1.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
2.
Latin
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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
3.
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
4.
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
5.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
6.
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
7.
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
8.
1 (number)
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1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few
9.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set
10.
Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime
11.
Magic square
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In recreational mathematics, a magic square is a n × n square grid filled with distinct positive integers in the range 1,2. N2 such that each contains a different integer and the sum of the integers in each row, column. The sum is called the constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n. In regard to magic sum, the problem of magic squares only requires the sum of row, column and diagonal to be equal. Thus, although magic squares may contain negative integers, they are just variations by adding or multiplying a number to every positive integer in the original square. Magic squares are called normal magic squares, in the sense that there are non-normal magic squares which integers are not restricted in 1,2. However, in places, magic squares is used as a general term to cover both the normal and non-normal ones, especially when non-normal ones are under discussion. Moreover, the term magic squares is also used to refer to various types of word squares. Magic squares have a history, dating back to at least 650 BC in China. At various times they have acquired magical or mythical significance, and have appeared as symbols in works of art, the constant that is the sum of every row, column and diagonal is called the magic constant or magic sum, M. Every normal magic square has a constant dependent on the n, calculated by the formula M = n /2. N2 is n 2 /2 which when divided by the n is the magic constant. For normal magic squares of orders n =3,4,5,6,7, and 8, the constants are, respectively,15,34,65,111,175. Normal magic squares of all sizes can be constructed except 2×2, any magic square can be rotated and reflected to produce 8 trivially distinct squares. In magic square theory, all of these are deemed equivalent. Excluding rotations and reflections, there is exactly one 3×3 magic square, exactly 880 4×4 magic squares, for the 6×6 case, there are estimated to be approximately 1.8 ×1019 squares. Then all magic squares of an order have the same moment of inertia as each other
12.
Riemann hypothesis
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In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. It was proposed by Bernhard Riemann, after whom it is named, the name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers, along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics. The Riemann zeta function ζ is a function whose argument s may be any complex number other than 1 and it has zeros at the negative even integers, that is, ζ =0 when s is one of −2, −4, −6. These are called its trivial zeros, However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros, the Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that, The real part of every non-trivial zero of the Riemann zeta function is 1/2. Thus, if the hypothesis is correct, all the non-trivial zeros lie on the line consisting of the complex numbers 1/2 + i t. There are several books on the Riemann hypothesis, such as Derbyshire, Rockmore. The books Edwards, Patterson, Borwein et al. and Mazur & Stein give mathematical introductions, while Titchmarsh, Ivić, furthermore, the book Open Problems in Mathematics, edited by John Forbes Nash Jr. and Michael Th. Rassias, features an essay on the Riemann hypothesis by Alain Connes. The convergence of the Euler product shows that ζ has no zeros in this region, the Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to continue the function to give it a definition that is valid for all complex s. This can be done by expressing it in terms of the Dirichlet eta function as follows. If the real part of s is greater than one, then the function satisfies ζ = ∑ n =1 ∞ n +1 n s =11 s −12 s +13 s − ⋯. However, the series on the right converges not just when the part of s is greater than one. Thus, this alternative series extends the function from Re >1 to the larger domain Re >0. The zeta function can be extended to these values, as well, by taking limits, in the strip 0 < Re <1 the zeta function satisfies the functional equation ζ =2 s π s −1 sin Γ ζ. If s is an even integer then ζ =0 because the factor sin vanishes
13.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
14.
Riemann zeta function
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More general representations of ζ for all s are given below. The Riemann zeta function plays a role in analytic number theory and has applications in physics, probability theory. As a function of a variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis. The values of the Riemann zeta function at even positive integers were computed by Euler, the first of them, ζ, provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ, the values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, the Riemann zeta function ζ is a function of a complex variable s = σ + it. It can also be defined by the integral ζ =1 Γ ∫0 ∞ x s −1 e x −1 d x where Γ is the gamma function. The Riemann zeta function is defined as the continuation of the function defined for σ >1 by the sum of the preceding series. Leonhard Euler considered the series in 1740 for positive integer values of s. The above series is a prototypical Dirichlet series that converges absolutely to a function for s such that σ >1. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠1, for s =1 the series is the harmonic series which diverges to +∞, and lim s →1 ζ =1. Thus the Riemann zeta function is a function on the whole complex s-plane. For any positive even integer 2n, ζ = n +1 B2 n 2 n 2, where B2n is the 2nth Bernoulli number. For odd positive integers, no simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers. For nonpositive integers, one has ζ = B n +1 n +1 for n ≥0 In particular, ζ = −12, Similarly to the above, this assigns a finite result to the series 1 +1 +1 +1 + ⋯. ζ ≈ −1.4603545 This is employed in calculating of kinetic boundary layer problems of linear kinetic equations, ζ =1 +12 +13 + ⋯ = ∞, if we approach from numbers larger than 1. Then this is the harmonic series, but its Cauchy principal value lim ε →0 ζ + ζ2 exists which is the Euler–Mascheroni constant γ =0. 5772…. ζ ≈2.612, This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems
15.
Proof that 0.999... equals 1
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In mathematics, the repeating decimal 0. 999… denotes a real number that can be shown to be the number one. In other words, the symbols 0. 999… and 1 represent the same number, more generally, every nonzero terminating decimal has an equal twin representation with infinitely many trailing 9s. The terminating decimal representation is preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases or in any representation of the real numbers. The equality of 0. 999… and 1 is closely related to the absence of nonzero infinitesimals in the number system. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals, the equality 0. 999… =1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of studies in mathematics education. Algebraic proofs showing that 0. 999… represents the number 1 use concepts such as fractions, long division, however, these proofs are not rigorous as they do not include a careful analytic definition of 0. 999…. One reason that infinite decimals are an extension of finite decimals is to represent fractions. Using long division, a division of integers like 1⁄9 becomes a recurring decimal,0. 111…. This decimal yields a quick proof for 0. 999… =1, If 0. 999… is to be consistent, it must equal 9⁄9 =1. 0.333 … =390.888 … =890.999 … =99 =1 When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 ×0. 999… equals 9. 999…, which is 9 greater than the original number, in introductory arithmetic, such proofs help explain why 0. 999… =1 but 0. 333… <0.34. In introductory algebra, the proofs help explain why the method of converting between fractions and repeating decimals works. Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can demonstrate that the decimals 0. 999… and 1. 000… both represent the same real number, it is built into the definition. Since the question of 0. 999… does not affect the development of mathematics
16.
Typewriter
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A typewriter is a mechanical or electromechanical machine for writing characters similar to those produced by printers movable type. A typewriter operates by means of keys that strike a ribbon to transmit ink or carbon impressions onto paper, typically, a single character is printed on each key press. The machine prints characters by making ink impressions of type elements similar to the used in movable type letterpress printing. At the end of the century, the term typewriter was also applied to a person who used a typing machine. After its invention in the 1860s, the quickly became an indispensable tool for practically all writing other than personal handwritten correspondence. It was widely used by writers, in offices. As with the automobile, telephone, and telegraph, a number of people contributed insights, historians have estimated that some form of typewriter was invented 52 times as thinkers tried to come up with a workable design. Some of the early typing instruments, In 1575 an Italian printmaker, Francesco Rampazzetto, invented the scrittura tattile, in 1714, Henry Mill obtained a patent in Britain for a machine that, from the patent, appears to have been similar to a typewriter. In 1802 Italian Agostino Fantoni developed a particular typewriter to enable his blind sister to write, in 1808 Italian Pellegrino Turri invented a typewriter. He also invented carbon paper to provide the ink for his machine, in 1823 Italian Pietro Conti di Cilavegna invented a new model of typewriter, the tachigrafo, also known as tachitipo. In 1829, William Austin Burt patented a machine called the Typographer which, the Science Museum describes it merely as the first writing mechanism whose invention was documented, but even that claim may be excessive, since Turris invention pre-dates it. Even in the hands of its inventor, this machine was slower than handwriting, Burt and his promoter John D. Sheldon never found a buyer for the patent, so the invention was never commercially produced, because the typographer used a dial, rather than keys, to select each character, it was called an index typewriter rather than a keyboard typewriter. Index typewriters of that era resemble the squeeze-style embosser from the 1960s more than they resemble the modern keyboard typewriter, by the mid-19th century, the increasing pace of business communication had created a need for mechanization of the writing process. Stenographers and telegraphers could take down information at rates up to 130 words per minute, from 1829 to 1870, many printing or typing machines were patented by inventors in Europe and America, but none went into commercial production. Charles Thurber developed multiple patents, of which his first in 1843 was developed as an aid to the blind, in 1855, the Italian Giuseppe Ravizza created a prototype typewriter called Cembalo scrivano o macchina da scrivere a tasti. It was a machine that let the user see the writing as it was typed. In 1861, Father Francisco João de Azevedo, a Brazilian priest, made his own typewriter with basic materials and tools, such as wood, in that same year the Brazilian emperor D
17.
Fractions
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
18.
Code page 437
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Code page 437 is the character set of the original IBM PC, or DOS. It is also known as CP437, OEM-US, OEM437, PC-8, the set includes ASCII codes 32–126, extended codes for accented letters, some Greek letters, icons, and line-drawing symbols. It is sometimes referred to as the OEM font or high ASCII and this character set remains the primary font in the core of any EGA and VGA-compatible graphics card. Text shown when a PC reboots, before any other font can be loaded from a storage medium, many file formats developed at the time of the IBM PC are based on code page 437 as well. The IBM Enhanced Graphics Adapter contained an 8×14 pixels-per-character version, all these display adapters have text modes in which each character cell contains an 8-bit character code point, giving 256 possible values for graphic characters. All 256 codes were assigned a character in ROM, including the codes from 0 to 31 that were reserved in ASCII for non-graphical control characters. Various Eastern European PCs used different character sets, sometimes user-selectable via jumpers or CMOS setup. These sets were designed to match 437 as much as possible, for sharing the code points for many of the line-drawing characters. A legacy of code page 437 and other DOS codepages is the set of number combinations used in Windows Alt keycodes introduced in the first versions of DOS, the user could enter a character by holding down the Alt key and entering the three-digit decimal Alt keycode on the numpad. The following tables show code page 437, each character is shown with its equivalent Unicode code point and its decimal code point. See also the notes below as there are multiple equivalent Unicode characters for some code points, the decimal codes are also known as Alt codes. Although the ROM provides a graphic for all 256 different possible 8-bit codes, some APIs will not print some code points, in particular the range 1-31, instead they will interpret them as control characters. For instance many methods of outputting text on the original IBM PC would interpret the codes for BEL, BS, CR, many printers were also unable to print these characters. Legend, When translating to Unicode it should be noted that some codes do not have a unique, single Unicode equivalent and we were also fascinated by dedicated word processors from Wang, because we believed that general-purpose machines could do that just as well. Thats why, when it time to design the keyboard for the IBM PC, we put the funny Wang character set into the machine—you know, smiley faces and boxes and triangles. We were thinking wed like to do a clone of Wang word-processing software someday, according to an interview with David J. Bradley the characters were decided upon during a four-hour meeting on a plane trip from Seattle to Atlanta by Andy Saenz, Lew Eggebrecht and himself. The selection of characters has some internal logic, Table rows 0 and 1. The isolated character 127 also belongs to this group, Table rows 2 to 7, codes 32 to 126, are the standard ASCII printable characters
19.
Unicode
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Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the worlds writing systems. As of June 2016, the most recent version is Unicode 9.0, the standard is maintained by the Unicode Consortium. Unicodes success at unifying character sets has led to its widespread, the standard has been implemented in many recent technologies, including modern operating systems, XML, Java, and the. NET Framework. Unicode can be implemented by different character encodings, the most commonly used encodings are UTF-8, UTF-16 and the now-obsolete UCS-2. UTF-8 uses one byte for any ASCII character, all of which have the same values in both UTF-8 and ASCII encoding, and up to four bytes for other characters. UCS-2 uses a 16-bit code unit for each character but cannot encode every character in the current Unicode standard, UTF-16 extends UCS-2, using one 16-bit unit for the characters that were representable in UCS-2 and two 16-bit units to handle each of the additional characters. Many traditional character encodings share a common problem in that they allow bilingual computer processing, Unicode, in intent, encodes the underlying characters—graphemes and grapheme-like units—rather than the variant glyphs for such characters. In the case of Chinese characters, this leads to controversies over distinguishing the underlying character from its variant glyphs. In text processing, Unicode takes the role of providing a unique code point—a number, in other words, Unicode represents a character in an abstract way and leaves the visual rendering to other software, such as a web browser or word processor. This simple aim becomes complicated, however, because of concessions made by Unicodes designers in the hope of encouraging a more rapid adoption of Unicode, the first 256 code points were made identical to the content of ISO-8859-1 so as to make it trivial to convert existing western text. For other examples, see duplicate characters in Unicode and he explained that he name Unicode is intended to suggest a unique, unified, universal encoding. In this document, entitled Unicode 88, Becker outlined a 16-bit character model, Unicode could be roughly described as wide-body ASCII that has been stretched to 16 bits to encompass the characters of all the worlds living languages. In a properly engineered design,16 bits per character are more than sufficient for this purpose, Unicode aims in the first instance at the characters published in modern text, whose number is undoubtedly far below 214 =16,384. By the end of 1990, most of the work on mapping existing character encoding standards had been completed, the Unicode Consortium was incorporated in California on January 3,1991, and in October 1991, the first volume of the Unicode standard was published. The second volume, covering Han ideographs, was published in June 1992, in 1996, a surrogate character mechanism was implemented in Unicode 2.0, so that Unicode was no longer restricted to 16 bits. The Microsoft TrueType specification version 1.0 from 1992 used the name Apple Unicode instead of Unicode for the Platform ID in the naming table, Unicode defines a codespace of 1,114,112 code points in the range 0hex to 10FFFFhex. Normally a Unicode code point is referred to by writing U+ followed by its hexadecimal number, for code points in the Basic Multilingual Plane, four digits are used, for code points outside the BMP, five or six digits are used, as required. Code points in Planes 1 through 16 are accessed as surrogate pairs in UTF-16, within each plane, characters are allocated within named blocks of related characters
20.
HTML
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Hypertext Markup Language is the standard markup language for creating web pages and web applications. With Cascading Style Sheets and JavaScript it forms a triad of cornerstone technologies for the World Wide Web, Web browsers receive HTML documents from a webserver or from local storage and render them into multimedia web pages. HTML describes the structure of a web page semantically and originally included cues for the appearance of the document, HTML elements are the building blocks of HTML pages. With HTML constructs, images and other objects, such as interactive forms and it provides a means to create structured documents by denoting structural semantics for text such as headings, paragraphs, lists, links, quotes and other items. HTML elements are delineated by tags, written using angle brackets, tags such as <img /> and <input /> introduce content into the page directly. Include explicit close tags for elements that permit content but are left empty, by carefully following the W3Cs compatibility guidelines, a user agent should be able to interpret the document equally as HTML or XHTML. For documents that are XHTML1.0 and have made compatible in this way. When delivered as XHTML, browsers should use an XML parser, HTML4 defined three different versions of the language, Strict, Transitional and Frameset. The Transitional and Frameset versions allow for presentational markup, which is omitted in the Strict version, instead, cascading style sheets are encouraged to improve the presentation of HTML documents. Because XHTML1 only defines an XML syntax for the language defined by HTML4, as this list demonstrates, the loose versions of the specification are maintained for legacy support. However, contrary to popular misconceptions, the move to XHTML does not imply a removal of this legacy support, rather the X in XML stands for extensible and the W3C is modularizing the entire specification and opening it up to independent extensions. The primary achievement in the move from XHTML1.0 to XHTML1.1 is the modularization of the entire specification, the strict version of HTML is deployed in XHTML1.1 through a set of modular extensions to the base XHTML1.1 specification. Likewise, someone looking for the loose or frameset specifications will find similar extended XHTML1.1 support, the modularization also allows for separate features to develop on their own timetable. So for example, XHTML1.1 will allow quicker migration to emerging XML standards such as MathML, in summary, the HTML4 specification primarily reined in all the various HTML implementations into a single clearly written specification based on SGML. XHTML1.0, ported this specification, as is, next, XHTML1.1 takes advantage of the extensible nature of XML and modularizes the whole specification. XHTML2.0 was intended to be the first step in adding new features to the specification in a standards-body-based approach. The WHATWG considers their work as living standard HTML for what constitutes the state of the art in major browser implementations by Apple, Google, Mozilla, Opera, hTML5 is specified by the HTML Working Group of the W3C following the W3C process. HTML lacks some of the found in earlier hypertext systems, such as source tracking, fat links
21.
Language
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Language is the ability to acquire and use complex systems of communication, particularly the human ability to do so, and a language is any specific example of such a system. The scientific study of language is called linguistics, questions concerning the philosophy of language, such as whether words can represent experience, have been debated since Gorgias and Plato in Ancient Greece. Thinkers such as Rousseau have argued that language originated from emotions while others like Kant have held that it originated from rational and logical thought, 20th-century philosophers such as Wittgenstein argued that philosophy is really the study of language. Major figures in linguistics include Ferdinand de Saussure and Noam Chomsky, estimates of the number of languages in the world vary between 5,000 and 7,000. However, any precise estimate depends on an arbitrary distinction between languages and dialects. Natural languages are spoken or signed, but any language can be encoded into secondary media using auditory, visual, or tactile stimuli – for example, in whistling, signed and this is because human language is modality-independent. All languages rely on the process of semiosis to relate signs to particular meanings, human language has the properties of productivity and displacement, and relies entirely on social convention and learning. Its complex structure affords a wider range of expressions than any known system of animal communication. Language is processed in different locations in the human brain. Humans acquire language through interaction in early childhood, and children generally speak fluently when they are approximately three years old. The use of language is deeply entrenched in human culture, a group of languages that descend from a common ancestor is known as a language family. The languages of the Dravidian family that are mostly in Southern India include Tamil. Academic consensus holds that between 50% and 90% of languages spoken at the beginning of the 21st century will probably have become extinct by the year 2100. The English word language derives ultimately from Proto-Indo-European *dn̥ǵʰwéh₂s tongue, speech, language through Latin lingua, language, tongue, and Old French language. The word is used to refer to codes, ciphers. Unlike conventional human languages, a language in this sense is a system of signs for encoding and decoding information. This article specifically concerns the properties of human language as it is studied in the discipline of linguistics. As an object of study, language has two primary meanings, an abstract concept, and a specific linguistic system, e. g. French
22.
English language
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English /ˈɪŋɡlɪʃ/ is a West Germanic language that was first spoken in early medieval England and is now the global lingua franca. Named after the Angles, one of the Germanic tribes that migrated to England, English is either the official language or one of the official languages in almost 60 sovereign states. It is the third most common language in the world, after Mandarin. It is the most widely learned second language and a language of the United Nations, of the European Union. It is the most widely spoken Germanic language, accounting for at least 70% of speakers of this Indo-European branch, English has developed over the course of more than 1,400 years. The earliest forms of English, a set of Anglo-Frisian dialects brought to Great Britain by Anglo-Saxon settlers in the century, are called Old English. Middle English began in the late 11th century with the Norman conquest of England, Early Modern English began in the late 15th century with the introduction of the printing press to London and the King James Bible, and the start of the Great Vowel Shift. Through the worldwide influence of the British Empire, modern English spread around the world from the 17th to mid-20th centuries, English is an Indo-European language, and belongs to the West Germanic group of the Germanic languages. Most closely related to English are the Frisian languages, and English, Old Saxon and its descendent Low German languages are also closely related, and sometimes Low German, English, and Frisian are grouped together as the Ingvaeonic or North Sea Germanic languages. Modern English descends from Middle English, which in turn descends from Old English, particular dialects of Old and Middle English also developed into a number of other English languages, including Scots and the extinct Fingallian and Forth and Bargy dialects of Ireland. English is classified as a Germanic language because it shares new language features with other Germanic languages such as Dutch, German and these shared innovations show that the languages have descended from a single common ancestor, which linguists call Proto-Germanic. Through Grimms law, the word for foot begins with /f/ in Germanic languages, English is classified as an Anglo-Frisian language because Frisian and English share other features, such as the palatalisation of consonants that were velar consonants in Proto-Germanic. The earliest form of English is called Old English or Anglo-Saxon, in the fifth century, the Anglo-Saxons settled Britain and the Romans withdrew from Britain. England and English are named after the Angles, Old English was divided into four dialects, the Anglian dialects, Mercian and Northumbrian, and the Saxon dialects, Kentish and West Saxon. Through the educational reforms of King Alfred in the century and the influence of the kingdom of Wessex. The epic poem Beowulf is written in West Saxon, and the earliest English poem, Modern English developed mainly from Mercian, but the Scots language developed from Northumbrian. A few short inscriptions from the period of Old English were written using a runic script. By the sixth century, a Latin alphabet was adopted, written with half-uncial letterforms and it included the runic letters wynn ⟨ƿ⟩ and thorn ⟨þ⟩, and the modified Latin letters eth ⟨ð⟩, and ash ⟨æ⟩