1.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
2.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
3.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
4.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
5.
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
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
Harmonic number
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In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers, H n =1 +12 +13 + ⋯ +1 n = ∑ k =1 n 1 k. Harmonic numbers are related to the mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers. Harmonic numbers were studied in antiquity and are important in various branches of number theory and they are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the series to provide a new proof of the infinity of prime numbers. His work was extended into the plane by Bernhard Riemann in 1859. When the value of a quantity of items has a Zipfs law distribution. This leads to a variety of surprising conclusions in the Long Tail, bertrands postulate entails that, except for the case n =1, the harmonic numbers are never integers. By definition, the harmonic numbers satisfy the recurrence relation H n = H n −1 +1 n, the harmonic numbers are connected to the Stirling numbers of the first kind, H n =1 n. The functions f n = x n n. satisfy the property f n ′ = f n −1, in particular f 1 = x is an integral of the logarithmic function. The harmonic numbers satisfy the series identity ∑ k =1 n H k =, the equality above is straightforward by the simple algebraic identity 1 − x n 1 − x =1 + x + ⋯ + x n −1. The nth harmonic number is about as large as the logarithm of n. The reason is that the sum is approximated by the integral ∫1 n 1 x d x whose value is ln. The values of the sequence Hn - ln decrease monotonically towards the limit lim n → ∞ = γ, a generating function for the harmonic numbers is ∑ n =1 ∞ z n H n = − ln 1 − z, where ln is the natural logarithm. An exponential generating function is ∑ n =1 ∞ z n n, H n = − e z ∑ k =1 ∞1 k k k. = e z Ein where Ein is the exponential integral. Note that Ein = E1 + γ + ln z = Γ + γ + ln z where Γ is the gamma function. The harmonic numbers have several interesting arithmetic properties and it is well-known that H n is an integer if and only if n =1, a result often attributed to Taeisinger
19.
Tree (graph theory)
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In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any connected graph is a tree. A forest is a disjoint union of trees, the various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree itself has been defined by authors as a directed graph. The term tree was coined in 1857 by the British mathematician Arthur Cayley, a tree is an undirected graph G that satisfies any of the following equivalent conditions, G is connected and has no cycles. Any two vertices in G can be connected by a simple path. If G has finitely many vertices, say n of them, then the statements are also equivalent to any of the following conditions. G has no simple cycles and has n −1 edges, an internal vertex is a vertex of degree at least 2. Similarly, a vertex is a vertex of degree 1. An irreducible tree is a tree in which there is no vertex of degree 2, a forest is an undirected graph, all of whose connected components are trees, in other words, the graph consists of a disjoint union of trees. Equivalently, a forest is an acyclic graph. As special cases, an empty graph, a tree. A polytree is an acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, a directed tree is a directed graph which would be a tree if the directions on the edges were ignored, i. e. a polytree. Some authors restrict the phrase to the case where the edges are all directed towards a particular vertex, a rooted tree is a tree in which one vertex has been designated the root. The edges of a tree can be assigned a natural orientation, either away from or towards the root. The tree-order is the ordering on the vertices of a tree with u < v if. A rooted tree which is a subgraph of some graph G is a tree if the ends of every edge in G are comparable in this tree-order whenever those ends are vertices of the tree
20.
Eisenstein prime
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In mathematics, an Eisenstein prime is an Eisenstein integer z = a + b ω that is irreducible in the ring-theoretic sense, its only Eisenstein divisors are the units, a + bω itself and its associates. The associates and the conjugate of any Eisenstein prime are also prime. It follows that the absolute value squared of every Eisenstein prime is a prime or the square of a natural prime. The first few Eisenstein primes that equal a natural prime 3n −1 are,2,5,11,17,23,29,41,47,53,59,71,83,89,101. Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes, some non-real Eisenstein primes are 2 + ω,3 + ω,4 + ω,5 + 2ω,6 + ω,7 + ω,7 + 3ω. Up to conjugacy and unit multiples, the primes listed above, as of March 2017, the largest known Eisenstein prime is the seventh largest known prime 10223 ×231172165 +1, discovered by Péter Szabolcs and PrimeGrid. All larger known primes are Mersenne primes, discovered by GIMPS, real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes are congruent to 1 mod 3, thus no Mersenne prime is an Eisenstein prime
21.
Sphenic number
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In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers. A sphenic number is a product pqr where p, q and this definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance,60 =22 ×3 ×5 has exactly 3 prime factors, the smallest sphenic number is 30 =2 ×3 ×5, the product of the smallest three primes. The first few numbers are 30,42,66,70,78,102,105,110,114,130,138,154,165. As of January 2016 the largest known number is × ×. It is the product of the three largest known primes, all sphenic numbers have exactly eight divisors. If we express the number as n = p ⋅ q ⋅ r, where p, q. For example,24 is not a number, but it has exactly eight divisors. All sphenic numbers are by definition squarefree, because the factors must be distinct. The Möbius function of any number is −1. The cyclotomic polynomials Φ n, taken over all sphenic numbers n, the first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17,1310 = 2×5×131, and 1311 = 3×19×23, there is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. The numbers 2013,2014, and 2015 are all sphenic, the next three consecutive sphenic years will be 2665,2666 and 2667. Semiprimes, products of two prime numbers
22.
Amicable number
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Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes a sequence of period 2. A related concept is that of a number, which is a number that equals the sum of its own proper divisors. Numbers that are members of a sequence with period greater than 2 are known as sociable numbers. The smallest pair of numbers is. The first ten amicable pairs are, and, Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by some of these numbers could be derived was invented circa 850 by the Iraqi mathematician Thābit ibn Qurra. Other Arab mathematicians who studied amicable numbers are al-Majriti, al-Baghdadi, the Iranian mathematician Muhammad Baqir Yazdi discovered the pair, though this has often been attributed to Descartes. Much of the work of Eastern mathematicians in this area has been forgotten, Thābit ibn Qurras formula was rediscovered by Fermat and Descartes, to whom it is sometimes ascribed, and extended by Euler. It was extended further by Borho in 1972, Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs, the second smallest pair, was discovered in 1866 by a then teenage B. Paganini, having been overlooked by earlier mathematicians, by 1946 there were 390 known pairs, but the advent of computers has allowed the discovery of many thousands since then. Exhaustive searches have been carried out to find all pairs less than a bound, this bound being extended from 108 in 1970, to 1010 in 1986,1011 in 1993,1017 in 2015. As of April 2016, there are over 1,000,000,000 known amicable pairs, while these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive. The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the century by the Arab mathematician Thābit ibn Qurra. This formula gives the pairs for n =2, for n =4, and for n =7, Numbers of the form 3×2n −1 are known as Thabit numbers. In order for Ibn Qurras formula to produce an amicable pair, to establish the theorem, Thâbit ibn Qurra proved nine lemmas divided into two groups. The first three deal with the determination of the aliquot parts of a natural integer
23.
Flash point
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The flash point is the lowest temperature at which vapours of a volatile material will ignite, when given an ignition source. The flash point may sometimes be confused with the autoignition temperature, the fire point is the lowest temperature at which the vapor will keep burning after being ignited and the ignition source removed. The fire point is higher than the point, because at the flash point the vapor may be reliably expected to cease burning when the ignition source is removed. The flash point is a characteristic that is used to distinguish between flammable liquids, such as petrol, and combustible liquids, such as diesel. It is also used to characterize the fire hazards of liquids, all liquids have a specific vapor pressure, which is a function of that liquids temperature and is subject to Boyles Law. As temperature increases, vapor pressure increases, as vapor pressure increases, the concentration of vapor of a flammable or combustible liquid in the air increases. Hence, temperature determines the concentration of vapor of the liquid in the air. The flash point is the lowest temperature at which there will be enough flammable vapor to induce ignition when a source is applied. There are two types of flash point measurement, open cup and closed cup. In open cup devices, the sample is contained in a cup which is heated and, at intervals. The measured flash point will vary with the height of the flame above the liquid surface and, at sufficient height. The best-known example is the Cleveland open cup, in both these types, the cups are sealed with a lid through which the ignition source can be introduced. Closed cup testers normally give lower values for the point than open cup and are a better approximation to the temperature at which the vapour pressure reaches the lower flammable limit. The flash point is an empirical measurement rather than a physical parameter. The measured value will vary with equipment and test protocol variations, including temperature ramp rate, time allowed for the sample to equilibrate, sample volume, methods for determining the flash point of a liquid are specified in many standards. For example, testing by the Pensky-Martens closed cup method is detailed in ASTM D93, IP34, ISO2719, DIN51758, JIS K2265 and AFNOR M07-019. Determination of flash point by the Small Scale closed cup method is detailed in ASTM D3828 and D3278, EN ISO3679 and 3680, cEN/TR15138 Guide to Flash Point Testing and ISO TR29662 Guidance for Flash Point Testing cover the key aspects of flash point testing. Gasoline is a used in a spark-ignition engine
24.
Paper
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Paper is a thin material produced by pressing together moist fibres of cellulose pulp derived from wood, rags or grasses, and drying them into flexible sheets. It is a material with many uses, including writing, printing, packaging, cleaning. The modern pulp and paper industry is global, with China leading its production, the oldest known archaeological fragments of the immediate precursor to modern paper, date to the 2nd century BC in China. The pulp papermaking process is ascribed to Cai Lun, a 2nd-century AD Han court eunuch, with paper as an effective substitute for silk in many applications, China could export silk in greater quantity, contributing to a Golden Age. Because of papers introduction to the West through the city of Baghdad, in the 19th century, industrial manufacture greatly lowered its cost, enabling mass exchange of information and contributing to significant cultural shifts. In 1844, the Canadian inventor Charles Fenerty and the German F. G. Keller independently developed processes for pulping wood fibres, before the industrialisation of the paper production the most common fibre source was recycled fibres from used textiles, called rags. The rags were from hemp, linen and cotton, a process for removing printing inks from recycled paper was invented by German jurist Justus Claproth in 1774. Today this method is called deinking and it was not until the introduction of wood pulp in 1843 that paper production was not dependent on recycled materials from ragpickers. The word paper is etymologically derived from Latin papyrus, which comes from the Greek πάπυρος, although the word paper is etymologically derived from papyrus, the two are produced very differently and the development of the first is distinct from the development of the second. Papyrus is a lamination of natural plant fibres, while paper is manufactured from fibres whose properties have changed by maceration. To make pulp from wood, a chemical pulping process separates lignin from cellulose fibres and this is accomplished by dissolving lignin in a cooking liquor, so that it may be washed from the cellulose, this preserves the length of the cellulose fibres. Paper made from chemical pulps are also known as wood-free papers–not to be confused with paper, this is because they do not contain lignin. The pulp can also be bleached to produce paper, but this consumes 5% of the fibres, chemical pulping processes are not used to make paper made from cotton. There are three main chemical pulping processes, the process dates back to the 1840s and it was the dominant method extent before the second world war. Most pulping operations using the process are net contributors to the electricity grid or use the electricity to run an adjacent paper mill. Another advantage is that this process recovers and reuses all inorganic chemical reagents, soda pulping is another specialty process used to pulp straws, bagasse and hardwoods with high silicate content. There are two major mechanical pulps, thermomechanical pulp and groundwood pulp, in the TMP process, wood is chipped and then fed into steam heated refiners, where the chips are squeezed and converted to fibres between two steel discs. In the groundwood process, debarked logs are fed into grinders where they are pressed against rotating stones to be made into fibres
25.
Square pyramidal number
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In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid. The first few square pyramidal numbers are,1,5,14,30,55,91,140,204,285,385,506,650,819 and this is a special case of Faulhabers formula, and may be proved by a mathematical induction. An equivalent formula is given in Fibonaccis Liber Abaci, in modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L of a polyhedron P is a polynomial that counts the number of points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid base is a unit square with integer coordinates. The square pyramidal numbers can also be expressed as sums of binomial coefficients, the smaller tetrahedral number represents 1 +3 +6 + ⋯ + T and the larger 1 +3 +6 + ⋯ + T. Offsetting the larger and adding, we arrive at 1,1 +3,3 +6 …, Square pyramidal numbers are also related to tetrahedral numbers in a different way, P n =14. The sum of two square pyramidal numbers is an octahedral number. Augmenting a pyramid whose base edge has n balls by adding to one of its faces a tetrahedron whose base edge has n −1 balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism and this geometric dissection leads to another relation, P n = n −. Besides 1, there is one other number that is both a square and a pyramid number,4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918, in the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes. Also, P n = − which is the difference of two pentatope numbers and this can be seen by expanding, n − n = n = n and dividing through by 24. A common mathematical puzzle involves finding the number of squares in a n by n square grid. This number can be derived as follows, The number of 1 ×1 boxes found in the grid is n2, the number of 2 ×2 boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of 2 ×2 boxes, the number of k × k boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of k × k boxes and it follows that the number of squares in an n × n square grid is, n 2 +2 +2 +2 + … +12 = n 6
26.
Star Trek
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Star Trek is an American science fiction media franchise based on the television series created by Gene Roddenberry. The first television series, simply called Star Trek and now referred to as The Original Series, debuted in 1966 and it followed the interstellar adventures of Captain James T. Kirk and his crew aboard the starship USS Enterprise, an exploration vessel. The Star Trek canon of the franchise include The Original Series, a series, four spin-off television series, its film franchise. In creating Star Trek, Roddenberry was inspired by the Horatio Hornblower novels, the satirical book Gullivers Travels and these adventures continued in the short-lived Star Trek, The Animated Series and six feature films. The adventures of The Next Generation crew continued in four feature films. In 2009, the franchise underwent a reboot set in an alternate timeline, or Kelvin Timeline. This film featured a new cast portraying younger versions of the crew from the show, their adventures were continued in the sequel film. The thirteenth film feature and sequel, Star Trek Beyond, was released to coincide with the franchises 50th anniversary, a new Star Trek TV series, titled Star Trek, Discovery, will premiere in May 2017 on the digital platform CBS All Access. Star Trek has been a phenomenon for decades. Fans of the franchise are called Trekkies or Trekkers, the franchise spans a wide range of spin-offs including games, figurines, novels, toys, and comics. Star Trek had an attraction in Las Vegas that opened in 1998. At least two museum exhibits of props travel the world, the series has its own full-fledged constructed language, Klingon. Several parodies have been made of Star Trek, in addition, viewers have produced several fan productions. As of July 2016, the franchise had generated $10 billion in revenue, Star Trek is noted for its cultural influence beyond works of science fiction. The franchise is also noted for its civil rights stances. The Original Series included one of televisions first multiracial casts, Star Trek references can be found throughout popular culture from movies such as the submarine thriller Crimson Tide to the animated series South Park. As early as 1964, Gene Roddenberry drafted a proposal for the series that would become Star Trek
27.
Intel 80286
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The Intel 80286 is a 16-bit microprocessor that was introduced on 1 February 1982. It was the first 8086 based CPU with separate, non-multiplexed, address and data buses and also the first with memory management, the 80286 was employed for the IBM PC/AT, introduced in 1984, and then widely used in most PC/AT compatible computers until the early 1990s. Although now long since obsolete for use in computers,80286 based processors are still widely used in embedded microcontroller applications. Intels first 80286 chips were specified for a maximum clockrate of 4,6 or 8 MHz, AMD and Harris later produced 16 MHz,20 MHz and 25 MHz parts, respectively. Intersil and Fujitsu also designed fully static CMOS versions of Intels original depletion-load nMOS implementation, the 6 MHz,10 MHz and 12 MHz models were reportedly measured to operate at 0.9 MIPS,1.5 MIPS and 2.66 MIPS respectively. The later E-stepping level of the 80286 was free of the several significant errata that caused problems for programmers, the 80286 was designed for multi-user systems with multitasking applications, including communications and real-time process control. It had 134,000 transistors and consisted of four independent units, address unit, bus unit, instruction unit, the significantly increased performance over the 8086 was primarily due to the non-multiplexed address and data buses, more address calculation hardware and a faster multiplier. It was produced in a 68-pin package including PLCC, LCC, the performance increase of the 80286 over the 8086 could be more than 100% per clock cycle in many programs. This was an increase, fully comparable to the speed improvements around a decade later when the i486 or the original Pentium were introduced. This was partly due to the address and data buses. They were performed by a unit in the 80286 while the older 8086 had to do effective address computation using its general ALU. Also, the 80286 was more efficient in the prefetch of instructions, buffering, execution of jumps, together with contemporary 80186,286 added new instructions, ENTER, LEAVE, BOUND, INS, OUTS, PUSHA, POPA, PUSH immediate, IMUL immediate, multi-bit shifts. The intel 80286 had a 24-bit address bus and was able to address up to 16 MB of RAM, however cost and initial rarity of software using the memory above 1 MB meant that 80286 computers were rarely shipped with more than one megabyte of RAM. Additionally, there was a performance penalty involved in accessing extended memory from real mode, the 286 was the first of the x86 CPU family to support protected mode. In addition, it was the first commercially available microprocessor with on-chip MMU capabilities and this would allow IBM compatibles to have advanced multitasking OSes for the first time and compete in the Unix-dominated server/workstation market. Several additional instructions were introduced in protected mode of 80286, which are helpful for multitasking operating systems, another important feature of 80286 is Prevention of Unauthorized Access. This is achieved by, Forming different segments for data, code, and stack, segment with lower privilege level cannot access the segment with higher privilege level. By design, the 286 could not revert from protected mode to the basic 8086-compatible real mode without a hardware-initiated reset, though it worked correctly, the method imposed a huge performance penalty
28.
Superfactorial
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In mathematics, the factorial of a non-negative integer n, denoted by n. is the product of all positive integers less than or equal to n. =5 ×4 ×3 ×2 ×1 =120, the value of 0. is 1, according to the convention for an empty product. The factorial operation is encountered in areas of mathematics, notably in combinatorics, algebra. Its most basic occurrence is the fact there are n. ways to arrange n distinct objects into a sequence. This fact was known at least as early as the 12th century, fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial, Now the nature of these methods is such, the factorial function is formally defined by the product n. = ∏ k =1 n k, or by the relation n. = {1 if n =0. The factorial function can also be defined by using the rule as n. All of the above definitions incorporate the instance 0, =1, in the first case by the convention that the product of no numbers at all is 1. This is convenient because, There is exactly one permutation of zero objects, = n. ×, valid for n >0, extends to n =0. It allows for the expression of many formulae, such as the function, as a power series. It makes many identities in combinatorics valid for all applicable sizes, the number of ways to choose 0 elements from the empty set is =0. More generally, the number of ways to choose n elements among a set of n is = n. n, the factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica, although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. There are n. different ways of arranging n distinct objects into a sequence, often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations from a set with n elements, one can obtain such a combination by choosing a k-permutation, successively selecting and removing an element of the set, k times, for a total of n k _ = n ⋯ possibilities. This however produces the k-combinations in an order that one wishes to ignore, since each k-combination is obtained in k. different ways. This number is known as the coefficient, because it is also the coefficient of Xk in n
29.
Untouchable number
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An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. For example, the number 4 is not untouchable as it is equal to the sum of the divisors of 9,1 +3 =4. Thus, if a number n can be written as a sum of two primes, then n+1 is not an untouchable number. Thus it appears that besides 2 and 5, all numbers are composite numbers. No perfect number is untouchable, since, at the very least, similarly, none of the amicable numbers or sociable numbers are untouchable. There are infinitely many numbers, a fact that was proven by Paul Erdős. According to Chen & Zhao, their density is at least d >0.06. No untouchable number is one more than a number, since if p is prime. Also, no number is three more than a prime number, except 5, since if p is an odd prime then the sum of the proper divisors of 2p is p +3. Aliquot sequence nontotient noncototient Weird number Richard K. Guy, Unsolved Problems in Number Theory, Springer Verlag,2004 ISBN 0-387-20860-7, sloanes A070015, Least m such that sum of aliquot parts of m equals n or 0 if no such number exists. The On-Line Encyclopedia of Integer Sequences
30.
Tetration
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In mathematics, tetration is the next hyperoperation after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra-, Tetration is used for the notation of very large numbers. The notation n a means a a ⋅ ⋅ a, the application of exponentiation n −1 times. Shown here are the first four hyperoperations, with tetration as the fourth, multiplication a × n = a + a + ⋯ + a ⏟ n n copies of a combined by addition. Exponentiation a n = a × a × ⋯ × a ⏟ n n copies of a combined by multiplication, Tetration n a = a a ⋅ ⋅ a ⏟ n n copies of a combined by exponentiation, right-to-left. The above example is read as the nth tetration of a, each operation is defined by iterating the previous one. Tetration is not an elementary function, analogously, tetration can be thought of as a chained power involving n numbers a. The parameter a may be called the base-parameter in the following, computer programmers refer to this choice as right-associative. When a and 10 are coprime, we can compute the last m decimal digits of n a using Eulers theorem, there are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale, the term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory, has gained dominance. It was also popularized in Rudy Ruckers Infinity and the Mind, the term superexponentiation was published by Bromer in his paper Superexponentiation in 1987. It was used earlier by Ed Nelson in his book Predicative Arithmetic, the term hyperpower is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence, when considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is misleading, since it is referring to tetration. The term power tower is used, in the form the power tower of order n for a a ⋅ ⋅ a ⏟ n. This is a misnomer, however, because tetration cannot be expressed with iterated power functions, the term snap is occasionally used in informal contexts, in the form a snap n for a a ⋅ ⋅ a ⏟ n. This term is not yet accepted, although it is used within select communities. It is believed to be a reference to Jounce, the derivative of position in physics
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Pseudoforest
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In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. A pseudotree is a connected pseudoforest, the names are justified by analogy to the more commonly studied trees and forests. Gabow and Tarjan attribute the study of pseudoforests to Dantzigs 1963 book on linear programming, pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. The name pseudoforest comes from Picard & Queyranne and we define an undirected graph to be a set of vertices and edges such that each edge has two vertices as endpoints. That is, we allow multiple edges and loops, a subgraph of a graph is the graph formed by any subsets of its vertices and edges such that each edge in the edge subset has both endpoints in the vertex subset. A connected component of a graph is the subgraph consisting of the vertices and edges that can be reached by following edges from a single given starting vertex. A graph is connected if every vertex or edge is reachable from every vertex or edge. A cycle in a graph is a connected subgraph in which each vertex is incident to exactly two edges, or is a loop. A pseudoforest is a graph in which each connected component contains at most one cycle. Equivalently, it is a graph in which each connected component has no more edges than vertices. The components that have no cycles are just trees, while the components that have a cycle within them are called 1-trees or unicyclic graphs. That is, a 1-tree is a graph containing exactly one cycle. A pseudoforest with a connected component is either a tree or a 1-tree. If one removes from a 1-tree one of the edges in its cycle, certain more specific types of pseudoforests have also been studied. A 1-forest, sometimes called a maximal pseudoforest, is a pseudoforest to which no more edges can be added without causing some component of the graph to contain multiple cycles, thus, the 1-forests are exactly the pseudoforests in which every component is a 1-tree. The spanning pseudoforests of an undirected graph G are the pseudoforest subgraphs of G that have all the vertices of G, such a pseudoforest need not have any edges, since for example the subgraph that has all the vertices of G and no edges is a pseudoforest. The maximal pseudoforests of G are the pseudoforest subgraphs of G that are not contained within any larger pseudoforest of G, a maximal pseudoforest of G is always a spanning pseudoforest, but not conversely. If G has no connected components that are trees, then its maximal pseudoforests are 1-forests, like an undirected graph, a directed graph consists of vertices and edges, but each edge is directed from one of its endpoints to the other endpoint
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Endofunction
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In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a map, f, V → V. In general, we can talk about endomorphisms in any category, in the category of sets, endomorphisms are functions from a set S to itself. In any category, the composition of any two endomorphisms of X is again an endomorphism of X and it follows that the set of all endomorphisms of X forms a monoid, denoted End. An invertible endomorphism of X is called an automorphism, the set of all automorphisms is a subset of End with a group structure, called the automorphism group of X and denoted Aut. In the following diagram, the arrows denote implication, Any two endomorphisms of a group, A, can be added together by the rule = f + g. Under this addition, the endomorphisms of a group form a ring. For example, the set of endomorphisms of ℤn is the ring of all n × n matrices with integer entries, the endomorphisms of a vector space or module also form a ring, as do the endomorphisms of any object in a preadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as a near-ring, depending on the additional structure defined for the category at hand, such operators can have properties like continuity, boundedness, and so on. More details should be found in the article about operator theory, an endofunction is a function whose domain is equal to its codomain. A homomorphic endofunction is an endomorphism, let S be an arbitrary set. Among endofunctions on S one finds permutations of S and constant functions associating to each x ∈ S a given c ∈ S, every permutation of S has the codomain equal to its domain and is bijective and invertible. A constant function on S, if S has more than 1 element, has an image that is a subset of its codomain, is not bijective. The function associating to each natural integer n the floor of n/2 has its image equal to its codomain and is not invertible, finite endofunctions are equivalent to directed pseudoforests. For sets of n there are nn endofunctions on the set. Particular bijective endofunctions are the involutions, i. e. the functions coinciding with their inverses
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Centered octagonal number
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A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered octagonal numbers are the same as the odd square numbers, thus, the nth centered octagonal number is given by the formula 2 =4 n 2 −4 n +1. The first few centered octagonal numbers are 1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089. Calculating Ramanujans tau function on an octagonal number yields an odd number
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12 (number)
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12 is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a highly composite number, divisible by 2,3,4. It is central to systems of counting, including the Western calendar and units of time. The word twelve is the largest number with a name in English. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance and it derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels Book of John. It has cognates in every Germanic language, whose Proto-Germanic ancestor has been reconstructed as *twaliƀi, from *twa and suffix *-lif- or *-liƀ- of uncertain meaning. It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19, every other Indo-European language instead uses a form of two+ten, such as the Latin duōdecim. The usual ordinal form is twelfth but dozenth or duodecimal is also used in some contexts, similarly, a group of twelve things is usually a dozen but may also be referred to as a duodecad. The adjective referring to a group of twelve is duodecuple, as with eleven, the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan, with the implicit meaning that two is left after having already counted to ten. The Lithuanian suffix is also considered to share a similar development, the suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten. While, as mentioned above,12 has its own name in Germanic languages such as English and German, it is a number in many other languages, e. g. Italian dodici. In Germany, according to an old tradition, the numbers 0 through 12 were spelt out, the Duden now calls this tradition outdated and no longer valid, but many writers still follow it. Another system spells out all numbers written in one or two words, Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1,2,3,4,6 and 12. Twelve is also a composite number, the next one being twenty-four. Twelve is also a highly composite number, the next one being sixty. It is the first composite number of the form p2q, a square-prime,12 has an aliquot sum of 16. Accordingly,12 is the first abundant number and demonstrates an 8-member aliquot sequence,12 is the 3rd composite number in the 3-aliquot tree, the only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant, Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number