1.
Integer
–
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
Integer
–
Algebraic structure → Group theory
Group theory
2.
Negative number
–
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
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
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
100 (number)
–
The
U.S. hundred-dollar bill, Series 2009.
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
Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.
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
Greek numerals
–
Numeral systems
Greek numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
6.
Roman numerals
–
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
Roman numerals
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Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
–
Numeral systems
Roman numerals
–
A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
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An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
7.
Binary number
–
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
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
8.
Ternary numeral system
–
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
Ternary numeral system
–
Numeral systems
9.
Quaternary numeral system
–
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
Quaternary numeral system
–
Numeral systems
10.
Quinary
–
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
Quinary
–
Numeral systems
11.
Senary
–
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
Senary
–
Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting
Senary
12.
Octal
–
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
Octal
–
Numeral systems
13.
Duodecimal
–
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
Duodecimal
–
Numeral systems
Duodecimal
–
A duodecimal multiplication table
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
Hexadecimal
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Numeral systems
Hexadecimal
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Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
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Hexadecimal finger-counting scheme.
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
Vigesimal
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Numeral systems
Vigesimal
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The
Maya numerals are a base-20 system.
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
Base 36
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Numeral systems
Base 36
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34 senary = 22 decimal, in senary finger counting
Base 36
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
Natural number
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The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
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Natural numbers can be used for counting (one
apple, two apples, three apples, …)
18.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime number
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The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
19.
Christian Broadcasting Network
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The Christian Broadcasting Network is an American Christian-oriented religious television network and production company. Founded by televangelist Pat Robertson, its headquarters and main studios are based in Virginia Beach, CBN was founded by televangelist Pat Robertson in 1961, using a religious variety program format that has been successfully used in religious broadcasting ever since. One of the companys mainstays is The 700 Club, the program in the variety format. CBN also operates online channels on its website, such as the CBN News Channel, the company also produces versions of The 700 Club aimed at Latin American and British audiences. CBN has broadcast programs in over 70 languages, on April 29,1977, CBN launched a religious cable network, the CBN Satellite Service. The channel was revamped as the CBN Cable Network in 1981. In September 1990, it rebranded as The Family Channel, IFE later sold it to News Corporation in 1997, which later sold it to The Walt Disney Company in 2001. The terms of the sale to International Family Entertainment stipulated that the channel continue carrying two CBN programs, including The 700 Club. It is often thought the deal stipulated that the channel maintain the word Family in its name in perpetuity, CBN and Regent University jointly produced the film First Landing. The secular commercial stations that continue to air The 700 Club in syndication air CBNs annual telethon during the last week of January. CBN entered into the industry in 1960, when Robertson founded WYAH-TV in Portsmouth. CBN later signed on WHAE-TV in Atlanta, Georgia in June 1971, finally, it signed on WXNE-TV in Boston in October 1977. CBN gradually sold its stations during the late 1980s and 1990s and it retained ownership of KXTX until 2000, when it sold the station to NBC, which converted it into a Telemundo owned-and-operated station. This station group was split up after CBN sold the licenses to separate owners, in addition, CBN planned to build a television station in Richmond, Virginia, WRNX on UHF channel 63. However, CBN sold the permit for that station to Capitol Christian Television in 1983. That station was sold and in 1986, converted into secular independent station WVRN. The 700 Club – a daily newsmagazine that debuted in 1966, one of the longest runs of any program within that genre, the program is hosted by Pat Robertson, Terry Meeuwsen and Gordon Robertson. Club 700 Hoy – a half-hour weekly Spanish-language version of The 700 Club that is syndicated throughout Latin America, the magazine-style formatted morning program features opinions on current issues, interviews, informative features, stories about people, places and music, and life advice
Christian Broadcasting Network
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Christian Broadcasting Network
20.
The 700 Club
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The 700 Club is the flagship television program of the Christian Broadcasting Network, airing in syndication throughout the United States and available worldwide on CBN. com. Airing each weekday, the magazine program features live guests, daily news, contemporary music, testimonies. Celebrities and other guests are interviewed, and Christian lifestyle issues are presented. The program additionally features major world news stories plus in-depth investigative reporting by the CBN News team and it is hosted by Pat Robertson, Gordon Robertson, Terry Meeuwsen, and Wendy Griffith. Previous co-hosts include Ben Kinchlow, Sheila Walsh, Danuta Rylko Soderman, Kristi Watts, tim Robertson served as host for a year, along with Kinchlow and actress Susan Howard, while Pat Robertson ran unsuccessfully for President of the United States in the 1988 campaign. In 1960, Pat Robertson, the son of former U. S, senator Absalom Willis Robertson, purchased the license for WTOV-TV, channel 27 in Portsmouth, Virginia, which had ceased operation because of poor viewership. Renamed WYAH-TV, the station began broadcasting Christian programming to the Hampton Roads area on October 1,1961, in 1962, the station suffered financially and almost closed. To keep the station on the air, WYAH produced a special edition of the show. For the telethon, Robertson set a goal of 700 members each contributing $10.00 per month, Robertson referred to these members as the 700 Club and the name stuck. The telethon was successful and is held annually. After the telethon in 1966, The 700 Club continued as a nightly, two-hour Christian variety program of music, preaching, group prayer, Bible study, the music was hymns, instrumental pieces, southern gospel music, and urban gospel music. The first permanent host of the program was Jim Bakker who, along with his then-wife Tammy Faye Bakker, the couple left CBN in 1972, reportedly Jim Bakker was fired by Pat Robertson over philosophical differences. The Bakkers then moved on to launch the Trinity Broadcasting Network before starting their own television ministry and signature show. After the Bakkers left, some staffers at the station responded by destroying the Bakkers sets and puppets. Pat Robertson took over as host, and evolved his 700 Club by cutting back on music, Robertson transformed the 700 Club from a nightly religious-themed telethon to a Christian talk show. The 700 Club originally aired only on WYAH-TV and other CBN-owned stations in Atlanta and Dallas, the roster of stations carrying the program grew to over 100 markets by 1976. In some markets, the show aired on stations, choosing between either the full 90-minute version or an edited 60-minute version. In 1977, The 700 Club received additional exposure nationally on the newly launched CBN Cable Network where, like CBNs broadcast outlets, it aired three times daily
The 700 Club
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The 700 Club
21.
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
Eisenstein prime
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Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3 n − 1. All others have an absolute value squared equal to a natural prime.
22.
Pronic number
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A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers, however, the rectangular number name has also been applied to the composite numbers. The first few numbers are,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462 …. The nth pronic number is also the difference between the odd square 2 and the st centered hexagonal number. The sum of the reciprocals of the numbers is a telescoping series that sums to 1,1 =12 +16 +112 ⋯ = ∑ i =1 ∞1 i. The partial sum of the first n terms in this series is ∑ i =1 n 1 i = n n +1, the nth pronic number is the sum of the first n even integers. It follows that all numbers are even, and that 2 is the only prime pronic number. It is also the only number in the Fibonacci sequence. The number of entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a number is present in only one of the factors n or n+1. Thus a pronic number is squarefree if and only if n and n +1 are also squarefree, the number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n +1. If 25 is appended to the representation of any pronic number. This is because 2 =100 n 2 +100 n +25 =100 n +25
Pronic number
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Overview
23.
Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime
Triangular number
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The first six triangular numbers
24.
Hexagonal number
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A hexagonal number is a figurate number. The formula for the nth hexagonal number h n =2 n 2 − n = n =2 n ×2. The first few numbers are,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,780,861,946. Every hexagonal number is a number, but only every other triangular number is a hexagonal number. Like a triangular number, the root in base 10 of a hexagonal number can only be 1,3,6. The digital root pattern, repeating every nine terms, is 166193139. Every even perfect number is hexagonal, given by the formula M p 2 p −1 = M p /2 = h /2 = h 2 p −1 where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal, for example, the 2nd hexagonal number is 2×3 =6, the 4th is 4×7 =28, the 16th is 16×31 =496, and the 64th is 64×127 =8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130, adrien-Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. Hexagonal numbers can be rearranged into rectangular numbers of n by. Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages, to avoid ambiguity, hexagonal numbers are sometimes called cornered hexagonal numbers. One can efficiently test whether a positive x is an hexagonal number by computing n =8 x +1 +14. If n is an integer, then x is the nth hexagonal number, if n is not an integer, then x is not hexagonal. The nth number of the sequence can also be expressed by using Sigma notation as h n = ∑ i =0 n −1 where the empty sum is taken to be 0. Centered hexagonal number Mathworld entry on Hexagonal Number
Hexagonal number
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Contents
25.
Northern Virginia
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With 2.8 million residents, it is the most populous region of Virginia and the Washington metropolitan area. Communities in the form the Virginia portion of the Washington metropolitan area. Notable features of the include the Pentagon and the Central Intelligence Agency, and the many companies which serve them. The areas attractions include various monuments and Colonial and Civil War-era sites such as Mount Vernon and Arlington National Cemetery and it is the most affluent region in the nation. g. Eastern United States vs. western Massachusetts, the name Northern Virginia does not seem to have been used in the early history of the area. The Fairfax line, surveyed in 1746, ran from the first spring of the Potomac to the first spring of the Rappahannock, at the head of the Conway River. The Northern Neck was composed of 5,282,000 acres, at some point, these eastern counties came to be called separately simply the Northern Neck, and, for the remaining area west of them, the term was no longer used. One of the most prominent early mentions of Northern Virginia as a title was the naming of the Confederate Army of Northern Virginia during the American Civil War. The most common definition of Northern Virginia includes those counties and independent cities on the Virginia side of the Washington-Baltimore-Northern Virginia Combined Statistical Area. Most narrowly defined, Northern Virginia consists of the counties of Arlington and Fairfax, as well as the independent cities of Alexandria, Falls Church, businesses, governments and non-profit agencies may define the area considered Northern Virginia differently for various purposes. Eight of his supporters were named, among them Thomas Culpeper. On February 25,1673, a new charter was given to Thomas Lord Culpeper, Lord Culpeper was named the Royal Governor of Virginia from 1677–1683. Culpeper County was later named for him when it was formed in 1749, however, in 1682 rioting in the colony forced him to return, but by the time he arrived, the riots were already quelled. After apparently misappropriating £9,500 from the treasury of the colony, he returned to England, during this tumultuous time, Culpepers erratic behavior meant that he had to rely increasingly on his cousin and Virginia agent, Col. Nicholas Spencer. Spencer succeeded Culpeper as acting Governor upon Lord Culpepers departure from the colony, for many years, Lord Culpepers descendants allowed men in Virginia to manage the properties. The lands of Lord Fairfax were defined as that between the Rappahannock and Potomac rivers, and were called the Northern Neck. In 1746 a back line was surveyed and established between the headwaters of the Potomac and Rappahannock rivers, defining the west end of the grants. According to documents held by the Handley Regional Library of the Winchester–Frederick County Historical Society, Lord Fairfax was a lifelong bachelor, and became one of the more well-known persons of the late colonial era
Northern Virginia
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Reston, an internationally known
planned community, seen from the
Dulles Toll Road
Northern Virginia
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Northern Virginia
megaprojects
Northern Virginia
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Map of the
Northern Neck Proprietary land grant c. 1737
Northern Virginia
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Mount Vernon, the plantation home of
George Washington
26.
Body mass index
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The body mass index or Quetelet index is a value derived from the mass and height of an individual. The BMI is defined as the body divided by the square of the body height. The BMI is an attempt to quantify the amount of mass in an individual. However, there is debate about where on the BMI scale the dividing lines between categories should be placed. Commonly accepted BMI ranges are underweight, under 18.5 kg/m2, normal weight,18.5 to 25, overweight,25 to 30, obese, over 30. The modern term body mass index for the ratio of body weight to squared height was coined in a paper published in the July 1972 edition of the Journal of Chronic Diseases by Ancel Keys. In this paper, Keys argued that what he termed the BMI was, BMI was explicitly cited by Keys as appropriate for population studies and inappropriate for individual evaluation. Nevertheless, due to its simplicity, it has come to be used for preliminary diagnosis. Additional metrics, such as waist circumference, can be more useful, the BMI is universally expressed in kg/m2, resulting from mass in kilograms and height in metres. If pounds and inches are used, a factor of 703 / must be applied. When the term BMI is used informally, the units are usually omitted, BMI was designed to be used as a simple means of classifying average sedentary populations, with an average body composition. Some athletes, such as linemen, have a high muscle to fat ratio. BMI is proportional to the mass and inversely proportional to the square of the height, so, if all body dimensions double, and mass scales naturally with the cube of the height, then BMI doubles instead of remaining the same. This results in people having a reported BMI that is uncharacteristically high. In comparison, the Ponderal index is based on the scaling of mass with the third power of the height. However, many people are not just scaled up short people. Nick Korevaar suggests that instead of squaring the body height or cubing the body height, carl Lavie has written that, The B. M. I. Tables are excellent for identifying obesity and body fat in large populations, a frequent use of the BMI is to assess how much an individuals body weight departs from what is normal or desirable for a persons height
Body mass index
–
A graph of body mass index as a function of body mass and body height. The dashed lines represent subdivisions within a major class.
27.
Charlotte, NC
–
Charlotte /ˈʃɑːrlət/ is the largest city in the state of North Carolina. It is the county seat of Mecklenburg County and the second-largest city in the southeastern United States, just behind Jacksonville, Charlotte is the third-fastest growing major city in the United States. In 2014 the estimated population of Charlotte according to the U. S. Census Bureau was 809,958, the Charlotte metropolitan area ranks 22nd-largest in the U. S. and had a 2014 population of 2,380,314. The Charlotte metropolitan area is part of a sixteen-county market region or combined statistical area with a 2014 U. S. Census population estimate of 2,537,990, residents of Charlotte are referred to as Charlotteans. It is listed as a global city by the Globalization. Charlotte Douglas International Airport is an international hub, and was ranked the 23rd-busiest airport in the world by passenger traffic in 2013. Charlotte has a subtropical climate. The city is located several miles east of the Catawba River and southeast of Lake Norman, Lake Wylie and Mountain Island Lake are two smaller man-made lakes located near the city. The Catawba Native Americans were the first to settle Mecklenburg County and were first recorded in European records around 1567, by 1759 half the Catawba tribe had been killed by smallpox. At the time of their largest population, Catawba people numbered 10,000, Mecklenburg County was initially part of Bath County of New Hanover Precinct, which became New Hanover County in 1729. The western portion of New Hanover split into Bladen County in 1734, Mecklenburg County formed from Anson County in 1762. Further apportionment was made in 1792, with Cabarrus County formed from Mecklenburg and these areas were all part of one of the original six judicial/military districts of North Carolina known as the Salisbury District. The area that is now Charlotte was settled by people of European descent around 1755, Thomas Polk, who later married Thomas Spratts daughter, built his house by the intersection of two Native American trading paths between the Yadkin and Catawba rivers. One path ran north–south and was part of the Great Wagon Road, within decades of Polks settling, the area grew to become Charlotte Town, incorporating in 1768. The crossroads, perched atop the Piedmont landscape, became the heart of Uptown Charlotte, in 1770, surveyors marked the streets in a grid pattern for future development. The east–west trading path became Trade Street, and the Great Wagon Road became Tryon Street, in honor of William Tryon, the intersection of Trade and Tryon—commonly known today as Trade & Tryon, or simply The Square—is more properly called Independence Square. While surveying the boundary between the Carolinas in 1772, William Moultrie stopped in Charlotte Town, whose five or six houses were very ordinary built of logs, local leaders came together in 1775 and signed the Mecklenburg Resolves, more popularly known as the Mecklenburg Declaration of Independence. While not a declaration of independence from British rule, it is among the first such declarations that eventually led to the American Revolution
Charlotte, NC
–
Clockwise:
UNC Charlotte,
Harvey B. Gantt Center for African-American Arts + Culture,
Duke Energy Center, Charlotte's skyline, First Presbyterian Church of Charlotte, Charlotte Main Library and
NASCAR Hall of Fame building
Charlotte, NC
–
View of the Old Court House, Charlotte, 1888.
Charlotte, NC
–
Uptown Charlotte's skyline
Charlotte, NC
–
Little Sugar Creek Greenway in winter
28.
Sphenic number
–
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
Sphenic number
–
Overview
29.
Telecommunications Relay Service
–
Originally, relay services were designed to be connected through a TDD, teletypewriter or other assistive telephone device. Services gradually have expanded to include almost any real-time text capable technology such as a computer, laptop, mobile phone, PDA. The first TTY was invented by deaf scientist Robert Weitbrecht in 1964, the first relay service was established in 1974 by Converse Communications of Connecticut. Depending on the technical and physical abilities and physical environments of users, once the most common type of TRS call, TTY calls involve a call from a deaf or hard-of-hearing person who utilizes a TTY to a hearing person. In this type of call, typed messages are relayed as voice messages by a TRS operator and this allows callers who are unable to use a regular telephone to be able to place calls to people who use a regular telephone and vice versa. This mode of communication has largely superseded by other modes of communications, including the utilization of IP relay, VPs, VRS. A common type of call is voice carry over, VCO and this allows a person who is hard of hearing or deaf but can speak to use their voice while receiving responses from a person who is hearing via the operators typed text. There are many variations of VCO, including two-line VCO and VCO with privacy, the operator will not hear the VCO users voiced messages and the VCO user does not need to voice GA. The operator will hear the person who is hearing, and the person who is hearing must give the GA each time to alert the operator it is the VCO users turn. The VCO user does not need to voice GA, because the VCO user types it or presses the VCO GA button on the VCO phone when its the voice users turn to talk. Two-line VCO allows a VCO user using a TTY or computer to call a TRS operator, who in turn calls the VCO user on a telephone line. The user puts the operator on a hold to initiate a three-way call with the hearing person. This method is used by people who are hard of hearing. With two-line VCO, the VCO user and the user can interrupt each other. VCO with Privacy cannot be used with two-line VCO, because the operator, VCO user, a less common call type is hearing carry over. HCO allows a person who is speech-disabled but can hear to use their hearing while sending responses to a person who is hearing via the HCO users typed text. The operator voices the HCO users typed messages, and then the HCO users picks up the handset, there are many variations of HCO, including two-line HCO and HCO with privacy. The operator will not hear the voice users voiced messages and the user does not need to voice GA
Telecommunications Relay Service
–
The examples and perspective in this article deal primarily with the United States and do not represent a
worldwide view of the subject. Please improve this article and discuss the issue on the
talk page. (September 2010)
30.
Houston, TX
–
Houston is the most populous city in the state of Texas and the fourth-most populous city in the United States. With a census-estimated 2014 population of 2.239 million within an area of 667 square miles, it also is the largest city in the southern United States and the seat of Harris County. Located in Southeast Texas near the Gulf of Mexico, it is the city of Houston–The Woodlands–Sugar Land. Houston was founded on August 28,1836, near the banks of Buffalo Bayou and incorporated as a city on June 5,1837. The city was named after former General Sam Houston, who was president of the Republic of Texas and had commanded, the burgeoning port and railroad industry, combined with oil discovery in 1901, has induced continual surges in the citys population. Houstons economy has an industrial base in energy, manufacturing, aeronautics. Leading in health care sectors and building equipment, Houston has more Fortune 500 headquarters within its city limits than any city except for New York City. The Port of Houston ranks first in the United States in international waterborne tonnage handled, the city has a population from various ethnic and religious backgrounds and a large and growing international community. Houston is the most diverse city in Texas and has described as the most diverse in the United States. It is home to cultural institutions and exhibits, which attract more than 7 million visitors a year to the Museum District. Houston has a visual and performing arts scene in the Theater District. In August 1836, two real estate entrepreneurs from New York, Augustus Chapman Allen and John Kirby Allen, purchased 6,642 acres of land along Buffalo Bayou with the intent of founding a city. The Allen brothers decided to name the city after Sam Houston, the general at the Battle of San Jacinto. The great majority of slaves in Texas came with their owners from the slave states. Sizable numbers, however, came through the slave trade. New Orleans was the center of trade in the Deep South. Thousands of enslaved African Americans lived near the city before the Civil War, many of them near the city worked on sugar and cotton plantations, while most of those in the city limits had domestic and artisan jobs. Houston was granted incorporation on June 5,1837, with James S. Holman becoming its first mayor, in the same year, Houston became the county seat of Harrisburg County and the temporary capital of the Republic of Texas
Houston, TX
–
Clockwise from top: Sam Houston monument,
Downtown Houston,
Houston Ship Channel,
The Galleria,
University of Houston, and the
Christopher C. Kraft Jr. Mission Control Center.
Houston, TX
–
Sam Houston
Houston, TX
–
Houston, c. 1873
Houston, TX
–
Union Station, Houston, Texas (postcard, c. 1911)
31.
Judaism
–
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
Judaism
–
Judaica (clockwise from top):
Shabbat candlesticks,
handwashing cup,
Chumash and
Tanakh,
Torah pointer,
shofar and
etrog box
Judaism
–
Silver case containing a handwritten
Torah (
Museum of Jewish Art and History, Paris)
Judaism
–
Glass platter inscribed with the Hebrew word zokhreinu – remember us
Judaism
–
A 19th-century silver
Macedonian Hanukkah menorah
32.
Mezuzah
–
A mezuzah comprises a piece of parchment called a klaft inscribed with specific Hebrew verses from the Torah. These verses consist of the Jewish prayer Shema Yisrael, beginning with the phrase, Hear, O Israel, the LORD our God, the LORD is One. In mainstream Rabbinic Judaism, a mezuzah is affixed to the doorpost of Jewish homes to fulfill the mitzvah to write the words of God on the gates and doorposts of your house. Some interpret Jewish law to require a mezuzah in every doorway in the home except bathrooms laundry rooms and closets, if they are too small to qualify as rooms. The klaft parchment is prepared by a scribe who has undergone many years of meticulous training. The parchment is then rolled up and placed inside the case and this article deals mainly with the mezuzah as it is used in Rabbinic Judaism. Karaite Judaism and Samaritanism have their own distinct traditions, in Karaite Judaism the deuteronomic verse And you shall write them on the doorposts of your houses and your gates is interpreted to be a metaphor and not as referring to the Rabbanite mezuzah. Thus Karaites do not traditionally use mezuzot, but put up a plaque in the shape of the two Tables of the Law with the Ten Commandments. In Israel, where they try not to make other Jews feel uncomfortable, many Karaites make an exception. The Karaite version of the mezuzah is fixed to the doorways of public buildings and sometimes to private buildings, the Samaritans interpret the deuteronomic commandment to mean displaying any select text from the Samaritan version of the five Books of Moses. This can contain a blessing or a holy or uplifting message. Nowadays a Samaritan mezuzah is usually made of marble, a wooden plate, or a sheet of parchment or high quality paper. This they place either above the door, or inside the house. These mezuzot are found in every Samaritan household as well as in the synagogue, today some Samaritans would also use a Jewish-style mezuzah case and place inside it a small written Samaritan scroll, i. e. a text from the Samaritan Torah, written in the Samaritan alphabet. The more such mezuzot there are in the house, the better it is considered to be. According to halakha, the mezuzah should be placed on the side of the door or doorpost, in the upper third of the doorpost. Care should be taken to not tear or damage the parchment or the wording on it, as this will invalidate the mezuzah, generally, halakha requires that mezuzot be affixed within 30 days of moving into a rented house or apartment. This applies to Jews living in the Diaspora, for a purchased home or apartment in the Diaspora, or a residence in Israel, the mezuzah is affixed immediately upon moving in
Mezuzah
–
The parchment of the mezuzah
Mezuzah
–
Mezuzah affixed to a door frame on South Street in Philadelphia.
Mezuzah
–
A Macedonian mezuzah
Mezuzah
–
Clear mezuzah case with Hebrew letter ש (Shin), Jerusalem, Israel
33.
Orange County, California
–
Orange County is a county in the U. S. state of California. As of the 2010 census, the population was 3,010,232 making it the third-most populous county in California, the sixth-most populous in the United States and its county seat is Santa Ana. It is the second most densely populated county in the state, the countys four largest cities, Anaheim, Santa Ana, Irvine, and Huntington Beach each have populations exceeding 200,000. Several of Orange Countys cities are on the Pacific coast, including Huntington Beach, Newport Beach, Laguna Beach, Orange County is included in the Los Angeles-Long Beach-Anaheim, CA Metropolitan Statistical Area. Thirty-four incorporated cities are located in the county, the newest is Aliso Viejo, Anaheim was the first city, incorporated in 1870, when the region was still part of neighboring Los Angeles County. Whereas most population centers in the United States tend to be identified by a major city and it is mostly suburban except for some traditionally urban areas at the centers of the older cities of Anaheim, Fullerton, Huntington Beach, Orange, and Santa Ana. There are several edge city-style developments such as Irvine Business Center, Newport Center, the county is famous for its tourism as the home of attractions like Disneyland, Knotts Berry Farm, and several beaches along its more than 40 miles of coastline. It is part of the Tech Coast, members of the Tongva, Juaneño, and Luiseño Native American groups long inhabited the area. After the 1769 expedition of Gaspar de Portolà, a Spanish expedition led by Junipero Serra named the area Valle de Santa Ana, on November 1,1776, Mission San Juan Capistrano became the areas first permanent European settlement. Among those who came with Portolá were José Manuel Nieto and José Antonio Yorba, both these men were given land grants—Rancho Los Nietos and Rancho Santiago de Santa Ana, respectively. The Nieto heirs were granted land in 1834, the Nieto ranches were known as Rancho Los Alamitos, Rancho Las Bolsas, and Rancho Los Coyotes. Yorba heirs Bernardo Yorba and Teodosio Yorba were also granted Rancho Cañón de Santa Ana and Rancho Lomas de Santiago, other ranchos in Orange County were granted by the Mexican government during the Mexican period in Alta California. A severe drought in the 1860s devastated the industry, cattle ranching. In 1887, silver was discovered in the Santa Ana Mountains, attracting settlers via the Santa Fe and this growth led the California legislature to divide Los Angeles County and create Orange County as a separate political entity on March 11,1889. The county is said to have named for the citrus fruit in an attempt to promote immigration by suggesting a semi-tropical paradise–a place where anything could grow. Other citrus crops, avocados, and oil extraction were important to the early economy. Orange County benefited from the July 4,1904 completion of the Pacific Electric Railway, the link made Orange County an accessible weekend retreat for celebrities of early Hollywood. It was deemed so significant that Pacific City changed its name to Huntington Beach in honor of Henry E. Huntington, president of the Pacific Electric, Transportation further improved with the completion of the State Route and U. S. Route 101 in the 1920s
Orange County, California
Orange County, California
Orange County, California
Orange County, California
34.
Home run
–
In modern baseball, the feat is typically achieved by hitting the ball over the outfield fence between the foul poles without first touching the ground, resulting in an automatic home run. There is also the home run, increasingly rare in modern baseball. When a home run is scored, the batter is credited with a hit and a run scored. Likewise, the pitcher is recorded as having given up a hit, a batted ball is also a home run if it touches either foul pole or its attached screen before touching the ground, as the foul poles are by definition in fair territory. A batted ball that goes over the wall after touching the ground is not a home run. A fielder is allowed to reach over the wall to attempt to catch the ball as long as his feet are on or over the field during the attempt. If the fielder successfully catches the ball while it is in flight the batter is out, however, since the fielder is not part of the field, a ball that bounces off a fielder and over the wall without touching the ground is still a home run. A home run accomplished in any of the above manners is a home run. This stipulation is in Approved Ruling of Rule 7.10, an inside-the-park home run occurs when a batter hits the ball into play and is able to circle the bases before the fielders can put him out. Unlike with a home run, the batter-runner and all preceding runners are liable to be put out by the defensive team at any time while running the bases. This can only happen if the ball does not leave the ballfield, with outfields much less spacious and more uniformly designed than in the games early days, inside-the-park home runs are now a rarity. They are usually the result of a ball being hit by a very fast runner, either way, this sends the ball into open space in the outfield and thereby allows the batter-runner to circle the bases before the defensive team can put him out. The speed of the runner is crucial as even triples are relatively rare in most modern ballparks, all runs scored on such a play, however, still count. An example of an unexpected bounce occurred during the 2007 Major League Baseball All-Star Game at AT&T Park in San Francisco on July 10,2007, by the time the ball was relayed, Ichiro had already crossed the plate standing up. This was the first inside-the-park home run in All-Star Game history, Home runs are often characterized by the number of runners on base at the time. A home run hit with the bases empty is seldom called a one-run homer, with one runner on base, two runs are scored and thus the home run is often called a two-run homer or two-run shot. Similarly, a home runs with two runners on base is a three-run homer or three-run shot, the term four-run homer is seldom used, instead, it is nearly always called a grand slam. Hitting a grand slam is the best possible result for the turn at bat
Home run
–
Barry Bonds holds the
all-time homerun record in
Major League Baseball
Home run
–
Sadaharu Oh, pictured here in 2006, holds the officially verified all-time world home-run record outside of MLB (Major League Baseball).
Home run
–
The
Polo Grounds left field foul line with guide rope, as seen from upper deck, 1917
35.
Babe Ruth
–
George Herman Babe Ruth Jr. was an American professional baseball player whose career in Major League Baseball spanned 22 seasons, from 1914 through 1935. Ruth established many MLB batting records, including home runs, runs batted in, bases on balls, slugging percentage, and on-base plus slugging. Ruth is regarded as one of the greatest sports heroes in American culture and is considered by many to be the greatest baseball player of all time, in 1936, Ruth was elected into the Baseball Hall of Fame as one of its first five inaugural members. At age seven, Ruth was sent to St, in 1914, Ruth was signed to play minor-league baseball for the Baltimore Orioles but was soon sold to the Red Sox. By 1916, he had built a reputation as a pitcher who sometimes hit long home runs. With regular playing time, he broke the MLB single-season home run record in 1919, after that season, Red Sox owner Harry Frazee sold Ruth to the Yankees amid controversy. The trade fueled Bostons subsequent 86 year championship drought and popularized the Curse of the Bambino superstition, in his 15 years with New York, Ruth helped the Yankees win seven American League championships and four World Series championships. As part of the Yankees vaunted Murderers Row lineup of 1927, Ruth hit 60 home runs and he retired in 1935 after a short stint with the Boston Braves. During his career, Ruth led the AL in home runs during a season twelve times, Ruths legendary power and charismatic personality made him a larger-than-life figure in the Roaring Twenties. During his career, he was the target of press and public attention for his baseball exploits. His often reckless lifestyle was tempered by his willingness to do good by visiting children at hospitals, after his retirement as a player, he was denied a managerial job in baseball, most likely due to poor behavior during parts of his playing career. In his final years, Ruth made many appearances, especially in support of American efforts in World War II. In 1946, he became ill with cancer, and died two years later, George Herman Ruth Jr. was born in 1895 at 216 Emory Street in Pigtown, a working-class section of Baltimore, Maryland, named for its meat-packing plants. Its population included recent immigrants from Ireland, Germany and Italy, Ruths parents, George Herman Ruth, Sr. and Katherine Schamberger, were both of German American ancestry. According to the 1880 census, his parents were born in Maryland, the paternal grandparents of Ruth, Sr. were from Prussia and Hanover. Ruth, Sr. had a series of jobs, including lightning rod salesman and streetcar operator, before becoming a counterman in a combination grocery. George Ruth Jr. was born in the house of his grandfather, Pius Schamberger. Only one of young Georges seven siblings, his younger sister Mamie, many details of Ruths childhood are unknown, including the date of his parents marriage
Babe Ruth
–
Ruth in his New York Yankees uniform, in 1920
Babe Ruth
–
Babe Ruth birthplace in Baltimore, Maryland
Babe Ruth
–
Ruth (top row, center) at St. Mary's Industrial School for Boys in Baltimore, Maryland, in 1912
Babe Ruth
–
Baseball card depicting Ruth as a
Baltimore Oriole, 1914
36.
Hank Aaron
–
Henry Louis Hank Aaron, nicknamed Hammer, or Hammerin Hank, is a retired American Major League Baseball right fielder who is currently the senior vice president of the Atlanta Braves. He played 21 seasons for the Milwaukee/Atlanta Braves in the National League and 2 seasons for the Milwaukee Brewers in the American League, Aaron held the MLB record for career home runs for 33 years, and he still holds several MLB offensive records. He hit 24 or more home runs every year from 1955 through 1973, in 1999, The Sporting News ranked Aaron fifth on its 100 Greatest Baseball Players list. Aaron was born and raised in and around Mobile, Alabama, Aaron had seven siblings, including Tommie Aaron, who later played in MLB with him. Aaron appeared briefly in the Negro American League and in minor league baseball before starting his major league career and he played late in Negro league history, by his final MLB season, Aaron was the last Negro league baseball player on a major league roster. Aaron played the vast majority of his MLB games in right field, in his last two seasons, he was primarily a designated hitter. Aaron was an NL All-Star for 20 seasons and an AL All-Star for 1 season, Aaron holds the record for the most seasons as an All-Star, the most All-Star Game selections, and is tied with Willie Mays and Stan Musial for the most All-Star Games played. He was a Gold Glove winner for three seasons, in 1957, he was the NL Most Valuable Player when the Milwaukee Braves won the World Series. He won the NL Player of the Month award in May 1958, Aaron holds the MLB records for the most career runs batted in, extra base hits, and total bases. Aaron is also in the top five for career hits and runs and he is one of only four players to have at least seventeen seasons with 150 or more hits. Aaron is in place in home runs and at-bats. At the time of his retirement, Aaron held most of the games key career power hitting records, since his retirement, Aaron has held front office roles with the Atlanta Braves. He was inducted into the National Baseball Hall of Fame in 1982, in 1999, MLB introduced the Hank Aaron Award to recognize the top offensive players in each league. He was awarded the Presidential Medal of Freedom in 2002 and he was named a 2010 Georgia Trustee by the Georgia Historical Society in recognition of accomplishments that reflect the ideals of Georgias founders. Aaron was born in Mobile, Alabama, to Herbert Aaron, Sr. Tommie Aaron, one of his brothers, also went on to play Major League Baseball. By the time Aaron retired, he and his brother held the record for most career home runs by a pair of siblings and they were also the first siblings to appear in a League Championship Series as teammates. While he was born in a section of Mobile referred to as Down the Bay, Aaron grew up in a poor family. His family could not afford baseball equipment, so he practiced by hitting bottle caps with sticks and he would create his own bats and balls out of materials he found on the streets
Hank Aaron
–
Aaron in 2013
Hank Aaron
–
The Braves' jersey Hank Aaron wore when he broke Babe Ruth's career home run record in 1974
Hank Aaron
–
The fence outside of Turner Field over which Hank Aaron hit his 715th career home run still exists.
Hank Aaron
–
Hank Aaron's Hall of Fame plaque at the
Baseball Hall of Fame in
Cooperstown, New York
37.
The Adventures of Tintin
–
The Adventures of Tintin is a series of 24 comic albums created by Belgian cartoonist Georges Remi, who wrote under the pen name Hergé. The series was one of the most popular European comics of the 20th century, by 2007, a century after Hergés birth in 1907, Tintin had been published in more than 70 languages with sales of more than 200 million copies. The series first appeared in French on 10 January 1929 in Le Petit Vingtième, the success of the series saw the serialised strips published in Belgiums leading newspaper Le Soir and spun into a successful Tintin magazine. In 1950, Hergé created Studios Hergé, which produced the canonical versions of ten Tintin albums, the Adventures of Tintin have been adapted for radio, television, theatre, and film. The series is set during a largely realistic 20th century and its hero is Tintin, a young Belgian reporter and adventurer. He is aided by his faithful dog Snowy, the series has been admired for its clean, expressive drawings in Hergés signature ligne claire style. Its well-researched plots straddle a variety of genres, swashbuckling adventures with elements of fantasy, mysteries, political thrillers, the stories feature slapstick humour, offset by dashes of sophisticated satire and political or cultural commentary. Georges Remi, best known under the pen name Hergé, was employed as an illustrator at Le Vingtième Siècle, run by the Abbé Norbert Wallez, the paper described itself as a Catholic Newspaper for Doctrine and Information and disseminated a far-right, fascist viewpoint. Wallez appointed Hergé editor of a new Thursday youth supplement, titled Le Petit Vingtième, propagating Wallezs socio-political views to its young readership, it contained explicitly pro-fascist and anti-Semitic sentiment. In addition to editing the supplement, Hergé illustrated Lextraordinaire aventure de Flup, Nénesse, Poussette et Cochonnet, dissatisfied with this, Hergé wanted to write and draw his own cartoon strip. He already had experience creating comic strips, from July 1926 he had written a strip about a Boy Scout patrol leader titled Les Aventures de Totor C. P. des Hannetons for the Scouting newspaper Le Boy Scout Belge. Totor was an influence on Tintin, with Hergé describing the latter as being like Totors younger brother. Although Hergé wanted to send Tintin to the United States, Wallez ordered him to set his adventure in the Soviet Union, the result, Tintin in the Land of the Soviets, was serialised in Le Petit Vingtième from January 1929 to May 1930. Popular in Francophone Belgium, Wallez organised a publicity stunt at the Gare du Nord station, the storys popularity led to an increase in sales, so Wallez granted Hergé two assistants. At Wallezs direction, in June he began serialisation of the story, Tintin in the Congo. The Adventures of Tintin had been syndicated to French Catholic magazine Cœurs Vaillants since 1930, and Hergé was soon receiving syndication requests from Swiss and Portuguese newspapers too. Hergé went on to pen a string of Adventures of Tintin, sending his character to real locations such as the Belgian Congo, the United States, Egypt, India, China, in May 1940, Nazi Germany invaded Belgium as World War II broke out across Europe. Although Hergé briefly fled to France and considered a self-imposed exile, for political reasons, the Nazi authorities closed down Le Vingtième Siècle, leaving Hergé unemployed
The Adventures of Tintin
–
The front page of the 1 May 1930 edition of Le Petit Vingtième, declaring " Tintin revient! " ("Tintin Returns!") from his adventure in the Soviet Union.
The Adventures of Tintin
–
The main characters of The Adventures of Tintin. In the centre are
Tintin and
Snowy.
The Adventures of Tintin
–
Tintin and Snowy, located on the roof of the former headquarters of
Le Lombard, close to
Gare du Midi, in
Brussels
The Adventures of Tintin
–
The early works of Tintin naively depicted controversial images. Later, Hergé called his actions "a transgression of my youth." Hergé substituted this sequence with one in which the rhino accidentally discharges Tintin's rifle.
38.
Binomial coefficient
–
In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a coefficient is indexed by a pair of integers n ≥ k ≥0 and is written. It is the coefficient of the xk term in the expansion of the binomial power n. The value of the coefficient is given by the expression n. k, arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascals triangle. The properties of binomial coefficients have led to extending the definition to beyond the case of integers n ≥ k ≥0. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier, the earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingalas Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī, alternative notations include C, nCk, nCk, Ckn, Cnk, and Cn, k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display, in this form the binomial coefficients are easily compared to k-permutations of n, written as P, etc. For natural numbers n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of n, the same coefficient also occurs in the binomial formula, which explains the name binomial coefficient. This shows in particular that is a number for any natural numbers n and k. Most of these interpretations are easily seen to be equivalent to counting k-combinations, several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations. It also follows from tracing the contributions to Xk in n−1, as there is zero Xn+1 or X−1 in n, one might extend the definition beyond the above boundaries to include =0 when either k > n or k <0. This recursive formula then allows the construction of Pascals triangle, surrounded by white spaces where the zeros, or the trivial coefficients, a more efficient method to compute individual binomial coefficients is given by the formula = n k _ k. = n ⋯ k ⋯1 = ∏ i =1 k n +1 − i i and this formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, the denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the limit of the product above to the smaller of k. This formula follows from the formula above by multiplying numerator and denominator by. As a consequence it involves many factors common to numerator and denominator and it is less practical for explicit computation unless common factors are first cancelled
Binomial coefficient
–
The binomial coefficients can be arranged to form
Pascal's triangle.
39.
Buffalo, NY
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Buffalo is a city in western New York state and the county seat of Erie County, on the eastern shores of Lake Erie at the head of the Niagara River. As of 2014, Buffalo is New York states 2nd-most populous city after New York City, the metropolitan area has a population of 1.13 million. After an economic downturn in the half of the 20th century, Buffalos economy has transitioned to sectors that include financial services, technology, biomedical engineering. Residents of Buffalo are called Buffalonians, the citys nicknames include The Queen City, The Nickel City and The City of Good Neighbors. The city of Buffalo received its name from a creek called Buffalo Creek. British military engineer Captain John Montresor made reference to Buffalo Creek in his journal of 1764, there are several theories regarding how Buffalo Creek received its name. In 1804, as principal agent opening the area for the Holland Land Company, Joseph Ellicott, designed a radial street and grid system that branches out from downtown like bicycle spokes similar to the street system he used in the nations capital. Although Ellicott named the settlement New Amsterdam, the name did not catch on, during the War of 1812, on December 30,1813, Buffalo was burned by British forces. The George Coit House 1818 and Samuel Schenck House 1823 are currently the oldest houses within the limits of the City of Buffalo, on October 26,1825, the Erie Canal was completed with Buffalo a port-of-call for settlers heading westward. At the time, the population was about 2,400, the Erie Canal brought about a surge in population and commerce, which led Buffalo to incorporate as a city in 1832. In 1845, construction began on the Macedonia Baptist Church, an important meeting place for the abolitionist movement, Buffalo was a terminus point of the Underground Railroad with many fugitive slaves crossing the Niagara River to Fort Erie, Ontario in search of freedom. During the 1840s, Buffalos port continued to develop, both passenger and commercial traffic expanded with some 93,000 passengers heading west from the port of Buffalo. Grain and commercial goods shipments led to repeated expansion of the harbor, in 1843, the worlds first steam-powered grain elevator was constructed by local merchant Joseph Dart and engineer Robert Dunbar. Darts Elevator enabled faster unloading of lake freighters along with the transshipment of grain in bulk from barges, canal boats, by 1850, the citys population was 81,000. At the dawn of the 20th century, local mills were among the first to benefit from hydroelectric power generated by the Niagara River, the city got the nickname City of Light at this time due to the widespread electric lighting. It was also part of the revolution, hosting the brass era car builders Pierce Arrow. President William McKinley was shot and mortally wounded by an anarchist at the Pan-American Exposition in Buffalo on September 6,1901, McKinley died in the city eight days later and Theodore Roosevelt was sworn in at the Wilcox Mansion as the 26th President of the United States. The Great Depression of 1929–39 saw severe unemployment, especially working class men
Buffalo, NY
Buffalo, NY
Buffalo, NY
Buffalo, NY
40.
Brooklyn, NY
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Brooklyn is the most populous borough of New York City, with a Census-estimated 2,636,735 residents in 2015. It borders the borough of Queens at the end of Long Island. Today, if New York City dissolved, Brooklyn would rank as the third-most populous city in the U. S. behind Los Angeles, the borough continues, however, to maintain a distinct culture. Many Brooklyn neighborhoods are ethnic enclaves, Brooklyns official motto, displayed on the Borough seal and flag, is Eendraght Maeckt Maght which translates from early modern Dutch as Unity makes strength. Since 2010, Brooklyn has evolved into a hub of entrepreneurship and high technology startup firms. The history of European settlement in Brooklyn spans more than 350 years, the neighborhood of Marine Park was home to North Americas first tidal mill. It was built by the Dutch, and the foundation can be seen today, however, the area was not formally settled as a town. Many incidents and documents relating to this period are in Gabriel Furmans early compilation, what is today Brooklyn left Dutch hands after the final English conquest of New Netherland in 1664, a prelude to the Second Anglo–Dutch War. The English reorganized the six old Dutch towns on southwestern Long Island as Kings County on November 1,1683 and this tract of land was recognized as a political entity for the first time, and the municipal groundwork was laid for a later expansive idea of Brooklyn identity. On August 27,1776 was fought the Battle of Long Island, the first major engagement fought in the American Revolutionary War after independence was declared, and the largest of the entire conflict. British troops forced Continental Army troops under George Washington off the heights near the sites of Green-Wood Cemetery, Prospect Park. The fortified American positions at Brooklyn Heights consequently became untenable and were evacuated a few days later, One result of the Treaty of Paris in 1783 was the evacuation of the British from New York City, celebrated by residents into the 20th century. The New York Navy Yard operated in Wallabout Bay for the entire 19th century, the first center of urbanization sprang up in the Town of Brooklyn, directly across from Lower Manhattan, which saw the incorporation of the Village of Brooklyn in 1817. Reliable steam ferry service across the East River to Fulton Landing converted Brooklyn Heights into a town for Wall Street. Ferry Road to Jamaica Pass became Fulton Street to East New York, Town and Village were combined to form the first, kernel incarnation of the City of Brooklyn in 1834. Industrial deconcentration in mid-century was bringing shipbuilding and other manufacturing to the part of the county. Each of the two cities and six towns in Kings County remained independent municipalities, and purposely created non-aligning street grids with different naming systems and it later became the most popular and highest circulation afternoon paper in America. The publisher changed to L. Van Anden on April 19,1842, on May 14,1849 the name was shortened to The Brooklyn Daily Eagle, on September 5,1938 it was further shortened to Brooklyn Eagle
Brooklyn, NY
–
Clockwise from top left:
Brooklyn Bridge, Brooklyn brownstones,
Soldiers' and Sailors' Arch,
Brooklyn Borough Hall,
Coney Island
Brooklyn, NY
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Currier and Ives print of Brooklyn, 1886.
Brooklyn, NY
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Brooklyn Museum - Hooker's Map of the Village of Brooklyn
Brooklyn, NY
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A typical dining table in the Dutch village of Brooklyn, c. 1664, from
The Brooklyn Museum.
41.
Bronx, NY
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The Bronx is the northernmost of the five boroughs of New York City, within the U. S. state of New York. Since 1914, the Bronx has had the boundaries as Bronx County, a county of New York. The Bronx is divided by the Bronx River into a section in the west, closer to Manhattan. East and west street addresses are divided by Jerome Avenue—the continuation of Manhattans Fifth Avenue, the West Bronx was annexed to New York City in 1874, and the areas east of the Bronx River in 1895. Bronx County was separated from New York County in 1914, about a quarter of the Bronxs area is open space, including Woodlawn Cemetery, Van Cortlandt Park, Pelham Bay Park, the New York Botanical Garden, and the Bronx Zoo in the boroughs north and center. These open spaces are situated primarily on land reserved in the late 19th century as urban development progressed north. The name Bronx originated with Jonas Bronck, who established the first settlement in the area as part of the New Netherland colony in 1639, the native Lenape were displaced after 1643 by settlers. This cultural mix has made the Bronx a wellspring of both Latin music and hip hop. The Bronx, particularly the South Bronx, saw a decline in population, livable housing, and the quality of life in the late 1960s. Since then the communities have shown significant redevelopment starting in the late 1980s before picking up pace from the 1990s until today, the Bronx was called Rananchqua by the native Siwanoy band of Lenape, while other Native Americans knew the Bronx as Keskeskeck. It was divided by the Aquahung River, the origin of Jonas Bronck is contested. Some sources claim he was a Swedish born emigrant from Komstad, Norra Ljunga parish in Småland, Sweden, who arrived in New Netherland during the spring of 1639. Bronck became the first recorded European settler in the now known as the Bronx and built a farm named Emmanus close to what today is the corner of Willis Avenue. He leased land from the Dutch West India Company on the neck of the mainland north of the Dutch settlement in Harlem. He eventually accumulated 500 acres between the Harlem River and the Aquahung, which known as Broncks River or the Bronx. Dutch and English settlers referred to the area as Broncks Land, the American poet William Bronk was a descendant of Pieter Bronck, either Jonas Broncks son or his younger brother. More recent research indicates that Pieter was probably Jonas nephew or cousin, the Bronx is referred to with the definite article as The Bronx, both legally and colloquially. The region was named after the Bronx River and first appeared in the Annexed District of The Bronx created in 1874 out of part of Westchester County
Bronx, NY
–
Yankee Stadium (center) and the
Grand Concourse to its left. To the right of the Stadium is
its former site.
Bronx, NY
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Flag
Bronx, NY
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Seal
Bronx, NY
–
Map of the Bronx in 1867
42.
6 (number)
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6 is the natural number following 5 and preceding 7. The SI prefix for 10006 is exa-, and for its reciprocal atto-,6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number, its proper divisors are 1,2 and 3, since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and S -perfect number. As a perfect number,6 is related to the Mersenne prime 3,6 is the only even perfect number that is not the sum of successive odd cubes. As a perfect number,6 is the root of the 6-aliquot tree, and is itself the sum of only one number. Six is the number that is both the sum and the product of three consecutive positive numbers. Unrelated to 6 being a number, a Golomb ruler of length 6 is a perfect ruler. Six is the first discrete biprime and the first member of the discrete biprime family, Six is the smallest natural number that can be written as the sum of two positive rational cubes which are not integers,6 =3 +3. Six is a perfect number, a harmonic divisor number and a superior highly composite number. The next superior highly composite number is 12,5 and 6 form a Ruth-Aaron pair under either definition. There are no Graeco-Latin squares with order 6, if n is a natural number that is not 2 or 6, then there is a Graeco-Latin square with order n. The smallest non-abelian group is the symmetric group S3 which has 3, s6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of mathematical objects such as the S Steiner system, the projective plane of order 4. This can also be expressed category theoretically, consider the category whose objects are the n element sets and this category has a non-trivial functor to itself only for n =6. 6 similar coins can be arranged around a central coin of the radius so that each coin makes contact with the central one. This makes 6 the answer to the kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the lattice in which each circle touches just six others. 6 is the largest of the four all-Harshad numbers, a six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane
6 (number)
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X-ray of a
polydactyl human hand with six
fingers
6 (number)
–
A standard
guitar has 6
strings
6 (number)
–
The cells of a
beehive are 6-sided
43.
Factorial
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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
Factorial
–
Amplitude and phase of factorial of complex argument
Factorial
–
Plot of the natural logarithm of the factorial
44.
Highly composite number
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A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan, the related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The initial or smallest 38 highly composite numbers are listed in the table below, the number of divisors is given in the column labeled d. The table below shows all the divisors of one of these numbers, the 15, 000th highly composite number can be found on Achim Flammenkamps website. Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, also, except in two special cases n =4 and n =36, the last exponent ck must equal 1. It means that 1,4, and 36 are the only square highly composite numbers, saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. Note, that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example,96 =25 ×3 satisfies the conditions and has 12 divisors but is not highly composite since there is a smaller number 60 which has the same number of divisors. If Q denotes the number of composite numbers less than or equal to x. The first part of the inequality was proved by Paul Erdős in 1944 and we have 1.13862 < lim inf log Q log log x ≤1.44 and lim sup log Q log log x ≤1.71. Highly composite numbers higher than 6 are also abundant numbers, one need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. It is false that all composite numbers are also Harshad numbers in base 10. The first HCN that is not a Harshad number is 245,044,800, which has a sum of 27. 10 of the first 38 highly composite numbers are highly composite numbers. The sequence of composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors. A positive integer n is a composite number if d ≥ d for all m ≤ n. The counting function QL of largely composite numbers satisfies c ≤ log Q L ≤ d for positive c, d with 0.2 ≤ c ≤ d ≤0.5. Because the prime factorization of a composite number uses all of the first k primes
Highly composite number
–
Plot of the number of divisors of integers from 1 to 1000. The first 15 highly composite numbers are in bold.
45.
Turn (geometry)
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A turn is a unit of plane angle measurement equal to 2π radians, 360° or 400 gon. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot, a turn can be subdivided in many different ways, into half turns, quarter turns, centiturns, milliturns, binary angles, points etc. A turn can be divided in 100 centiturns or 1000 milliturns, with each corresponding to an angle of 0. 36°. A protractor divided in centiturns is normally called a percentage protractor, binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, the binary degree, also known as the binary radian, is 1⁄256 turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte, other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n. The notion of turn is used for planar rotations. Two special rotations have acquired appellations of their own, a rotation through 180° is commonly referred to as a half-turn, the word turn originates via Latin and French from the Greek word τόρνος. In 1697, David Gregory used π/ρ to denote the perimeter of a divided by its radius. However, earlier in 1647, William Oughtred had used δ/π for the ratio of the diameter to perimeter, the first use of the symbol π on its own with its present meaning was in 1706 by the Welsh mathematician William Jones. Euler adopted the symbol with that meaning in 1737, leading to its widespread use, percentage protractors have existed since 1922, but the terms centiturns and milliturns were introduced much later by Sir Fred Hoyle. The German standard DIN1315 proposed the unit symbol pla for turns, since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g, one turn is equal to 2π radians. In 1958, Albert Eagle proposed the Greek letter tau τ as a symbol for 1/2π and his proposal used a pi with three legs symbol to denote the constant. In 2010, Michael Hartl proposed to use tau to represent Palais circle constant, τ=2π. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed directly, for instance. Second, τ visually resembles π, whose association with the constant is unavoidable. Hartls Tau Manifesto gives many examples of formulas that are simpler if tau is used instead of pi, however, a rebuttal was given in The Pi Manifesto, stating a variety of reasons tau should not supplant pi
Turn (geometry)
–
Counterclockwise
rotations about the center point where a complete rotation is equal to 1 turn
46.
Centered hexagonal number
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The nth centered hexagonal number is given by the formula n 3 −3 =3 n +1. Expressing the formula as 1 +6 shows that the centered hexagonal number for n is 1 more than 6 times the th triangular number. The first few centered hexagonal numbers are,1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919. In base 10 one can notice that the hexagonal numbers rightmost digits follow the pattern 1–7–9–7–1, the sum of the first n centered hexagonal numbers is n3. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. In particular, prime centered hexagonal numbers are cuban primes, the difference between 2 and the nth centered hexagonal number is a number of the form 3n2 + 3n −1, while the difference between 2 and the nth centered hexagonal number is a pronic number. Hexagonal number Magic hexagon Star number
Centered hexagonal number
47.
Eight queens puzzle
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The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other. Thus, a solution requires that no two share the same row, column, or diagonal. Chess composer Max Bezzel published the eight queens puzzle in 1848, franz Nauck published the first solutions in 1850. Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n × n squares, since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized n-queens version. In 1874, S. Gunther proposed a method using determinants to find solutions, in 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming. It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids brute-force computational techniques, generating permutations further reduces the possibilities to just 40,320, which are then checked for diagonal attacks. Martin Richards published a program to count solutions to the problem using bitwise operations. The eight queens puzzle has 92 distinct solutions, if solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions. These are called solutions, representatives of each are shown below. A fundamental solution usually has eight variants obtained by rotating 90,180, or 270°, however, should a solution be equivalent to its own 90° rotation, that fundamental solution will have only two variants. Should a solution be equivalent to its own 180° rotation, it will have four variants, if n >1, it is not possible for a solution to be equivalent to its own reflection because that would require two queens to be facing each other. The different fundamental solutions are presented below, Solution 10 has the property that no three queens are in a straight line. These brute-force algorithms to count the number of solutions are computationally manageable for n =8, if the goal is to find a single solution then explicit solutions exist for all n ≥4, requiring no combinatorial search whatsoever. The explicit solutions exhibit stair-stepped patterns, as in the examples for n =8,9 and 10. Let be the square in column i and row j on the n × n chessboard, if n is even and n ≠ 6k +2, then place queens at and for i =1,2. If n is even and n ≠ 6k, then place queens at, if n is odd, then use one of the patterns above for and add a queen at. If the remainder is 2, swap 1 and 3 in odd list, if the remainder is 3, move 2 to the end of even list and 1,3 to the end of odd list. Append odd list to the even list and place queens in the rows given by these numbers, for N =8 this results in fundamental solution 1 above
Eight queens puzzle
48.
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
Square number
–
m = 1 2 = 1
49.
Cube (arithmetics)
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In arithmetic and algebra, the cube of a number n is its third power, the result of the number multiplied by itself twice, n3 = n × n × n. It is also the number multiplied by its square, n3 = n × n2 and this is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n and it determines the side of the cube of a given volume. It is also n raised to the one-third power, both cube and cube root are odd functions,3 = −. The cube of a number or any other mathematical expression is denoted by a superscript 3, a cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The perfect cubes up to 603 are, Geometrically speaking, an integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger. For example,27 small cubes can be arranged into one larger one with the appearance of a Rubiks Cube, the difference between the cubes of consecutive integers can be expressed as follows, n3 −3 = 3n +1. There is no minimum perfect cube, since the cube of an integer is negative. For example, −4 × −4 × −4 = −64, unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25,75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6, some cube numbers are also square numbers, for example,64 is a square number and a cube number. This happens if and only if the number is a perfect sixth power, the last digits of each 3rd power are, It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1,8 or 9. That is their values modulo 9 may be only −1,1 and 0, every positive integer can be written as the sum of nine positive cubes. The equation x3 + y3 = z3 has no solutions in integers. In fact, it has none in Eisenstein integers, both of these statements are also true for the equation x3 + y3 = 3z3. The sum of the first n cubes is the nth triangle number squared,13 +23 + ⋯ + n 3 =2 =2. Proofs Charles Wheatstone gives a simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. Indeed, he begins by giving the identity n 3 = + + + ⋯ + ⏟ n consecutive odd numbers, kanim provides a purely visual proof, Benjamin & Orrison provide two additional proofs, and Nelsen gives seven geometric proofs
Cube (arithmetics)
–
y = x 3 for values of 0 ≤ x ≤ 25.
50.
9 (number)
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9 is the natural number following 8 and preceding 10. In the NATO phonetic alphabet, the digit 9 is called Niner, five-digit produce PLU codes that begin with 9 are organic. Common terminal digit in psychological pricing, Nine is a number that appears often in Indian Culture and mythology. Nine influencers are attested in Indian astrology, in the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements, Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. Navaratri is a festival dedicated to the nine forms of Durga. Navaratna, meaning 9 jewels may also refer to Navaratnas - accomplished courtiers, Navratan - a kind of dish, according to Yoga, the human body has nine doors - two eyes, two ears, the mouth, two nostrils, and the openings for defecation and procreation. In Indian aesthetics, there are nine kinds of Rasa, Nine is considered a good number in Chinese culture because it sounds the same as the word long-lasting. Nine is strongly associated with the Chinese dragon, a symbol of magic, there are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales –81 yang and 36 yin, all three numbers are multiples of 9 as well as having the same digital root of 9. The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City, the name of the area called Kowloon in Hong Kong literally means, nine dragons. The nine-dotted line delimits certain island claims by China in the South China Sea, the nine-rank system was a civil service nomination system used during certain Chinese dynasties. 9 Points of the Heart / Heart Master Channels in Traditional Chinese Medicine, the nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. The Ennead is a group of nine Egyptian deities, who, in versions of the Osiris myth. The Nine Worthies are nine historical, or semi-legendary figures who, in Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil. The nine Muses in Greek mythology are Calliope, Clio, Erato, Euterpe, Melpomene, Polyhymnia, Terpsichore, Thalia and it takes nine days to fall from heaven to earth, and nine more to fall from earth to Tartarus—a place of torment in the underworld. Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo, according to Georges Ifrah, the origin of the 9 integers can be attributed to ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0. In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot, the Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, as time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller
9 (number)
–
A
Nine-ball rack with the 9 ball at the center
9 (number)
9 (number)
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Playing cards showing the 9 of all four suits