1.
6 (number)
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6 is the natural number following 5 and preceding 7. The SI prefix for 10006 is exa-, and for its reciprocal atto-,6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number, its proper divisors are 1,2 and 3, since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and S -perfect number. As a perfect number,6 is related to the Mersenne prime 3,6 is the only even perfect number that is not the sum of successive odd cubes. As a perfect number,6 is the root of the 6-aliquot tree, and is itself the sum of only one number. Six is the number that is both the sum and the product of three consecutive positive numbers. Unrelated to 6 being a number, a Golomb ruler of length 6 is a perfect ruler. Six is the first discrete biprime and the first member of the discrete biprime family, Six is the smallest natural number that can be written as the sum of two positive rational cubes which are not integers,6 =3 +3. Six is a perfect number, a harmonic divisor number and a superior highly composite number. The next superior highly composite number is 12,5 and 6 form a Ruth-Aaron pair under either definition. There are no Graeco-Latin squares with order 6, if n is a natural number that is not 2 or 6, then there is a Graeco-Latin square with order n. The smallest non-abelian group is the symmetric group S3 which has 3, s6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of mathematical objects such as the S Steiner system, the projective plane of order 4. This can also be expressed category theoretically, consider the category whose objects are the n element sets and this category has a non-trivial functor to itself only for n =6. 6 similar coins can be arranged around a central coin of the radius so that each coin makes contact with the central one. This makes 6 the answer to the kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the lattice in which each circle touches just six others. 6 is the largest of the four all-Harshad numbers, a six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane
2.
8 (number)
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8 is the natural number following 7 and preceding 9. 8 is, a number, its proper divisors being 1,2. It is twice 4 or four times 2, a power of two, being 23, and is the first number of the form p3, p being an integer greater than 1. The first number which is neither prime nor semiprime, the base of the octal number system, which is mostly used with computers. In octal, one digit represents 3 bits, in modern computers, a byte is a grouping of eight bits, also called an octet. A Fibonacci number, being 3 plus 5, the next Fibonacci number is 13. 8 is the only positive Fibonacci number, aside from 1, the order of the smallest non-abelian group all of whose subgroups are normal. The dimension of the octonions and is the highest possible dimension of a division algebra. The first number to be the sum of two numbers other than itself, the discrete biprime 10, and the square number 49. It has a sum of 7 in the 4 member aliquot sequence being the first member of 7-aliquot tree. All powers of 2, have a sum of one less than themselves. A number is divisible by 8 if its last 3 digits,8 and 9 form a Ruth–Aaron pair under the second definition in which repeated prime factors are counted as often as they occur. There are a total of eight convex deltahedra, a polygon with eight sides is an octagon. Figurate numbers representing octagons are called octagonal numbers, a polyhedron with eight faces is an octahedron. A cuboctahedron has as faces six equal squares and eight regular triangles. Sphenic numbers always have exactly eight divisors, the number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example, if O is the limit of the inclusions of real orthogonal groups O ↪ O ↪ … ↪ O ↪ …. Clifford algebras also display a periodicity of 8, for example, the algebra Cl is isomorphic to the algebra of 16 by 16 matrices with entries in Cl
3.
-1 (number)
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In mathematics, −1 is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element,0. It is the negative integer greater than two and less than 0. Negative one bears relation to Eulers identity since eπi = −1, in software development, −1 is a common initial value for integers and is also used to show that a variable contains no useful information. Negative one has some similar but slightly different properties to positive one, multiplying a number by −1 is equivalent to changing the sign on the number. The square of −1, i. e. −1 multiplied by −1, as a consequence, a product of two negative real numbers is positive. For an algebraic proof of this result, start with the equation 0 = −1 ⋅0 = −1 ⋅ The first equality follows from the above result. The second follows from the definition of −1 as additive inverse of 1, it is precisely that number that when added to 1 gives 0. Now, using the law, we see that 0 = −1 ⋅ = −1 ⋅1 + ⋅ = −1 + ⋅ The second equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies ⋅ =1 The above arguments hold in any ring, the complex number i satisfies x2 = −1, and as such can be considered as a square root of −1. The only other complex number x satisfying the equation x2 = −1 is −i, in the algebra of quaternions, containing the complex plane, the equation x2 = −1 has an infinity of solutions. Exponentiation of a real number can be extended to negative integers. We make the definition that x−1 = 1/x, meaning that we define raising a number to the power −1 to have the effect as taking its reciprocal. This definition then extended to negative integers preserves the exponential law xaxb = x for real numbers a and b, for example, f−1 is the inverse of f, or sin−1 is a notation of arcsine function. The Inductive dimension of the empty set is defined to be −1, most computer systems represent negative integers using twos complement. In such systems, −1 is represented using a bit pattern of all ones, for example, an 8-bit signed integer using twos complement would represent −1 as the bitstring 11111111, or FF in hexadecimal. If interpreted as an integer, the same bitstring of n ones represents 2n −1. For example, the 8-bit string 11111111 above represents 28 −1 =255
4.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone
5.
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
6.
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
7.
3 (number)
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3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
8.
4 (number)
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4 is a number, numeral, and glyph. It is the number following 3 and preceding 5. Four is the only cardinal numeral in the English language that has the number of letters as its number value. Four is the smallest composite number, its divisors being 1 and 2. Four is also a composite number. The next highly composite number is 6, Four is the second square number, the second centered triangular number. 4 is the smallest squared prime and the even number in this form. It has a sum of 3 which is itself prime. The aliquot sequence of 4 has 4 members and is accordingly the first member of the 3-aliquot tree, a number is a multiple of 4 if its last two digits are a multiple of 4. For example,1092 is a multiple of 4 because 92 =4 ×23, only one number has an aliquot sum of 4 and that is squared prime 9. Four is the smallest composite number that is equal to the sum of its prime factors, however, it is the only composite number n for which. It is also a Motzkin number, in bases 6 and 12,4 is a 1-automorphic number. In addition,2 +2 =2 ×2 =22 =4, continuing the pattern in Knuths up-arrow notation,2 ↑↑2 =2 ↑↑↑2 =4, and so on, for any number of up arrows. A four-sided plane figure is a quadrilateral which include kites, rhombi, a circle divided by 4 makes right angles and four quadrants. Because of it, four is the number of plane. Four cardinal directions, four seasons, duodecimal system, and vigesimal system are based on four, a solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces of a polyhedron. The regular tetrahedron is the simplest Platonic solid, a tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only regular polyhedron
9.
5 (number)
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5 is a number, numeral, and glyph. It is the number following 4 and preceding 6. Five is the prime number. Because it can be written as 221 +1, five is classified as a Fermat prime, therefore a regular polygon with 5 sides is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also the number that is part of more than one pair of twin primes. Five is conjectured to be the only odd number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being 2 plus 3,5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation. Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers,5 is the length of the hypotenuse of the smallest integer-sided right triangle. In bases 10 and 20,5 is a 1-automorphic number,5 and 6 form a Ruth–Aaron pair under either definition. There are five solutions to Známs problem of length 6 and this is related to the fact that the symmetric group Sn is a solvable group for n ≤4 and not solvable for n ≥5. While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar, K5, Five is also the number of Platonic solids. A polygon with five sides is a pentagon, figurate numbers representing pentagons are called pentagonal numbers. Five is also a square pyramidal number, Five is the only prime number to end in the digit 5, because all other numbers written with a 5 in the ones-place under the decimal system are multiples of five. As a consequence of this,5 is in base 10 a 1-automorphic number, vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions, unlike expansions with all other prime denominators, because they are prime factors of ten, the base. When written in the system, all multiples of 5 will end in either 5 or 0
10.
7 (number)
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7 is the natural number following 6 and preceding 8. Seven, the prime number, is not only a Mersenne prime. It is also a Newman–Shanks–Williams prime, a Woodall prime, a prime, a lucky prime, a happy number, a safe prime. Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers, Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree. N =7 is the first natural number for which the statement does not hold, Two nilpotent endomorphisms from Cn with the same minimal polynomial. 7 is the only number D for which the equation 2n − D = x2 has more than two solutions for n and x natural, in particular, the equation 2n −7 = x2 is known as the Ramanujan–Nagell equation. 7 is the dimension, besides the familiar 3, in which a vector cross product can be defined. 7 is the lowest dimension of an exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere. 999,999 divided by 7 is exactly 142,857, for example, 1/7 =0.142857142857. and 2/7 =0.285714285714. In fact, if one sorts the digits in the number 142857 in ascending order,124578, the remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example,628 ÷7 =89 5/7, here 5 is the remainder, so in this case,628 ÷7 =89.714285. Another example,5238 ÷7 =748 2/7, hence the remainder is 2, in this case,5238 ÷7 =748.285714. A seven-sided shape is a heptagon, the regular n-gons for n ≤6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons are called heptagonal numbers, Seven is also a centered hexagonal number. Seven is the first integer reciprocal with infinitely repeating sexagesimal representation, There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers. There are seven types of catastrophes. When rolling two standard six-sided dice, seven has a 6 in 36 probability of being rolled, the greatest of any number, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved, in quaternary,7 is the smallest prime with a composite sum of digits
11.
9 (number)
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9 is the natural number following 8 and preceding 10. In the NATO phonetic alphabet, the digit 9 is called Niner, five-digit produce PLU codes that begin with 9 are organic. Common terminal digit in psychological pricing, Nine is a number that appears often in Indian Culture and mythology. Nine influencers are attested in Indian astrology, in the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements, Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. Navaratri is a festival dedicated to the nine forms of Durga. Navaratna, meaning 9 jewels may also refer to Navaratnas - accomplished courtiers, Navratan - a kind of dish, according to Yoga, the human body has nine doors - two eyes, two ears, the mouth, two nostrils, and the openings for defecation and procreation. In Indian aesthetics, there are nine kinds of Rasa, Nine is considered a good number in Chinese culture because it sounds the same as the word long-lasting. Nine is strongly associated with the Chinese dragon, a symbol of magic, there are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales –81 yang and 36 yin, all three numbers are multiples of 9 as well as having the same digital root of 9. The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City, the name of the area called Kowloon in Hong Kong literally means, nine dragons. The nine-dotted line delimits certain island claims by China in the South China Sea, the nine-rank system was a civil service nomination system used during certain Chinese dynasties. 9 Points of the Heart / Heart Master Channels in Traditional Chinese Medicine, the nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. The Ennead is a group of nine Egyptian deities, who, in versions of the Osiris myth. The Nine Worthies are nine historical, or semi-legendary figures who, in Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil. The nine Muses in Greek mythology are Calliope, Clio, Erato, Euterpe, Melpomene, Polyhymnia, Terpsichore, Thalia and it takes nine days to fall from heaven to earth, and nine more to fall from earth to Tartarus—a place of torment in the underworld. Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo, according to Georges Ifrah, the origin of the 9 integers can be attributed to ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0. In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot, the Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, as time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller
12.
10 (number)
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10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十
13.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
14.
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
15.
20 (number)
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20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants
16.
30 (number)
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30 is the natural number following 29 and preceding 31. Thirty is the sum of the first four squares, which makes it a square pyramidal number and it is a primorial and is the smallest Giuga number. 30 is the smallest sphenic number, and the smallest of the form 2 ×3 × r,30 has an aliquot sum of 42, the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises Thirty has but one number for which it is the aliquot sum, adding up some subsets of its divisors gives 30, hence 30 is a semiperfect number. 30 is the largest number such that all smaller than itself. A polygon with thirty sides is called a triacontagon, the icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a graph with 30 vertices. The atomic number of zinc is 30 Messier object M30, a magnitude 8, the duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series began on June 19,1803 BC. The duration of Saros series 30 was 1316.2 years, Thirty is, Used to indicate the end of a newspaper story, a copy editors typographical notation. S. Judas Iscariot betrayed Jesus for 30 pieces of silver, one of the rallying-cries of the 1960s student/youth protest movement was the slogan, Dont trust anyone over thirty. In Franz Kafkas novel The Trial Joseph wakes up on the morning of his birthday to find himself under arrest for an unspecified crime. After making many attempts to find the nature of the crime or the name of his accuser. The number of uprights that formed the Sarsen Circle at Stonehenge, western Christianitys most prolific 20th century essayist, F. W. Also in that essay Boreham writes It was said of Keats, in tennis, the number 30 represents the second point gained in a game. Under NCAA rules for basketball, the offensive team has 30 seconds to attempt a shot. As of 2012, three of the four major leagues in the United States and Canada have 30 teams each. The California Angels baseball team retired the number in honor of its most notable wearer, Nolan Ryan, the San Francisco Giants extended the same honor to Orlando Cepeda
17.
40 (number)
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Despite being related to the word four, the modern spelling of 40 is forty. The archaic form fourty is now considered a misspelling, the modern spelling possibly reflects a pronunciation change due to the horse–hoarse merger. Forty is a number, an octagonal number, and as the sum of the first four pentagonal numbers. Adding up some subsets of its divisors gives 40, hence 40 is a semiperfect number, given 40, the Mertens function returns 0. 40 is the smallest number n with exactly 9 solutions to the equation φ = n, Forty is the number of n-queens problem solutions for n =7. Since 402 +1 =1601 is prime,40 is a Størmer number,40 is a repdigit in base 3 and a Harshad number in base 10. Negative forty is the temperature at which the Fahrenheit and Celsius scales correspond. It is referred to as either minus forty or forty below, the planet Venus forms a pentagram in the night sky every eight years with it returning to its original point every 40 years with a 40-day regression. The duration of Saros series 40 was 1280.1 years, lunar eclipse series which began on -1387 February 12 and ended on -71 April 12. The duration of Saros series 40 was 1316.2 years, the number 40 is used in Jewish, Christian, Islamic, and other Middle Eastern traditions to represent a large, approximate number, similar to umpteen. In the Hebrew Bible, forty is often used for periods, forty days or forty years. Rain fell for forty days and forty nights during the Flood, spies explored the land of Israel for forty days. The Hebrew people lived in the Sinai desert for forty years and this period of years represents the time it takes for a new generation to arise. Moses life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, and his return to lead his people out, several Jewish leaders and kings are said to have ruled for forty years, that is, a generation. Examples include Eli, Saul, David, and Solomon, goliath challenged the Israelites twice a day for forty days before David defeated him. He went up on the day of Tammuz to beg forgiveness for the peoples sin. He went up on the first day of Elul and came down on the day of Tishrei. A mikvah consists of 40 seah of water 40 lashes is one of the punishments meted out by the Sanhedrin, One of the prerequisites for a man to study Kabbalah is that he is forty years old
18.
60 (number)
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60 is the natural number following 59 and preceding 61. Being three times 20, it is called three score in older literature. It is a number, with divisors 1,2,3,4,5,6,10,12,15,20,30. Because it is the sum of its divisors, it is a unitary perfect number. Being ten times a number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6 and it is the smallest number with exactly 12 divisors. It is the sum of a pair of twin primes and the sum of four consecutive primes and it is adjacent to two primes. It is the smallest number that is the sum of two odd primes in six ways, the smallest non-solvable group has order 60. There are four Archimedean solids with 60 vertices, the icosahedron, the rhombicosidodecahedron, the snub dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs, there are also two Archimedean solids with 60 edges, the snub cube and the icosidodecahedron. The skeleton of the forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares, in geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, a number system with base 60 is called sexagesimal. It is the smallest positive integer that is written only the smallest. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule and this ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 is an isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz
19.
80 (number)
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80 is the natural number following 79 and preceding 81. 80 is, the sum of Eulers totient function φ over the first sixteen integers, a semiperfect number, since adding up some subsets of its divisors gives 80. Palindromic in bases 3,6,9,15,19 and 39, a repdigit in bases 3,9,15,19 and 39. A Harshad number in bases 2,3,4,5,6,7,9,10,11,13,15 and 16 The Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes. Every solvable configuration of the Fifteen puzzle can be solved in no more than 80 single-tile moves, the atomic number of mercury According to Exodus 7,7, Moses was 80 years old when he initially spoke to Pharaoh on behalf of his people. Today,80 years of age is the age limit for cardinals to vote in papal elections. Jerry Rice wore the number 80 for the majority of his NFL career
20.
90 (number)
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90 is the natural number preceded by 89 and followed by 91. In English speech, the numbers 90 and 19 are often confused, when carefully enunciated, they differ in which syllable is stressed,19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in such as 1999, and when contrasting numbers in the teens and when counting, such as 17,18,19. 90 is, a perfect number because it is the sum of its unitary divisors. A semiperfect number because it is equal to the sum of a subset of its divisors, a Perrin number, preceded in the sequence by 39,51,68. Palindromic and a repdigit in bases 14,17,29, a Harshad number since 90 is divisible by the sum of its base 10 digits. In normal space, the angles of a rectangle measure 90 degrees each. Also, in a triangle, the angle opposing the hypotenuse measures 90 degrees. Thus, an angle measuring 90 degrees is called a right angle, ninety is, the atomic number of thorium, an actinide. As an atomic weight,90 identifies an isotope of strontium, the latitude in degrees of the North and the South geographical poles. NFL, New York Jets Dennis Byrds #90 is retired +90 is the code for international direct dial phone calls to Turkey,90 is the code for the French département Belfort
21.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
22.
Numeral system
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A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
23.
Septenary
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This is a list of numeral systems, that is, writing systems for expressing numbers. Numeral systems are classified here as to whether they use positional notation, the common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system and this turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
24.
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
25.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
26.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
27.
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
28.
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
29.
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
30.
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
31.
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
32.
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
33.
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
34.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
35.
Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
36.
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
37.
Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
38.
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
39.
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
40.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
41.
Base 36
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
42.
Greek numeral
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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
43.
Zeta
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Zeta is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7 and it was derived from the Phoenician letter zayin. Letters that arose from zeta include the Roman Z and Cyrillic З, unlike the other Greek letters, this letter did not take its name from the Phoenician letter from which it was derived, it was given a new name on the pattern of beta, eta and theta. The word zeta is the ancestor of zed, the name of the Latin letter Z in British English. Swedish and many Romanic languages, do not distinguish between the Greek and Roman forms of the letter, zeta is used to refer to the Roman letter Z as well as the Greek letter, the letter ζ represents the voiced alveolar fricative /z/ in Modern Greek. The sound represented by zeta in Classical Greek is disputed, see Ancient Greek phonology and Pronunciation of Ancient Greek in teaching. Most handbooks agree on attributing to it the pronunciation /zd/, PIE *zd becomes ζ in Greek. Without there would be an empty space between and in the Greek sound system, and a voiced affricate would not have a voiceless correspondent, alcman, Sappho, Alcaeus and Theocritus have σδ for Attic-Ionic ζ. Contra, The tradition would not have invented this special digraph for these poets if was the pronunciation in all Greek. Furthermore, this convention is not found in inscriptions. Thus, σδ indicates only a different pronunciation from Hellenistic Greek, the grammarians Dionysius Thrax and Dionysius of Halicarnassus class ζ with the double letters ψ, ξ and analyse it as σ + δ. Contra, The Roman grammarian Verrius Flaccus believed in the sequence, δ + σ. This suggests that different dialects had different pronunciations, the Greek inscriptions almost never write ζ in words like ὅσδε, τούσδε or εἰσδέχται, so there must have been a difference between this sound and the sound of ἵζω, Ἀθήναζε. It seems improbable that Greek would invent a special symbol for the bisegmental combination, furthermore, it is not clear that ζ was pronounced when it was originally invented. Mycenean Greek had a symbol to denote some sort of affricate or palatal consonant, ζ may have been invented for this sound. Boeotian, Elean, Laconian and Cretan δδ are more easily explained as a development from *dz than through an intermediary *zd. Greek in South Italy has preserved until modern times, contra, a) this may be a later development from or under the influence of Italian, b) even if it is derived from an ancient, it may be a dialectal pronunciation. Vulgar Latin inscriptions use the Greek letter Z for indigenous affricates, Italian, similarly, has consistently used Z for and
44.
Amharic language
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Amharic is an Afro-Asiatic language of the Semitic branch. It is spoken as a mother tongue by the Amhara in Ethiopia, the language serves as the official working language of Ethiopia, and is also the official or working language of several of the states within the federal system. Amharic is the second-most widely spoken Semitic language in the world after Arabic and it is written using Amharic Fidel, ፊደል, which grew out of the Geez abugida—called, in Ethiopian Semitic languages, ፊደል fidel and አቡጊዳ abugida. There is no agreed way of transliterating Amharic into Roman characters, the Amharic examples in the sections below use one system that is common, though not universal, among linguists specializing in Ethiopian Semitic languages. Amharic is spoken by 22 million native speakers in Ethiopia and 15 million secondary speakers in Ethiopia, additionally,3 million emigrants outside of Ethiopia speak the language. Most of the Ethiopian Jewish communities in Ethiopia and Israel speak Amharic, in Washington DC, Amharic became one of the six non-English languages in the Language Access Act of 2004, which allows government services and education in Amharic. Furthermore, Amharic is considered a language by the Rastafari religion and is widely used among its followers worldwide. It is the most widely spoken language in the Horn of Africa, the Amharic ejective consonants correspond to the Proto-Semitic emphatic consonants, usually transcribed with a dot below the letter. The consonant and vowel tables give these symbols in parentheses where they differ from the standard IPA symbols, the Amharic script is an abugida, and the graphemes of the Amharic writing system are called fidel. Each character represents a sequence, but the basic shape of each character is determined by the consonant. Some consonant phonemes are written by more than one series of characters, /ʔ/, /s/, /sʼ/ and this is because these fidel originally represented distinct sounds, but phonological changes merged them. The citation form for each series is the form, i. e. the first column of the fidel. The Amharic script is included in Unicode, and glyphs are included in fonts available with major operating systems, as in most other Ethiopian Semitic languages, gemination is contrastive in Amharic. That is, consonant length can distinguish words from one another, for example, alä he said, allä there is, yǝmätall he hits, gemination is not indicated in Amharic orthography, but Amharic readers typically do not find this to be a problem. This property of the system is analogous to the vowels of Arabic and Hebrew or the tones of many Bantu languages. Punctuation includes the following, ፠ section mark ፡ word separator ። full stop ፣ comma ፤ semicolon ፥ colon ፦ Preface colon, question mark ፨ paragraph separator Simple Amharic sentences One may construct simple Amharic sentences by using a subject and a predicate. Here are a few sentences, ኢትዮጵያ አፍሪቃ ውስጥ ናት ʾItyop̣p̣ya ʾAfriqa wǝsṭ nat Ethiopia is in Africa. Lǝǧu is the boy The boy is asleep, አየሩ ደስ ይላል Ayyäru däss yǝlall
45.
Arabic
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Arabic is a Central Semitic language that was first spoken in Iron Age northwestern Arabia and is now the lingua franca of the Arab world. Arabic is also the language of 1.7 billion Muslims. It is one of six languages of the United Nations. The modern written language is derived from the language of the Quran and it is widely taught in schools and universities, and is used to varying degrees in workplaces, government, and the media. The two formal varieties are grouped together as Literary Arabic, which is the language of 26 states. Modern Standard Arabic largely follows the standards of Quranic Arabic. Much of the new vocabulary is used to denote concepts that have arisen in the post-Quranic era, Arabic has influenced many languages around the globe throughout its history. During the Middle Ages, Literary Arabic was a vehicle of culture in Europe, especially in science, mathematics. As a result, many European languages have borrowed many words from it. Many words of Arabic origin are found in ancient languages like Latin. Balkan languages, including Greek, have acquired a significant number of Arabic words through contact with Ottoman Turkish. Arabic has also borrowed words from languages including Greek and Persian in medieval times. Arabic is a Central Semitic language, closely related to the Northwest Semitic languages, the Ancient South Arabian languages, the Semitic languages changed a great deal between Proto-Semitic and the establishment of the Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include, The conversion of the suffix-conjugated stative formation into a past tense, the conversion of the prefix-conjugated preterite-tense formation into a present tense. The elimination of other prefix-conjugated mood/aspect forms in favor of new moods formed by endings attached to the prefix-conjugation forms, the development of an internal passive. These features are evidence of descent from a hypothetical ancestor. In the southwest, various Central Semitic languages both belonging to and outside of the Ancient South Arabian family were spoken and it is also believed that the ancestors of the Modern South Arabian languages were also spoken in southern Arabia at this time. To the north, in the oases of northern Hijaz, Dadanitic and Taymanitic held some prestige as inscriptional languages, in Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested
46.
Persian language
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Persian, also known by its endonym Farsi, is one of the Western Iranian languages within the Indo-Iranian branch of the Indo-European language family. It is primarily spoken in Iran, Afghanistan, and Tajikistan and it is mostly written in the Persian alphabet, a modified variant of the Arabic script. Its grammar is similar to that of many contemporary European languages, Persian gets its name from its origin at the capital of the Achaemenid Empire, Persis, hence the name Persian. A Persian-speaking person may be referred to as Persophone, there are approximately 110 million Persian speakers worldwide, with the language holding official status in Iran, Afghanistan, and Tajikistan. For centuries, Persian has also been a cultural language in other regions of Western Asia, Central Asia. It also exerted influence on Arabic, particularly Bahrani Arabic. Persian is one of the Western Iranian languages within the Indo-European family, other Western Iranian languages are the Kurdish languages, Gilaki, Mazanderani, Talysh, and Balochi. Persian is classified as a member of the Southwestern subgroup within Western Iranian along with Lari, Kumzari, in Persian, the language is known by several names, Western Persian, Parsi or Farsi has been the name used by all native speakers until the 20th century. Since the latter decades of the 20th century, for reasons, in English. Tajiki is the variety of Persian spoken in Tajikistan and Uzbekistan by the Tajiks, according to the Oxford English Dictionary, the term Persian as a language name is first attested in English in the mid-16th century. Native Iranian Persian speakers call it Fārsi, Farsi is the Arabicized form of Pārsi, subsequent to Muslim conquest of Persia, due to a lack of the phoneme /p/ in Standard Arabic. The origin of the name Farsi and the place of origin of the language which is Fars Province is the Arabicized form of Pārs, in English, this language has historically been known as Persian, though Farsi has also gained some currency. Farsi is encountered in some literature as a name for the language. In modern English the word Farsi refers to the language while Parsi describes Zoroastrians, some Persian language scholars such as Ehsan Yarshater, editor of Encyclopædia Iranica, and University of Arizona professor Kamran Talattof, have also rejected the usage of Farsi in their articles. The international language-encoding standard ISO 639-1 uses the code fa, as its system is mostly based on the local names. The more detailed standard ISO 639-3 uses the name Persian for the dialect continuum spoken across Iran and Afghanistan and this consists of the individual languages Dari and Iranian Persian. Currently, Voice of America, BBC World Service, Deutsche Welle, Radio Free Europe/Radio Liberty also includes a Tajik service and an Afghan service. This is also the case for the American Association of Teachers of Persian, The Centre for Promotion of Persian Language and Literature, Persian is an Iranian language belonging to the Indo-Iranian branch of the Indo-European family of languages
47.
Central Kurdish
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Central Kurdish, also called Sorani is a Kurdish language spoken in Iraq, mainly in Iraqi Kurdistan, as well as the Kurdistan Province of western Iran. Central Kurdish is one of the two languages of Iraq, along with Arabic, and is in political documents simply referred to as Kurdish. In Sulaymaniyah, the Ottoman Empire had created a secondary school and this allowed Central Kurdish, which was spoken in Silêmanî, to progressively replace Hawrami dialects as the literary vehicle for Kurdish. Since the fall of the Arab Socialist Baath Party – Iraq Region, as a result, Central Kurdish has become the dominant written form of Kurdish. Central Kurdish is written with a modified Arabic alphabet and this is in contrast to the other main Kurdish language, Northern Kurdish, which is spoken mainly in Turkey and is usually written in the Latin alphabet. However, during the past decade, official TV in Iraqi Kurdistan has mainly used the Latin script for Central Kurdish. The exact number of Soriani speakers is difficult to determine, but it is thought that Soriani is spoken by about 6 to 7 million people in Iraq. It is the most widespread speech of Kurds in Iran and Iraq, in particular, it is spoken by, Around 3 million Kurds in Iranian Kurdistan. Located south of Lake Urmia that stretches roughly to the outside of Kermanshah, around 3 million Kurds in Iraqi Kurdistan, including the Soriani tribe. Most of the Kurds who use it are found in the vicinity of Hewlêr, Sulaymaniyah, Kirkuk, following includes the traditional internal variants of Soriani. This region is known as Mukriyan. Ardalani, spoken in the cities of Sanandaj, Marivan, Kamyaran, Divandarreh and Dehgolan in Kordestan province and this region is known as Ardalan. Garmiani, in and around Kirkuk Hawlari, spoken in and around the city of Hawler in Iraqi Kurdistan and its main distinction is changing the consonant /l/ into /r/ in many words. Babani, spoken in and around the city of Sulaymaniya in Iraq, jafi, spoken in the towns of Javanroud, Ravansar and some villages around Sarpole Zahab and Paveh. A recent proposal was made for Central Kurdish to be the language of the Kurdistan Regional Government. This idea has been favoured by some Central Kurdish-speakers but has disappointed Northern Kurdish speakers, Sorani Kurdish has a rich consonant inventory and a fairly rich vowel inventory as well. This section makes use of the International Phonetic Alphabet, the following table contains the vowels of Sorani Kurdish. Vowels in parentheses are not phonemic, but have included in the table below because of their ubiquity in the language
48.
Urdu
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Urdu is a persianized standard register of the Hindustani language. It is the language and lingua franca of Pakistan. It is also one of the 22 official languages recognized in the Constitution of India, hyderabad, Rampur, Bhopal and Lucknow are noted Urdu-speaking cities of India. Urdu is historically associated with the Muslims of the northern Indian subcontinent, apart from specialized vocabulary, Urdu is mutually intelligible with Standard Hindi, another recognized register of Hindustani. Urdu, like Hindi, is a form of Hindustani, Urdu developed under the influence of the Persian and Arabic languages, both of which have contributed a significant amount of vocabulary to formal speech. Around 99% of Urdu verbs have their roots in Sanskrit and Prakrit, Urdu words originating from Chagatai and Arabic were borrowed through Persian and hence are Persianized versions of the original words. For instance, the Arabic ta marbuta changes to he or te, nevertheless, contrary to popular belief, Urdu did not borrow from the Turkish language, but from Chagatai. Urdu and Turkish borrowed from Arabic and Persian, hence the similarity in pronunciation of many Urdu, Arabic influence in the region began with the late first-millennium Arab invasion of India in the 7th century. The Persian language was introduced into the subcontinent a few centuries later by various Persianized Central Asian Turkic and Afghan dynasties including that of the Delhi Sultanate. With the advent of the British Raj, Persian was no longer the language of administration but Hindustani, still written in the Persian script, the name Urdu was first used by the poet Ghulam Hamadani Mushafi around 1780. From the 13th century until the end of the 18th century Urdu was commonly known as Hindi, the language was also known by various other names such as Hindavi and Dehlavi. The communal nature of the language lasted until it replaced Persian as the language in 1837 and was made co-official. Urdu was promoted in British India by British policies to counter the previous emphasis on Persian and this triggered a Brahman backlash in northwestern India, which argued that the language should be written in the native Devanagari script. At independence, Pakistan established a highly Persianized literary form of Urdu as its national language, English has exerted a heavy influence on both as a co-official language. Owing to interaction with other languages, Urdu has become localized wherever it is spoken, similarly, the Urdu spoken in India can also be distinguished into many dialects like Dakhni of South India, and Khariboli of the Punjab region since recent times. Because of Urdus similarity to Hindi, speakers of the two languages can understand one another if both sides refrain from using specialized vocabulary. The syntax, morphology, and the vocabulary are essentially identical. Thus linguists usually count them as one language and contend that they are considered as two different languages for socio-political reasons
49.
Bengali language
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Bengali, also known by its endonym Bangla, is an Indo-Aryan language spoken in South Asia. With over 210 million speakers, Bengali is the seventh most spoken language in the world. Dominant in the last group was Persian, which was also the source of some grammatical forms, more recent studies suggest that the use of native and foreign words has been increasing, mainly because of the preference of Bengali speakers for the colloquial style. Today, Bengali is the language spoken in Bangladesh and the second most spoken language in India. Both the national anthems of Bangladesh and India were composed in Bengali, in 1952, the Bengali Language Movement successfully pushed for the languages official status in the Dominion of Pakistan. In 1999, UNESCO recognized 21 February as International Mother Language Day in recognition of the movement in East Pakistan. Language is an important element of Bengali identity and binds together a diverse region. Sanskrit was spoken in Bengal since the first millennium BCE, during the Gupta Empire, Bengal was a hub of Sanskrit literature. The Middle Indo-Aryan dialects were spoken in Bengal in the first millennium when the region was a part of the Magadha Realm and these dialects were called Magadhi Prakrit. They eventually evolved into Ardha Magadhi, Ardha Magadhi began to give way to what are called Apabhraṃśa languages at the end of the first millennium. Along with other Eastern Indo-Aryan languages, Bengali evolved circa 1000–1200 AD from Sanskrit, for example, Ardhamagadhi is believed to have evolved into Abahatta around the 6th century, which competed with the ancestor of Bengali for some time. Proto-Bengali was the language of the Pala Empire and the Sena dynasty, during the medieval period, Middle Bengali was characterized by the elision of word-final অ ô, the spread of compound verbs and Arabic and Persian influences. Bengali was a court language of the Sultanate of Bengal. Muslim rulers promoted the development of Bengali as part of efforts to Islamize. Bengali became the most spoken language in the Sultanate. This period saw borrowing of Perso-Arabic terms into Bengali vocabulary, major texts of Middle Bengali include Chandidas Shreekrishna Kirtana. The modern literary form of Bengali was developed during the 19th and early 20th centuries based on the dialect spoken in the Nadia region, a west-central Bengali dialect. Bengali presents a case of diglossia, with the literary
50.
Chinese numeral
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Chinese numerals are words and characters used to denote numbers in Chinese. Today speakers of Chinese use three written numeral systems, the system of Arabic numerals used worldwide, and two indigenous systems, the more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These are shared with languages of the Chinese cultural sphere such as Japanese, Korean. The other indigenous system is the Suzhou numerals, or huama, a positional system and these were once used by Chinese mathematicians, and later in Chinese markets, such as those in Hong Kong before the 1990s, but have been gradually supplanted by Arabic numerals. The Chinese character numeral system consists of the Chinese characters used by the Chinese written language to write spoken numerals, similar to spelling-out numbers in English, it is not an independent system per se. Since it reflects spoken language, it not use the positional system as in Arabic numerals. There are characters representing the numbers zero through nine, and other characters representing larger numbers such as tens, hundreds, thousands, there are two sets of characters for Chinese numerals, one for everyday writing and one for use in commercial or financial contexts known as dàxiě. A forger could easily change the everyday characters 三十 to 五千 just by adding a few strokes and that would not be possible when writing using the financial characters 參拾 and 伍仟. They are also referred to as bankers numerals, anti-fraud numerals, for the same reason, rod numerals were never used in commercial records. T denotes Traditional Chinese characters, S denotes Simplified Chinese characters, in the PLA, some numbers will have altered names when used for clearer radio communications. They are,0, renamed 洞 lit, hole 1, renamed 幺 lit. small 2, renamed 两 lit. Double 7, renamed 拐 lit. cane, kidnap, turn 9, hook For numbers larger than 10,000, similarly to the long and short scales in the West, there have been four systems in ancient and modern usage. The original one, with names for all powers of ten up to the 14th, is ascribed to the Yellow Emperor in the 6th century book by Zhen Luan. To avoid problems arising from the ambiguity, the PRC government never uses this character in official documents, the ROC government in Taiwan uses 兆 to mean 1012 in official documents. Numerals beyond 載 zài come from Buddhist texts in Sanskrit, but are found in ancient texts. Some of the words are still being used today. The following are characters used to denote small order of magnitude in Chinese historically, with the introduction of SI units, some of them have been incorporated as SI prefixes, while the rest have fallen into disuse. In the Peoples Republic of China, the translations for the SI prefixes in 1981 were different from those used today, the Republic of China defined 百萬 as the translation for mega
51.
Telugu language
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Telugu is a Dravidian language native to India. It is also spoken by significant minorities in Karnataka, Tamil Nadu, Maharashtra, Chhattisgarh, Odisha, the Andaman and Nicobar Islands and it is one of six languages designated a classical language of India by the Government of India. Telugu ranks third by the number of speakers in India. It is one of the scheduled languages of the Republic of India. Approximately 10,000 inscriptions exist in the Telugu language, the speakers of the language call it Telugu or Tenugu. The older forms of the name include Teluṅgu, Tenuṅgu and Teliṅga, the etymology of Telugu is not certain. Some historical scholars have suggested a derivation from Sanskrit triliṅgam, as in Trilinga Desa, tradition holds that Shiva descended as a lingam on three mountains, Kaleshwaram, Srisailam, and Bhimeswaram, which are said to have marked the boundaries of the Trilinga Desa. Atharvana Acharya in the 13th century wrote a grammar of Telugu, Appa Kavi in the 17th century explicitly wrote that Telugu was derived from Trilinga. Scholar Charles P. Brown comments that it was a strange notion as all the predecessors of Appa Kavi had no knowledge of such a derivation. George Abraham Grierson and other linguists doubt this derivation, holding rather that Telugu was the older term, another view holds that tenugu is derived from the proto-Dravidian word ten– south to mean the people who lived in the south/southern direction. The name telugu then, is a result of n -> l alternation established in Telugu, according to the natve tradition Telugu grammar has a hoary past. Sage Kanva was said to be the first grammarian of Telugu, a Rajeswara Sarma discussed the hisoricity and content of Kanvas grammar written in Sanskrit. He cited twenty grammatical aphorisms ascribed to Kanva, and concluded that Kanva wrote an ancient Telugu Grammar which was lost, according to the Russian linguist M. S. Andronov, Telugu split from the Proto-Dravidian languages between 1500 and 1000 BC. According to linguist Bhadriraju Krishnamurti, Telugu, as a Dravidian language, descends from Proto-Dravidian, linguistic reconstruction suggests that Proto-Dravidian was spoken around the third millennium BC, possibly in the region around the lower Godavari river basin in peninsular India. The material evidence suggests that the speakers of Proto-Dravidian were of the associated with the Neolithic societies of South India. A legend gives the Lepakshi town a significant place in the Ramayana — this was where the bird Jatayu fell, when Sri Rama reached the spot, he saw the bird and said compassionately, “Le Pakshi” — ‘rise, bird’ in Telugu. This indicates the presence of Telugu Language during Ramayana period, there is a mention of Telugu people or Telugu country in ancient Tamil literature as Telunka Nadu. Telugu words were found in Hebrew literature, inscriptions with Telugu words dating back to 400 BC to 100 BC have been discovered in Bhattiprolu in the Guntur district of Andhra Pradesh
52.
Tamil language
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Tamil is a Dravidian language predominantly spoken by the Tamil people of India and Sri Lanka, and also by the Tamil diaspora, Sri Lankan Moors, Burghers, Douglas, and Chindians. Tamil is a language of two countries, Singapore and Sri Lanka. It has official status in the Indian state of Tamil Nadu and it is also used as one of the languages of education in Malaysia, along with English, Malay and Mandarin. Tamil is also spoken by significant minorities in the four other South Indian states of Kerala, Karnataka, Andhra Pradesh and Telangana and it is one of the 22 scheduled languages of India. Tamil is one of the classical languages in the world. Tamil-Brahmi inscriptions from 500 BC have been found on Adichanallur and 2 and it has been described as the only language of contemporary India which is recognizably continuous with a classical past. The variety and quality of classical Tamil literature has led to it being described as one of the classical traditions. A recorded Tamil literature has been documented for over 2000 years, the earliest period of Tamil literature, Sangam literature, is dated from ca.300 BC – AD300. It has the oldest extant literature among other Dravidian languages, the earliest epigraphic records found on rock edicts and hero stones date from around the 3rd century BC. More than 55% of the inscriptions found by the Archaeological Survey of India are in the Tamil language. Tamil language inscriptions written in Brahmi script have been discovered in Sri Lanka, the two earliest manuscripts from India, acknowledged and registered by the UNESCO Memory of the World register in 1997 and 2005, were written in Tamil. In 1578, Portuguese Christian missionaries published a Tamil prayer book in old Tamil script named Thambiraan Vanakkam, the Tamil Lexicon, published by the University of Madras, was one of the earliest dictionaries published in the Indian languages. According to a 2001 survey, there were 1,863 newspapers published in Tamil, Tamil belongs to the southern branch of the Dravidian languages, a family of around 26 languages native to the Indian subcontinent. It is also classified as being part of a Tamil language family, the closest major relative of Tamil is Malayalam, the two began diverging around the 9th century CE. According to linguists like Bhadriraju Krishnamurti, Tamil, as a Dravidian language, descends from Proto-Dravidian, linguistic reconstruction suggests that Proto-Dravidian was spoken around the third millennium BC, possibly in the region around the lower Godavari river basin in peninsular India. The material evidence suggests that the speakers of Proto-Dravidian were of the associated with the Neolithic complexes of South India. The next phase in the reconstructed proto-history of Tamil is Proto-South Dravidian, the linguistic evidence suggests that Proto-South Dravidian was spoken around the middle of the second millennium BC, and that proto-Tamil emerged around the 3rd century BC. The earliest epigraphic attestations of Tamil are generally taken to have been shortly thereafter
53.
Hebrew (language)
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Hebrew is a language native to Israel, spoken by over 9 million people worldwide, of whom over 5 million are in Israel. Historically, it is regarded as the language of the Israelites and their ancestors, the earliest examples of written Paleo-Hebrew date from the 10th century BCE. Hebrew belongs to the West Semitic branch of the Afroasiatic language family, Hebrew is the only living Canaanite language left, and the only truly successful example of a revived dead language. Hebrew had ceased to be a spoken language somewhere between 200 and 400 CE, declining since the aftermath of the Bar Kokhba revolt. Aramaic and to a lesser extent Greek were already in use as international languages, especially among elites and it survived into the medieval period as the language of Jewish liturgy, rabbinic literature, intra-Jewish commerce, and poetry. Then, in the 19th century, it was revived as a spoken and literary language, and, according to Ethnologue, had become, as of 1998, the language of 5 million people worldwide. After Israel, the United States has the second largest Hebrew-speaking population, with 220,000 fluent speakers, Modern Hebrew is one of the two official languages of the State of Israel, while premodern Hebrew is used for prayer or study in Jewish communities around the world today. Ancient Hebrew is also the tongue of the Samaritans, while modern Hebrew or Arabic is their vernacular. For this reason, Hebrew has been referred to by Jews as Leshon Hakodesh, the modern word Hebrew is derived from the word Ivri, one of several names for the Israelite people. It is traditionally understood to be a based on the name of Abrahams ancestor, Eber. This name is based upon the root ʕ-b-r meaning to cross over. Interpretations of the term ʕibrim link it to this verb, cross over, in the Bible, the Hebrew language is called Yәhudit because Judah was the surviving kingdom at the time of the quotation. In Isaiah 19,18 it is called the Language of Canaan, Hebrew belongs to the Canaanite group of languages. In turn, the Canaanite languages are a branch of the Northwest Semitic family of languages, according to Avraham ben-Yosef, Hebrew flourished as a spoken language in the Kingdoms of Israel and Judah during about 1200 to 586 BCE. Scholars debate the degree to which Hebrew was a vernacular in ancient times following the Babylonian exile. In July 2008 Israeli archaeologist Yossi Garfinkel discovered a ceramic shard at Khirbet Qeiyafa which he claimed may be the earliest Hebrew writing yet discovered, dating around 3000 years ago. The Gezer calendar also dates back to the 10th century BCE at the beginning of the Monarchic Period, classified as Archaic Biblical Hebrew, the calendar presents a list of seasons and related agricultural activities. The Gezer calendar is written in an old Semitic script, akin to the Phoenician one that through the Greeks, the Gezer calendar is written without any vowels, and it does not use consonants to imply vowels even in the places where later Hebrew spelling requires it
54.
Khmer numerals
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Khmer numerals are the numerals used in the Khmer language. They have been in use since at least the early 7th century, with the earliest known use being on a stele dated to AD604 found in Prasat Bayang, Cambodia, having been derived from the Hindu numerals, modern Khmer numerals also represent a decimal positional notation system. It is the script with the first extant material evidence of zero as a figure, dating its use back to the seventh century. However, Old Khmer, or Angkorian Khmer, also possessed separate symbols for the numbers 10,20 and this inconsistency with its decimal system suggests that spoken Angkorian Khmer used a vigesimal system. For example,6 is formed from 5 plus 1, with the exception of the number 0, which stems from Sanskrit, the etymology of the Khmer numbers from 1 to 5 is of proto-Mon–Khmer origin. For details of the various alternative romanization systems, see Romanization of Khmer, some authors may alternatively mark as the pronunciation for the word two, and either or for the word three. In neighbouring Thailand the number three is thought to bring good luck, however, in Cambodia, taking a picture with three people in it is considered bad luck, as it is believed that the person situated in the middle will die an early death. As mentioned above, the numbers from 6 to 9 may be constructed by adding any number between 1 and 4 to the base number 5, so that 7 is literally constructed as 5 plus 2. Beyond that, Khmer uses a base, so that 14 is constructed as 10 plus 4, rather than 2 times 5 plus 4. In constructions from 6 to 9 that use 5 as a base, /pram/ may alternatively be pronounced, giving and this is especially true in dialects which elide /r/, but not necessarily restricted to them, as the pattern also follows Khmers minor syllable pattern. The numbers from thirty to ninety in Khmer bear many resemblances to both the modern Thai and Cantonese numbers, informally, a speaker may choose to omit the final and the number is still understood. For example, it is possible to say instead of the full, Language Comparisons, Words in parenthesis indicate literary pronunciations, while words preceded with an asterisk mark are non-productive. The standard Khmer numbers starting from one hundred are as follows, Although មួយកោដិ is most commonly used to mean ten million, in some areas this is also colloquially used to refer to one billion. In order to avoid confusion, sometimes ដប់លាន is used to mean ten million, along with មួយរយលាន for one hundred million, different Cambodian dialects may also employ different base number constructions to form greater numbers above one thousand. As a result of prolonged literary influence from both the Sanskrit and Pali languages, Khmer may occasionally use borrowed words for counting. One reason for the decline of numbers is that a Khmer nationalism movement. The Khmer Rouge also attempted to cleanse the language by removing all words which were considered politically incorrect, Khmer ordinal numbers are formed by placing the word ទី in front of a cardinal number. This is similar to the use of ที่ thi in Thai and it is generally assumed that the Angkorian and pre-Angkorian numbers also represented a dual base system, with both base 5 and base 20 in use
55.
Thai numerals
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The Thai language lacks grammatical number. A count is expressed in the form of an uninflected noun followed by a number. In Thai, counting is kannap, the classifier, laksananam Variations to this pattern do occur, a partial list of Thai words that also classify nouns can be found in Wiktionary category, Thai classifiers. Thai sūn is written as oval 0 when using Arabic numerals, but a small circle ๐ when using traditional numerals and it is from Sanskrit śūnya, as are the alternate names for numbers one to four given below, but not the counting 1. Thai names for N +1 and the regular digits 2 through 9 as shown in the table, below, resemble those in Chinese varieties as spoken in Southern China, Thai and Lao words for numerals are almost identical, however, the numerical digits vary somewhat in shape. Shown below is a comparison between three languages using Cantonese and Minnan characters and pronunciations, the Thai transliteration uses the Royal Thai General System of Transcription. Sanskrit lakh designates the place value of a digit, which are named for the powers of ten, the place is lak nuai, tens place, lak sip, hundreds place, lak roi. The number one following any multiple of sip becomes et, the number ten is the same as Minnan 十. Numbers from twenty to twenty nine begin with yi sip, names of the lak sip for 30 to 90, and for the lak of 100,1000,10,000,100,000 and million, are almost identical to those of the like Khmer numerals. For the numbers twenty-one through twenty-nine, the part signifying twenty, yi sip, see the alternate numbers section below. The hundreds are formed by combining roi with the tens and ones values, for example, two hundred and thirty-two is song roi sam sip song. The words roi, phan, muen, and saen should occur with a preceding numeral, nueng never precedes sip, so song roi nueng sip is incorrect. Native speakers will sometimes use roi nueng with different tones on nueng to distinguish one hundred from one hundred, however, such distinction is often not made, and ambiguity may follow. To resolve this problem, if the number 101 is intended, numbers above a million are constructed by prefixing lan with a multiplier. For example, ten million is sip lan, and a trillion is lan lan, colloquially, decimal numbers are formed by saying chut where the decimal separator is located. For example,1.01 is nueng chut sun nueng, fractional numbers are formed by placing nai between the numerator and denominator or using x suan y to clearly indicate. For example, ⅓ is nueng nai sam or nueng suan sam, the word set can be omitted. The word khrueng is used for half and it precedes the measure word if used alone, but it follows the measure word when used with another number
56.
Saraiki dialect
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Saraiki is an Indo-Aryan language of the Lahnda group, spoken in the south-western half of the province of Punjab in Pakistan. Saraiki is to a degree mutually intelligible with Standard Punjabi and shares with it a large portion of its vocabulary. At the same time in its phonology it is radically different, the Saraiki language identity arose in the 1960s, encompassing more local earlier identities, and distinguishing itself from broader ones like that of Punjabi. Saraiki is the first language of 20 million people in Pakistan, across the South Punjab, southern Khyber Pakhtunkhwa, the present extent of the meaning of Sirāikī is a recent development, and the term most probably gained its currency during the nationalist movement of the 1960s. In this context, the term can most plausibly be explained as originally having had the meaning the language of the north, from the Sindhi word siro up-river, north. This name can refer to the northern dialects of Sindhi. An alternative hypothesis is that Sarākī originated in the word sauvīrā, or Sauvira, the most common rendering of the name is Saraiki. However, Seraiki and Siraiki have also used in academia until recently. Precise spelling aside, the name was first adopted in the 1960s by regional social and political leaders, an organization named Saraiki Academy was founded in Multan on 6 April 1962 and gave the name of universal application to the dialect. Saraiki is a member of the Indo-Aryan subdivision of the Indo-Iranian branch of the Indo-European language family, Standard Punjabi and Saraiki are mutually intelligible. Subsequent Indo-Aryanist linguists have confirmed the reality of this dialect cluster, Grierson also maintained that Lahnda was his novel designation for various dialects up to then called Western Punjabi, spoken north, west, and south of Lahore. The local dialect of Lahore is the Majhi dialect of Punjabi, however, outside of Indo-Aryanist circles, the concept of Lahnda is still found in compilations of the worlds languages. The following dialects have been proposed for Saraiki, Central Saraiki, including Multani, spoken in the districts of Dera Ghazi Khan, Muzaffargarh, Leiah, Multan. Southern Saraiki, prevalent in the districts of Rajanpur and Rahimyar Khan, Sindhi Saraiki, dispersed throughout the province of Sindh. Northern Saraiki, or Thali, spoken in the district of Dera Ismail Khan, Eastern Saraiki, transitional to Punjabi and spoken in the Bar region along the boundary with the eastern Majhi dialect. This group includes the dialects of Jhangi and Shahpuri, the historical inventory of names for the dialects now called Saraiki is a confusion of overlapping or conflicting ethnic, local, and regional designations. Hindki and Hindko – refer to various Saraiki and even non-Saraiki dialects in Punjab Province and farther north within the country, due to the fact they were applied by arrivals from Afghanistan. One historical name for Saraiki, Jaṭki, means of the Jaṭṭs, a northern South Asian ethnic group, only a small minority of Saraiki speakers are Jaṭṭs, and not all Saraiki speaking Jaṭṭs necessarily speak the same dialect of Saraiki
57.
Kannada language
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The language has roughly 40 million native speakers who are called Kannadigas, and a total of 50.8 million speakers according to a 2001 census. It is one of the languages of India and the official. The Kannada language is written using the Kannada script, which evolved from the 5th-century Kadamba script, Kannada is attested epigraphically for about one and a half millennia, and literary Old Kannada flourished in the 6th-century Ganga dynasty and during the 9th-century Rashtrakuta Dynasty. Kannada has a literary history of over a thousand years. Based on the recommendations of the Committee of Linguistic Experts, appointed by the ministry of culture, in July 2011, a centre for the study of classical Kannada was established as part of the Central Institute of Indian Languages at Mysore to facilitate research related to the language. Kannada is a Southern Dravidian language, and according to Dravidian scholar Sanford B, steever, its history can be conventionally divided into three periods, Old Kannada from 450–1200 CE, Middle Kannada from 1200–1700, and Modern Kannada from 1700 to the present. Kannada is influenced to an extent by Sanskrit. Influences of other such as Prakrit and Pali can also be found in the Kannada language. Literary Prakrit seems to have prevailed in Karnataka since ancient times, the vernacular Prakrit-speaking people may have come into contact with Kannada speakers, thus influencing their language, even before Kannada was used for administrative or liturgical purposes. Kannada phonetics, morphology, vocabulary, grammar and syntax show significant influence from these languages, some examples of naturalised words of Prakrit origin in Kannada are, baṇṇa derived from vaṇṇa, hunnime from puṇṇivā. Examples of naturalized Sanskrit words in Kannada are, varṇa, arasu from rajan, paurṇimā, Kannada has numerous borrowed words such as dina, kopa, surya, mukha, nimiṣa and anna. Pre-old Kannada was the language of Banavasi in the early Common Era, the Ashoka rock edict found at Brahmagiri has been suggested to contain words in identifiable Kannada. According to Jain tradition, Brahmi, the daughter of Rishabhadeva, the first Tirthankara of Jainism, invented 18 alphabets, including Kannada, which points to the antiquity of the language. Supporting this tradition, an inscription of about the 9th century CE, containing specimens of different alphabets and it has been claimed that the Greek dramatists of the 5th–4th century BCE were familiar with the Kannada country and language. This would show a far more intimate contact of the Greeks with Kannada culture than with Indian culture elsewhere, the palm manuscripts contained texts written not only in Greek, Latin and Hebrew, but also in Sanskrit and Kannada. In the 150 CE Prakrit book Gaathaa Saptashati, written by Haala Raja, Kannada words like tIr, tuppa, on the Pallava Prakrit inscription of 250 CE of Hire Hadagalis Shivaskandavarman, the Kannada word kOTe transforms into koTTa. In the 350 CE Chandravalli Prakrit inscription, words of Kannada origin like punaaTa, in one more Prakrit inscription of 250 CE found in Malavalli, Kannada towns like vEgooraM, kundamuchchaMDi find a reference. Pliny the Elder was a naval and army commander in the early Roman Empire and he writes about pirates between Muziris and Nitrias
58.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
59.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
60.
Cubic number
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In arithmetic and algebra, the cube of a number n is its third power, the result of the number multiplied by itself twice, n3 = n × n × n. It is also the number multiplied by its square, n3 = n × n2 and this is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n and it determines the side of the cube of a given volume. It is also n raised to the one-third power, both cube and cube root are odd functions,3 = −. The cube of a number or any other mathematical expression is denoted by a superscript 3, a cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The perfect cubes up to 603 are, Geometrically speaking, an integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger. For example,27 small cubes can be arranged into one larger one with the appearance of a Rubiks Cube, the difference between the cubes of consecutive integers can be expressed as follows, n3 −3 = 3n +1. There is no minimum perfect cube, since the cube of an integer is negative. For example, −4 × −4 × −4 = −64, unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25,75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6, some cube numbers are also square numbers, for example,64 is a square number and a cube number. This happens if and only if the number is a perfect sixth power, the last digits of each 3rd power are, It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1,8 or 9. That is their values modulo 9 may be only −1,1 and 0, every positive integer can be written as the sum of nine positive cubes. The equation x3 + y3 = z3 has no solutions in integers. In fact, it has none in Eisenstein integers, both of these statements are also true for the equation x3 + y3 = 3z3. The sum of the first n cubes is the nth triangle number squared,13 +23 + ⋯ + n 3 =2 =2. Proofs Charles Wheatstone gives a simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. Indeed, he begins by giving the identity n 3 = + + + ⋯ + ⏟ n consecutive odd numbers, kanim provides a purely visual proof, Benjamin & Orrison provide two additional proofs, and Nelsen gives seven geometric proofs
61.
Cross product
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In mathematics and vector algebra, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. Given two linearly independent vectors a and b, the product, a × b, is a vector that is perpendicular to both a and b and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and it should not be confused with dot product. If two vectors have the direction or if either one has zero length, then their cross product is zero. The cross product is anticommutative and is distributive over addition, the space R3 together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, the cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used, if the vectors a and b are parallel, by the above formula, the cross product of a and b is the zero vector 0. Then, the n is coming out of the thumb. Using this rule implies that the cross-product is anti-commutative, i. e. b × a = −. By pointing the forefinger toward b first, and then pointing the finger toward a. Using the cross product requires the handedness of the system to be taken into account. If a left-handed coordinate system is used, the direction of the n is given by the left-hand rule. This, however, creates a problem because transforming from one arbitrary reference system to another, the problem is clarified by realizing that the cross product of two vectors is not a vector, but rather a pseudovector. See cross product and handedness for more detail, in 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced both the dot product and the cross product using a period and an x, respectively, to denote them. These alternative names are widely used in the literature. Both the cross notation and the cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a ⋅ b involves multiplications between corresponding components of a and b, as explained below, the cross product can be expressed in the form of a determinant of a special 3 ×3 matrix
62.
Seven-dimensional cross product
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In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in R7 a vector a × b also in R7, like the cross product in three dimensions, the seven-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the cross product is unique up to a sign. The seven-dimensional cross product has the relationship to the octonions as the three-dimensional product does to the quaternions. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results, the product can be given by a multiplication table, such as the one here. This table, due to Cayley, gives the product of basis vectors ei, for example, from the table e 1 × e 2 = e 3 = − e 2 × e 1 The table can be used to calculate the product of any two vectors. This can be repeated for the six components. There are 480 such tables, one for each of the products satisfying the definition, the top left 3 ×3 corner of this table gives the cross product in three dimensions. The first property states that the product is perpendicular to its arguments, a third statement of the magnitude condition is | x × y | = | x | | y | if =0. Given the properties of bilinearity, orthogonality and magnitude, a cross product exists only in three and seven dimensions. This can be shown by postulating the properties required for the cross product, in zero dimensions there is only the zero vector, while in one dimension all vectors are parallel, so in both these cases the product must be identically zero. The restriction to 0,1,3 and 7 dimensions is related to Hurwitzs theorem, in contrast the three-dimensional cross product, which is unique, there are many possible binary cross products in seven dimensions. Unlike in three dimensions, x × y = a × b does not imply that a and b lie in the plane as x and y. One possible multiplication table is described in the Example section, unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product. Once we have established a multiplication table, it is applied to general vectors x and y by expressing x and y in terms of the basis. More compactly this rule can be written as e i × e i +1 = e i +3 with i =1.7 modulo 7, together with anticommutativity this generates the product. This rule directly produces the two immediately adjacent to the diagonal of zeros in the table. Also, from an identity in the subsection on consequences, e i × = − e i +1 = e i × e i +3, which produces diagonals further out, and so on
63.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
64.
999,999 (number)
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One million or one thousand thousand is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione, from mille, thousand and it is commonly abbreviated as m or M, further MM, mm, or mn in financial contexts. In scientific notation, it is written as 1×106 or 106, physical quantities can also be expressed using the SI prefix mega, when dealing with SI units, for example,1 megawatt equals 1,000,000 watts. The meaning of the word million is common to the scale and long scale numbering systems, unlike the larger numbers. Information, Not counting spaces, the text printed on 136 pages of an Encyclopædia Britannica, length, There are one million millimeters in a kilometer, and roughly a million sixteenths of an inch in a mile. A typical car tire might rotate a million times in a 1, 200-mile trip, fingers, If the width of a human finger is 2.2 cm, then a million fingers lined up would cover a distance of 22 km. If a person walks at a speed of 4 km/h, it would take approximately five. A city lot 70 by 100 feet is about a million square inches, volume, The cube root of one million is only one hundred, so a million objects or cubic units is contained in a cube only a hundred objects or linear units on a side. A million grains of salt or granulated sugar occupies only about 64 ml. One million cubic inches would be the volume of a room only 8 1⁄3 feet long by 8 1⁄3 feet wide by 8 1⁄3 feet high. Mass, A million cubic millimeters of water would have a volume of one litre, a million millilitres or cubic centimetres of water has a mass of a million grams or one tonne. Weight, A million 80-milligram honey bees would weigh the same as an 80 kg person, landscape, A pyramidal hill 600 feet wide at the base and 100 feet high would weigh about a million tons. Computer, A display resolution of 1,280 by 800 pixels contains 1,024,000 pixels, money, A USD bill of any denomination weighs 1 gram. There are 454 grams in a pound, one million $1 bills would weigh 2,204.62 pounds, or just over 1 ton. Time, A million seconds is 11.57 days, in Indian English and Pakistani English, it is also expressed as 10 lakh or 10 Lac. Lakh is derived from laksh for 100,000 in Sanskrit
65.
Vulgar fraction
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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
66.
Denominator
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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