1.
80 (number)
–
80 is the natural number following 79 and preceding 81. 80 is, the sum of Eulers totient function φ over the first sixteen integers, a semiperfect number, since adding up some subsets of its divisors gives 80. Palindromic in bases 3,6,9,15,19 and 39, a repdigit in bases 3,9,15,19 and 39. A Harshad number in bases 2,3,4,5,6,7,9,10,11,13,15 and 16 The Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes. Every solvable configuration of the Fifteen puzzle can be solved in no more than 80 single-tile moves, the atomic number of mercury According to Exodus 7,7, Moses was 80 years old when he initially spoke to Pharaoh on behalf of his people. Today,80 years of age is the age limit for cardinals to vote in papal elections. Jerry Rice wore the number 80 for the majority of his NFL career
80 (number)
–
Element 80: Mercury (Hg)
2.
83 (number)
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83 is the natural number following 82 and preceding 84. 83 is, the sum of three consecutive primes, the sum of five consecutive primes. The 23rd prime number, following 79 and preceding 89, an Eisenstein prime with no imaginary part and real part of the form 3n −1. The duration of Saros series 83 was 1262.1 years, the Saros number of the lunar eclipse series which began on -197 August 22 and ended on 1318 February. The duration of Saros series 83 was 1514.5 years, when someone reaches 83 they may celebrate a second bar mitzvah M83 is the debut album of the French electronic music group M8383 is a song written by John Mayer in the Room for Squares album. As an example, the television station CIVIC-TV managed by the James Woods character Max Renn in the 1983 film Videodrome was on Channel 83. Eighty-three is also, The year AD83,83 BC, or 1983 The TI-83 series and this symbology is also known to be used by many non-racist Christians and non-denominational Churches. An emoticon based on,3 with wide-open eyes
83 (number)
–
TI-83 calculator
3.
84 (number)
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84 is the natural number following 83 and preceding 85. 84 is, the sum of the first seven triangular numbers, a semiperfect number, being thrice a perfect number. A palindromic number and a repdigit in bases 11,13,20,27, the lim sup of the largest finite subgroup of the mapping class group of a genus g surface divided by g. A hepteract is a hypercube with 84 penteract 5-faces. Messier object M84, a magnitude 11, the duration of Saros series 84 was 1280.1 years, and it contained 72 solar eclipses. Further, the number of the lunar eclipse series began on -96 September 13. The duration of Saros series 84 was 1496.5 years, eighty-four is also, The year AD84,84 BC, or 1984. The film 84 Charing Cross Road starring Anne Bancroft and Anthony Hopkins KKNX Radio 84 in Eugene, how many Earth years it takes Uranus to orbit the sun once List of highways numbered 84
84 (number)
–
A
hepteract is a seven-
dimensional hypercube with 84
penteract 5-faces
4.
85 (number)
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85 is the natural number following 84 and preceding 86. 85 is, the product of two numbers, and is therefore a biprime, specifically, the 24th biprime not counting perfect squares. Together with 86 and 87, it forms the second cluster of three consecutive biprimes, the smallest number that can be expressed as a sum of two squares, with all squares greater than 1, in two ways,85 =92 +22 =72 +62. The length of the hypotenuse of four pythagorean triangles, a palindromic number in bases 2,4,7, and 16. A repdigit in bases 4 and 16, the aliquot sum of 85 is 23 within the aliquot sequence,85 being the second composite number in the 23-aliquot tree. Messier object M85 is a magnitude 10, the specifics vary by the two Division I football subdivisions, In the top-level FBS, each player provided with a scholarship may, and almost always does, receive a full scholarship. In the second-level FCS, schools are allowed to provide football-related athletic aid equivalent to 63 full scholarships, the year AD85,85 BC, or 1985. The Muslim calendar year 85 AH
85 (number)
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Published in Brazil: ISBN 85..
5.
88 (number)
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88 is the natural number following 87 and preceding 89. An Erdős–Woods number, since it is possible to find sequences of 88 consecutive integers such that each member shares a factor with either the first or the last member. A palindromic number in bases 5,10,21, and 43, a repdigit in bases 10,21 and 43. The atomic number of the element radium, the number of constellations in the sky as defined by the International Astronomical Union. Messier object M88, a magnitude 11.0 spiral galaxy in the constellation Coma Berenices, the New General Catalogue object NGC88, a spiral galaxy in the constellation Phoenix, and a member of Roberts Quartet. Space Shuttle Mission 88, launched and completed in December,1998, the Saros number of the solar eclipse series which began on -246 October 6 and ended on 1233 March. The duration of Saros series 88 was 1478.4 years, further, the Saros number of the lunar eclipse series which began on 38 July and ended on 1336 August. The duration of Saros series 88 was 1298.1 years, approximately the number of days it takes Mercury to complete its orbit. Number 88 symbolizes fortune and good luck in Chinese culture, since the word 8 sounds similar to the word fā, the number 8 is considered to be the luckiest number in Chinese culture, and prices in Chinese supermarkets often contain many 8s. The shape of the Chinese character for 8 implies that a person will have a great, wide future as the character starts narrow, the Chinese government has been auctioning auto license plates containing many 8s for tens of thousands of dollars. The 2008 Beijing Olympics opened on 8/8/08 at 8 p. m, in amateur radio,88 is used as shorthand for love and kisses when signing a message or ending an exchange. It is used in word, Morse code, and in various digital modes. It is considered more intimate than 73, which means best regards. The two may be used together, sometimes either expression is pluralized by appending an -s. These number codes originate with the 92 Code adopted by Western Union in 1859, neo-Nazis use the number 88 as an abbreviation for the Nazi salute Heil Hitler. The letter H is eighth in the alphabet, whereby 88 becomes HH, often, this number is associated with the number 14, e. g. 14/88, 14-88, or 1488, this number symbolizes the Fourteen Words coined by David Lane, a prominent white nationalist. Example uses of 88 include the song 88 Rock n Roll Band by Landser, currently, the Michigan-based ANP uses 14 in its domain name and 88 as part of a radio sign-off. Professional golfer Kathy Whitworth, throughout her career won 88 LPGA Tour tournaments
88 (number)
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Olds 88
6.
89 (number)
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89 is the natural number following 88 and preceding 90. 89 is, the 24th prime number, following 83 and preceding 97, the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms. An Eisenstein prime with no part and real part of the form 3n −1. A Fibonacci number and thus a Fibonacci prime as well, the first few digits of its reciprocal coincide with the Fibonacci sequence due to the identity 189 = ∑ n =1 ∞ F ×10 − =0.011235955 …. A Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers, M89 is the 10th Mersenne prime. Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse, among the known non-Lychrel numbers in the first 10000 integers, no other number requires that many or more iterations. The palindrome reached is also unusually large, eighty-nine is, The atomic number of actinium. Messier object M89, a magnitude 11.5 elliptical galaxy in the constellation Virgo, the New General Catalogue object NGC89, a magnitude 13.5 peculiar spiral galaxy in the constellation Phoenix and a member of Roberts Quartet. The Oklahoma Redhawks, an American minor league team, were formerly known as the Oklahoma 89ers. The number alludes to the Land Run of 1889, when the Unassigned Lands of Oklahoma were opened to white settlement, the teams home of Oklahoma City was founded during this event. In Rugby, an 89 or eight-nine move is a following a scrum, in which the number 8 catches the ball. The Elite 89 Award is presented by the U. S. NCAA to the participant in each of the NCAAs 89 championship finals with the highest grade point average. The jersey number 89 has been retired by three National Football League teams in honor of past playing greats, The Baltimore Colts, for Hall of Famer Gino Marchetti, the franchise continues to honor the number in its current identity as the Indianapolis Colts. The Boston Patriots, for Bob Dee, the franchise, now the New England Patriots, continues to honor the number. The Chicago Bears, for Mike Ditka, eighty-nine is also, The designation of Interstate 89, a freeway that runs from New Hampshire to Vermont The designation of U. S. The number of units of each colour in the board game Blokus The number of the French department Yonne Information Is Beautiful cites eighty-nine as one of the words censored on the Chinese internet
89 (number)
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TI-89
7.
90 (number)
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90 is the natural number preceded by 89 and followed by 91. In English speech, the numbers 90 and 19 are often confused, when carefully enunciated, they differ in which syllable is stressed,19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in such as 1999, and when contrasting numbers in the teens and when counting, such as 17,18,19. 90 is, a perfect number because it is the sum of its unitary divisors. A semiperfect number because it is equal to the sum of a subset of its divisors, a Perrin number, preceded in the sequence by 39,51,68. Palindromic and a repdigit in bases 14,17,29, a Harshad number since 90 is divisible by the sum of its base 10 digits. In normal space, the angles of a rectangle measure 90 degrees each. Also, in a triangle, the angle opposing the hypotenuse measures 90 degrees. Thus, an angle measuring 90 degrees is called a right angle, ninety is, the atomic number of thorium, an actinide. As an atomic weight,90 identifies an isotope of strontium, the latitude in degrees of the North and the South geographical poles. NFL, New York Jets Dennis Byrds #90 is retired +90 is the code for international direct dial phone calls to Turkey,90 is the code for the French département Belfort
90 (number)
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Interstate 90 is a freeway that runs from
Washington to
Massachusetts.
8.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
9.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
10.
20 (number)
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20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants
20 (number)
–
An
icosahedron has 20
faces
11.
30 (number)
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30 is the natural number following 29 and preceding 31. Thirty is the sum of the first four squares, which makes it a square pyramidal number and it is a primorial and is the smallest Giuga number. 30 is the smallest sphenic number, and the smallest of the form 2 ×3 × r,30 has an aliquot sum of 42, the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises Thirty has but one number for which it is the aliquot sum, adding up some subsets of its divisors gives 30, hence 30 is a semiperfect number. 30 is the largest number such that all smaller than itself. A polygon with thirty sides is called a triacontagon, the icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a graph with 30 vertices. The atomic number of zinc is 30 Messier object M30, a magnitude 8, the duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series began on June 19,1803 BC. The duration of Saros series 30 was 1316.2 years, Thirty is, Used to indicate the end of a newspaper story, a copy editors typographical notation. S. Judas Iscariot betrayed Jesus for 30 pieces of silver, one of the rallying-cries of the 1960s student/youth protest movement was the slogan, Dont trust anyone over thirty. In Franz Kafkas novel The Trial Joseph wakes up on the morning of his birthday to find himself under arrest for an unspecified crime. After making many attempts to find the nature of the crime or the name of his accuser. The number of uprights that formed the Sarsen Circle at Stonehenge, western Christianitys most prolific 20th century essayist, F. W. Also in that essay Boreham writes It was said of Keats, in tennis, the number 30 represents the second point gained in a game. Under NCAA rules for basketball, the offensive team has 30 seconds to attempt a shot. As of 2012, three of the four major leagues in the United States and Canada have 30 teams each. The California Angels baseball team retired the number in honor of its most notable wearer, Nolan Ryan, the San Francisco Giants extended the same honor to Orlando Cepeda
30 (number)
–
For other uses, see
The Thirty.
12.
40 (number)
–
Despite being related to the word four, the modern spelling of 40 is forty. The archaic form fourty is now considered a misspelling, the modern spelling possibly reflects a pronunciation change due to the horse–hoarse merger. Forty is a number, an octagonal number, and as the sum of the first four pentagonal numbers. Adding up some subsets of its divisors gives 40, hence 40 is a semiperfect number, given 40, the Mertens function returns 0. 40 is the smallest number n with exactly 9 solutions to the equation φ = n, Forty is the number of n-queens problem solutions for n =7. Since 402 +1 =1601 is prime,40 is a Størmer number,40 is a repdigit in base 3 and a Harshad number in base 10. Negative forty is the temperature at which the Fahrenheit and Celsius scales correspond. It is referred to as either minus forty or forty below, the planet Venus forms a pentagram in the night sky every eight years with it returning to its original point every 40 years with a 40-day regression. The duration of Saros series 40 was 1280.1 years, lunar eclipse series which began on -1387 February 12 and ended on -71 April 12. The duration of Saros series 40 was 1316.2 years, the number 40 is used in Jewish, Christian, Islamic, and other Middle Eastern traditions to represent a large, approximate number, similar to umpteen. In the Hebrew Bible, forty is often used for periods, forty days or forty years. Rain fell for forty days and forty nights during the Flood, spies explored the land of Israel for forty days. The Hebrew people lived in the Sinai desert for forty years and this period of years represents the time it takes for a new generation to arise. Moses life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, and his return to lead his people out, several Jewish leaders and kings are said to have ruled for forty years, that is, a generation. Examples include Eli, Saul, David, and Solomon, goliath challenged the Israelites twice a day for forty days before David defeated him. He went up on the day of Tammuz to beg forgiveness for the peoples sin. He went up on the first day of Elul and came down on the day of Tishrei. A mikvah consists of 40 seah of water 40 lashes is one of the punishments meted out by the Sanhedrin, One of the prerequisites for a man to study Kabbalah is that he is forty years old
40 (number)
–
The number on the logo for the American-Japanese hard rock band Crush 40.
13.
60 (number)
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60 is the natural number following 59 and preceding 61. Being three times 20, it is called three score in older literature. It is a number, with divisors 1,2,3,4,5,6,10,12,15,20,30. Because it is the sum of its divisors, it is a unitary perfect number. Being ten times a number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6 and it is the smallest number with exactly 12 divisors. It is the sum of a pair of twin primes and the sum of four consecutive primes and it is adjacent to two primes. It is the smallest number that is the sum of two odd primes in six ways, the smallest non-solvable group has order 60. There are four Archimedean solids with 60 vertices, the icosahedron, the rhombicosidodecahedron, the snub dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs, there are also two Archimedean solids with 60 edges, the snub cube and the icosidodecahedron. The skeleton of the forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares, in geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, a number system with base 60 is called sexagesimal. It is the smallest positive integer that is written only the smallest. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule and this ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 is an isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz
60 (number)
–
There are 60 seconds in a minute, and 60 minutes in an hour
60 (number)
–
The
icosidodecahedron has 60 edges, all equivalent.
14.
100 (number)
–
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.
15.
Factorization
–
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.
16.
Divisor
–
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
Divisor
–
The divisors of 10 illustrated with
Cuisenaire rods: 1, 2, 5, and 10
17.
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
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Numeral systems
Greek numerals
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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.
18.
Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
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Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
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Numeral systems
Roman numerals
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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
19.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
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
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Gottfried Leibniz
Binary number
–
George Boole
20.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
Ternary numeral system
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Numeral systems
21.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
Quaternary numeral system
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Numeral systems
22.
Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
Quinary
–
Numeral systems
23.
Senary
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Senary
–
Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting
Senary
24.
Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
Octal
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Numeral systems
25.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
Duodecimal
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Numeral systems
Duodecimal
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A duodecimal multiplication table
26.
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
–
Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
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Hexadecimal finger-counting scheme.
27.
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
–
Numeral systems
Vigesimal
–
The
Maya numerals are a base-20 system.
28.
Base 36
–
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
–
Numeral systems
Base 36
–
34 senary = 22 decimal, in senary finger counting
Base 36
29.
Natural number
–
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
–
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
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
30.
Square number
–
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
31.
9 (number)
–
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)
–
Playing cards showing the 9 of all four suits
32.
3 (number)
–
3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
3 (number)
–
The
Shield of the Trinity is a diagram of the Christian doctrine of the Trinity
33.
Heptagonal number
–
A heptagonal number is a figurate number that represents a heptagon. The n-th heptagonal number is given by the formula 5 n 2 −3 n 2, like square numbers, the digital root in base 10 of a heptagonal number can only be 1,4,7 or 9. Five times a number, plus 1 equals a triangular number. A generalized heptagonal number is obtained by the formula T n + T ⌊ n 2 ⌋, where Tn is the nth triangular number. The first few generalized heptagonal numbers are,1,4,7,13,18,27,34,46,55,70,81,99,112, besides 1 and 70, no generalized heptagonal numbers are also Pell numbers. The heptagonal root of x is given by the formula n =40 x +9 +310
Heptagonal number
–
The first five heptagonal numbers.
34.
Centered octagonal number
–
A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered octagonal numbers are the same as the odd square numbers, thus, the nth centered octagonal number is given by the formula 2 =4 n 2 −4 n +1. The first few centered octagonal numbers are 1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089. Calculating Ramanujans tau function on an octagonal number yields an odd number
Centered octagonal number
–
See also [edit]
35.
Open meandric number
–
In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number of bridges, the line and curve together form a meandric system. Two meanders are said to be equivalent if there is a homeomorphism of the plane that takes L to itself. The meander of order 1 intersects the line twice, The meanders of order 2 intersect the line four times, the first fifteen meandric numbers are given below. The curve is oriented upward at the intersection labelled 1, the cyclic permutation with no fixed points is obtained by following the oriented curve through the labelled intersection points. In the diagram on the right, the order 4 meandric permutation is given by and this is a permutation written in cyclic notation and not to be confused with one-line notation. If π is a permutation, then π2 consists of two cycles, one containing of all the even symbols and the other all the odd symbols. Permutations with this property are called alternate permutations, since the symbols in the original permutation alternate between odd and even integers, however, not all alternate permutations are meandric because it may not be possible to draw them without introducing a self-intersection in the curve. For example, the order 3 alternate permutation, is not meandric, two open meanders are said to be equivalent if they are homeomorphic in the plane. The open meander of order 1 intersects the line once, The open meander of order 2 intersects the line twice, the first fifteen open meandric numbers are given below. Two semi-meanders are said to be equivalent if they are homeomorphic in the plane, the semi-meander of order 1 intersects the ray once, The semi-meander of order 2 intersects the ray twice, The number of distinct semi-meanders of order n is the semi-meandric number Mn. The first fifteen semi-meandric numbers are given below
Open meandric number
–
Contents
36.
Messier object
–
The Messier objects are a set of over 100 astronomical objects first listed by French astronomer Charles Messier in 1771. The number of objects in the lists he published reached 103, a similar list had been published in 1654 by Giovanni Hodierna, but attracted attention only recently and was probably not known to Messier. The first edition covered 45 objects numbered M1 to M45, the first such addition came from Nicolas Camille Flammarion in 1921, who added Messier 104 after finding a note Messier made in a copy of the 1781 edition of the catalogue. M105 to M107 were added by Helen Sawyer Hogg in 1947, M108 and M109 by Owen Gingerich in 1960, M102 was observed by Méchain, who communicated his notes to Messier. Méchain later concluded that this object was simply a re-observation of M101, though sources suggest that the object Méchain observed was the galaxy NGC5866. Messiers final catalogue was included in the Connaissance des Temps for 1784 and these objects are still known by their Messier number from this list. Messier lived and did his work at the Hôtel de Cluny. The list he compiled contains only objects found in the sky area he could observe and he did not observe or list objects visible only from farther south, such as the Large and Small Magellanic Clouds. A summary of the astrophysics of each Messier object can be found in the Concise Catalog of Deep-sky Objects, in early spring, astronomers sometimes gather for Messier marathons, when all of the objects can be viewed over a single night
Messier object
–
All Messier objects. The pictures were taken and put together by an amateur astronomer
37.
Messier 81
–
Messier 81 is a spiral galaxy about 12 million light-years away in the constellation Ursa Major. Due to its proximity to Earth, large size and active galactic nucleus, the galaxys large size and relatively high brightness also make it a popular target for amateur astronomers. Messier 81 was first discovered by Johann Elert Bode on December 31,1774, consequently, the galaxy is sometimes referred to as Bodes Galaxy. In 1779, Pierre Méchain and Charles Messier reidentified Bodes object, most of the emission at infrared wavelengths originates from interstellar dust. This interstellar dust is found primarily within the spiral arms. Only one supernova has been detected in Messier 81, the supernova, named SN 1993J, was discovered on 28 March 1993 by F. García in Spain. At the time, it was the second brightest supernova observed in the 20th century, the spectral characteristics of the supernova changed over time. Moreover, the variations in SN 1993Js luminosity over time were not like the variations observed in other type II supernova, hence, the supernova has been classified as a type IIb, a transitory class between type II and type Ib. The supernova was also used to estimate a distance of 8.5 ±1.3 Mly to Messier 81, as a local galaxy, the Central Bureau for Astronomical Telegrams tracks novae in M81 along with M31 and M33. Messier 81 is the largest galaxy in the M81 Group, a group of 34 galaxies located in the constellation Ursa Major. At approximately 11.7 Mly from the Earth, it makes this group, gravitational interactions of M81 with M82 and NGC3077 have stripped hydrogen gas away from all three galaxies, forming gaseous filamentary structures in the group. Moreover, these interactions have allowed interstellar gas to fall into the centers of M82 and NGC3077, Messier 81 is located approximately 10° northwest of Alpha Ursae Majoris along with several other galaxies in the Messier 81 Group. Messier 81 and Messier 82 can both be viewed easily using binoculars and small telescopes, telescopes with apertures of 8 inches or larger are needed to distinguish structures in the galaxy. Its far northern declination makes it visible for observers in the northern hemisphere. It is not visible to most observers in the southern hemisphere, except those in a narrow latitude range immediately south of the equator
Messier 81
–
Messier 81
Messier 81
–
An
infrared image of Messier 81 taken by the
Spitzer Space Telescope. The blue colors represent stellar emission observed at 3.6
μm. The green colors represent 8 μm emission originating primarily from
polycyclic aromatic hydrocarbons in the
interstellar medium. The red colors represent 24 μm emission originating from heated dust in the interstellar medium.
Messier 81
–
M81 (left) and
M82 (right). M82 is one of two galaxies strongly influenced gravitationally by M81. The other,
NGC 3077, is located off the top edge of this image.
38.
Visual magnitude
–
The apparent magnitude of a celestial object is a number that is a measure of its brightness as seen by an observer on Earth. The brighter an object appears, the lower its magnitude value, the Sun, at apparent magnitude of −27, is the brightest object in the sky. It is adjusted to the value it would have in the absence of the atmosphere, furthermore, the magnitude scale is logarithmic, a difference of one in magnitude corresponds to a change in brightness by a factor of 5√100, or about 2.512. The measurement of apparent magnitudes or brightnesses of celestial objects is known as photometry, apparent magnitudes are used to quantify the brightness of sources at ultraviolet, visible, and infrared wavelengths. An apparent magnitude is measured in a specific passband corresponding to some photometric system such as the UBV system. In standard astronomical notation, an apparent magnitude in the V filter band would be denoted either as mV or often simply as V, the scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes. The brightest stars in the sky were said to be of first magnitude, whereas the faintest were of sixth magnitude. Each grade of magnitude was considered twice the brightness of the following grade and this rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest, and is generally believed to have originated with Hipparchus. This implies that a star of magnitude m is 2.512 times as bright as a star of magnitude m +1 and this figure, the fifth root of 100, became known as Pogsons Ratio. The zero point of Pogsons scale was defined by assigning Polaris a magnitude of exactly 2. However, with the advent of infrared astronomy it was revealed that Vegas radiation includes an Infrared excess presumably due to a disk consisting of dust at warm temperatures. At shorter wavelengths, there is negligible emission from dust at these temperatures, however, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the scale was extrapolated to all wavelengths on the basis of the black body radiation curve for an ideal stellar surface at 11000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance for the zero magnitude point, with the modern magnitude systems, brightness over a very wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30, astronomers have developed other photometric zeropoint systems as alternatives to the Vega system. The AB magnitude zeropoint is defined such that an objects AB, the dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Since an increase of 5 magnitudes corresponds to a decrease in brightness by a factor of exactly 100, each magnitude increase implies a decrease in brightness by the factor 5√100 ≈2.512. Inverting the above formula, a magnitude difference m1 − m2 = Δm implies a brightness factor of F2 F1 =100 Δ m 5 =100.4 Δ m ≈2.512 Δ m
Visual magnitude
–
Asteroid
65 Cybele and two stars, with their magnitudes labeled
Visual magnitude
–
30 Doradus image taken by
ESO 's
VISTA. This
nebula has an apparent magnitude of 8.
39.
Spiral galaxy
–
A spiral galaxy is a type of galaxy originally described by Edwin Hubble in his 1936 work The Realm of the Nebulae and, as such, forms part of the Hubble sequence. Spiral galaxies consist of a flat, rotating disk containing stars, gas and dust, and these are surrounded by a much fainter halo of stars, many of which reside in globular clusters. Spiral galaxies are named for the structures that extend from the center into the galactic disc. The spiral arms are sites of ongoing star formation and are brighter than the surrounding disc because of the young, hot OB stars that inhabit them. Roughly two-thirds of all spirals are observed to have a component in the form of a bar-like structure, extending from the central bulge. Our own Milky Way has recently confirmed to be a barred spiral. The most convincing evidence for its existence comes from a recent survey, performed by the Spitzer Space Telescope, together with irregular galaxies, spiral galaxies make up approximately 60% of galaxies in the local Universe. They are mostly found in low-density regions and are rare in the centers of galaxy clusters, Spiral arms are regions of stars that extend from the center of spiral and barred spiral galaxies. These long, thin regions resemble a spiral and thus give spiral galaxies their name, naturally, different classifications of spiral galaxies have distinct arm-structures. Sc and SBc galaxies, for instance, have very loose arms, whereas Sa, either way, spiral arms contain many young, blue stars, which make the arms so bright. A bulge is a huge, tightly packed group of stars, the term commonly refers to the central group of stars found in most spiral galaxies. Using the Hubble classification, the bulge of Sa galaxies is usually composed of Population II stars, further, the bulge of Sa and SBa galaxies tends to be large. In contrast, the bulges of Sc and SBc galaxies are much smaller and are composed of young, some bulges have similar properties to those of elliptical galaxies, others simply appear as higher density centers of disks, with properties similar to disk galaxies. Many bulges are thought to host a supermassive black hole at their centers, such black holes have never been directly observed, but many indirect proofs exist. In our own galaxy, for instance, the object called Sagittarius A* is believed to be a black hole. There is a correlation between the mass of the black hole and the velocity dispersion of the stars in the bulge. However, some stars inhabit a spheroidal halo or galactic spheroid, the orbital behaviour of these stars is disputed, but they may describe retrograde and/or highly inclined orbits, or not move in regular orbits at all. The galactic halo also contains many globular clusters, due to their irregular movement around the center of the galaxy—if they do so at all—these stars often display unusually high proper motion
Spiral galaxy
–
An example of a spiral galaxy, the
Pinwheel Galaxy (also known as Messier 101 or NGC 5457)
Spiral galaxy
–
Barred spiral galaxy UGC 12158.
Spiral galaxy
–
NGC 1300 in
infrared light.
Spiral galaxy
–
Spiral galaxy NGC 1345
40.
Constellation
–
A constellation is formally defined as a region of the celestial sphere, with boundaries laid down by the International Astronomical Union. The constellation areas mostly had their origins in Western-traditional patterns of stars from which the constellations take their names, in 1922, the International Astronomical Union officially recognized the 88 modern constellations, which cover the entire sky. They began as the 48 classical Greek constellations laid down by Ptolemy in the Almagest, Constellations in the far southern sky are late 16th- and mid 18th-century constructions. 12 of the 88 constellations compose the zodiac signs, though the positions of the constellations only loosely match the dates assigned to them in astrology. The term constellation can refer to the stars within the boundaries of that constellation. Notable groupings of stars that do not form a constellation are called asterisms, when astronomers say something is “in” a given constellation they mean it is within those official boundaries. Any given point in a coordinate system can unambiguously be assigned to a single constellation. Many astronomical naming systems give the constellation in which an object is found along with a designation in order to convey a rough idea in which part of the sky it is located. For example, the Flamsteed designation for bright stars consists of a number, the word constellation seems to come from the Late Latin term cōnstellātiō, which can be translated as set of stars, and came into use in English during the 14th century. It also denotes 88 named groups of stars in the shape of stellar-patterns, the Ancient Greek word for constellation was ἄστρον. Colloquial usage does not draw a distinction between constellation in the sense of an asterism and constellation in the sense of an area surrounding an asterism. The modern system of constellations used in astronomy employs the latter concept, the term circumpolar constellation is used for any constellation that, from a particular latitude on Earth, never sets below the horizon. From the North Pole or South Pole, all constellations south or north of the equator are circumpolar constellations. In the equatorial or temperate latitudes, the term equatorial constellation has sometimes been used for constellations that lie to the opposite the circumpolar constellations. They generally include all constellations that intersect the celestial equator or part of the zodiac, usually the only thing the stars in a constellation have in common is that they appear near each other in the sky when viewed from the Earth. In galactic space, the stars of a constellation usually lie at a variety of distances, since stars also travel on their own orbits through the Milky Way, the star patterns of the constellations change slowly over time. After tens to hundreds of thousands of years, their familiar outlines will become unrecognisable, the terms chosen for the constellation themselves, together with the appearance of a constellation, may reveal where and when its constellation makers lived. The earliest direct evidence for the constellations comes from inscribed stones and it seems that the bulk of the Mesopotamian constellations were created within a relatively short interval from around 1300 to 1000 BC
Constellation
Constellation
Constellation
–
Babylonian tablet recording
Halley's comet in 164 BC.
Constellation
–
Chinese star map with a cylindrical projection (
Su Song)
41.
Ursa Major
–
Ursa Major is a constellation in the northern celestial hemisphere. One of the 48 constellations listed by Ptolemy, it one of the 88 modern constellations. It can be throughout the year in most of the northern hemisphere. The Big Dipper and the constellation as a whole have mythological significance in world cultures. Ursa Major occupies an area in the northern celestial hemisphere. Its also the namesake of its family, which includes all the constellations it borders except for Leo. The three-letter abbreviation for the constellation, as adopted by the IAU in 1922, is UMa, the Big Dipper is an asterism within Ursa Major composed of seven bright stars that together comprise one of the best-known patterns in the sky. β Ursae Majoris, called Merak, with a magnitude of 2.37, γ Ursae Majoris, known as either Phecda or Phad, with a magnitude of 2.44. δ Ursae Majoris, or Megrez, meaning root of the tail, Alioth is the brightest star of Ursa Major and the 33rd-brightest in the sky, with a magnitude of 1.76. It is also the brightest of the peculiar A stars, magnetic stars whose chemical elements are either depleted or enhanced, ζ Ursae Majoris, Mizar, the second star in from the end of the handle of the Big Dipper, and the constellations fourth-brightest star. Mizar, which means girdle, forms a double star, with its optical companion Alcor. The ability to resolve the two stars with the eye is often quoted as a test of eyesight, although even people with quite poor eyesight can see the two stars. η Ursae Majoris, known as either Alkaid or Benetnash, both meaning the end of the tail, with a magnitude of 1.85, Alkaid is the third-brightest star of Ursa Major. Except for Dubhe and Alkaid, the stars of the Big Dipper all have proper motions heading toward a point in Sagittarius. A few other such stars have been identified, and together they are called the Ursa Major Moving Group, the stars Merak and Dubhe are known as the pointer stars because they are helpful for finding Polaris, also known as the North Star or Pole Star. By visually tracing a line from Merak through Dubhe and continuing, ones eye will land on Polaris, W Ursae Majoris is the prototype of a class of contact binary variable stars, and ranges between 7. 75m and 8. 48m. 47 Ursae Majoris is a Sun-like star with a three-planet system,47 Ursae Majoris b, discovered in 1996, orbits every 1078 days and is 2.53 times the mass of Jupiter. 47 Ursae Majoris c, discovered in 2001, orbits every 2391 days and is 0.54 times the mass of Jupiter
Ursa Major
–
Visible at latitudes between +
90 ° and −
30 °. Best visible at 21:00 (9 p.m.) during the month of April. The
Big Dipper or Plough.
Ursa Major
–
List of stars in Ursa Major
Ursa Major
–
The constellation Ursa Major as it can be seen by the unaided eye.
Ursa Major
–
Ursa Major and Ursa Minor in relation to Polaris
42.
M81 Group
–
The approximate center of the group is located at a distance of 3.6 Mpc, making it one of the nearest groups to the Local Group. The group is estimated to have a mass of ×1012M☉. The M81 Group, the Local Group, and other groups all lie within the Virgo Supercluster. The table below lists galaxies that have identified as associated with the M81 Group by I. D. Karachentsev. Note that the names used in the above table differ from the names used by Karachentsev. NGC, IC, UGC, and PGC numbers have been used in cases to allow for easier referencing. Messier 81, Messier 82, and NGC3077 are all strongly interacting with each other, the gravitational interactions have stripped some hydrogen gas away from all three galaxies, leading to the formation of filamentary gas structures within the group. M81 Group @ SEDS M81 Group from An Atlas of The Universe
M81 Group
–
M81 Group
M81 Group
–
Galaxy UGC 8201 is a dwarf irregular galaxy member of the M81 galaxy group.
M81 Group
–
Amateur picture
Messier 81 +
82 and
NGC 3077 all of the M81 group, 33 frames stacked of 1 minute each.
43.
New General Catalogue
–
The NGC contains 7,840 objects, known as the NGC objects. It is one of the largest comprehensive catalogues, as it includes all types of space objects and is not confined to, for example. Dreyer also published two supplements to the NGC in 1895 and 1908, known as the Index Catalogues, describing a further 5,386 astronomical objects. Objects in the sky of the southern hemisphere are catalogued somewhat less thoroughly, the Revised New General Catalogue and Index Catalogue was compiled in 2009 by Wolfgang Steinicke. The original New General Catalogue was compiled during the 1880s by John Louis Emil Dreyer using observations from William Herschel and his son John, Dreyer had already published a supplement to Herschels General Catalogue of Nebulae and Clusters, containing about 1,000 new objects. In 1886, he suggested building a second supplement to the General Catalogue and this led to the publication of the New General Catalogue in the Memoirs of the Royal Astronomical Society in 1888. Assembling the NGC was a challenge, as Dreyer had to deal with many contradicting and unclear reports, while he did check some himself, the sheer number of objects meant Dreyer had to accept them as published by others for the purpose of his compilation. Dreyer was a careful transcriber and made few errors himself, and he was very thorough in his referencing, which allowed future astronomers to review the original references and publish corrections to the original NGC. The first major update to the NGC is the Index Catalogue of Nebulae and Clusters of Stars and it serves as a supplement to the NGC, and contains an additional 5,386 objects, collectively known as the IC objects. It summarizes the discoveries of galaxies, clusters and nebulae between 1888 and 1907, most of them made possible by photography, a list of corrections to the IC was published in 1912. The Revised New Catalogue of Nonstellar Astronomical Objects was compiled by Jack W. Sulentic and William G. Tifft in the early 1970s, and was published in 1973, as an update to the NGC. However, because the update had to be completed in just three summers, it failed to incorporate several previously-published corrections to the NGC data, and even introduced new errors. NGC2000.0 is a 1988 compilation of the NGC and IC made by Roger W. Sinnott and it incorporates several corrections and errata made by astronomers over the years. However, it too ignored the original publications and favoured modern corrections, the NGC/IC Project is a collaboration formed in 1993. It aims to identify all NGC and IC objects, and collect images, the Revised New General Catalogue and Index Catalogue is a compilation made by Wolfgang Steinicke in 2009. It is considered one of the most comprehensive and authoritative treatments of the NGC, messier object Catalogue of Nebulae and Clusters of Stars The Interactive NGC Catalog Online Adventures in Deep Space, Challenging Observing Projects for Amateur Astronomers
New General Catalogue
–
Spiral Galaxy NGC 3982 displays numerous spiral arms filled with bright stars, blue star clusters, and dark dust lanes. It spans about 30,000 light years, lies about 68 million light years from Earth and can be seen with a small telescope in the constellation of Ursa Major.
New General Catalogue
–
Four different
planetary nebulae. Clockwise starting from the top left:
NGC 6543,
NGC 7662,
NGC 6826, and
NGC 7009.
44.
Andromeda (constellation)
–
Andromeda is one of the 48 constellations listed by the 2nd-century Greco-Roman astronomer Ptolemy and remains one of the 88 modern constellations. Located north of the equator, it is named for Andromeda, daughter of Cassiopeia, in the Greek myth. Andromeda is most prominent during autumn evenings in the Northern Hemisphere, because of its northern declination, Andromeda is visible only north of 40° south latitude, for observers farther south it lies below the horizon. It is one of the largest constellations, with an area of 722 square degrees. This is over 1,400 times the size of the moon, 55% of the size of the largest constellation, Hydra. Its brightest star, Alpha Andromedae, is a star that has also been counted as a part of Pegasus, while Gamma Andromedae is a colorful binary. Only marginally dimmer than Alpha, Beta Andromedae is a red giant, the constellations most obvious deep-sky object is the naked-eye Andromeda Galaxy, the closest spiral galaxy to the Milky Way and one of the brightest Messier objects. Several fainter galaxies, including M31s companions M110 and M32, as well as the more distant NGC891, the Blue Snowball Nebula, a planetary nebula, is visible in a telescope as a blue circular object. Andromeda is the location of the radiant for the Andromedids, a meteor shower that occurs in November. The uranography of Andromeda has its roots most firmly in the Greek tradition, the stars that make up Pisces and the middle portion of modern Andromeda formed a constellation representing a fertility goddess, sometimes named as Anunitum or the Lady of the Heavens. Andromeda is known as the Chained Lady or the Chained Woman in English and it was known as Mulier Catenata in Latin and al-Marat al Musalsalah in Arabic. Offended at her remark, the nymphs petitioned Poseidon to punish Cassiopeia for her insolence, Andromedas panicked father, Cepheus, was told by the Oracle of Ammon that the only way to save his kingdom was to sacrifice his daughter to Cetus. Perseus and Andromeda then married, the myth recounts that the couple had nine children together – seven sons, after Andromedas death Athena placed her in the sky as a constellation, to honor her. Several of the neighboring constellations also represent characters in the Perseus myth and it is connected with the constellation Pegasus. Andromeda was one of the original 48 constellations formulated by Ptolemy in his 2nd-century Almagest, in which it was defined as a specific pattern of stars. She is typically depicted with α Andromedae as her head, ο and λ Andromedae as her chains, and δ, π, μ, Β, however, there is no universal depiction of Andromeda and the stars used to represent her body, head, and chains. Arab astronomers were aware of Ptolemys constellations, but they included a second constellation representing a fish at Andromedas feet, several stars from Andromeda and most of the stars in Lacerta were combined in 1787 by German astronomer Johann Bode to form Frederici Honores. It was designed to honor King Frederick II of Prussia, in 1922, the IAU defined its recommended three-letter abbreviation, And
Andromeda (constellation)
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Johannes Hevelius 's depiction of Andromeda, from the 1690 edition of his Uranographia. As was conventional for
celestial atlases of the time, the constellation is a mirror image of modern maps as it was drawn from a perspective outside the
celestial sphere.
Andromeda (constellation)
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List of stars in Andromeda
Andromeda (constellation)
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Andromeda as depicted in
Urania's Mirror, a set of constellation cards published in London c. 1825, showing the constellation from the inside of the celestial sphere
Andromeda (constellation)
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Photo of the constellation Andromeda, as it appears to the naked eye. Lines have been added for clarity.
45.
Shogi
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The earliest predecessor of the game, chaturanga, originated in India in the 6th century. Shogi was the earliest chess variant to allow captured pieces to be returned to the board by the capturing player, david Pritchard compares this rule to the practice of 16th century mercenaries switching loyalties when captured. Two players, Sente 先手 and Gote 後手, play on a board composed of rectangles in a grid of 9 ranks by 9 files, the rectangles are undifferentiated by marking or color. The board is always rectangular, square boards are uncommon. Pairs of dots mark the promotion zones. Each player has a set of 20 wedge-shaped pieces of different sizes. Except for the kings, opposing pieces are undifferentiated by marking or color, pieces face forward, this shows who controls the piece during play. Each piece has its name written on its surface in the form of two kanji, usually in black ink, following is a table of the pieces with their Japanese representations and English equivalents. The abbreviations are used for game notation and often referring to the pieces in speech in Japanese. * The kanji 竜 is a form of 龍. English speakers sometimes refer to promoted bishops as horses and promoted rooks as dragons, after their Japanese names, silver generals and gold generals are commonly referred to simply as silvers and golds. The characters inscribed on the sides of the pieces to indicate promotion may be in red ink. The characters on the backs of the pieces that promote to gold generals are cursive variants of 金 gold and these cursive forms have these equivalents in print, 全 for promoted silver, 今 for promoted knight, 仝 for promoted lance, and 个 for promoted pawn. The suggestion that the Japanese characters have deterred Western players from learning shogi has led to Westernized or international pieces which use iconic symbols instead of characters. Most players soon learn to recognize the characters, however, partially because the pieces are already iconic by size. As a result, Westernized pieces have never become popular, bilingual pieces with both Japanese characters and English captions have been developed as have pieces with animal cartoons. Each player sets up his pieces facing forward and that is, the first rank is or In the second rank, each player places, the bishop in the same file as the left knight, the rook in the same file as the right knight. In the third rank, the nine pawns are placed one per file, traditionally, the order of placing the pieces on the board is determined
Shogi
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A game of shogi using a magnetic travel set. Captured pieces in the tray (bottom-center) can be dropped into play on the board by the capturer.
Shogi
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A traditional shōgi-ban (shogi board) displaying a set of koma (pieces). The pieces on the far side are turned to show their promoted values. The stands on either side are komadai used to hold captured pieces. The board itself is raised for the comfort of players seated on tatami mats (background), and is hollowed underneath to produce a pleasing sound when the pieces are moved.
Shogi
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Closeup of shogi pieces. Top: +R, R, K (reigning), K (challenging), B, +B. Bottom: +L, L, +S, S, G, N, +N, P, +P.
Shogi
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The king
46.
1981
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January – The subterranean Sarawak Chamber is discovered in Borneo. January 1 Greece enters the European Community, which becomes the European Union. January 10 – Salvadoran Civil War, The FMLN launches its first major offensive, gaining control of most of Morazán, january 15 – Pope John Paul II receives a delegation led by Solidarity leader Lech Wałęsa at the Vatican. January 16 – Loyalists shoot and seriously wound nationalist activist Bernadette Devlin McAliskey, january 17 – Philippine President Ferdinand Marcos lifts martial law. January 19 – United States and Iranian officials sign an agreement to release 52 American hostages after 14 months of captivity. January 20 – Iran releases the 52 Americans held for 444 days minutes after Ronald Reagan is sworn in as President of the United States, ending the Iran hostage crisis. January 21 – The first DeLorean DMC-12 automobile, a stainless steel car with gull-wing doors, rolls off the production line in Dunmurry. January 23 – An earthquake of 6.8 magnitude in Sichuan, january 25 Jiang Qing is sentenced to death in the Peoples Republic of China. In South Africa the largest part of the town Laingsburg is swept away within minutes by one of the strongest floods ever experienced in the Great Karoo, january 27 – The Indonesian passenger ship Tamponas 2 catches fire and capsizes in the Java Sea, killing 580. February 4 – Gro Harlem Brundtland becomes Prime Minister of Norway, february 9 – Polish Prime Minister Józef Pińkowski resigns and is replaced by General Wojciech Jaruzelski. February 14 Stardust fire, A fire at the Stardust nightclub in Artane, Dublin, Ireland in the early hours kills 48 people, Australia withdraws recognition of the Pol Pot regime in Cambodia. February 17--February 22 – Pope John Paul II visit to the Philippines, the coup détat fails thanks to King Juan Carlos. March 1 – Bobby Sands, a Provisional Irish Republican Army member and he died May 5, the first of 10 men. March 11 – Chilean military dictator Augusto Pinochet is sworn in as President of Chile for another 8-year term, March 17 – In Italy the Propaganda Due Masonic Lodge is discovered. March 19 – Three workers are killed and 5 injured during a test of the Space Shuttle Columbia, March 29 – The first London Marathon starts with 7,500 runners. March 30 – U. S. President Ronald Reagan is shot in the chest outside a Washington, D. C. hotel by John Hinckley, Jr.2 police officers, April 1 – Daylight saving time is introduced in the Soviet Union. April 4 – UK pop group Bucks Fizz song Making Your Mind Up wins the 1981 Eurovision Song Contest in Dublin, April 6 – Eventual 2016 presidential candidate Bernie Sanders becomes the Mayor of Burlington, Vermont and was in office until April 4,1989. 26 years later he got elected as Democratic U. S, Senator from Vermont, and still remains the states current Senator
1981
–
April 12: First
Space Shuttle launch:
Columbia, April 12, 1981.
1981
–
Pitbull
1981
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Alicia Keys
1981
–
Elijah Wood
47.
Atomic number
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The atomic number or proton number of a chemical element is the number of protons found in the nucleus of an atom of that element. It is identical to the number of the nucleus. The atomic number identifies a chemical element. In an uncharged atom, the number is also equal to the number of electrons. The atomic number Z, should not be confused with the mass number A and this number of neutrons, N, completes the weight, A = Z + N. Atoms with the atomic number Z but different neutron numbers N. Historically, it was these atomic weights of elements that were the quantities measurable by chemists in the 19th century. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge, loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, and so they can be numbered in order. Dmitri Mendeleev claimed that he arranged his first periodic tables in order of atomic weight, however, in consideration of the elements observed chemical properties, he changed the order slightly and placed tellurium ahead of iodine. This placement is consistent with the practice of ordering the elements by proton number, Z. A simple numbering based on periodic table position was never entirely satisfactory and this central charge would thus be approximately half the atomic weight. This proved eventually to be the case, the experimental position improved dramatically after research by Henry Moseley in 1913. To do this, Moseley measured the wavelengths of the innermost photon transitions produced by the elements from aluminum to gold used as a series of movable anodic targets inside an x-ray tube. The square root of the frequency of these photons increased from one target to the next in an arithmetic progression and this led to the conclusion that the atomic number does closely correspond to the calculated electric charge of the nucleus, i. e. the element number Z. Among other things, Moseley demonstrated that the series must have 15 members—no fewer. After Moseleys death in 1915, the numbers of all known elements from hydrogen to uranium were examined by his method. There were seven elements which were not found and therefore identified as still undiscovered, from 1918 to 1947, all seven of these missing elements were discovered. By this time the first four transuranium elements had also been discovered, in 1915 the reason for nuclear charge being quantized in units of Z, which were now recognized to be the same as the element number, was not understood
Atomic number
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An explanation of the superscripts and subscripts seen in atomic number notation. Atomic number is the number of protons, and therefore also the total positive charge, in the atomic nucleus.
Atomic number
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Russian chemist Dmitri Mendeleev created a periodic table of the elements that ordered them numerically by atomic weight, yet occasionally used chemical properties in contradiction to weight.
Atomic number
–
Niels Bohr's 1913
Bohr model of the atom required van den Broek's atomic number of nuclear charges, and Bohr believed that Moseley's work contributed greatly to the acceptance of the model.
Atomic number
–
Henry Moseley helped develop the concept of atomic number by showing experimentally (1913) that Van den Broek's 1911 hypothesis combined with the
Bohr model nearly correctly predicted atomic X-ray emissions.
48.
Thallium
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Thallium is a chemical element with symbol Tl and atomic number 81. This soft gray post-transition metal is not found free in nature, when isolated, thallium resembles tin, but discolors when exposed to air. Chemists William Crookes and Claude-Auguste Lamy discovered thallium independently in 1861, both used the newly developed method of flame spectroscopy, in which thallium produces a notable green spectral line. Thallium, from Greek θαλλός, thallos, meaning a green shoot or twig, was named by Crookes and it was isolated by both Lamy and Crookes in 1862, Lamy by electrolysis and Crookes by precipitation and melting of the resultant powder. Crookes exhibited it as a powder precipitated by zinc at the International exhibition which opened on 1 May, Thallium tends to oxidize to the +3 and +1 oxidation states as ionic salts. The +3 state resembles that of the elements in group 13. Commercially, however, thallium is produced not from potassium ores, approximately 60–70% of thallium production is used in the electronics industry, and the remainder is used in the pharmaceutical industry and in glass manufacturing. It is also used in infrared detectors, the radioisotope thallium-201 is used in small, nontoxic amounts as an agent in a nuclear medicine scan, during one type of nuclear cardiac stress test. Soluble thallium salts are toxic, and they were used in rat poisons. Use of these compounds has been restricted or banned in many countries, notably, thallium poisoning results in hair loss. Because of its popularity as a murder weapon, thallium has gained notoriety as the poisoners poison. A thallium atom has 81 electrons, arranged in the electron configuration 4f145d106s26p1, of these, however, due to the inert pair effect, the 6s electron pair is relativistically stabilised and it is more difficult to get them involved in chemical bonding than for the heavier elements. Thallium is malleable and sectile enough to be cut with a knife at room temperature and it has a metallic luster that, when exposed to air, quickly tarnishes to a bluish-gray tinge, resembling lead. It may be preserved by immersion in oil, a heavy layer of oxide builds up on thallium if left in air. In the presence of water, thallium hydroxide is formed, sulfuric and nitric acid dissolve thallium rapidly to make the sulfate and nitrate salts, while hydrochloric acid forms an insoluble thallium chloride layer. Thallium has 25 isotopes which have atomic masses that range from 184 to 210, 203Tl and 205Tl are the only stable isotopes and make up nearly all of natural thallium. 204Tl is the most stable radioisotope, with a half-life of 3.78 years and it is made by the neutron activation of stable thallium in a nuclear reactor. It is the most popular isotope used for thallium nuclear cardiac stress tests, Thallium compounds resemble the corresponding aluminium compounds
Thallium
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Thallium, 81 Tl
Thallium
–
Crystals of
hutchinsonite (TlPbAs 5 S 9)
Thallium
–
Corroded thallium rod
49.
Hells Angels
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The Hells Angels Motorcycle Club is a worldwide one-percenter motorcycle club whose members typically ride Harley-Davidson motorcycles. The organization is predominantly male and is considered an organized crime syndicate by the United States Department of Justice. In the United States and Canada, the Hells Angels are incorporated as the Hells Angels Motorcycle Corporation, common nicknames for the club are the H. A. The Hells Angels website denies the suggestion that any misfit or malcontent troops are connected with the motorcycle club. The website also notes that the name was suggested by Arvid Olsen, an associate of the founders, who had served in the Flying Tigers Hells Angels squadron in China during World War II. In 1930, the Howard Hughes film Hells Angels displayed extraordinary and dangerous feats of aviation, some of the early history of the HAMC is not clear, and accounts differ. One of the lesser known clubs existed in North Chino/South Pomona, California, the Frisco Hells Angels were reorganized in 1955 with thirteen charter members, Frank Sadilek serving as President, and using the smaller, original logo. The Oakland chapter, at the time headed by Barger, used a version of the Deaths Head patch nicknamed the Barger Larger. It later became the club standard, the Hells Angels are often depicted in semi-mythical romantic fashion like the 19th-century James–Younger Gang, free-spirited, iconic, bound by brotherhood and loyalty. At other times, such as in the 1966 Roger Corman film The Wild Angels, they are depicted as violent and nihilistic, little more than a violent criminal gang, the club launched the career of Gonzo journalist Hunter S. Thompson. The Hells Angels official website attributes the deaths head insignia design to Frank Sadilek. The colors and shape of the early-style jacket emblem were copied from the insignias of the 85th Fighter Squadron, the Hells Angels utilize a system of patches similar to military medals. Although the specific meaning of each patch is not publicly known, the official colors of the Hells Angels are red lettering displayed on a white background—hence the clubs nickname The Red and White. These patches are worn on leather or denim jackets and vests, Red and white are also used to display the number 81 on many patches, as in Support 81, Route 81. The 8 and 1 stand for the positions in the alphabet of H and A. These are used by friends and supporters of the club in deference to club rules, the diamond-shaped one-percenter patch is also used, displaying 1% in red on a white background with a red merrowed border. The AMA has no record of such a statement to the press, most members wear a rectangular patch identifying their respective chapter locations. Another similarly designed patch reads Hells Angels, when applicable, members of the club wear a patch denoting their position or rank within the organization
Hells Angels
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Hells Angels MC
Hells Angels
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This B-17F, tail number 41-24577, was named Hell's Angels after the 1930
Howard Hughes movie about
World War I fighter pilots.
Hells Angels
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Insignia of the Hells Angels from
Karlsruhe chapter, with the '1%' patch
Hells Angels
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New York Hells Angels patch.
50.
Short film
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A short film, is a cinema format, denoting any film not long enough to be considered a feature film. The term featurette originally applied to a longer than a short subject. The increasingly rare term short subject means approximately the same thing and it is an industry term which carries more of an assumption that the film is shown as part of a presentation along with a feature film. Short is an abbreviation for either term, Short films are often screened at local, national, or international film festivals and made by independent filmmakers for non profit, either with a low budget or no budget at all. They are usually funded by grants, non profit organizations. Short films are used by filmmakers to gain experience or prove their talent in order to gain funding for future films from private investors, entertainment companies. Longer and shorter films coexisted with similar popularity throughout the days of film. However, comedy films were produced in large numbers compared to lengthy features such as D. W. Griffiths The Birth of a Nation. By the 1920s, a ticket purchased a varied program including a feature and several supporting works from such as second feature, short comedy, 5–10 minute cartoon, travelogue. Short comedies were popular, and typically came in a serial or series. Even though there was no set release schedule, these series could be considered somewhat like a modern TV sitcom – lower in status than feature films. Animated cartoons came principally as short subjects, in the 1930s, the distribution system changed in many countries owing to the Great Depression. Instead of the cinema owner assembling a program of their own choice, the studios sold a package centered on a main and supporting feature, with the rise of the double feature, two reel shorts went into decline as a commercial category. Hal Roach, for example, moved Laurel and Hardy full-time into feature films after 1935, by the mid-1950s, with the rise of television, the commercial live-action short was virtually dead, The Three Stooges being the last major series of 2-reelers, ending in 1959. Short films had become a medium for student, independent and specialty work, cartoon shorts had a longer life, due in part to the implementation of lower-cost limited animation techniques, but also declined in this period. Warner Bros. one of the most prolific of the golden era, the Pink Panther was the last regular theatrical cartoon short series, having begun in 1964 and ended in 1980. By the 1960s, the market for animated shorts had largely shifted to television, a few animated shorts continue within mainstream commercial distribution. For instance, Pixar has screened a short along with each of its feature films during its theatrical run since 1995
Short film
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William Garwood starred in numerous short films, many of which were only 20 minutes in length
Short film