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.
Semiprime
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In mathematics, a semiprime is a natural number that is the product of two prime numbers. The semiprimes less than 100 are 4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94, and 95. Semiprimes that are not perfect squares are called discrete, or distinct, by definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its factors are 1,2,13. The total number of prime factors Ω for a n is two, by definition. A semiprime is either a square of a prime or square-free, the square of any prime number is a semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a number is a semiprime without knowing the two factors. A composite n non-divisible by primes ≤ n 3 is semiprime, various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits. These are considered novelties, since their construction method might prove vulnerable to factorization, for a semiprime n = pq the value of Eulers totient function is particularly simple when p and q are distinct, φ = = p q − +1 = n − +1. If otherwise p and q are the same, φ = φ = p = p2 − p = n − p and these methods rely on the fact that finding two large primes and multiplying them together is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of specific large semiprimes, the most recent such challenge closed in 2007. In practical cryptography, it is not sufficient to choose just any semiprime, the factors p and q of n should both be very large, around the same order of magnitude as the square root of n, this makes trial division and Pollards rho algorithm impractical. At the same time they should not be too close together, or else the number can be quickly factored by Fermats factorization method. The number may also be chosen so that none of p −1, p +1, q −1, or q +1 are smooth numbers, protecting against Pollards p −1 algorithm or Williams p +1 algorithm. However, these checks cannot take future algorithms or secret algorithms into account, in 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. It consisted of 1679 binary digits intended to be interpreted as a 23×73 bitmap image, the number 1679 = 23×73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns. Chens theorem Weisstein, Eric W. Semiprime

19.
38 (number)
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38 is the natural number following 37 and preceding 39. 38 is the 11th distinct semiprime and the 7th in the family and it is the initial member of the third distinct semiprime pair. 38 has a sum of 22 which is itself a distinct semiprime In fact 38 is the first number to be at the head of a chain of four distinct semiprimes in its 8-member aliquot sequence. 38 is the 8th member of the 7-aliquot tree, −1 yields 523022617466601111760007224100074291199999999, which is the 16th factorial prime. There is no answer to the equation φ =38, making 38 a nontotient,38 is the sum of the squares of the first three primes. 37 and 38 are the first pair of positive integers not divisible by any of their digits. 38 is the largest even number which cannot be written as the sum of two odd composite numbers, there are only two normal magic hexagons, order 1 and order 3. The sum of row of an order 3 magic hexagon is 38. The duration of Saros series 38 was 1298.1 years, the lunar eclipse series which began on -1408 April 16 and ended on -111 June 3. The duration of Saros series 38 was 1298.1 years, the New General Catalogue object NGC38, a spiral galaxy in the constellation Pisces Thirty-eight is also, The 38th parallel north is the pre-Korean War boundary between North Korea and South Korea

20.
Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number

21.
Solar day
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Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of time is the day. Two types of time are apparent solar time and mean solar time. Fix a tall pole vertically in the ground, at some instant on any day the shadow will point exactly north or south. That instant is local apparent noon,12,00 local apparent time, about 24 hours later the shadow will again point north/south, the Sun seeming to have covered a 360-degree arc around the Earths axis. When the Sun has covered exactly 15 degrees, local apparent time is 13,00 exactly, after 15 more degrees it will be 14,00 exactly. As explained in the equation of time article, this is due to the eccentricity of the Earths orbit, the effect of this is that a clock running at a constant rate – e. g. This is mean solar time, which is not perfectly constant from one century to the next but is close enough for most purposes. Currently a mean solar day is about 86,400.002 SI seconds, the two kinds of solar time are among the three kinds of time reckoning that were employed by astronomers until the 1950s. By the 1950s it had become clear that the Earths rotation rate was not constant, so astronomers developed ephemeris time, the apparent sun is the true sun as seen by an observer on Earth. Apparent solar time or true solar time is based on the apparent motion of the actual Sun and it is based on the apparent solar day, the interval between two successive returns of the Sun to the local meridian. Solar time can be measured by a sundial. The equivalent on other planets is termed local true solar time, the length of a solar day varies through the year, and the accumulated effect produces seasonal deviations of up to 16 minutes from the mean. The effect has two main causes, first, Earths orbit is an ellipse, not a circle, so the Earth moves faster when it is nearest the Sun and slower when it is farthest from the Sun. Second, due to Earths axial tilt, the Suns annual motion is along a circle that is tilted to Earths celestial equator. In June and December when the sun is farthest from the equator a given shift along the ecliptic corresponds to a large shift at the equator. So apparent solar days are shorter in March and September than in June or December and these lengths will change slightly in a few years and significantly in thousands of years. Mean solar time is the angle of the mean Sun plus 12 hours

22.
Mean tropical year
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Because of the precession of the equinoxes, the seasonal cycle does not remain exactly synchronized with the position of the Earth in its orbit around the Sun. As a consequence, the year is about 20 minutes shorter than the time it takes Earth to complete one full orbit around the Sun as measured with respect to the fixed stars. Since antiquity, astronomers have progressively refined the definition of the tropical year, the entry for year, tropical in the Astronomical Almanac Online Glossary 2015 states, the period of time for the ecliptic longitude of the Sun to increase 360 degrees. The mean tropical year is approximately 365 days,5 hours,48 minutes,45 seconds, an equivalent, more descriptive, definition is The natural basis for computing passing tropical years is the mean longitude of the Sun reckoned from the precessionally moving equinox. Whenever the longitude reaches a multiple of 360 degrees the mean Sun crosses the vernal equinox, the mean tropical year in 2000 was 365.24219 ephemeris days, each ephemeris day lasting 86,400 SI seconds. This is 365.24217 mean solar days, the word tropical comes from the Greek tropikos meaning turn. Thus, the tropics of Cancer and Capricorn mark the north and south latitudes where the Sun can appear directly overhead. Because of this connection between the tropics and the cycle of the apparent position of the Sun, the word tropical also lent its name to the tropical year. The early Chinese, Hindus, Greeks, and others made approximate measures of the tropical year, in the 2nd century BC Hipparchus measured the time required for the Sun to travel from an equinox to the same equinox again. He reckoned the length of the year to be 1/300 of a day less than 365.25 days, Hipparchus used this method because he was better able to detect the time of the equinoxes, compared to that of the solstices. He reckoned the value as 1° per century, a value that was not improved upon until about 1000 years later, since this discovery a distinction has been made between the tropical year and the sidereal year. During the Middle Ages and Renaissance a number of progressively better tables were published that allowed computation of the positions of the Sun, Moon, an important application of these tables was the reform of the calendar. The length of the year was given as 365 solar days 5 hours 49 minutes 16 seconds. This length was used in devising the Gregorian calendar of 1582, in the 16th century Copernicus put forward a heliocentric cosmology. This was actually less accurate than the value of the Alfonsine Tables. Major advances in the 17th century were made by Johannes Kepler, in 1609 and 1619 Kepler published his three laws of planetary motion. In 1627, Kepler used the observations of Tycho Brahe and Waltherus to produce the most accurate tables up to that time and he evaluated the mean tropical year as 365 solar days,5 hours,48 minutes,45 seconds. Newtons three laws of dynamics and theory of gravity were published in his Philosophiæ Naturalis Principia Mathematica in 1687 and these effects did not begin to be understood until Newtons time

23.
Ontario
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Ontario, one of the 13 provinces and territories of Canada, is located in east-central Canada. It is Canadas most populous province by a margin, accounting for nearly 40 percent of all Canadians. Ontario is fourth-largest in total area when the territories of the Northwest Territories and it is home to the nations capital city, Ottawa, and the nations most populous city, Toronto. There is only about 1 km of land made up of portages including Height of Land Portage on the Minnesota border. Ontario is sometimes divided into two regions, Northern Ontario and Southern Ontario. The great majority of Ontarios population and arable land is located in the south, in contrast, the larger, northern part of Ontario is sparsely populated with cold winters and is heavily forested. The province is named after Lake Ontario, a thought to be derived from Ontarí, io, a Huron word meaning great lake, or possibly skanadario. Ontario has about 250,000 freshwater lakes, the province consists of three main geographical regions, The thinly populated Canadian Shield in the northwestern and central portions, which comprises over half the land area of Ontario. Although this area mostly does not support agriculture, it is rich in minerals and in part covered by the Central and Midwestern Canadian Shield forests, studded with lakes, Northern Ontario is subdivided into two sub-regions, Northwestern Ontario and Northeastern Ontario. The virtually unpopulated Hudson Bay Lowlands in the north and northeast, mainly swampy. Southern Ontario which is further sub-divided into four regions, Central Ontario, Eastern Ontario, Golden Horseshoe, the highest point is Ishpatina Ridge at 693 metres above sea level located in Temagami, Northeastern Ontario. In the south, elevations of over 500 m are surpassed near Collingwood, above the Blue Mountains in the Dundalk Highlands, the Carolinian forest zone covers most of the southwestern region of the province. A well-known geographic feature is Niagara Falls, part of the Niagara Escarpment, the Saint Lawrence Seaway allows navigation to and from the Atlantic Ocean as far inland as Thunder Bay in Northwestern Ontario. Northern Ontario occupies roughly 87 percent of the area of the province. Point Pelee is a peninsula of Lake Erie in southwestern Ontario that is the southernmost extent of Canadas mainland, Pelee Island and Middle Island in Lake Erie extend slightly farther. All are south of 42°N – slightly farther south than the border of California. The climate of Ontario varies by season and location, the effects of these major air masses on temperature and precipitation depend mainly on latitude, proximity to major bodies of water and to a small extent, terrain relief. In general, most of Ontarios climate is classified as humid continental, Ontario has three main climatic regions

24.
Driver's license
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The laws relating to the licensing of drivers vary between jurisdictions. In some jurisdictions, a license is issued after the recipient has passed a driving test, while in others, different categories of license often exist for different types of motor vehicles, particularly large trucks and passenger vehicles. The difficulty of the driving test varies considerably between jurisdictions, as do such as age and the required level of practice. Up until the start of the 20th century, European authorities issued licenses to drive motor vehicles similarly ad hoc, the first mandatory license requirement for driving was introduced in the Motor Car Act 1903 in the United Kingdom. Every car owner had to register his automobile with his local government authority, the minimum qualifying age was set at 17. The license gave its holder freedom of the road with a maximum 20 mph speed limit, compulsory testing was introduced in 1934, with the passing of the Road Traffic Act. Prussia, then a state within the German Empire, introduced compulsory licensing on September 29,1903, a test on mechanical aptitude had to be passed and the Dampfkesselüberwachungsverein was charged with conducting these tests. In 1910, the German imperial government mandated the licensing of drivers on a scale, establishing a system of tests. As automobile-related fatalities soared in North America, public outcry provoked legislators to begin studying the French, on August 1,1910, North Americas first licensing law for motor vehicles went into effect in the US state of New York, though it initially applied only to professional chauffeurs. In July 1913, the state of New Jersey became the first to all drivers to pass a mandatory examination before receiving a license. Many countries, including Australia, New Zealand, Canada, the United Kingdom, as many people have drivers licenses, they are often accepted as de facto proof of identity. Most identity cards and drivers licenses are credit card size—the ID-1 size, many European countries require drivers to produce their license on demand when driving. Some European countries require adults to carry proof of identity at all times, in the United Kingdom drivers are not required to carry their licence. A driver may be required by a constable or vehicle examiner to produce their licence, but may provide it in a police station within seven days. In Spain, Sweden and Finland, the license number is the same as the citizens ID number. In Bulgaria, Italy, Poland, Romania and Spain, a vehicle registration card. In Hong Kong a driving license carries the number as the holders ID card. Upon inspection both must be presented, plans to make the newly phased in Smart ID contain driving license information have been shelved

25.
Judaism
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Judaism encompasses the religion, philosophy, culture and way of life of the Jewish people. Judaism is an ancient monotheistic Abrahamic religion, with the Torah as its text, and supplemental oral tradition represented by later texts such as the Midrash. Judaism is considered by religious Jews to be the expression of the relationship that God established with the Children of Israel. With between 14.5 and 17.4 million adherents worldwide, Judaism is the tenth-largest religion in the world, Judaism includes a wide corpus of texts, practices, theological positions, and forms of organization. Modern branches of Judaism such as Humanistic Judaism may be nontheistic, today, the largest Jewish religious movements are Orthodox Judaism, Conservative Judaism and Reform Judaism. Major sources of difference between groups are their approaches to Jewish law, the authority of the Rabbinic tradition. Orthodox Judaism maintains that the Torah and Jewish law are divine in origin, eternal and unalterable, Conservative and Reform Judaism are more liberal, with Conservative Judaism generally promoting a more traditional interpretation of Judaisms requirements than Reform Judaism. A typical Reform position is that Jewish law should be viewed as a set of guidelines rather than as a set of restrictions and obligations whose observance is required of all Jews. Historically, special courts enforced Jewish law, today, these still exist. Authority on theological and legal matters is not vested in any one person or organization, the history of Judaism spans more than 3,000 years. Judaism has its roots as a religion in the Middle East during the Bronze Age. Judaism is considered one of the oldest monotheistic religions, the Hebrews and Israelites were already referred to as Jews in later books of the Tanakh such as the Book of Esther, with the term Jews replacing the title Children of Israel. Judaisms texts, traditions and values strongly influenced later Abrahamic religions, including Christianity, Islam, many aspects of Judaism have also directly or indirectly influenced secular Western ethics and civil law. Jews are a group and include those born Jewish and converts to Judaism. In 2015, the world Jewish population was estimated at about 14.3 million, Judaism thus begins with ethical monotheism, the belief that God is one and is concerned with the actions of humankind. According to the Tanakh, God promised Abraham to make of his offspring a great nation, many generations later, he commanded the nation of Israel to love and worship only one God, that is, the Jewish nation is to reciprocate Gods concern for the world. He also commanded the Jewish people to one another, that is. These commandments are but two of a corpus of commandments and laws that constitute this covenant, which is the substance of Judaism

26.
613 mitzvot
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The tradition that 613 commandments is the number of mitzvot in the Torah, began in the 3rd century CE, when Rabbi Simlai mentioned it in a sermon that is recorded in Talmud Makkot 23b. These principles of Biblical law are sometimes called connections or commandments and referred to collectively as the Law of Moses, Mosaic Law, Sinaitic Law, the word mitzvot is plural, singular is mitzvah. Although there have been attempts to codify and enumerate the commandments contained in the Torah. The 613 commandments include positive commandments, to perform an act, though the number 613 is mentioned in the Talmud, its real significance increased in later medieval rabbinic literature, including many works listing or arranged by the mitzvot. Three types of negative commandments fall under the self-sacrificial principle yehareg veal yaavor and these are murder, idolatry, and forbidden sexual relations. The 613 mitzvot have been divided also into three categories, mishpatim, edot, and chukim. Mishpatim include commandments that are deemed to be self-evident, such as not to murder, edot commemorate important events in Jewish history. For example, the Shabbat is said to testify to the story that Hashem created the world in six days and rested on the seventh day, chukim are commandments with no known rationale, and are perceived as pure manifestations of the Divine will. Many of the mitzvot cannot be observed now, following the destruction of the Second Temple, According to one standard reckoning, there are 77 positive and 194 negative commandments that can be observed today, of which there are 26 commands that apply only within the Land of Israel. Furthermore, there are some time-related commandments from which women are exempt, some depend on the special status of a person in Judaism, while others apply only to men or only to women. 33,04 is to be interpreted to mean that Moses transmitted the Torah from God to the Israelites, Moses commanded us the Torah as an inheritance for the community of Jacob. The Talmud attributes the number 613 to Rabbi Simlai, but other classical sages who hold this view include Rabbi Simeon ben Azzai and it is quoted in Midrash Shemot Rabbah 33,7, Bamidbar Rabbah 13, 15–16,18,21 and Talmud Yevamot 47b. Many Jewish philosophical and mystical works find allusions and inspirational calculations relating to the number of commandments. The tzitzit of the tallit are connected to the 613 commandments by interpretation, principal Torah commentator Rashi bases the number of knots on a gematria, Each tassel has eight threads and five sets of knots, totalling 13. The sum of all numbers is 613 and this reflects the concept that donning a garment with tzitzit reminds its wearer of all Torah commandments. Rabbinic support for the number of commandments being 613 is not without dissent and, even as the number gained acceptance, some rabbis declared that this count was not an authentic tradition, or that it was not logically possible to come up with a systematic count. No early work of Jewish law or Biblical commentary depended on the 613 system, the classical Biblical commentator and grammarian Rabbi Abraham ibn Ezra denied that this was an authentic rabbinic tradition. Nahmanides held that this particular counting was a matter of controversy

27.
Enoch (ancestor of Noah)
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Enoch is a figure in Biblical literature. In the seventh generation from Adam, he was considered the author of the Book of Enoch, in addition to an appearance in the Book of Genesis of the Hebrew Bible, Enoch is the subject of many Jewish and Christian writings. Enoch was the son of Jared, the father of Methuselah, at 65 years old, he begot Methuselah. Regim and Gaidad are also mentioned as his sons according to 2 Enoch, the Bible says that Enoch lived 365 years before he was taken by God. The text reads that Enoch walked with God, and he was no more, for God took him and this Enoch is not to be confused with Cains son Enoch. The Christian New Testament has three references to Enoch from the lineage of Seth, Enoch appears in the Book of Genesis of the Pentateuch as the seventh of the ten pre-Deluge Patriarchs. Genesis recounts that each of the pre-Flood Patriarchs lived for several centuries, Genesis 5 provides a genealogy of these ten figures, providing the age at which each fathered the next, and the age of each figure at death. Enoch is considered by many to be the exception, who is said to not see death. Furthermore, Genesis 5, 22–29 states that Enoch lived 365 years which is short in the context of his peers. The brief account of Enoch in Genesis 5 ends with the note that he not, 3rd Book of Enoch, a Rabbinic text in Hebrew usually dated to the fifth century AD. These recount how Enoch was taken up to Heaven and was appointed guardian of all the treasures, chief of the archangels. He was subsequently taught all secrets and mysteries and, with all the angels at his back, fulfils of his own accord whatever comes out of the mouth of God, executing His decrees. Much esoteric literature like the 3rd Book of Enoch identifies Enoch as the Metatron, the Book of Giants resembles the Book of Enoch, a pseudepigraphical Jewish work from the 3rd century BC. At least six and as many as eleven copies were found among the Dead Sea Scrolls collections, the third-century BC translators who produced the Greek Septuagint rendered the phrase God took him with the Greek verb metatithemi meaning moving from one place to another. Sirach 44,16, from about the period, states that Enoch pleased God and was translated into paradise that he may give repentance to the nations. The Greek word used here for paradise, paradeisos, was derived from an ancient Persian word meaning enclosed garden, later, however, the term became synonymous for heaven, as is the case here. In classical Rabbinical literature, there are views of Enoch. One view regarding Enoch was that found in the Targum Pseudo-Jonathan, which thought of Enoch as a man, taken to Heaven

28.
Entering heaven alive
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Entering Heaven alive is a belief held to be true by multiple religions. Unlike the other entries in this article, this paragraph does not, in the view of most Christians, Jesus is considered by the vast majority of Christians to have died before being resurrected and ascending to heaven. The minority views that Jesus didnt die are known as the Swoon hypothesis, in the Roman Catholic Church, the Ascension of the Lord is a Holy Day of Obligation. In the Eastern Orthodox Church the Ascension is one of twelve Great Feasts, in the Reformed churches tradition of Calvinism, belief in the ascension of Christ is included in the Westminster Confession of Faith, the Heidelberg Catechism and the Second Helvetic Confession. The Rapture is a reference to being caught up as found in 1 Thessalonians 4,17, when the dead in Christ and we who are alive and remain will be caught up in the clouds to meet the Lord. The Roman Catholic Church distinguishes between The Ascension, in which Christ rose to heaven by his own power, and The Assumption in which Mary, mother of Jesus, was raised to heaven by Gods power. On November 1,1950, Pope Pius XII, acting ex cathedra, issued Munificentissimus Deus, the doctrine is based on Sacred Tradition that Mary, mother of Jesus, was bodily assumed into heaven. For centuries before that, the assumption was celebrated in art, the proclamation leaves open whether or not Mary died before assumption into heaven. Some theologians have argued that Mary didnt die, but the dogma itself doesnt say this, the Eastern Orthodox Church teaches that three other persons were taken bodily into heaven, Enoch, Elijah and the Theotokos. Similar to the Western Assumption of Mary, the Orthodox celebrate the Dormition of the Theotokos on August 15. The church teaches that the Apostles received a revelation during which the Theotokos appeared to them and told them she had been resurrected by Jesus and taken body and soul into heaven. The Orthodox teach that Mary already enjoys the fullness of heavenly bliss that the saints will experience only after the Last Judgment. According to Revelation, they will be resurrected and ascend again to heaven, simon Magus, a first-century Gnostic who claimed to be an incarnation of God reportedly had the ability to levitate, along with many other magical powers. As a dissenter from the Proto-orthodox Christianity of the time, this was branded by Christians as evil magic and he is said to have attempted to levitate to the heavens from the Roman Forum, but fell back to earth and injured himself. Apollonius of Tyana was said to have been assumed into Elysium by Philostratus, yudhishthira of the Mahabharata is believed to be the only human to cross the plane between mortals and heaven in his mortal body. Sant Tukaram was taken to Vaikunta on Garuda Vahan which was witnessed by all the village people, chaitanya Mahaprabhu disappeared after entering the temple deity room of Lord Jagannath. Ramalinga Swamigal, a great Sage revered by his teaching, ramalinga supposedly attained the Supreme Body of the Godhead when Divinity itself merged with him. He was reported to have disappeared after deciding to de-materialize his immortal body by his own free will and she believed that she could create for herself a new kind of “light body”

29.
Abraxas
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Abraxas was a word of mystic meaning in the system of the Gnostic Basilides, being there applied to the Great Archon, the princeps of the 365 spheres. The word is found in Gnostic texts such as the Holy Book of the Great Invisible Spirit and it was engraved on certain antique gemstones, called on that account Abraxas stones, which were used as amulets or charms. As the initial spelling on stones was Abrasax, the spelling of Abraxas seen today probably originates in the confusion made between the Greek letters Sigma and Xi in the Latin transliteration, the seven letters spelling its name may represent each of the seven classic planets. The word may be related to Abracadabra, although other explanations exist, opinions abound on Abraxas, who in recent centuries has been claimed to be both an Egyptian god and a demon. It is uncertain what the role and function of Abraxas was in the Basilidian system. They in turn originate a series, who create a second heaven. The process continues in like manner until 365 heavens are in existence, the ruler of the 365 heavens is Abraxas, and for this reason he contains within himself 365 numbers. The name occurs in the Refutation of all Heresies by Hippolytus, the author of the appendix to Tertullian De Praescr. Nothing can be built on the allusions of Jerome, according to whom Abraxas meant for Basilides the greatest God, the highest God, the Almighty God. The notices in Theodoret, Augustine, and Praedestinatus, have no independent value, with the availability of primary sources, such as those in the Nag Hammadi library, the identity of Abrasax remains unclear. The Holy Book of the Great Invisible Spirit, for instance, refers to Abrasax as an Aeon dwelling with Sophia and he further indicated the Basilidians attributed to Abraxas the rule over 365 skies and 365 virtues. In a final statement on Basilidians, de Plancy states that their view was that Jesus Christ was merely a benevolent ghost sent on Earth by Abracax, a vast number of engraved stones are in existence, to which the name Abrasax-stones has long been given. One particularly fine example was included as part of the Thetford treasure from fourth century Norfolk, the subjects are mythological, and chiefly grotesque, with various inscriptions, in which ΑΒΡΑΣΑΞ often occurs, alone or with other words. Sometimes the whole space is taken up with the inscription, the meaning of the legends is seldom intelligible, but some of the gems are amulets, and the same may be the case with nearly all. The Abrasax-image alone, without external Iconisms, and either without, or but a simple, inscription. The Abrasax-imago proper is found with a shield, a sphere or wreath and whip, a sword or sceptre, a cocks head, the body clad with armor. There are, however, innumerable modifications of these figures, Lions, hawks, and eagles skins, with or without mottos, with or without a trident and star, Abrasax combined with other Gnostic Powers. The name ΙΑΩ, to which ΣΑΒΑΩΘ is sometimes added, is found with this even more frequently than ΑΒΡΑΣΑΞ

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.
Heaven
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According to the beliefs of some religions, heavenly beings can descend to earth or incarnate, and earthly beings can ascend to Heaven in the afterlife, or in exceptional cases enter Heaven alive. Some believe in the possibility of a Heaven on Earth in a World to Come, another belief is in an axis mundi or world tree which connects the heavens, the terrestrial world, and the underworld. In Indian religions, Heaven is considered as Svarga loka, and this cycle can be broken after a soul achieves Moksha or Nirvana. Any place of existence, either of humans, souls or deities, the modern English word heaven is derived from the earlier heven, this in turn was developed from the previous Old English form heofon. By about 1000, heofon was being used in reference to the Christianized place where God dwells, all of these have been derived from a reconstructed Proto-Germanic form *Hemina-. In Ancient Egyptian religion, belief in an afterlife is much more stressed than in ancient Judaism, Heaven was a physical place far above the Earth in a dark area of space where there were no stars, basically beyond the Universe. Their heart would finally be weighed with the feather of truth, almost nothing is known of Bronze Age Canaanite views of Heaven, and the archeological findings at Ugarit have not provided information. The 1st century Greek author Philo of Byblos may preserve elements of Iron Age Phoenician religion in his Sanchuniathon, in the Middle Hittite myths, Heaven is the abode of the gods. In the Song of Kumarbi, Alalu was king in Heaven for nine years before giving birth to his son, Anu was himself overthrown by his son, Kumarbi. The Baháí Faith regards the description of Heaven as a specific place as symbolic. The Baháí writings describe Heaven as a condition where closeness to God is defined as Heaven. For Baháís, entry into the life has the potential to bring great joy. Baháulláh likened death to the process of birth and he explains, The world beyond is as different from this world as this world is different from that of the child while still in the womb of its mother. Accordingly, Baháís view life as a stage, where one can develop. The key to progress is to follow the path outlined by the current Manifestation of God. Baháulláh wrote, Know thou, of a truth, that if the soul of man hath walked in the ways of God, it will, assuredly return, in Buddhism there are several Heavens, all of which are still part of samsara. Those who accumulate good karma may be reborn in one of them, however, their stay in Heaven is not eternal—eventually they will use up their good karma and will undergo rebirth into another realm, as a human, animal or other being. Because Heaven is temporary and part of samsara, Buddhists focus more on escaping the cycle of rebirth, Nirvana is not a heaven but a mental state