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.
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
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
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
19.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
20.
Abundant number
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In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number and its proper divisors are 1,2,3,4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance, the number 12 has an abundance of 4, for example. A number n for which the sum of divisors σ>2n, or, equivalently, the sum of proper divisors s>n. The first 28 abundant numbers are,12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,100,102,104,108,112,114,120, …. For example, the divisors of 24 are 1,2,3,4,6,8. Because 36 is more than 24, the number 24 is abundant and its abundance is 36 −24 =12. The smallest odd abundant number is 945, the smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5,7,11,13,17,19,23, and 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If A represents the smallest abundant number not divisible by the first k primes then for all ϵ >0 we have,2 − ϵ < ln A <2 + ϵ for sufficiently large k, infinitely many even and odd abundant numbers exist. The set of abundant numbers has a natural density, marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480. Every multiple of a number is abundant. For example, every multiple of 6 is abundant because the divisors include 1, n/2, n/3, every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10, every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number which is not a number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The abundancy index of n is the ratio σ/n, distinct numbers n1, n2. with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 =12 corresponds to the first abundant number, if p = is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant
21.
Pronic number
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A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers, however, the rectangular number name has also been applied to the composite numbers. The first few numbers are,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462 …. The nth pronic number is also the difference between the odd square 2 and the st centered hexagonal number. The sum of the reciprocals of the numbers is a telescoping series that sums to 1,1 =12 +16 +112 ⋯ = ∑ i =1 ∞1 i. The partial sum of the first n terms in this series is ∑ i =1 n 1 i = n n +1, the nth pronic number is the sum of the first n even integers. It follows that all numbers are even, and that 2 is the only prime pronic number. It is also the only number in the Fibonacci sequence. The number of entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a number is present in only one of the factors n or n+1. Thus a pronic number is squarefree if and only if n and n +1 are also squarefree, the number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n +1. If 25 is appended to the representation of any pronic number. This is because 2 =100 n 2 +100 n +25 =100 n +25
22.
NASCAR
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The National Association for Stock Car Auto Racing is an American family-owned and operated business venture that sanctions and governs multiple auto-racing sports events. Bill France Sr. founded the company in 1948 and his grandson Brian France became its CEO in 2003, NASCAR is motorsports preeminent stock-car racing organization. The three largest racing-series sanctioned by this company are the Monster Energy NASCAR Cup Series, the Xfinity Series, the company also oversees NASCAR Local Racing, the Whelen Modified Tour, the Whelen All-American Series, and the NASCAR iRacing. com Series. NASCAR sanctions over 1,500 races at over 100 tracks in 39 of the 50 US states as well as in Canada. NASCAR has presented exhibition races at the Suzuka and Motegi circuits in Japan, the Autódromo Hermanos Rodríguez in Mexico, NASCAR has its official headquarters in Daytona Beach, Florida, and also maintains offices in the North Carolina cities of Charlotte, Concord, and Conover. Regional offices are located in New York City and Los Angeles, with offices in Mexico City. Owing to NASCARs Southern roots, all but a handful of NASCAR teams are based in North Carolina. NASCAR is second to the National Football League among professional sports franchises in terms of television viewers, internationally, its races are broadcast on television in over 150 countries. In 2004, NASCARs Director of Security stated that the company holds 17 of the Top 20 regularly attended single-day sporting events in the world, fortune 500 companies sponsor NASCAR more than any other motor sport, although this sponsorship has declined since the early-2000s. By the time the Bonneville Salt Flats became the location for pursuit of land speed records. Drivers raced on a 4. 1-mile course, consisting of a 1. 5–2. 0-mile stretch of beach as one straightaway, the two straights were connected by two tight, deeply rutted and sand covered turns at each end. Stock car racing in the United States has its origins in bootlegging during Prohibition, bootleggers needed to distribute their illicit products, and they typically used small, fast vehicles to better evade the police. Many of the drivers would modify their cars for speed and handling, as well as increased cargo capacity, the cars continued to improve, and by the late 1940s, races featuring these cars were being run for pride and profit. These races were popular entertainment in the rural Southern United States, most races in those days were of modified cars. Street vehicles were lightened and reinforced, mechanic William France Sr. moved to Daytona Beach, Florida, from Washington, D. C. in 1935 to escape the Great Depression. He was familiar with the history of the area from the speed record attempts. France entered the 1936 Daytona event, finishing fifth and he took over running the course in 1938. He promoted a few races before World War II, France had the notion that people would enjoy watching stock cars race
23.
Coca-Cola 600
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The Coca-Cola 600 is an annual 600-mile Monster Energy NASCAR Cup Series points race held at the Charlotte Motor Speedway in Concord, North Carolina during Memorial Day weekend. The event, when first held in 1960, became the first race to be held at the Charlotte Motor Speedway, run since 1960, it is the longest race on NASCARs schedule at 600 miles. It is also unique for the fact that the race changes drastically from start to finish and it starts around 6,20 PM and the track is bathed in sunlight for the first third of the race. The second third happens at dusk, and the third under the lights. Turner thought he could borrow money to build a $750,000 track with 45,000 permanent seats on his property in Cabarrus County. Afterward, he learned that a group led by Bruton Smith had an idea to build a track near Pineville. Smith and Turner formed an alliance to build the track, once the construction crew broke ground, they found a layer of granite under the topsoil, making the construction costly. The area for the first turn alone used $70,000 worth of dynamite, in the spring of 1960, Turner begged for a six-week postponement for the race after a snow storm delayed the pouring on concrete. With two weeks remaining until the race, the paving subcontractor threatened to leave the job site for lack of payment. To solve the problem, Turner and one of his friends threatened the paving subcontractor with a shotgun, the first event at the recently completed Charlotte Motor Speedway was held on June 19,1960. The event was started as an attempt by NASCAR to stage a Memorial Day weekend event to compete with the open-wheel Indianapolis 500 and it was not until 1974, however, that both races competed head-to-head on the same day. In fact, the first World 600 was not held on the Memorial Day weekend, the 2009 race, postponed by rain from its original May 24 date, was the first race to have run on Memorial Day itself. With the installation of lights in 1992, fans complained to management to have the race start later in the day because of the notorious North Carolina heat. They wanted to follow The Winstons popularity the previous week and switch the race to a finish to create cooler temperatures for spectators. The start time was moved several times throughout the 1990s. With the new starting time came new challenges, not only do race teams have to deal with the blistering Carolina heat, but the considerable temperature change at night make track conditions completely different. The nighttime portion of the race is lit with a system that uses parabolic reflectors so that dangerous glare that would otherwise be in the eyes is minimized. Experts disagreed over whether, for health and safety reasons, anyone should be allowed to race 1100 miles in one day, from 2005 to 2010, the issue became moot when the state of Indiana finally decided to go to daylight saving time
24.
Fiat 600
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The Fiat 600 is a city car produced by the Italian manufacturer Fiat from 1955 to 1969. Measuring only 3.22 m long, it was the first rear-engined Fiat, the total number produced from 1955 to 1969 at the Mirafiori plant in Turin was 2,695,197. Codenamed Progetto 100, the Fiat 600 mirrored the layout of the Volkswagen Beetle, a total of 5 prototypes were built between 1952 and 1954, which all differed from one another. Chassis number 000001 with engine number 000002 is believed to be the remaining example. It was powered by an innovative single-cam V2-cylinder engine designed to simplify maintenance, at the official launch in 1955, FIAT engineer, Dante Giacosa declared that the aim had been to create something new, both in the interest of progress and simplification. This prototype, however, did not become the chosen design, the car had hydraulic drum brakes on all four wheels. All 600 models had 3-synchro 4-speed transaxles, all models of the 600 had generators with mechanical external regulators. The top speed ranged from 95 km/h empty with the 633 cc inline-four engine to 110 km/h with the 767 cc version, the car had good ventilation and defrosting systems. A year after its debut, in 1956, a version was introduced. It was a precursor of current multi-purpose vehicles, at the time when the millionth car was produced, the manufacturer reported it was producing the car at the then remarkable rate of 1,000 a day. As of 2011 there are only 65 left in the UK that are road legal, in Spain, the 600 model was made under the make of SEAT, from 1957 to 1973. Up to 797,319 SEAT600 were made, the Spanish car maker exported them to a number of countries worldwide. This car motorised Spain after the Spanish Civil War, the most interesting version produced between 1964 and 1967 by SEAT is the SEAT800, the sole four-door derivative of the 600 model which received a longer wheelbase. It was developed in-house by SEAT and produced exclusively by the Spanish car maker without any equivalent model in Fiats range, the Fiat 600 was also manufactured at Fiat Neckar in Germany between 1956 and 1967. Presented in a first time as Jagst 600, in 1960 with the release of Fiat 600D it became Jagst 770, the model was manufactured until the end of 1967, more than 172,000 copies. It was produced by the Zastava factory in Kragujevac, Serbia, from the early 1960s until 1985, during time it played a major role in motorisation of the country. The 600 was built as the Fiat 600 R by Sevel in Argentina from 1960 to 1982, with operations also taking place in Uruguay by Ayax S. A. At first, Someca S. A. built the 600 with rear-hinged doors, as a new plant was constructed in the Ferreyra, a suburb of Córdoba, the local parts content steadily increased
25.
Latin language
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
26.
Cohort (military unit)
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A cohort was the standard tactical unit of a Roman legion and was composed of 360 soldiers. A Cohort is considered to be the equivalent of a military battalion. The Cohort unit replaced the system following the reforms traditionally attributed to Gaius Marius in 107 BC. Until the middle of the first century AD,10 cohorts made up a Roman Legion, during the 1st century AD, the command structure and make-up of the legions was formally laid down, in a form that would endure for centuries. The first cohort was now made up of five double-strength centuries totalling 800 men and this century was known as the primus pilii, and its centurion was known as the primus pilus. The Primus Pilus could be promoted to Praefectus Castrorum, or Camp Prefect, the Praefectus Castrorum would be in charge of the daily running of a legion. The other cohort consisted of approximately 480 men in six centuriae of 80 men, at various times prior to the reforms, a century might have meant a unit of 60 to 80. The cohort had no permanent commander, it is assumed that in combat, in order of seniority, the six centurions were titled hastatus posterior, hastatus prior, princeps posterior, princeps prior, pilus posterior and pilus prior. The legion at this time numbered about 5,400 men, including officers, engineers, auxiliary cohorts could be quinquagenaria or milliaria. The term was first used to refer to the bodyguard of a general during the Republic, later, cohors togata was a unit of the Praetorian guard in civilian dress tasked with duties within the pomerium. Cohortes urbanae, urban cohort, military police unit patrolling in the capital, cohortes vigilum, watchmen, unit of the police force which also was the fire brigade in the capital. Cohors Germanorum, the unit of Germani custodes corporis, furthermore, the Latin word cohors was used in a looser way to describe a rather large company of people. Auxiliaries List of Roman auxiliary regiments
27.
Phoenix, AZ
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Phoenix is the capital and most populous city of the U. S. state of Arizona. Phoenix is the anchor of the Phoenix metropolitan area, also known as the Valley of the Sun, the metropolitan area is the 12th largest by population in the United States, with approximately 4.3 million people as of 2010. Settled in 1867 as a community near the confluence of the Salt and Gila Rivers. Located in the reaches of the Sonoran Desert, Phoenix has a subtropical desert climate. Despite this, its canal system led to a farming community, many of the original crops remaining important parts of the Phoenix economy for decades, such as alfalfa, cotton, citrus. The city averaged a four percent annual growth rate over a 40-year period from the mid-1960s to the mid-2000s. This growth rate slowed during the Great Recession of 2007–09, and has rebounded slowly, Phoenix is the cultural center of the Valley of the Sun, as well as the entire state. For more than 2,000 years, the Hohokam people occupied the land that would become Phoenix, the Hohokam created roughly 135 miles of irrigation canals, making the desert land arable. Paths of these canals would later used for the modern Arizona Canal, Central Arizona Project Canal. The Hohokam also carried out trade with the nearby Anasazi, Mogollon and Sinagua. It is believed that between 1300 and 1450, periods of drought and severe floods led to the Hohokam civilizations abandonment of the area. After the departure of the Hohokam, groups of Akimel Oodham, Tohono Oodham and Maricopa tribes began to use the area, as well as segments of the Yavapai and Apache. The Oodham were offshoots of the Sobaipuri tribe, who in turn were thought to be the descendants of the formerly urbanized Hohokam and their crops included corn, beans, and squash for food, while cotton and tobacco were also cultivated. Mostly a peaceful group, they did together with the Maricopa for their mutual protection against incursions by both the Yuma and Apache tribes. The Tohono Oodham lived in the region as well, but their concentration was to the south. Living in small settlements, the Oodham were seasonal farmers who took advantage of the rains and they also hunted local game such as deer, rabbit, and javalina for meat. When the Mexican–American War ended in 1848, Mexico ceded its northern zone to the United States, the Phoenix area became part of the New Mexico Territory. In 1863 the mining town of Wickenburg was the first to be established in what is now Maricopa County, at the time Maricopa County had not yet been incorporated, the land was within Yavapai County, which included the major town of Prescott to the north of Wickenburg
28.
Area code 480
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North American area code 480 is a telephone area code in Arizona serving the eastern and northern portions of the Phoenix metropolitan area. It was created on April 1,1999 in a split of area code 602. Generally,480 is coextensive with the East Valley, while most of the West Valley is area code 623, original plans called for the 480 code to be an overlay code for the entire Phoenix metro area. However, overlays were a new concept at the time, and met considerable resistance due to potential geographic ambiguity caused by overlay codes. A three-way geographic split was chosen instead, with 480 being assigned to the East Valley, also in area code 480 are pagers for the Iridium satellite telephone service. As of October 2016, area code 480 is expected to reach exhaustion by the quarter of 2020, however, as of February 2016. NANPA Area Code Map of Arizona