria-Hungary delivered an ultimatum to the Kingdom of Serbia. Within weeks, the powers were at war and the conflict soon spread around the world. On 25 July Russia began mobilisation and on 28 July, the Austro-Hungarians declared war on Serbia, Germany presented an ultimatum to Russia to demobilise, and when this was refused, declared war on Russia on 1 August. Germany then invaded neutral Belgium and Luxembourg before moving towards France, after the German march on Paris was halted, what became known as the Western Front settled into a battle of attrition, with a trench line that changed little until 1917. On the Eastern Front, the Russian army was successful against the Austro-Hungarians, in November 1914, the Ottoman Empire joined the Central Powers, opening fronts in the Caucasus, Mesopotamia and the Sinai. In 1915, Italy joined the Allies and Bulgaria joined the Central Powers, Romania joined the Allies in 1916, after a stunning German offensive along the Western Front in the spring of 1918, the Allies rallied and drove back the Germans in a series of successful offensives. By the end of the war or soon after, the German Empire, Russian Empire, Austro-Hungarian Empire, national borders were redrawn, with several independent nations restored or created, and Germanys colonies were parceled out among the victors. During the Paris Peace Conference of 1919, the Big Four imposed their terms in a series of treaties, the League of Nations was formed with the aim of preventing any repetition of such a conflict. This effort failed, and economic depression, renewed nationalism, weakened successor states, and feelings of humiliation eventually contributed to World War II. From the time of its start until the approach of World War II, at the time, it was also sometimes called the war to end war or the war to end all wars due to its then-unparalleled scale and devastation. In Canada, Macleans magazine in October 1914 wrote, Some wars name themselves, during the interwar period, the war was most often called the World War and the Great War in English-speaking countries. Will become the first world war in the sense of the word. These began in 1815, with the Holy Alliance between Prussia, Russia, and Austria, when Germany was united in 1871, Prussia became part of the new German nation. Soon after, in October 1873, German Chancellor Otto von Bismarck negotiated the League of the Three Emperors between the monarchs of Austria-Hungary, Russia and Germany
World War I
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Clockwise from the top: The aftermath of shelling during the
Battle of the Somme,
Mark V tanks cross the
Hindenburg Line,
HMS Irresistible sinks after hitting a
mine in the
Dardanelles, a British
Vickers machine gun crew wears
gas masks during the Battle of the Somme,
Albatros D.III fighters of
Jagdstaffel 11
World War I
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Sarajevo citizens reading a poster with the proclamation of the
Austrian annexation in 1908.
World War I
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This picture is usually associated with the arrest of
Gavrilo Princip, although some believe it depicts Ferdinand Behr, a bystander.
World War I
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Serbian Army
Blériot XI "Oluj", 1915.
41.
184th Fighter Squadron
–
The 184th Attack Squadron is a unit of the Arkansas Air National Guard 188th Fighter Wing located at Fort Smith Air National Guard Station, Fort Smith, Arkansas. The 184th is equipped with the MQ-9 Reaper, in June 2014 the squadron transitioned from A-10C to the MQ-9. Authorized by the National Guard Bureau in 1953 at the 184th Tactical Reconnaissance Squadron, organized at Fort Smith Regional Airport, Arkansas and extended recognition as a new unit on 15 October 1953. The squadron was assigned to the Tennessee ANG 118th Tactical Reconnaissance Group, Berry Field, Nashville, for administration, the 184th TRS was initially equipped with World War II-era RB-26C Invader night photo-reconnaissance aircraft. The black-painted RB-26s were originally medium bombers that were modified for reconnaissance in the late 1940s. Most of the aircraft received were Korean War veterans, were unarmed and carried cameras, in 1956, the B-26 was reaching the end of its operational USAF service, and the squadron was re-equipped with RF-80A Shooting Star daylight reconnaissance aircraft that were also nearly obsolescent. In January 1957, the 184th retired its worn-out RF-80s and received new RF-84F Thunderstreak reconnaissance aircraft, on 22 August 1962, the 184th was authorized to expand to a group level, and the 188th Tactical Reconnaissance Group was established by the National Guard Bureau. The 184th TRS becoming the flying squadron. Other squadrons assigned into the group were the 188th Headquarters, 188th Material Squadron, 188th Combat Support Squadron, in 1970 with the winding-down of the Vietnam War, the 184th began receiving McDonnell RF-101C Voodoos, replacing the RF-84Fs the unit had been flying for over a decade. Following their withdrawal from the Vietnam War, numerous USAF F-100D Super Sabres were turned over to the Air National Guard, Tactical Air Command realigned the 151st into a Tactical Fighter Group in 1972, and equipping the unit with Vietnam Veteran F-100D and twin-seat F-100F Trainers. In 1979, the Super Sabre was being retired and the 184th TFS began receiving F-4C Phantom IIs to be used in an air defense role, in 1988, as part of the retirement of the Phantom II, the squadron began receiving Block 15 F-16A Fighting Falcons. The first F-16 delivery to the squadron was on 1 July 1988, on 15 March 1992 the 184th dropped the Tactical name from the squadron as the parent 184th converted to the USAF Objective organization. In early 2001 the 184th FS began to retire its F-16A/B block 15s to AMARC in exchange for F-16C block 32s, because the squadron flew the rarely seen block 32, the squadron became a source for spare F-16s for the USAF Thunderbirds flight demonstration team. In the end the 184th FS never had to give up any of their aircraft as the Thunderbirds took needed aircraft from home based Nellis 57th Fighter Wing, deployed to Prince Sultan AB, Saudi Arabia in support of Operation Southern Watch. In 2005, the 188th deployed nearly 300 Airmen and multiple F-16C Fighting Falcons to Balad Air Base, Iraq, BRAC2005 initially decided to inactivate the 188th Fighter Wing and close Fort Smith ANGB. With a great deal of effort by Arkansas leaders caused the BRAC panel to change its decision on the 184th FS, the squadron would still lose its F-16s but in their place would get a total of eighteen A-10 Thunderbolt II ground attack aircraft. One of the factors was Fort Smiths location near Fort Chaffee. On 18 October 2006 the 184th FS began giving up F-16s when two departed for the 194th Fighter Squadron located at Fresno Air National Guard Base, California
184th Fighter Squadron
–
Lt. Col. Brian Burger, an A-10 Thunderbolt II pilot and the 188th Fighter Wing operations group commander, Ft. Smith Air National Guard fires off a flare while banking into a high angle firing position during a training exercise on Razorback Range located at Fort Chaffee maneuver training center, June 4, 2012.
184th Fighter Squadron
–
184th Fighter Squadron emblem
184th Fighter Squadron
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Lockheed RF-80A-5-LO Shooting Star 45-8371, about 1955
184th Fighter Squadron
–
McDonnell F-4C-15-MC Phantom II fighter (s/n 63-7411) of the 188th Tactical Fighter Group, Arkansas Air National Guard, prepares for take-off at Nellis Air Force Base, Nevada (USA), during the "
Gunsmoke '85" exercise on 6 October 1985.
42.
Arkansas Air National Guard
–
The Arkansas Air National Guard is the air force militia of the State of Arkansas, United States of America. It is, along with the Arkansas Army National Guard, an element of the Arkansas National Guard, as state militia units, the units in the Arkansas Air National Guard are not typically in the normal United States Air Force chain of command unless federalized. They are under the jurisdiction of the Governor of Arkansas through the office of the Arkansas Adjutant General unless they are federalized by order of the President of the United States. The Arkansas Air National Guard is headquartered at North Little Rock, under the Total Force concept, Arkansas Air National Guard units are an Air Reserve Components of the United States Air Force. Arkansas ANG units are trained and equipped by the Air Force and are gained by a Major Command of the USAF if federalized. State missions include disaster relief in times of earthquakes, hurricanes, floods and forest fires, search and rescue, protection of public services. The unit operates the C-130 Tactical Airlift Instructor School, where instructors are trained to they can return to their units. The Arkansas Air National Guard origins date to 28 August 1917 with the establishment of the 154th Aero Squadron as part of the World War I American Expeditionary Force, the 154th served in France on the Western Front, then after the 1918 Armistice with Germany was demobilized in 1919. The Militia Act of 1903 established the present National Guard system, units raised by the states but paid for by the Federal Government, if federalized by Presidential order, they fall under the regular military chain of command. On 1 June 1920, the Militia Bureau issued Circular No.1 on organization of National Guard air units. The unit was reorganized with the establishment of a permanent air service in 1920 as the 154th Observation Squadron on 24 October 1925 and it is one of the 29 original National Guard Observation Squadrons of the United States Army National Guard formed before World War II. The 154th Observation Squadron was activated for one year of training on 16 September 1940, the unit completed its one-year training and returned to state control, but was recalled to active duty on 7 December 1941 as a result of the Japanese attack on Pearl Harbor. These unit designations were allotted and transferred to various State National Guard bureaus to provide them unit designations to re-establish them as Air National Guard units, the modern Arkansas ANG received federal recognition on 27 May 1946 as the 154th Fighter Squadron at Adams Field, Little Rock. It was equipped with F-51D Mustangs and its mission was the air defense of the state, on 2 October 1950, the 154th Fighter Squadron, along with detachment B, 237th Air Services Group and the 154th Utility Flight reported to active duty for service in Korea. The unit went to Langley Air Force Base, VA where it was re-equipped with the F-84E fighter, the 154th flew its first combat sortie 2 May 1951. Initially operating out of Itaeke, Japan the unit moved to Taegu. The 154th returned to Arkansas and was relieved from active duty 1 July 1952, on 22 August 1962, the 184th Tactical Reconnaissance Squadron was authorized to expand to a group level, and the 188th Tactical Reconnaissance Group was established by the National Guard Bureau. The 184th TRS becoming the flying squadron
Arkansas Air National Guard
–
154th Training Squadron C-130 Hercules flies over the state capitol in Little Rock, Arkansas. The 154th TS is the oldest unit in the Arkansas Air National Guard, having over 80 years of service to the state and nation
Arkansas Air National Guard
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Members of the 154th Observation Squadron, 1925
Arkansas Air National Guard
–
The Arkansas ANG 154th Fighter-Bomber Squadron was deployed to Tageu Air Base, South Korea during 1951/1952 during the Korean War. Arkansas Air National Guardsmen flew numerous combat missions in support of the United Nations forces.
43.
184th Infantry Regiment (United States)
–
The 184th Infantry Regiment is an infantry regiment of the United States Army consisting of soldiers from the California Army National Guard. Only the regiments 1st Battalion remains a military unit. The battalion supports state and federal missions in the State of California, United States of America, the 184th Infantry Regiment can trace its lineage to the mid-19th century. The regiments 1st Battalion is currently one of two battalions in the 79th Infantry Brigade Combat Team. The battalion is currently conducting training and maintenance in order to prepare for future state, the 184th Infantry Regiment is a descendant of a number of unofficial militias that were formed in California in the mid-to-late-19th century. Bravo Company is the descendant of the Sarsfield Grenadier Guards. Charlie Company is the descendant of the Auburn Greys. In its dual role in service to both the United States of America and the State of California, the regiment can be mobilized to serve under state control, the 184th Infantry Regiment was formed on 20 October 1924 from what had been known as the 2d Infantry Regiment. On 9 December 1895, the 2d Infantry Regiment was formed by consolidating the 1st Artillery Regiment with the 8th Infantry Regiment, the headquarters of the 2d Infantry Regiment was in Sacramento and the unit was part of the California National Guard. The lineage of two units is described below. On 19 March 1880, the 1st Artillery Regiment was formed from the consolidation of the Sacramento Light Artillery, the headquarters of the 1st Artillery Regiment was in Sacramento and the unit was part of the California National Guard. Sacramento Light Artillery. In 1864, the Sacramento Light Artillery was formed from precursor militia elements in Sacramento as part of the California Militia. On 15 March 1872, the 4th Infantry Regiment was formed from precursor militia elements, including the Sarsfield Grenadier Guards as G company, from 1875 through 1877, the 1st Infantry Battalion was formed through the reduction, reorganization, and redesignation of the 4th Infantry Regiment. The headquarters of the 4th Infantry Regiment was in Sacramento and the unit was part of the California National Guard, on 15 February 1890, the 8th Infantry Battalion was formed from precursor militia elements with headquarters in Chico, as part of the California National Guard. On 31 October 1891, the 8th Infantry Regiment was formed through the expansion, reorganization, the headquarters of the 8th Infantry Regiment was in Chico and the unit was part of the California National Guard. Although the units were mobilized for service, they never left the United States. From 28 January 1899 through 6 February 1899, the three companies were demobilized in Washington and California and returned to the 2d Infantry Regiment, on 18 June 1916, the 2d Infantry Regiment was mobilized for border security duty during the Mexican Border Crisis. On 15 November 1916, the unit was demobilized, on 26 March 1917, the 2d Infantry Regiment was mobilized into federal service for World War I
184th Infantry Regiment (United States)
–
Coat of arms
184th Infantry Regiment (United States)
–
Soldiers from the 184th Infantry advance on an enemy position at Dagami,
Leyte
184th Infantry Regiment (United States)
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Distinctive unit insignia
184th Infantry Regiment (United States)
–
Soldiers from the 184th Infantry conduct
Air Assault Training at
Camp Roberts in June 2010.
44.
United States Army
–
The United States Army is the largest branch of the United States Armed Forces and performs land-based military operations. After the Revolutionary War, the Congress of the Confederation created the United States Army on 3 June 1784, the United States Army considers itself descended from the Continental Army, and dates its institutional inception from the origin of that armed force in 1775. As a uniformed service, the Army is part of the Department of the Army. As a branch of the forces, the mission of the U. S. The branch participates in conflicts worldwide and is the major ground-based offensive and defensive force of the United States, the United States Army serves as the land-based branch of the U. S. Section 3062 of Title 10, U. S, the army was initially led by men who had served in the British Army or colonial militias and who brought much of British military heritage with them. As the Revolutionary War progressed, French aid, resources, a number of European soldiers came on their own to help, such as Friedrich Wilhelm von Steuben, who taught Prussian Army tactics and organizational skills. The army fought numerous pitched battles and in the South in 1780–81 sometimes used the Fabian strategy and hit-and-run tactics, hitting where the British were weakest, to wear down their forces. Washington led victories against the British at Trenton and Princeton, but lost a series of battles in the New York and New Jersey campaign in 1776, with a decisive victory at Yorktown, and the help of the French, the Continental Army prevailed against the British. After the war, though, the Continental Army was quickly given land certificates, State militias became the new nations sole ground army, with the exception of a regiment to guard the Western Frontier and one battery of artillery guarding West Points arsenal. However, because of continuing conflict with Native Americans, it was realized that it was necessary to field a trained standing army. The War of 1812, the second and last war between the United States and Great Britain, had mixed results. After taking control of Lake Erie in 1813, the U. S. Army seized parts of western Upper Canada, burned York and defeated Tecumseh, which caused his Western Confederacy to collapse. Following U. S. victories in the Canadian province of Upper Canada, British troops, were able to capture and burn Washington, which was defended by militia, in 1814. Two weeks after a treaty was signed, Andrew Jackson defeated the British in the Battle of New Orleans and Siege of Fort St. Philip, U. S. troops and sailors captured HMS Cyane, Levant, and Penguin in the final engagements of the war. Per the treaty, both sides, the United States and Great Britain, returned to the status quo. Both navies kept the warships they had seized during the conflict, the armys major campaign against the Indians was fought in Florida against Seminoles. It took long wars to defeat the Seminoles and move them to Oklahoma
United States Army
–
Storming of Redoubt #10 in the
Siege of Yorktown during the
American Revolutionary War prompted the British government to begin negotiations, resulting in the
Treaty of Paris and British recognition of the United States of America.
United States Army
–
Emblem of the United States Department of the Army
United States Army
–
General
Andrew Jackson stands on the parapet of his makeshift defenses as his troops repulse attacking
Highlanders during the
defense of New Orleans, the final major battle of the War of 1812
United States Army
–
The
Battle of Gettysburg, the turning point of the American Civil War
45.
Regiment
–
A regiment is a military unit. Their role and size varies markedly, depending on the country, in Medieval Europe, the term regiment denoted any large body of front-line soldiers, recruited or conscripted in one geographical area, by a leader who was often also the feudal lord of the soldiers. By the 17th century, a regiment was usually about a thousand personnel. In many armies, the first role has been assumed by independent battalions, battlegroups, task forces, brigades and other, similarly-sized operational units. By the beginning of the 18th century, regiments in most European continental armies had evolved into permanent units with distinctive titles and uniforms, when at full strength, an infantry regiment normally comprised two field battalions of about 800 men each or 8–10 companies. In some armies, an independent regiment with fewer companies was labelled a demi-regiment, a cavalry regiment numbered 600 to 900 troopers, making up a single entity. With the widespread adoption of conscription in European armies during the nineteenth century, the regimental system underwent modification. Prior to World War I, a regiment in the French, German, Russian. As far as possible, the battalions would be garrisoned in the same military district, so that the regiment could be mobilized. A cavalry regiment by contrast made up an entity of up to 1,000 troopers. Usually, the regiment is responsible for recruiting and administering all of a military career. Depending upon the country, regiments can be either combat units or administrative units or both and this is often contrasted to the continental system adopted by many armies. Generally, divisions are garrisoned together and share the same installations, thus, in divisional administration, soldiers and officers are transferred in and out of divisions as required. Some regiments recruited from specific areas, and usually incorporated the place name into the regimental name. In other cases, regiments would recruit from an age group within a nation. In other cases, new regiments were raised for new functions within an army, e. g. the Fusiliers, the Parachute Regiment, a key aspect of the regimental system is that the regiment or battalion is the fundamental tactical building block. This flows historically from the period, when battalions were widely dispersed and virtually autonomous. For example, a regiment might include different types of battalions of different origins, within the regimental system, soldiers, and usually officers, are always posted to a tactical unit of their own regiment whenever posted to field duty
Regiment
–
The
Royal Regiment of Fusiliers on parade in England
Regiment
–
Standard
NATO symbol for a regiment of several battalions, indicated by the III. The shape, colour and pattern indicate friendly infantry.
Regiment
–
Regimental badge of the
Scots Guards.
Regiment
–
The Puerto Rican
65th Infantry Regiment 's
bayonet charge against a
Chinese division during the Korean War.
46.
California
–
California is the most populous state in the United States and the third most extensive by area. Located on the western coast of the U. S, California is bordered by the other U. S. states of Oregon, Nevada, and Arizona and shares an international border with the Mexican state of Baja California. Los Angeles is Californias most populous city, and the second largest after New York City. The Los Angeles Area and the San Francisco Bay Area are the nations second- and fifth-most populous urban regions, California also has the nations most populous county, Los Angeles County, and its largest county by area, San Bernardino County. The Central Valley, an agricultural area, dominates the states center. What is now California was first settled by various Native American tribes before being explored by a number of European expeditions during the 16th and 17th centuries, the Spanish Empire then claimed it as part of Alta California in their New Spain colony. The area became a part of Mexico in 1821 following its war for independence. The western portion of Alta California then was organized as the State of California, the California Gold Rush starting in 1848 led to dramatic social and demographic changes, with large-scale emigration from the east and abroad with an accompanying economic boom. If it were a country, California would be the 6th largest economy in the world, fifty-eight percent of the states economy is centered on finance, government, real estate services, technology, and professional, scientific and technical business services. Although it accounts for only 1.5 percent of the states economy, the story of Calafia is recorded in a 1510 work The Adventures of Esplandián, written as a sequel to Amadis de Gaula by Spanish adventure writer Garci Rodríguez de Montalvo. The kingdom of Queen Calafia, according to Montalvo, was said to be a land inhabited by griffins and other strange beasts. This conventional wisdom that California was an island, with maps drawn to reflect this belief, shortened forms of the states name include CA, Cal. Calif. and US-CA. Settled by successive waves of arrivals during the last 10,000 years, various estimates of the native population range from 100,000 to 300,000. The Indigenous peoples of California included more than 70 distinct groups of Native Americans, ranging from large, settled populations living on the coast to groups in the interior. California groups also were diverse in their organization with bands, tribes, villages. Trade, intermarriage and military alliances fostered many social and economic relationships among the diverse groups, the first European effort to explore the coast as far north as the Russian River was a Spanish sailing expedition, led by Portuguese captain Juan Rodríguez Cabrillo, in 1542. Some 37 years later English explorer Francis Drake also explored and claimed a portion of the California coast in 1579. Spanish traders made unintended visits with the Manila galleons on their trips from the Philippines beginning in 1565
California
–
A forest of redwood trees in
Redwood National Park
California
–
Flag
California
–
Mount Shasta
California
–
Aerial view of the
California Central Valley
47.
United States National Guard
–
All members of the National Guard of the United States are also members of the militia of the United States as defined by 10 U. S. C. National Guard units are under the control of the state. The majority of National Guard soldiers and airmen hold a civilian job full-time while serving part-time as a National Guard member, local militias were formed from the earliest English colonization of the Americas in 1607. The first colony-wide militia was formed by Massachusetts in 1636 by merging small older local units, the various colonial militias became state militias when the United States became independent. The title National Guard was used from 1824 by some New York State militia units, National Guard became a standard nationwide militia title in 1903, and specifically indicated reserve forces under mixed state and federal control from 1933. The first muster of militia forces in what is today the United States took place on September 16,1565, appropriately enough, this muster occurred in the shadow of an oncoming hurricane. This Spanish militia tradition and the English tradition that would be established to the north would provide the nucleus for Colonial defense in the New World. The militia tradition continued with the first permanent English settlements in the New World, Jamestown Colony and Plymouth Colony both had militia forces, which initially consisted of every able bodied adult male. By the mid-1600s every town had at least one militia company, as a result of the Spanish–American War, Congress was called upon to reform and regulate the training and qualification of state militias. The first national laws regulating the militia were the Militia acts of 1792, in 1903, with passage of the Dick Act, the predecessor to the modern-day National Guard was formed. It required the states to divide their militias into two sections, the law recommended the title National Guard for the first section, known as the organized militia, and Reserve Militia for all others. During World War I, Congress passed the National Defense Act of 1916, Congress also authorized the states to maintain Home Guards, which were reserve forces outside the National Guards being deployed by the Federal Government. The National Guard of the states, territories, and the District of Columbia serves as part of the first-line of defense for the United States. C. Where the National Guard operates under the President of the United States or his designee, the governors exercise control through the state adjutants general. The National Guard may be called up for duty by the governors to help respond to domestic emergencies and disasters, such as hurricanes, floods. The National Guard is administered by the National Guard Bureau, which is a joint activity of the Army, the National Guard Bureau provides a communication channel for state National Guards to the DoD. S. C. The National Guard Bureau is headed by the Chief of the National Guard Bureau, prior to 2008, the functions of Agricultural Development Teams were within Provincial Reconstruction Teams of the US Government. Today, ADTs consist of soldiers and airmen from the Army National Guard, today, ADTs bring an effective platform for enhanced dialogue, building confidence, sharing interests, and increasing cooperation amongst the disparate peoples and tribes of Afghanistan
United States National Guard
–
Army National Guard soldiers at New York City's
Penn Station in 2004.
United States National Guard
–
National Guard of the United States
United States National Guard
–
First Muster, Spring 1637, Massachusetts Bay Colony.
United States National Guard
–
A National Guardsman in 1917.
48.
184th Intelligence Wing
–
The 184th Intelligence Wing is a unit of the Kansas Air National Guard, stationed at McConnell Air Force Base, Wichita, Kansas. If activated to service, the Wing is gained by the United States Air Force Intelligence, Surveillance and Reconnaissance Agency. The 127th Command and Control Squadron assigned to the Wings 184th Regional Support Group, is a descendant organization of the 127th Observation Squadron and it is one of the 29 original National Guard Observation Squadrons of the United States Army National Guard formed before World War II. In April 2008 the 184th Intelligence Wing became the first Intelligence Wing in the Air National Guard, the Wing encompasses a variety of missions at the federal, state, and community levels. Federal, providing support for our nation—Guardsmen from our wing provide wartime support in the form of collecting and analyzing intelligence. Some unit members deploy overseas to augment active duty forces, state, support to civil authority—The units primary responsibility is to the state of Kansas to assist civil authorities during natural disasters and civil strife in our state. The 127th TFS becoming the flying squadron. Other squadrons assigned into the group were the 184th Headquarters, 184th Material Squadron, 184th Combat Support Squadron, as a result of this federalization, the 184th TFG was placed in a non-operational status. The 127th was released from duty in June 1969, being returned to Kansas state control. When returned to the 184th TFG, the group was returned to operational status, on 25 March 1971, the 184th was designated the 184th Tactical Fighter Training Group and acquired the F-105 Thunderchief aircraft, receiving Vietnam War returning aircraft. As the USAF Combat Crew Training School, the unit conducted pilot training in the F-105 for nine years, on 1 October 1973, the 184th assumed the responsibility of operating and maintaining the Smoky Hill Weapons Range at Salina, Kansas. With over 36,000 acres, Smoky Hill is the Air National Guards largest weapons range, on 7 August 1979, the unit received its first F-4D Phantom II, and on 8 October 1979, was designated as the 184th Tactical Fighter Group, equipped with 50 F-4Ds. In April 1982, the 184th was tasked to develop a F-4D Fighter Weapons Instructor Course to meet the needs of the Air Reserve Forces and the USAF Tactical Air Command. In January 1987, the 184th was tasked to activate a squadron of F-16A/B Fighting Falcon aircraft, on 8 July 1987, the 161st Tactical Fighter Training Squadron was established as the third flying squadron at the 184th TFG. Formal activation ceremonies for the 161st occurred on 12 September 1987, with the unit flying 10 F-16s, a third training squadron, the 177th Tactical Fighter Training Squadron was activated on 1 February 1984. In August 1988, the 127th Tactical Fighter Squadron graduated its final Fighter Weapons Instructor Course Class, the 127th TFS converted as the second F-16 training squadron. The last F-4D departed from the 184th TFG on 31 March 1990, the first F-16C Fighting Falcon arrived at the 184th TFG in July 1990. In July 1993, the 184th Fighter Group changed gaining commands and became part of the new Air Education, in July 1994, the 184th Fighter Group was designated at the 184th Bomb Wing and again became part of the Air Combat Command, flying the B-1B Lancer
184th Intelligence Wing
–
127th TFS F-100C Super Sabre 54-1913, 1968
184th Intelligence Wing
–
184th Intelligence Wing emblem
184th Intelligence Wing
–
184th TFG F-4D Phantom II 66-0710, 1986
184th Intelligence Wing
–
184th Bomb Wing Rockwell B-1B Lancer Lot IV 85-0081, 2000
49.
Air National Guard
–
When Air National Guard units are used under the jurisdiction of the state governor they are fulfilling their militia role. However, if federalized by order of the President of the United States and they are jointly administered by the states and the National Guard Bureau, a joint bureau of the Army and Air Force that oversees the National Guard of the United States. The ANG of the territories of Guam and the Virgin Islands have no aircraft assigned, ANG units typically operate under Title 32 USC. However, when operating under Title 10 USC all ANG units are operationally-gained by an active duty USAF major command. ANG units of the Combat Air Forces based in the Continental United States, conversely, CONUS-based ANG units in the Mobility Air Forces, plus the Puerto Rico ANGs airlift wing and the Virgin Islands ANGs civil engineering squadron are gained by the Air Mobility Command. The vast majority of ANG units fall under either ACC or AMC, established under Title 10 and Title 32 of the U. S. S. When not in a status, the Air National Guard operates under their respective state. The exception to rule is the District of Columbia Air National Guard. Because both state Air National Guard and the Air National Guard of the United States relatively go hand-in-hand, Air National Guard of the United States units or members may be called up for federal active duty in times of Congressionally sanctioned war or national emergency. The United States Air National Guard has about 110,000 men and women in service, even traditional part-time air guardsmen, especially pilots, navigators/combat systems officers, air battle managers and enlisted aircrew, often serve 100 or more man-days annually. As such, the concept of Air National Guard service as representing only one weekend a month, the Georgia Air National Guard and the Kansas Air National Guard previously flew the B-1B Lancer prior to converting to the E-8 Joint STARS and KC-135R Stratotanker, respectively. In addition, the 131st Fighter Wing of the Missouri Air National Guard transitioned from flying the F-15C/D Eagle at St and these proposals were eventually overruled and cancelled by the U. S. Congress. As state militia units, the units in the Air National Guard are not in the normal United States Air Force chain of command and they are under the jurisdiction of the United States National Guard Bureau unless they are federalized by order of the President of the United States. Air National Guard units are trained and equipped by the United States Air Force, the state ANG units, depending on their mission, are operationally gained by a major command of the USAF if federalized. Air National Guard personnel are expected to adhere to the moral and physical standards as their full-time active duty Air Force. The same ranks and insignia of the U. S. Air Force are used by the Air National Guard, the Air National Guard also bestows a number of state awards for local services rendered in a service members home state or equivalent. The creation of the regiments was caused by the perceived need to defend the Bay Colony against American Indians. This organization formed the basis of subsequent colonial and, post-independence, state and this distinction accounts for why there are no National Guard components in the U. S. Navy, U. S. Marine Corps or U. S. Coast Guard
Air National Guard
–
Air National Guard
F-16 Fighting Falcon fighters fly over
Gunsan,
South Korea. The tailflashes denote the aircraft from being from the
New Mexico,
Colorado and
Montana ANGs. Wisconsin Air National Guard
F-16s over
Madison, Wisconsin
Air National Guard
Air National Guard
–
A Galludet Tractor biplane which the New York National Guard aviators rented in 1915.
Air National Guard
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Captain Charles A. Lindbergh, Missouri National Guard, and members of his National Guard unit, 110th Observation Squadron, after he flew solo across the Atlantic Ocean, 1927.
50.
McConnell AFB
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McConnell Air Force Base is a United States Air Force base located four miles southeast of the central business district of Wichita, a city in Sedgwick County, Kansas, United States. The base was named in honor of Wichita brothers Fred and Thomas McConnell and it is the home of Air Mobility Commands 22d Air Refueling Wing, Air Force Reserve Commands 931st Air Refueling Group, and the Kansas Air National Guards 184th Intelligence Wing. McConnells primary mission is to global reach by conducting air refueling and airlift where. The Commander of the 22d Air Refueling Wing is Colonel Albert G. Miller, the Vice-Commander is Colonel Bruce P. Heseltine, Jr. rather, the group and squadron provide additional personnel to augment the active duty personnel of the 22d ARW. Tenant Units, 184th Intelligence Wing 127th Command and Control Squadron McConnell Air Force Base was known during the first part of its existence as the Wichita Municipal Airport. Although the field was designed originally to serve only municipal civil airport needs, mcConnells history began in October 1924, when the city of Wichita hosted more than 100,000 people for the National Air Congress. The event was used by city planners to raise funds for a proposed Wichita Municipal Airport, the event was a success and ground-breaking ceremonies for the airport were held on 28 June 1929. In August 1941, the Kansas National Guard 127th Observation Squadron was activated as the first military unit assigned to the Wichita airport, the units limited equipment included one BC-1A, one C-47, and four L-1 aircraft. This was the start of a relationship between the people of Wichita and military aviation. On 6 October 1941, the unit was ordered to extended active duty and remained a part of the United States Army Air Corps until 6 October 1945, with duty assignments in Tennessee. The airport, at time, was located about six miles from the city of Wichita. The runways were adequate, were five runways each 150 feet wide, two were 7,500 feet, one 7,100 feet, one 6,000 feet, all had a wheel load capacity of 60,000 pounds. A parking apron with dimensions of 8,373 by 931 feet, other facilities at the airport, however, were meager. The field could boast of one hangar and three small warehouses. No facilities were either for troop housing or troop messing. No fuel storage facilities existed, and all supplies were handled by commercial contract. A lease between the government and the city was concluded and on 1 March 1942, the AAF Materiel Center. As soon as permitted, the headquarters of the district was established in the administration building of the municipal airport
McConnell AFB
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Boeing KC-135 Stratotankers based at McConnell in formation as they taxi down a runway.
McConnell AFB
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Kansas Aviation Museum, formerly Wichita Municipal Airport from 1935 to 1951
McConnell AFB
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Boeing-Wichita B-29 Assembly Line – 1944
McConnell AFB
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Newly manufactured B-29s on the ramp at Boeing-Wichita awaiting delivery to operational units, 1945