1.
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

2.
Brazil
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Brazil, officially the Federative Republic of Brazil, is the largest country in both South America and Latin America. As the worlds fifth-largest country by area and population, it is the largest country to have Portuguese as an official language. Its Amazon River basin includes a vast tropical forest, home to wildlife, a variety of ecological systems. This unique environmental heritage makes Brazil one of 17 megadiverse countries, Brazil was inhabited by numerous tribal nations prior to the landing in 1500 of explorer Pedro Álvares Cabral, who claimed the area for the Portuguese Empire. Brazil remained a Portuguese colony until 1808, when the capital of the empire was transferred from Lisbon to Rio de Janeiro, in 1815, the colony was elevated to the rank of kingdom upon the formation of the United Kingdom of Portugal, Brazil and the Algarves. Independence was achieved in 1822 with the creation of the Empire of Brazil, a state governed under a constitutional monarchy. The ratification of the first constitution in 1824 led to the formation of a bicameral legislature, the country became a presidential republic in 1889 following a military coup détat. An authoritarian military junta came to power in 1964 and ruled until 1985, Brazils current constitution, formulated in 1988, defines it as a democratic federal republic. The federation is composed of the union of the Federal District, the 26 states, Brazils economy is the worlds ninth-largest by nominal GDP and seventh-largest by GDP as of 2015. A member of the BRICS group, Brazil until 2010 had one of the worlds fastest growing economies, with its economic reforms giving the country new international recognition. Brazils national development bank plays an important role for the economic growth. Brazil is a member of the United Nations, the G20, BRICS, Unasul, Mercosul, Organization of American States, Organization of Ibero-American States, CPLP. Brazil is a power in Latin America and a middle power in international affairs. One of the worlds major breadbaskets, Brazil has been the largest producer of coffee for the last 150 years and it is likely that the word Brazil comes from the Portuguese word for brazilwood, a tree that once grew plentifully along the Brazilian coast. In Portuguese, brazilwood is called pau-brasil, with the word brasil commonly given the etymology red like an ember, formed from Latin brasa and the suffix -il. As brazilwood produces a red dye, it was highly valued by the European cloth industry and was the earliest commercially exploited product from Brazil. The popular appellation eclipsed and eventually supplanted the official Portuguese name, early sailors sometimes also called it the Land of Parrots. In the Guarani language, a language of Paraguay, Brazil is called Pindorama

3.
Focke-Wulf Fw 190
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The Focke-Wulf Fw 190 Würger is a German single-seat, single-engine fighter aircraft designed by Kurt Tank in the late 1930s and widely used during World War II. Along with its counterpart, the Messerschmitt Bf 109, the Fw 190 became the backbone of the Luftwaffes Jagdwaffe. The Fw 190A started flying operationally over France in August 1941, and quickly proved superior in all but turn radius to the Royal Air Forces main front-line fighter, V, especially at low and medium altitudes. The 190 maintained superiority over Allied fighters until the introduction of the improved Spitfire Mk, in the opinion of German pilots who flew both the Bf 109 and the Fw 190, the latter provided increased firepower and, at low to medium altitude, manoeuvrability. The Fw 190A series performance decreased at high altitudes, which reduced its effectiveness as a high-altitude interceptor, problems with the turbocharger installations on the -B and -C subtypes meant only the D model would see service, entering service in September 1944. While these long nose versions gave them parity with Allied opponents, the Fw 190 was well-liked by its pilots. Some of the Luftwaffes most successful fighter aces claimed a great many of their kills while flying it, including Otto Kittel, Walter Nowotny, between 1934 and 1935 the German Ministry of Aviation ran a contest to produce a modern fighter for the rearming Luftwaffe. Kurt Tank entered the parasol-winged Fw 159 into the contest, against the Arado Ar 80, Heinkel He 112, the Fw 159 was hopelessly outclassed, and was soon eliminated from the competition along with the Ar 80. The He 112 and Bf 109 were generally similar in design, on 12 March 1936 the 109 was declared the winner. Kurt Tank responded with a number of designs, most based around a liquid-cooled inline engine, however, it was not until a design was presented using the air-cooled, 14-cylinder BMW139 radial engine that the Ministry of Aviations interest was aroused. As this design used an engine, it would not compete with the inline-powered Bf 109 for engines. This was not the case for competing designs like the Heinkel He 100 or twin-engined Focke-Wulf Fw 187, after the war, Tank denied a rumour that he had to fight a battle with the Ministry to convince them of the radial engines merits. Tank was not convinced of this, having witnessed the use of radial engines by the U. S. Navy. The hottest points on any air-cooled engine are the cylinder heads, in order to provide sufficient air to cool the engine, airflow had to be maximized at this outer edge. This was normally accomplished by leaving the majority of the front face of the open to the air. During the late 1920s, NACA led development of an improvement by placing an airfoil-shaped ring around the outside of the cylinder heads. The shaping accelerated the air as it entered the front of the cowl, increasing the total airflow, Tank introduced a further refinement to this basic concept. He suggested placing most of the components on the propeller

4.
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

5.
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

6.
Emergency telephone number
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In many countries the public switched telephone network has a single emergency telephone number that allows a caller to contact local emergency services for assistance. The emergency number differs from country to country, it is typically a number so that it can be easily remembered and dialed quickly. Some countries have a different emergency number for each of the different emergency services, see List of emergency telephone numbers. The emergency telephone number is a case in the countrys telephone number plan. In the past, calls to the telephone number were often routed over special dedicated circuits. Though with the advent of electronic exchanges these calls are now mixed with ordinary telephone traffic. Often the system is set up so that once a call is made to a telephone number. Should the caller abandon the call, the line may still be held until the emergency service answers, an emergency telephone number call may be answered by either a telephone operator or an emergency service dispatcher. The nature of the emergency is then determined, if the call has been answered by a telephone operator, they then connect the call to the appropriate emergency service, who then dispatches the appropriate help. In the case of services being needed on a call. Emergency dispatchers are trained to control the call in order to help in an appropriate manner. The emergency dispatcher may find it necessary to give urgent advice in life-threatening situations, some dispatchers have special training in telling people how to perform first aid or CPR. In many parts of the world, a service can identify the telephone number that a call has been placed from. This is normally done using the system that the company uses to bill calls. For an individual fixed landline telephone, the number can often be associated with the callers address. However, with phones and business telephones, the address may be a mailing address rather than the callers location. The latest enhanced systems, such as Enhanced 911, are able to provide the location of mobile telephones. This is often specifically mandated in a countrys legislation, when an emergency happened in the pre-dial telephone era, the user simply picked up the telephone receiver and waited for the operator to answer number, please

7.
Goznak
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It incorporates 7 factories and 1 R&D institute involved in different stages of the development, research, manufacturing cycle. Goznak also controls mints, which manufacture circulation coins, orders, decorations and it also manufactures credit cards, banking cards, phone cards. Goznak not only prints Russian money, but also prints banknotes of foreign countries, including Lebanon, Yemen, Guatemala, Rwanda, Angola and others. In 1838, a Russian academician Moritz von Jacobi, employed at this Department, in the 1890s, an employee of the Department Ivan Orlov invented and developed a new printing method called Orlovs printing. Also, he built multicolor printing presses, which would serve as a prototype for modern multicolor printing presses, Orlovs machines were still in use in some countries in the 1970s. After the October Revolution of 1917, the Department of State Currency Production was reorganized and renamed Goznak. Although the initial scope of Goznak had been the production of notes, the production of coins was added to its field of operation in 1941. Goznak had its own All-union Research Institute in Moscow, in the 1920s, a Goznak employee and a prominent Soviet sculptor Ivan Shadr created the first samples of the Soviet money. Thus, all of the paper and printing factories and mints of the Goznak were equipped with the counting machines, after the revolution of 1917 the Saint-Petersburg Mint of Goznak was renamed into the Leningrad Mint. Its original name was returned in the 1990s, the Saint-Petersburg Paper Mill of Goznak was also called the Leningrad Paper Mill during the Soviet period. In 1997, Perm Printing Factory launched telecards for public telephones, during the first year of the new site more than one million cards were produced. The quality of cards is fully in line with the international standard, in 1999, the Moscow Mint, for the first time in the history of Russia won the tender for the manufacture of currency for India. In October 1999, they signed an agreement with India for the manufacture of copper-nickel coins two and five rupees, in 2006 the Association of the State Companies Goznak has been transformed into the Federal State Unitary Enterprise Goznak. Goznak includes 8 branches,2 Printing Factories,2 Paper Mills,2 Mints, Printing House, at the beginning of 2006, the Moscow Printing Factory of Goznak started with the personalization of the first ePassport. The personalization center is capable of processing of more than 5 million passports per year, on March 3,2008 Goznak has become a strategic enterprise of Russia

8.
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

9.
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

10.
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

11.
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

12.
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