1.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
2.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
3.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
4.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
5.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
6.
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
7.
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
8.
Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
9.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
10.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
11.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
12.
Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
13.
Senary
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
14.
Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
15.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
16.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
17.
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
18.
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
19.
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
20.
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
21.
Star number
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A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on. The nth star number is given by the formula Sn = 6n +1, the digital root of a star number is always 1 or 4, and progresses in the sequence 1,4,1. The last two digits of a number in base 10 are always 01,13,21,33,37,41,53,61,73,81. Unique among the numbers is 35113, since its prime factors are also consecutive star numbers. Infinitely many star numbers are triangular numbers, the first four being S1 =1 = T1, S7 =253 = T22, S91 =49141 = T313. Infinitely many star numbers are also numbers, the first four being S1 =12, S5 =121 =112, S45 =11881 =1092. A star prime is a number that is prime. The first few star primes are 13,37,73,181,337,433,541,661,937, the term star number or stellate number is occasionally used to refer to octagonal numbers
22.
Hexagram
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A hexagram or sexagram is a six-pointed geometric star figure with the Schläfli symbol,2, or. It is the compound of two equilateral triangles, the intersection is a regular hexagon. It is used in historical, religious and cultural contexts, for example in Hanafism, Jewish identity, in mathematics, the root system for the simple Lie group G2 is in the form of a hexagram, with six long roots and six short roots. A six-pointed star, like a hexagon, can be created using a compass. Without changing the radius of the compass, set its pivot on the circles circumference, with the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked. With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles. It is possible that as a geometric shape, like for example the triangle, circle, or square. The hexagram is a symbol called satkona yantra or sadkona yantra found on ancient South Indian Hindu temples. It symbolizes the nara-narayana, or perfect meditative state of balance achieved between Man and God, and if maintained, results in moksha, or nirvana, some researchers have theorized that the hexagram represents the astrological chart at the time of Davids birth or anointment as king. The hexagram is also known as the Kings Star in astrological circles, in antique papyri, pentagrams, together with stars and other signs, are frequently found on amulets bearing the Jewish names of God, and used to guard against fever and other diseases. Curiously the hexagram is not found among these signs, in the Greek Magical Papyri at Paris and London there are 22 signs side by side, and a circle with twelve signs, but neither a pentagram nor a hexagram. Six-pointed stars have also found in cosmological diagrams in Hinduism, Buddhism. The reasons behind this symbols common appearance in Indic religions and the West are unknown, one possibility is that they have a common origin. The other possibility is that artists and religious people from several cultures independently created the hexagram shape, within Indic lore, the shape is generally understood to consist of two triangles—one pointed up and the other down—locked in harmonious embrace. The two components are called Om and the Hrim in Sanskrit, and symbolize mans position between earth and sky, the downward triangle symbolizes Shakti, the sacred embodiment of femininity, and the upward triangle symbolizes Shiva, or Agni Tattva, representing the focused aspects of masculinity. The mystical union of the two triangles represents Creation, occurring through the union of male and female. The two locked triangles are known as Shanmukha—the six-faced, representing the six faces of Shiva & Shaktis progeny Kartikeya. This symbol is also a part of several yantras and has significance in Hindu ritual worship
23.
Chinese checkers
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Chinese Checkers or Chinese chequers is a strategy board game of German origin which can be played by two, three, four, or six people, playing individually or with partners. The game is a modern and simplified variation of the American game Halma, the remaining players continue the game to establish second-, third-, fourth-, fifth-, and last-place finishers. The rules are simple, so young children can play. Despite its name, the game is not a variation of checkers, the game was invented in Germany in 1892 under the name Stern-Halma as a variation of the older American game Halma. The Stern refers to the star shape. The name Chinese Checkers originated in the United States as a scheme by Bill. The Pressman companys game was originally called Hop Ching Checkers, the game was introduced to Chinese-speaking regions mostly by the Japanese. The aim is to all ones pieces into the star corner on the opposite side of the board before opponents do the same. The destination corner is called home, each player has 10 pieces, except in games between two players when 15 are sometimes used. In hop across, the most popular variation, each starts with their colored pieces on one of the six points or corners of the star. A player may not combine hopping with a single-step move – a move consists of one or the other, there is no capturing in Chinese Checkers, so hopped pieces remain active and in play. Turns proceed clockwise around the board, in the diagram, Green might move the topmost piece one space diagonally forward as shown. A hop consists of jumping over an adjacent piece, either ones own or an opponents. Red might advance the indicated piece by a chain of three hops in a single move and it is not mandatory to make the most number of hops possible. Can be played all versus all, or three teams of two, when playing teams, teammates usually sit at opposite corners of the star, with each team member controlling their own colored set of pieces. The first team to both sets to their home destination corners is the winner. The remaining players usually play to determine second- and third-place finishers. The four-player game is the same as the game for six players, in a three-player game, all players control either one or two sets of pieces each
24.
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
25.
Thue-Morse sequence
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The first few steps of this procedure yield the strings 0 then 01,0110,01101001,0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. There are several equivalent ways of defining the Thue–Morse sequence, to compute the nth element tn, write the number n in binary. If the number of ones in this expansion is odd then tn =1. For this reason John H. Conway et al. call numbers n satisfying tn =1 odious numbers and numbers for which tn =0 evil numbers. In other words, tn =0 if n is an evil number, if this bit is at an even index, tn differs from tn −1, and otherwise it is the same as tn −1. The resulting algorithm takes constant time to each sequence element. The Thue–Morse sequence is the sequence tn satisfying the relation for all non-negative integers n. So, the first element is 0, then once the first 2n elements have been specified, forming a string s, then the next 2n elements must form the bitwise negation of s. Now we have defined the first 2n+1 elements, and we recurse, spelling out the first few steps in detail, We start with 0. The bitwise negation of 0 is 1, combining these, the first 2 elements are 01. The bitwise negation of 01 is 10, combining these, the first 4 elements are 0110. The bitwise negation of 0110 is 1001, combining these, the first 8 elements are 01101001. The sequence can also be defined by, ∏ i =0 ∞ = ∑ j =0 ∞ t j x j and that is, there are many instances of XX, where X is some string. For instance, with k =0, we have A = T0 =0, however, there are no cubes, instances of XXX. There are also no overlapping squares, instances of 0X0X0 or 1X1X1, notice that T2n is palindrome for any n >1. Further, let qn be a word obtain from T2n by counting ones between consecutive zeros, for instance, q1 =2 and q2 =2102012 and so on. The words Tn do not contain overlapping squares in consequence, the words qn are palindrome squarefree words, the easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps. Then nX can be set to any multiple of m that is larger than twice the length of X, but the Morse sequence is uniformly recurrent without being periodic, not even eventually periodic
26.
Square-free integer
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In mathematics, a square-free, or quadratfrei integer, is an integer which is divisible by no other perfect square than 1. For example,10 is square-free but 18 is not, as 18 is divisible by 9 =32. The smallest positive square-free numbers are 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39. The radical of an integer is its largest square-free factor, an integer is square-free if and only if it is equal to its radical. Any arbitrary positive integer n can be represented in a way as the product of a powerful number and a square-free integer. The square-free factor is the largest square-free divisor k of n that is coprime with n/k, a positive integer n is square-free if and only if in the prime factorization of n, no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor p of n, also n is square-free if and only if in every factorization n = ab, the factors a and b are coprime. An immediate result of this definition is that all numbers are square-free. A positive integer n is square-free if and only if all abelian groups of n are isomorphic. This follows from the classification of finitely generated abelian groups, a integer n is square-free if and only if the factor ring Z / nZ is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if, for every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice and it is a Boolean algebra if and only if n is square-free. A positive integer n is square-free if and only if μ ≠0, a positive integer n is squarefree if and only if ∑ d 2 ∣ n μ =1. This results from the properties of Möbius function, and the fact that this sum is equal to ∑ d ∣ m μ, where m is the largest divisor of n such that m2 divides n. The Dirichlet generating function for the numbers is ζ ζ = ∑ n =1 ∞ | μ | n s where ζ is the Riemann zeta function. This is easily seen from the Euler product ζ ζ = ∏ p = ∏ p, let Q denote the number of square-free integers between 1 and x. For large n, 3/4 of the positive integers less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Under the Riemann hypothesis, the term can be further reduced to yield Q = x ζ + O =6 x π2 + O
27.
Negative base
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A negative base may be used to construct a non-standard positional numeral system. The need to store the information normally contained by a sign often results in a negative-base number being one digit longer than its positive-base equivalent. Negative numerical bases were first considered by Vittorio Grünwald in his work Giornale di Matematiche di Battaglini, Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later rediscovered by A. J. Kempner in 1936 and Zdzisław Pawlak. Negabinary was implemented in the early Polish computer BINEG, built 1957–59, based on ideas by Z. Pawlak, implementations since then have been rare. The base −r expansion of a is given by the string dndn-1…d1d0. Negative-base systems may thus be compared to signed-digit representations, such as balanced ternary, some numbers have the same representation in base −r as in base r. For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal, similarly,17 =24 +20 =4 +0 and is represented by 10001 in binary and 10001 in negabinary. The base −r expansion of a number can be found by repeated division by −r, recording the non-negative remainders of 0,1, …, r −1, and concatenating those remainders, starting with the last. Note that if a / b = c, remainder d, then bc + d = a, to arrive at the correct conversion, the value for c must be chosen such that d is non-negative and minimal. This is exemplified in the line of the following example wherein –5 ÷ –3 must be chosen to equal 2 remainder 1 instead of 1 remainder –2. Note that in most programming languages, the result of dividing a number by a negative number is rounded towards 0. In such a case we have a = c + d = c + d − r + r = +, because |d| < r, is the positive remainder. The conversion from integer to some negative base, Visual Basic implementation, The conversion to negabinary allows a remarkable shortcut, the bitwise XOR portion is originally due to Schroeppel. Adding negabinary numbers proceeds bitwise, starting from the least significant bits, while adding two negabinary numbers, every time a carry is generated an extra carry should be propagated to next bit. Unary negation, −x, can be computed as binary subtraction from zero,0 − x, shifting to the left multiplies by −2, shifting to the right divides by −2. To multiply, multiply like normal decimal or binary numbers, but using the rules for adding the carry. It is possible to compare negabinary numbers by slightly adjusting a normal unsigned binary comparator, when comparing the numbers A and B, invert each odd positioned bit of both numbers
28.
Nonary
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
29.
Numeral system
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A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
30.
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
31.
Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
32.
5 (number)
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5 is a number, numeral, and glyph. It is the number following 4 and preceding 6. Five is the prime number. Because it can be written as 221 +1, five is classified as a Fermat prime, therefore a regular polygon with 5 sides is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also the number that is part of more than one pair of twin primes. Five is conjectured to be the only odd number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being 2 plus 3,5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation. Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers,5 is the length of the hypotenuse of the smallest integer-sided right triangle. In bases 10 and 20,5 is a 1-automorphic number,5 and 6 form a Ruth–Aaron pair under either definition. There are five solutions to Známs problem of length 6 and this is related to the fact that the symmetric group Sn is a solvable group for n ≤4 and not solvable for n ≥5. While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar, K5, Five is also the number of Platonic solids. A polygon with five sides is a pentagon, figurate numbers representing pentagons are called pentagonal numbers. Five is also a square pyramidal number, Five is the only prime number to end in the digit 5, because all other numbers written with a 5 in the ones-place under the decimal system are multiples of five. As a consequence of this,5 is in base 10 a 1-automorphic number, vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions, unlike expansions with all other prime denominators, because they are prime factors of ten, the base. When written in the system, all multiples of 5 will end in either 5 or 0
33.
31 (number)
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31 is the natural number following 30 and preceding 32. As a Mersenne prime,31 is related to the perfect number 496,31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a triangular number, the lowest prime centered pentagonal number. For the Steiner tree problem,31 is the number of possible Steiner topologies for Steiner trees with 4 terminals, at 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31,31 is a repdigit in base 5, and base 2. The numbers 31,331,3331,33331,333331,3333331, for a time it was thought that every number of the form 3w1 would be prime. Here,31 divides every fifteenth number in 3w1, the atomic number of gallium Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is visible to the naked eye in a modestly dark sky. The New General Catalogue object NGC31, a galaxy in the constellation Phoenix The Saros number of the solar eclipse series which began on -1805 January 31. The duration of Saros series 31 was 1316.2 years, the Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, the jersey number 31 has been retired by several North American sports teams in honor of past playing greats, In Major League Baseball, The San Diego Padres, for Dave Winfield. The Chicago Cubs, for Ferguson Jenkins and Greg Maddux, the Atlanta Braves, also for Maddux. The New York Mets, for Mike Piazza, in the NBA, The Boston Celtics, for Cedric Maxwell. The Indiana Pacers, for Reggie Miller, in the NHL, The Edmonton Oilers, for Grant Fuhr. The New York Islanders, for Billy Smith, in the NFL, The Atlanta Falcons, for William Andrews. The New Orleans Saints, for Jim Taylor, NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company. In ice hockey goaltenders often wear the number 31, in football the number 31 has been retired by Queens Park Rangers F. C.31 from the Prime Pages
34.
37 (number)
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37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37
35.
Saskatchewan, Canada
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Saskatchewan is a prairie and boreal province in west-central Canada, the only province without natural borders. It has an area of 651,900 square kilometres, nearly 10 percent of which is water, composed mostly of rivers, reservoirs. As of December 2013, Saskatchewans population was estimated at 1,114,170, residents primarily live in the southern prairie half of the province, while the northern boreal half is mostly forested and sparsely populated. Of the total population, roughly half live in the provinces largest city, Saskatoon, or the provincial capital, other notable cities include Prince Albert, Moose Jaw, Yorkton, Swift Current, North Battleford, and the border city Lloydminster. Saskatchewan is a province with large distances to moderating bodies of waters. As a result, its climate is continental, rendering severe winters throughout the province. Southern areas have very warm or hot summers, Midale and Yellow Grass near the U. S. border are tied for the highest ever recorded temperatures in Canada with 45 °C observed at both locations on July 5,1937. In winter, temperatures below −45 °C are possible even in the south during extreme cold snaps, Saskatchewan has been inhabited for thousands of years by various indigenous groups, and first explored by Europeans in 1690 and settled in 1774. It became a province in 1905, carved out from the vast North-West Territories, in the early 20th century the province became known as a stronghold for Canadian social democracy, North Americas first social-democratic government was elected in 1944. The provinces economy is based on agriculture, mining, and energy, Saskatchewans current premier is Brad Wall and its lieutenant-governor is Vaughn Solomon Schofield. In 1992, the federal and provincial governments signed a land claim agreement with First Nations in Saskatchewan. The First Nations received compensation and were permitted to buy land on the market for the tribes, they have acquired about 3,079 square kilometres. Some First Nations have used their settlement to invest in urban areas and its name derived from the Saskatchewan River. The river was known as kisiskāciwani-sīpiy in the Cree language, as Saskatchewans borders largely follow the geographic coordinates of longitude and latitude, the province is roughly a quadrilateral, or a shape with four sides. However the 49th parallel boundary and the 60th northern border appear curved on globes, additionally, the eastern boundary of the province is partially crooked rather than following a line of longitude, as correction lines were devised by surveyors prior to the homestead program. S. States of Montana and North Dakota, Saskatchewan has the distinction of being the only Canadian province for which no borders correspond to physical geographic features. Along with Alberta, Saskatchewan is one of only two land-locked provinces, the overwhelming majority of Saskatchewans population is located in the southern third of the province, south of the 53rd parallel. Saskatchewan contains two natural regions, the Canadian Shield in the north and the Interior Plains in the south
36.
181 West Madison Street
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181 West Madison Street is a skyscraper located in Chicago. Built in 1990, the building is 680 feet tall and contains 50 floors and it is architect Cesar Pellis first and only completed tower in the city. The glassy office towers most distinctive feature is its recessed crown, the top of the building is illuminated white at the corners, as well as other various colors depending on the holiday. In 1989, the combination of developer and architect envisioned the Miglin-Beitler Skyneedle nearby. The 2,000 foot and 125 story building would have been the tallest skyscraper in the world if completed, but plans were scrapped because of a sluggish real estate market
37.
Canadian Expeditionary Force
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The Canadian Expeditionary Force was the designation of the field force created by Canada for service overseas in the First World War. The force fielded several combat formations on the Western Front in France and Belgium, the Canadian Cavalry Brigade and the Canadian Independent Force, which were independent of the Canadian Corps, also fought on the Western Front. The CEF also had a reserve and training organization in England. The Germans went so far as to call them storm troopers for their combat efficiency. In August 1918, the CEFs Canadian Siberian Expeditionary Force travelled to revolution-torn Russia and it reinforced an anti-Bolshevik garrison in Vladivostok during the winter of 1918–19. At this time, another force of Canadian soldiers were placed in Archangel, the Canadian Expeditionary Force was mostly volunteers, as conscription was not enforced until the end of the war when call-ups began in January 1918. Ultimately, only 24,132 conscripts arrived in France before the end of the war, Canada was the senior Dominion in the British Empire and automatically at war with Germany upon the British declaration. According to Canadian historian Dr. Serge Durflinger at the Canadian War Museum, of the first contingent formed at Valcartier, Quebec in 1914, fully two-thirds were men born in the United Kingdom. By the end of the war in 1918, at least fifty per cent of the CEF consisted of British-born men, many British nationals from the United Kingdom or other territories who were resident in Canada also joined the CEF. As several CEF battalions were posted to the Bermuda Garrison before proceeding to France, although the Bermuda Militia Artillery and Bermuda Volunteer Rifle Corps both sent contingents to the Western Front, the first would not arrive there til June 1915. By then, many Bermudians had already been serving on the Western Front in the CEF for months, Bermudians in the CEF enlisted under the same terms as Canadians, and all male British Nationals resident in Canada became liable for conscription under the Military Service Act,1917. Two tank battalions were raised in 1918 but did not see service, most of the infantry battalions were broken up and used as reinforcements, with a total of fifty being used in the field, including the mounted rifle units, which were re-organized as infantry. The artillery and engineering units underwent significant re-organization as the war progressed, a distinct entity within the Canadian Expeditionary Force was the Canadian Machine Gun Corps. It consisted of several machine gun battalions, the Eatons, Yukon, and Borden Motor Machine Gun Batteries. During the summer of 1918, these units were consolidated into four machine gun battalions, the Canadian Corps with its four infantry divisions comprised the main fighting force of the CEF. The Canadian Cavalry Brigade also served in France, the 1915 Battle of Ypres, the first engagement of Canadian forces in the Great War, changed the Canadian perspective on war. Ypres exposed Canadian soldiers and their commanders to modern war and they had already experienced the effects of shellfire and developed a reputation for aggressive trench raiding despite their lack of formal training and generally inferior equipment. In April 1915, they were introduced to yet another facet of modern war, the Germans employed chlorine gas to create a hole in the French lines adjacent to the Canadian force and poured troops into the gap
38.
181st Airlift Squadron
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The 181st Airlift Squadron is a unit of the 136th Airlift Wing of the Texas Air National Guard stationed at Naval Air Station Joint Reserve Base Fort Worth, Texas. The 181st is equipped with the Lockheed C-130H Hercules, the squadron was first activated during World War II as the 395th Fighter Squadron. It served in the European Theater of Operations as a bomber unit, earning a Distinguished Unit Citation. After the war it became part of the forces in Germany until it was inactivated in August 1946. The squadron was allotted to the United States National Guard as the 181st Fighter Squadron and was activated in 1947, the squadron was first activated in June 1943 as the 395th Fighter Squadron, a Republic P-47 Thunderbolt fighter squadron. It trained under I Fighter Command on Long Island, New York, the 395th moved to England, arriving in January 1944. The unit began operations with IX Fighter Command on 14 March and it supported the landings in Normandy in June 1944 and began operations from the Continent later the same month. It operated with the Allied forces that pushed across the Rhine, during its operations the squadron earned a Distinguished Unit Citation and was cited twice in the Order of the day of the Belgian Army, earning the Belgian Fourragère. After V-E Day, the served with the occupation forces. It was inactivated in Germany on 20 August 1946 and its personnel and equipment were transferred to the 82d Fighter Squadron, the wartime 395th Fighter Squadron was redesignated the 181st Fighter Squadron and allotted to the National Guard the day after it was inactivated in Germany. It was organized at Love Field, Dallas, Texas and was extended federal recognition on 27 February 1947. The squadron was assigned to its World War II headquarters, which had also assigned to the National Guard as the 136th Fighter Group. The mission of the squadron was to train for air defense, the 181st Fighter Squadron remained in the Texas Air National Guard and was assigned directly to its headquarters. The 181st was re-equipped with the Very Long Range F-51H Mustang, the F-51H would allow the squadron to intercept any unidentified aircraft over any part of Texas. In September 1952, the became the 181st Fighter-Interceptor Squadron With the 136th Fighter-Bomber Groups release from active duty in July 1952. It became the 181st Fighter-Bomber Squadron the following January, despite this name change, the squadron remained focused on the air defense mission. It wasnt until January 1955 that the received its first jets. In July it was redesignated the 181st Fighter-Interceptor Squadron, the 181st was selected by ADC to man an alert program on an around the clock basis, with armed fighters ready to scramble at a moments notice
39.
Texas Air National Guard
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The Texas Air National Guard is the air force militia of the State of Texas, United States of America. It is, along with the Texas Army National Guard, an element of the Texas National Guard, No element of the Texas Air National Guard is under United States Air Force command. They are under the jurisdiction of the Governor of Texas through the office of the Texas Adjutant General unless they are federalized by order of the President of the United States. The Texas Air National Guard is headquartered at Camp Mabry, Austin, under the Total Force concept, Texas Air National Guard units are considered to be Air Reserve Components of the United States Air Force. Texas 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. Gained by, Air Mobility Command The 136th AW mission is tactical airlift, the aircraft is capable of operating from rough, dirt strips and is the prime transport for air dropping troops and equipment into hostile areas. Its combat support sorties provide theater and national-level leadership with critical real-time Intelligence, Surveillance, airCC units are embedded with their parent Texas Air National Guard units in San Antonio, Austin, Ft. Worth, Garland, Houston and La Porte. S. 204th Security Forces Squadron The 204th Security Forces Squadron located at Biggs Army Airfield, Fort Bliss and they are the only heavy weapons security forces unit in the Air National Guard. Since the September 11 attacks, members of the 204th SFS have seen duty in central and southwest Asia, in Africa and they have served on installations in several states in the U. S. and taught military base defense in Latin American countries. 217th Training Squadron The 217th TRS is a training unit that is subordinate to the 149th Fighter Wing at Lackland AFB. The 217th TRS is a GSU and is located on Goodfellow AFB,217 TRS instructors are integrated into the existing courses taught within the 17 TRG - primarily the 315 TRS. Although some 217th TRS instructors are drill status guardsmen, most are full-time air technicians, lieutenant Colonel James W. Marrs became the first commander of the newly formed 217th Training Squadron upon its activation. Additionally, the 217 TRS will soon be a major training source for Incident Awareness and Assessment, the unit will be responsible for training any Air National Guard units that require it in order for them to be better prepared to respond to local and national disasters. 221st Combat Communications Squadron The 221st Combat Communications Squadron is co-located in Grand Prairie with their command unit, the function of the 221st Combat Communications Squadron is to provide communications in a deployed environment. Such requirements may include establishing a Local Area Network, Telephone Network, Wide Area Network, all this while ensuring reliable connectivity for those parties serviced and maintaining mission effectiveness. Securing the AF gateways against IO attacks and it is subordinate to the 149th Fighter Wing located on Lackland AFB. The Texas Air National Guard origins date to 14 August 1917 with the establishment of the 111th Aero Squadron as part of the World War I United States Army Air Service
40.
181st Infantry Brigade (United States)
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The 181st Infantry Brigade is an infantry brigade of the United States Army based at Fort McCoy, Wisconsin. As an Active Component/Reserve Component brigade, the unit primarily in a training role for other units of the US armed forces. The brigade is subordinate to the First United States Army, headquartered at Rock Island Arsenal, the unit is responsible for training selected United States Army Reserve and Army National Guard units in the Central-Northern United States. The unit was designated as 2nd Brigade, 85th Division. The brigade was redesignated and re-missioned several times, such as in 1999, the 181st Infantry Brigade currently falls under the 1st Armys Division West, headquartered at Fort Hood, Texas. Headquarters, 181st Brigade 361st Infantry Regiment 362nd Infantry Regiment 347th Machine Gun Battalion 91st Reconnaissance Troop Headquarters and Headquarters Company, Fort McCoy, the 181st Infantry Brigade trained for 10 months at Camp Lewis prior to being deployed to France in August 1918. After four months of peacekeeping operations in liberated Belgium, the Brigade returned to the United States, the Brigade was transferred on 2 April 1919 to Camp Merritt, New Jersey. It proceeded to Camp Kearny, California, where it was demobilized on 19 April 1919, the Brigade was reconstituted in the Organized Reserve on 24 June 1921, still assigned to the 91st Division, and allotted to the Ninth Corps Area. The Brigade was redesignated Headquarters & Headquarters Company, 181st Brigade on 23 March 1925 and again redesignated HHC, the unit conducted summer training most years at Del Monte, California, from 1922–40. The 91st Reconnaissance Troop participated in the Rome-Arno, North Apennines, in July 1944, during the Arno Campaign of the Second World War, the 91st Reconnaissance Troop spearheaded Task Force Williamson under the command of Brigadier General E. S. Williamson, Assistant Division Commander for the 91st Division, the 2nd Platoon of the 91st Reconnaissance Troop and the 1st Battalion, 363rd Infantry were the first to enter Leghorn on its way to liberating Pisa. It was reactivated in 1947 as a cavalry reconnaissance troop, redesignated in 1949 as the 91st Reconnaissance Company. The 181st Infantry Brigade was reactivated at Fort McCoy, Wisconsin in December 2006, symbolism, The red and white of the background are the colors used in flags for Armies. The letter A represents Army and is also the first letter of the alphabet suggesting First Army. Background, A black letter A was approved as the authorized insignia by the Commanding General, American Expedition Force, on 16 November 1918, the background was added on 17 November 1950. Attached below the device a red scroll inscribed DOCERE BELLUM ET PAX PACIS in Silver, symbolism, The diagonal separation of colors denotes a line not crossed. The clevis symbolizes the long history and knowledge as being a key to winning the battle. The crossed rifle and saber allude to the Brigades mission during World War II as the 91st Reconnaissance Cavalry Company, the motto translates to To Win War and Peace
41.
United States Army
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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
42.
Fort McCoy, Wisconsin
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Fort McCoy is a United States Army installation. It is located on 60,000 acres between Sparta and Tomah, Wisconsin, in Monroe County, since its creation in 1909, the post has been used primarily as a military training center. A part of Fort McCoy is also used by the Wisconsin State Patrol as a training facility, the post has been in virtually constant use since it was first formed as the Sparta Maneuver Tract on 14,000 acres in 1909. At first, the tract was made up of two camps, Camp Emory Upton and Camp Robinson and these were separated by a line of the Chicago, Milwaukee, St. Paul and Pacific Railroad that ran across the land from east to west. In 1926, the name of the post was shortened to Camp McCoy, in 1938, the United States began a major expansion of the camp. This included the addition of over 45,000 acres to the post and this increased the camps capacity to 35,000 soldiers. In all, the project was estimated to have cost about $30 million, the expansion was officially concluded with a new inauguration on August 30,1942. During World War II, Fort McCoy was used as a center for approximately 170 Japanese and 120 German and Italian American civilians arrested as potentially dangerous enemy aliens in 1942. The post was used as a prisoner-of-war camp during the conflict, holding 4,000 Japanese. Fort McCoys POWs were featured in the 2011 film Fort McCoy, the camp was briefly deactivated following World War II, but with the advent of the Korean War in 1950, it was once again used for training. This continued until 1953, when the camp was again deactivated and it was then used to house various small national, state and civilian projects, and served as a training center for the National Guard and the Job Corps. In response, a Milwaukee official proposed that the camp be used as a landfill for Milwaukee garbage, in 1973, the Army reactivated Camp McCoy as a permanent training center, and on September 30,1974, it was officially re-designated as Fort McCoy. In the 1990s, a major construction project was undertaken. Today, Fort McCoy serves as a Total Force Training Center, around 100,000 members of the military are trained at the fort every year, and the total number has exceeded 149,000 in the past. Fort McCoy also is the headquarters of Naval Mobile Construction Battalion-25 which served a tour in Iraq. The 181st Infantry Brigade is the largest unit stationed at Fort McCoy, Fort McCoy was used as a mobilization station during Operation Desert Shield and Operation Desert Storm. This was the first time units had mobilized at Fort McCoy since the Korean War,74 military units deployed through Fort McCoy, totaling over 9,000 Soldiers, 8% of the reserve forces activated during the Persian Gulf War. Volk Field Air National Guard Base was used as the point of departure
43.
181st Intelligence Wing
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The 181st Intelligence Wing is a unit of the Indiana Air National Guard, stationed at Terre Haute Air National Guard Base, Indiana. If activated to service, the Wing is gained by the United States Air Force Intelligence, Surveillance and Reconnaissance Agency. The Wings 113th Air Support Operations Squadron is a descendant organization of the World War I 113th Aero Squadron, the airmen process, exploit, and disseminate the video feed, providing actionable intelligence to the ground commanders and war-fighting forces. The 113th TFS becoming the flying squadron. Other squadrons assigned into the group were the 181st Headquarters, 181st Material Squadron, 181st Combat Support Squadron, the 113th TFS was temporarily equipped with RF-84F Thunderstreaks photo-reconnaissance aircraft to allow its pilots to maintain proficiency. In 1964, the squadron received F-84F Thunderstreak tactical fighter-bombers and this deployment required two over-water air refuelings in either direction. In addition, the 113th deployed to Vincent AFB, Arizona for extensive gunnery, rocketry, the F-84F remained with the 181st until December 1971, when they were retired to AMARC and replaced by North American F-100C/D Super Sabres following their withdrawal from the Vietnam War. The F-100 remained with the squadron until 1979 and participated in numerous deployments, in April 1976, the squadron deployed to RAF Lakenheath, England as part of Cornet Prize, and was awarded an Air Force Outstanding Unit Award for the period October 1975 to May 1976. The unit had the honor to fly the last active United States Military F-100 mission when it flew F-100D 56-2979 to MASDC, Davis-Monthan AFB, Arizona, in the summer of 1979 the unit had begun conversion to the F-4C Phantom II. By 1 April 1988 the unit had completed its conversion to more the advanced F-4E version of the Phantom II, however, the squadron was not assigned the specialized Wild Weasel mission, and it operated its F-4Cs in the conventional strike role. With the receipt of the Phantoms in 1979, the 113th began using Tactical Air Command Tail Code HF on their aircraft, the 113th initially operated the F-4Cs in a tactical role. In addition, they served in the air defense role as part of the Air National Guard taking over the mission of the inactivated Aerospace Defense Command for continental air defense. In the air defense role, the squadron operated under Air Defense, Tactical Air Command, with the changeover to the F-16, the squadron changed its Tail Code to TH. In 1992, the designation changed to 113th Fighter Squadron, 181st Fighter Group. On 1 October 1995 the 181st Fighter Group was changed in status to a Wing, in mid-1996, the Air Force, in response to budget cuts, and changing world situations, began experimenting with Air Expeditionary organizations. The Air Expeditionary Force concept was developed that would mix Active-Duty, Reserve, additionally, the Unit received exceptional ratings on a number of higher headquarters evaluations. The 113th swapped their Block 25 F-16C/Ds for Block 30s in July/August 1995, the units vipers were equipped with the LITENING targeting pod, a precision targeting pod system designed for Air Force Reserves and Air National Guards F-16 Block 25/30/32 Fighting Falcons. The 181st Fighter Wing drastically increased its operations tempo during the early 2000s to guard Americas skies, in 2005, the Base Realignment and Closure commission mandated the end of the flying era for the 181st
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United States Air Force
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The United States Air Force is the aerial warfare service branch of the United States Armed Forces and one of the seven American uniformed services. Initially part of the United States Army, the USAF was formed as a branch of the military on 18 September 1947 under the National Security Act of 1947. It is the most recent branch of the U. S. military to be formed, the U. S. Air Force is a military service organized within the Department of the Air Force, one of the three military departments of the Department of Defense. The Air Force is headed by the civilian Secretary of the Air Force, who reports to the Secretary of Defense, the U. S. Air Force provides air support for surface forces and aids in the recovery of troops in the field. As of 2015, the service more than 5,137 military aircraft,406 ICBMs and 63 military satellites. It has a $161 billion budget with 313,242 active duty personnel,141,197 civilian employees,69,200 Air Force Reserve personnel, and 105,500 Air National Guard personnel. According to the National Security Act of 1947, which created the USAF and it shall be organized, trained, and equipped primarily for prompt and sustained offensive and defensive air operations. The stated mission of the USAF today is to fly, fight, and win in air, space and we will provide compelling air, space, and cyber capabilities for use by the combatant commanders. We will excel as stewards of all Air Force resources in service to the American people, while providing precise and reliable Global Vigilance, Reach and it should be emphasized that the core functions, by themselves, are not doctrinal constructs. The purpose of Nuclear Deterrence Operations is to operate, maintain, in the event deterrence fails, the US should be able to appropriately respond with nuclear options. Dissuading others from acquiring or proliferating WMD, and the means to deliver them, moreover, different deterrence strategies are required to deter various adversaries, whether they are a nation state, or non-state/transnational actor. Nuclear strike is the ability of forces to rapidly and accurately strike targets which the enemy holds dear in a devastating manner. Should deterrence fail, the President may authorize a precise, tailored response to terminate the conflict at the lowest possible level, post-conflict, regeneration of a credible nuclear deterrent capability will deter further aggression. Finally, the Air Force regularly exercises and evaluates all aspects of operations to ensure high levels of performance. Nuclear surety ensures the safety, security and effectiveness of nuclear operations, the Air Force, in conjunction with other entities within the Departments of Defense or Energy, achieves a high standard of protection through a stringent nuclear surety program. The Air Force continues to pursue safe, secure and effective nuclear weapons consistent with operational requirements, adversaries, allies, and the American people must be highly confident of the Air Forces ability to secure nuclear weapons from accidents, theft, loss, and accidental or unauthorized use. This day-to-day commitment to precise and reliable nuclear operations is the cornerstone of the credibility of the NDO mission, positive nuclear command, control, communications, effective nuclear weapons security, and robust combat support are essential to the overall NDO function. OCA is the method of countering air and missile threats, since it attempts to defeat the enemy closer to its source
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Terre Haute, Indiana
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Terre Haute is a city in and the county seat of Vigo County, Indiana, United States, near the states western border with Illinois. As of the 2010 census, the city had a population of 60,785. Located along the Wabash River, Terre Haute is the capital of the Wabash Valley. The city is home to higher education institutions, including Indiana State University, Saint Mary-of-the-Woods College. Terre Haute is notable for being the home of Socialist Party of America leader and five-time presidential nominee, Debs and the Federal Correctional Complex. Terre Haute is located alongside the bank of the Wabash River in western Indiana. The city lies about 75 miles west of Indianapolis, according to the 2010 census, Terre Haute has a total area of 35.272 square miles, of which 34.54 square miles is land and 0.732 square miles is water. The Wabash River dominates the geography of the city, forming its western border. Small bluffs on the east side of city mark the edge of the flood plain. Lost Creek and Honey Creek drain the northern and southern sections of the city, in the late 19th century, several oil and mineral wells were productive in and near the center of the city. That well produced oil into the 1920s, Terre Haute is at the intersection of two major roadways, U. S.40 from California to Maryland and US41 from Copper Harbor, Michigan to Miami, Florida. Terre Haute is located 77 miles southwest of Indianapolis and within 185 miles of Chicago, St. Louis, Louisville, Climate is characterized by relatively high temperatures and evenly distributed precipitation throughout the year. The Köppen Climate Classification subtype for this climate is Dfa, Terre Hautes name was derived from the French phrase terre haute, meaning Highland. It was likely named by French explorers in the area in the early 18th century to describe the unique location above the Wabash River, at the time the area was claimed by the French and British, these highlands were considered the border between Canada and Louisiana. The construction of Fort Harrison in 1811 marked the beginning of a permanent population of European-Americans. A Wea Indian village already existed near the fort, and the orchards, the village of Terre Haute, then a part of Knox County, Indiana, was platted in 1816. Growth really began when the founders won the bid to make it the county seat when Vigo County was formed in March 1818. When the villages 1,000 residents voted to incorporate in 1832, Terre Haute became a town, early Terre Haute was a center of farming, milling and pork processing
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APQ-181 radar
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The AN/APQ-181 is an all-weather, low probability of intercept radar system designed by Hughes Aircraft for the U. S. Air Force B-2A Spirit bomber aircraft. The system was developed in the mid-1980s and entered service in 1993, the APQ-181 provides a number of precision targeting modes, and also supports terrain-following radar and terrain avoidance. The radar operates in the Ku band, the original design uses a TWT-based transmitter with a 2-dimensional passive electronically scanned array antenna. In 2002, Raytheon was awarded a contract to develop a new and this upgrade will improve system reliability, and will also eliminate potential conflicts in frequency usage between the B-2 and commercial satellite systems that also use the J band. In 2008 the Federal Communications Commission accidentally sold the APQ-181 frequency to a commercial user resulting in the need for installing new radar arrays at a cost of over $1 billion, all B-2 aircraft are expected to have the upgraded radar by 2010. List of radars Joint Electronics Type Designation System Raytheon product description spacedaily. com article
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Radar
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Radar is an object-detection system that uses radio waves to determine the range, angle, or velocity of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, Radio waves from the transmitter reflect off the object and return to the receiver, giving information about the objects location and speed. Radar was developed secretly for military use by several nations in the period before, the term RADAR was coined in 1940 by the United States Navy as an acronym for RAdio Detection And Ranging or RAdio Direction And Ranging. The term radar has since entered English and other languages as a common noun, high tech radar systems are associated with digital signal processing, machine learning and are capable of extracting useful information from very high noise levels. Other systems similar to make use of other parts of the electromagnetic spectrum. One example is lidar, which uses ultraviolet, visible, or near infrared light from lasers rather than radio waves, as early as 1886, German physicist Heinrich Hertz showed that radio waves could be reflected from solid objects. In 1895, Alexander Popov, an instructor at the Imperial Russian Navy school in Kronstadt. The next year, he added a spark-gap transmitter, in 1897, while testing this equipment for communicating between two ships in the Baltic Sea, he took note of an interference beat caused by the passage of a third vessel. In his report, Popov wrote that this phenomenon might be used for detecting objects, the German inventor Christian Hülsmeyer was the first to use radio waves to detect the presence of distant metallic objects. In 1904, he demonstrated the feasibility of detecting a ship in dense fog and he obtained a patent for his detection device in April 1904 and later a patent for a related amendment for estimating the distance to the ship. He also got a British patent on September 23,1904 for a radar system. It operated on a 50 cm wavelength and the radar signal was created via a spark-gap. In 1915, Robert Watson-Watt used radio technology to advance warning to airmen. Watson-Watt became an expert on the use of direction finding as part of his lightning experiments. As part of ongoing experiments, he asked the new boy, Arnold Frederic Wilkins, Wilkins made an extensive study of available units before selecting a receiver model from the General Post Office. Its instruction manual noted that there was fading when aircraft flew by, in 1922, A. Hoyt Taylor and Leo C. Taylor submitted a report, suggesting that this might be used to detect the presence of ships in low visibility, eight years later, Lawrence A. Australia, Canada, New Zealand, and South Africa followed prewar Great Britain, and Hungary had similar developments during the war. Hugon, began developing a radio apparatus, a part of which was installed on the liner Normandie in 1935