1.
54 (number)
–
54 is the natural number following 53 and preceding 55. Twice the third power of three,54 is a Leyland number,54 can be written as the sum of three squares in three different ways,72 +22 +12 =62 +32 +32 =52 +52 +22 =54. It is the smallest number with this property, like all other multiples of 6, it is a semiperfect number. In base 10,54 is a Harshad number, the Holt graph has 54 edges. The sine of an angle of 54 degrees is half the golden ratio, the factorial of 54 is 230843697339241380472092742683027581083278564571807941132288000000000000, or approximately 2. 30843697339241×1071. The atomic number of xenon is 54, messier object M54, a magnitude 8. The Saros number of the solar eclipse series began on 25 July 1285 BC. The duration of Saros series 54 was 1316.2 years, the Saros number of the lunar eclipse series which began on -964 May 14 and ended on 334 July. The duration of Saros series 54 was 1298.1 years, fewest points in an NBA playoff game, Chicago, Utah, June 7,1998 The New York Rangers won the Stanley Cup in 1994,54 years after their previous Cup win. It is the longest drought in the trophys history, for years car number 54 was driven by NASCARs Lennie Pond. More recently, it is known as the Nationwide Series car number for Kyle Busch, a score of 54 in golf is colloquially referred to as a perfect round. This score has never achieved in competition. The number used when a player is defeated 3 games in a row in racquetball,54 is also, The number of milligrams of caffeine in 12 ounces of Mountain Dew
54 (number)
–
A
Rubik's Cube has 54 colored squares
2.
52 (number)
–
52 is the natural number following 51 and preceding 53. Fifty-two is the 6th Bell number and a decagonal number and it is an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is never the answer to the equation x − φ. The atomic number of tellurium Messier object M52, a magnitude 8.0 open cluster in the constellation Cassiopeia, the New General Catalogue object NGC52, a spiral galaxy in the constellation Pegasus. The Saros number of the solar eclipse series began on -1378 August 14. The duration of Saros series 52 was 1532.5 years, the Saros number of the lunar eclipse series which began on -1026 May 27 and ended on 204 June. The duration of Saros series 52 was 1280.1 years, U. S. Route 52 that runs from South Carolina to North Dakota Fifty-two is, The approximate number of weeks in a year. 52 weeks is 364 days, while the year is 365.24 days long. According to ISO8601, most years have 52 weeks while some have 53, the New 52 is a 2011 revamp and relaunch by DC Comics of its entire line of ongoing monthly superhero books. 52 is the car number of retired NASCAR driver Jimmy Means 52 American hostages were held in the Iran hostage crisis 52 BC, AD52,1952,2052, etc
52 (number)
–
The
piano has 52 white keys
3.
53 (number)
–
53 is the natural number following 52 and preceding 54. Fifty-three is the 16th prime number and it is also an Eisenstein prime, and a Sophie Germain prime. The sum of the first 53 primes is 5830, which is divisible by 53,53 written in hexadecimal is 35, that is, the same characters used in the decimal representation, but reversed. Four multiples of 53 share this property,371 =17316,5141 =141516,99481 =1849916, and 8520280 =82025816,53 cannot be expressed as the sum of any integer and its base-10 digits, making 53 a self number. 53 is the smallest prime number that does not divide the order of any sporadic group, the duration of Saros series 53 was 1514.5 years, and it contained 85 solar eclipses. The Saros number of the lunar eclipse series began on June 5,993 BC. The duration of Saros series 53 was 1280.1 years, fictional 53rd Precinct in the Bronx was found in the TV comedy Car 54, Where Are You. UDP and TCP port number for the Domain Name System protocol, 53-TET is a musical temperament that has a fifth that is closer to pure than our current system. 53 More Things To Do In Zero Gravity is a mentioned in The Hitchhikers Guide to the Galaxy. 53 a number used on the hand of the tulip in Infinity Train
53 (number)
–
A fan-built
Herbie
4.
59 (number)
–
59 is the natural number following 58 and preceding 60. Fifty-nine is the 17th smallest prime number, the next is sixty-one, with which it comprises a twin prime. 59 is an prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no part and real part of the form 3n −1. +1 is divisible by 59 but 59 is not one more than a multiple of 15,59 is a Pillai prime and it is also a highly cototient number. There are 59 stellations of the icosahedron,59 is one of the factors that divides the smallest composite Euclid number. In this case 59 divides the Euclid number 13# +1 =2 ×3 ×5 ×7 ×11 ×13 +1 =59 ×509, the duration of Saros series 59 was 1280.1 years, and it contained 72 solar eclipses. The Saros number of the lunar eclipse series began in March,729 BC
59 (number)
–
The
TI-59 was a
programmable calculator
59 (number)
–
A regular
icosahedron has
59 stellations
5.
60 (number)
–
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.
6.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
7.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
8.
20 (number)
–
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
9.
30 (number)
–
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.
10.
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.
11.
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)
12.
90 (number)
–
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)
–
Interstate 90 is a freeway that runs from
Washington to
Massachusetts.
13.
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.
14.
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.
15.
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
16.
Greek numerals
–
Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek numerals
–
Numeral systems
Greek numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
17.
Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
–
Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
–
Numeral systems
Roman numerals
–
A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
–
An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
18.
Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary number
–
Numeral systems
Binary number
–
Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
19.
Ternary numeral system
–
The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
Ternary numeral system
–
Numeral systems
20.
Quaternary numeral system
–
Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
Quaternary numeral system
–
Numeral systems
21.
Quinary
–
Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
Quinary
–
Numeral systems
22.
Senary
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Senary
–
Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting
Senary
23.
Octal
–
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
Octal
–
Numeral systems
24.
Duodecimal
–
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
Duodecimal
–
Numeral systems
Duodecimal
–
A duodecimal multiplication table
25.
Hexadecimal
–
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
–
Numeral systems
Hexadecimal
–
Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
–
Hexadecimal finger-counting scheme.
26.
Vigesimal
–
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.
27.
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
28.
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, …)
29.
Fibonacci number
–
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
Fibonacci number
–
A page of
Fibonacci 's
Liber Abaci from the
Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.
Fibonacci number
–
A tiling with squares whose side lengths are successive Fibonacci numbers
30.
Triangular number
–
A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime
Triangular number
–
The first six triangular numbers
31.
Square pyramidal number
–
In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid. The first few square pyramidal numbers are,1,5,14,30,55,91,140,204,285,385,506,650,819 and this is a special case of Faulhabers formula, and may be proved by a mathematical induction. An equivalent formula is given in Fibonaccis Liber Abaci, in modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L of a polyhedron P is a polynomial that counts the number of points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid base is a unit square with integer coordinates. The square pyramidal numbers can also be expressed as sums of binomial coefficients, the smaller tetrahedral number represents 1 +3 +6 + ⋯ + T and the larger 1 +3 +6 + ⋯ + T. Offsetting the larger and adding, we arrive at 1,1 +3,3 +6 …, Square pyramidal numbers are also related to tetrahedral numbers in a different way, P n =14. The sum of two square pyramidal numbers is an octahedral number. Augmenting a pyramid whose base edge has n balls by adding to one of its faces a tetrahedron whose base edge has n −1 balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism and this geometric dissection leads to another relation, P n = n −. Besides 1, there is one other number that is both a square and a pyramid number,4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918, in the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes. Also, P n = − which is the difference of two pentatope numbers and this can be seen by expanding, n − n = n = n and dividing through by 24. A common mathematical puzzle involves finding the number of squares in a n by n square grid. This number can be derived as follows, The number of 1 ×1 boxes found in the grid is n2, the number of 2 ×2 boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of 2 ×2 boxes, the number of k × k boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of k × k boxes and it follows that the number of squares in an n × n square grid is, n 2 +2 +2 +2 + … +12 = n 6
Square pyramidal number
–
A pyramid of
cannonballs in the
Musée historique de Strasbourg. The number of balls in the pyramid can be calculated as the fifth square pyramidal number, 55.
Square pyramidal number
–
Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30.
32.
Square (algebra)
–
In mathematics, a square is the result of multiplying a number by itself. The verb to square is used to denote this operation, squaring is the same as raising to the power 2, and is denoted by a superscript 2, for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the adjective which corresponds to squaring is quadratic. The square of an integer may also be called a number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, for instance, the square of the linear polynomial x +1 is the quadratic polynomial x2 + 2x +1. One of the important properties of squaring, for numbers as well as in other mathematical systems, is that. That is, the function satisfies the identity x2 =2. This can also be expressed by saying that the function is an even function. The squaring function preserves the order of numbers, larger numbers have larger squares. In other words, squaring is a function on the interval. Hence, zero is its global minimum, the only cases where the square x2 of a number is less than x occur when 0 < x <1, that is, when x belongs to an open interval. This implies that the square of an integer is never less than the original number, every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of one number, itself. For this reason, it is possible to define the square root function, no square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. There are several uses of the squaring function in geometry. The name of the squaring function shows its importance in the definition of the area, the area depends quadratically on the size, the area of a shape n times larger is n2 times greater. The squaring function is related to distance through the Pythagorean theorem and its generalization, Euclidean distance is not a smooth function, the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance, which has a paraboloid as its graph, is a smooth, the dot product of a Euclidean vector with itself is equal to the square of its length, v⋅v = v2
Square (algebra)
–
The
composition of the tiling
Image:ConformId.jpg (understood as a function on the complex plane) with the complex square function
Square (algebra)
–
5⋅5, or 5 2 (5 squared), can be shown graphically using a
square. Each block represents one unit, 1⋅1, and the entire square represents 5⋅5, or the area of the square.
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 nonagonal number
–
A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for n is given by the formula N c =2. Thus, the first few centered nonagonal numbers are 1,10,28,55,91,136,190,253,325,406,496,595,703,820,946, the list above includes the perfect numbers 28 and 496. All even perfect numbers are numbers whose index is an odd Mersenne prime. Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, in 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven
Centered nonagonal number
–
See also [edit]
35.
Caesium
–
Caesium or cesium is a chemical element with symbol Cs and atomic number 55. It is a soft, silvery-gold alkali metal with a point of 28.5 °C. Caesium is a metal and has physical and chemical properties similar to those of rubidium and potassium. The metal is extremely reactive and pyrophoric, reacting with water even at −116 °C, Caesium is one of the most reactive elements of all, even more reactive than fluorine, the most reactive nonmetal. It is the least electronegative element, with a value of 0.79 on the Pauling scale and it has only one stable isotope, caesium-133. Caesium is mined mostly from pollucite, while the radioisotopes, especially caesium-137, the German chemist Robert Bunsen and physicist Gustav Kirchhoff discovered caesium in 1860 by the newly developed method of flame spectroscopy. The first small-scale applications for caesium were as a getter in vacuum tubes, since then, caesium has been widely used in highly accurate atomic clocks. The radioactive isotope caesium-137 has a half-life of about 30 years and is used in applications, industrial gauges. Although the element is only toxic, the metal is a hazardous material. It is a ductile, pale metal, which darkens in the presence of trace amounts of oxygen. When in the presence of oil, it loses its metallic lustre and takes on a duller. It has a point of 28.4 °C, making it one of the few elemental metals that are liquid near room temperature. Mercury is the elemental metal with a known melting point lower than caesium. In addition, the metal has a low boiling point,641 °C. Its compounds burn with a blue or violet colour, Caesium forms alloys with the other alkali metals, gold, and mercury. At temperatures below 650 °C, it does not alloy with cobalt, iron, molybdenum, nickel, platinum, tantalum and it forms well-defined intermetallic compounds with antimony, gallium, indium, and thorium, which are photosensitive. It mixes with all the alkali metals, the alloy with a molar distribution of 41% caesium, 47% potassium. A few amalgams have been studied, CsHg 2 is black with a metallic lustre, while CsHg is golden-coloured
Caesium
–
Caesium, 55 Cs
Caesium
–
High-purity caesium-133 preserved under argon
Caesium
–
Secure caesium sample for teaching
Caesium
–
Monoatomic caesium halide wires grown inside double-wall
carbon nanotubes (
TEM image).
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 55
–
Messier 55 is a globular cluster in the constellation Sagittarius. It was discovered by Nicolas Louis de Lacaille in June 16,1752 while observing from what today is South Africa and he finally observed and catalogued it in 1778. The cluster can be seen with a pair of 50 mm binoculars, M55 is at a distance of about 17,600 light-years away from Earth. It has a mass of about 269,000 times that of the Sun, as with other Milky Way globular clusters, it has a low abundance of elements other than hydrogen and helium compared to the Sun—what astronomers term the metallicity of the cluster. This quantity is normally listed as the logarithm of the proportion relative to the Sun, for NGC6809 the metallicity is given by. Taking this exponent to the powers of 10 yields an abundance equal to 1. 1% of the proportion of elements in the Sun. Only about 55 variable stars have been discovered in the part of M55. Messier 55, SEDS Messier pages Messier 55, Galactic Globular Clusters Database page Messier 55 on WikiSky, DSS2, SDSS, GALEX, IRAS, Hydrogen α, X-Ray, Astrophoto, Sky Map, Articles and images
Messier 55
–
Messier 55
Messier 55
–
Messier 55 with amateur telescope
38.
Globular cluster
–
A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes, the name of this category of star cluster is derived from the Latin globulus—a small sphere. A globular cluster is known more simply as a globular. Globular clusters are found in the halo of a galaxy and contain considerably more stars and are older than the less dense open clusters. Globular clusters are common, there are about 150 to 158 currently known globular clusters in the Milky Way. These globular clusters orbit the Galaxy at radii of 40 kiloparsecs or more, larger galaxies can have more, Andromeda Galaxy, for instance, may have as many as 500. Some giant elliptical galaxies such as M87, have as many as 13,000 globular clusters, every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters. The Sagittarius Dwarf galaxy and the disputed Canis Major Dwarf galaxy appear to be in the process of donating their associated globular clusters to the Milky Way and this demonstrates how many of this galaxys globular clusters might have been acquired in the past. Although it appears that globular clusters contain some of the first stars to be produced in the galaxy, their origins, the first known globular cluster, now called M22, was discovered in 1665 by Abraham Ihle, a German amateur astronomer. However, given the small aperture of early telescopes, individual stars within a cluster were not resolved until Charles Messier observed M4 in 1764. The first eight globular clusters discovered are shown in the table, subsequently, Abbé Lacaille would list NGC104, NGC4833, M55, M69, and NGC6397 in his 1751–52 catalogue. The M before a number refers to Charles Messiers catalogue, while NGC is from the New General Catalogue by John Dreyer, when William Herschel began his comprehensive survey of the sky using large telescopes in 1782 there were 34 known globular clusters. Herschel discovered another 36 himself and was the first to virtually all of them into stars. He coined the term globular cluster in his Catalogue of a Second Thousand New Nebulae, the number of globular clusters discovered continued to increase, reaching 83 in 1915,93 in 1930 and 97 by 1947. A total of 152 globular clusters have now discovered in the Milky Way galaxy. These additional, undiscovered globular clusters are believed to be hidden behind the gas, beginning in 1914, Harlow Shapley began a series of studies of globular clusters, published in about 40 scientific papers. He examined the RR Lyrae variables in the clusters and would use their period–luminosity relationship for distance estimates, later, it was found that RR Lyrae variables are fainter than Cepheid variables, which caused Shapley to overestimate the distance to the clusters. Of the globular clusters within the Milky Way, the majority are found in a halo around the core
Globular cluster
–
The
Messier 80 globular cluster in the constellation
Scorpius is located about 30,000
light-years from the Sun and contains hundreds of thousands of stars.
Globular cluster
–
NGC 7006 is a highly concentrated, Class I globular cluster.
Globular cluster
–
Globular star cluster
Messier 54.
Globular cluster
–
Djorgovski 1 's stars contain hydrogen and helium, but not much else. In astronomical terms, they are described as "metal-poor".
39.
Sagittarius (constellation)
–
Sagittarius is one of the constellations of the zodiac. It is one of the 48 constellations listed by the 2nd-century astronomer Ptolemy and its name is Latin for the archer, and its symbol is, a stylized arrow. Sagittarius is commonly represented as a centaur pulling-back a bow and it lies between Scorpius and Ophiuchus to the west and Capricornus to the east. The center of the Milky Way lies in the westernmost part of Sagittarius, as seen from the northern hemisphere, the constellations brighter stars form an easily recognizable asterism known as the Teapot. The stars δ Sgr, ε Sgr, ζ Sgr, and φ Sgr form the body of the pot, λ Sgr is the point of the lid, γ2 Sgr is the tip of the spout and these same stars originally formed the bow and arrow of Sagittarius. Marking the bottom of the teapots handle (or the shoulder area of the archer, are the bright star Zeta Sagittarii, named Ascella, and the fainter Tau Sagittarii. The constellation as a whole is often depicted as having the appearance of a stick-figure archer drawing its bow. Sagittarius famously points its arrow at the heart of Scorpius, represented by the reddish star Antares, following the direct line formed by Delta Sagittarii and Gamma2 Sagittarii leads nearly directly to Antares. Fittingly, Sagittarii is Alnasl, the Arabic word for arrowhead, and Delta Sagittarii is called Kaus Media, Kaus Media bisects Lambda Sagittarii and Epsilon Sagittarii, whose names Kaus Borealis and Kaus Australis refer to the northern and southern portions of the bow, respectively. α Sgr despite having the alpha appellation, is not the brightest star of the constellation, instead, the brightest star is Epsilon Sagittarii, at magnitude 1.85. Sigma Sagittarii is the constellations second-brightest star at magnitude 2.08, Nunki is a B2V star approximately 260 light years away. Nunki is a Babylonian name of origin, but thought to represent the sacred Babylonian city of Eridu on the Euphrates. Zeta Sagittarii, with apparent magnitude 2.61 of A2 spectra, is actually a star whose two components have magnitudes 3.3 and 3.5. Delta Sagittarii, is a K2 spectra star with magnitude 2.71 and only 85 light years from Earth. Eta Sagittarii is a star with component magnitudes of 3.18 and 10, while Pi Sagittarii is actually a triple system whose components have magnitudes 3.7,3.8. The Bayer designation Beta Sagittarii is shared by two systems, β¹ Sagittarii, with apparent magnitude 3.96, and β² Sagittarii. The two stars are separated by 0. 36° in the sky and are 378 light years from earth, Beta Sagittarii, located at a position associated with the forelegs of the centaur, has the traditional name Arkab, meaning achilles tendon. Nova Sagittarii 2015 No.2 was discovered on March 15,2015, by John Seach of Chatsworth Island, NSW and it lies near the center of the constellation
Sagittarius (constellation)
–
The "Teapot" asterism is in Sagittarius. The Milky Way is the "steam" coming from the spout.
Sagittarius (constellation)
–
List of stars in Sagittarius
Sagittarius (constellation)
–
The constellation Sagittarius as it can be seen with the naked eye.
Sagittarius (constellation)
–
The Omega Nebula, also known as the Horseshoe or Swan Nebula.
40.
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.
41.
Barred spiral galaxy
–
A barred spiral galaxy is a spiral galaxy with a central bar-shaped structure composed of stars. Bars are found in approximately two-thirds of all spiral galaxies, bars generally affect both the motions of stars and interstellar gas within spiral galaxies and can affect spiral arms as well. The Milky Way Galaxy, where our own Solar System is located, is classified as a spiral galaxy. Edwin Hubble classified spiral galaxies of this type as SB in his Hubble sequence, sBa types feature tightly bound arms, while SBc types are at the other extreme and have loosely bound arms. SB0 is a lenticular galaxy. Among other types in Hubbles classifications for the galaxies are, spiral galaxy, elliptical galaxy, Barred galaxies are apparently predominant, with surveys showing that up to two-thirds of all spiral galaxies contain a bar. The current hypothesis is that the bar structure acts as a type of stellar nursery, the bar is thought to act as a mechanism that channels gas inwards from the spiral arms through orbital resonance, in effect funneling the flow to create new stars. This process is thought to explain why many barred spiral galaxies have active galactic nuclei. The creation of the bar is thought to be the result of a density wave radiating from the center of the galaxy whose effects reshape the orbits of the inner stars. This effect builds over time to stars orbiting further out, which creates a bar structure. Bars are thought to be temporary phenomena in the lives of spiral galaxies, past a certain size the accumulated mass of the bar compromises the stability of the overall bar structure. Barred spiral galaxies with high mass accumulated in their center tend to have short, since so many spiral galaxies have bar structures, it is likely that they are recurring phenomena in spiral galaxy development. The oscillating evolutionary cycle from spiral galaxy to barred spiral galaxy is thought to take on the average about two billion years, recent studies have confirmed the idea that bars are a sign of galaxies reaching full maturity as the formative years end. The sub-categories are based on how open or tight the arms of the spiral are, sBa types feature tightly bound arms. SBc types are at the extreme and have loosely bound arms. SBm describes somewhat irregular barred spirals, sB0 is a barred lenticular galaxy. Galaxy morphological classification Galaxy formation and evolution Lenticular galaxy Spiral galaxy Firehose instability Britt, Milky Way’s Central Structure Seen with Fresh Clarity. An article about the Spitzer Space Telescopes Milky Way discovery Devitt, galactic survey reveals a new look for the Milky Way
Barred spiral galaxy
–
NGC 1300, viewed nearly face-on;
Hubble Space Telescope image
Barred spiral galaxy
–
Milky Way Galaxy Spiral Arms - based on
WISE data.
Barred spiral galaxy
–
NGC 2787
Barred spiral galaxy
–
NGC 4314
42.
Sculptor (constellation)
–
Sculptor is a small and faint constellation in the southern sky. It was introduced by Nicolas Louis de Lacaille in the 18th century and he originally named it Apparatus Sculptoris, but the name was later shortened. The region to the south of Cetus and Aquarius had been named by Aratus in 270 BCE as The Waters – an area of scattered faint stars with two brighter stars standing out, professor of astronomy Bradley Schaefer has proposed that these stars were most likely Alpha and Delta Sculptoris. He named all but one in honour of instruments that symbolised the Age of Enlightenment, Sculptor is a small constellation bordered by Aquarius and Cetus to the north, Fornax to the east, Phoenix to the south, Grus to the southwest, and Piscis Austrinus to the west. The bright star Fomalhaut is nearby, the three-letter abbreviation for the constellation, as adopted by the International Astronomical Union in 1922, is Scl. The official constellation boundaries, as set by Eugène Delporte in 1930, are defined by a polygon of 6 segments. In the equatorial coordinate system, the right ascension coordinates of these borders lie between 23h 06. 4m and 01h 45. 5m, while the coordinates are between −24. 80° and −39. 37°. The whole constellation is visible to observers south of latitude 50°N, no stars brighter than 3rd magnitude are located in Sculptor. This is explained by the fact that Sculptor contains the south pole where stellar density is very low. Overall, there are 52 stars within the constellations borders brighter than or equal to apparent magnitude 6.5, the brightest star is Alpha Sculptoris, an SX Arietis-type variable star with a spectral type B7IIIp and an apparent magnitude of 4.3. It is 780 ±30 light-years distant from Earth, R Sculptoris is a red giant that has been found to be surrounded by spirals of matter likely ejected around 1800 years ago. The constellation also contains the Sculptor Dwarf, a galaxy which is a member of the Local Group, as well as the Sculptor Group. The Sculptor Galaxy, a spiral galaxy and the largest member of the group. Another prominent member of the group is the irregular galaxy NGC55, one unique galaxy in Sculptor is the Cartwheel Galaxy, at a distance of 500 million light-years. The result of a merger around 300 million years ago, the Cartwheel Galaxy has a core of older, yellow stars, and a ring of younger, blue stars. The smaller galaxy in the collision is now incorporated into the core, the shock waves from the collision sparked extensive star formation in the outer ring. Sculptor was a United States Navy Crater class cargo ship named after the constellation, notes Citations Sources Sculptor Constellation at Constellation Guide
Sculptor (constellation)
–
Curious spiral around red giant star
R Sculptoris.
Sculptor (constellation)
–
List of stars in Sculptor
43.
Kasabian
–
Kasabian are an English rock band formed in Leicester in 1997. The bands original members consisted of vocalist Tom Meighan, guitarist and vocalist Sergio Pizzorno, guitarist Chris Karloff, the bands line-up was completed by drummer Ian Matthews in 2004 after a string of session drummers. Karloff left the band in 2006 and founded a new band called Black Onassis, Jay Mehler joined as touring lead guitarist in 2006. Mehler left the band for Liam Gallaghers Beady Eye in 2013, in 2010 and 2014, Kasabian won the Q Awards for Best Act in the World Today, while they were also named Best Live Act at the 2014 Q Awards and the 2007 NME Awards. The bands music is described as indie rock, but Pizzorno has said he hates indie bands. Kasabian have released five studio albums – Kasabian, Empire, West Ryder Pauper Lunatic Asylum and they are due to release their sixth album, For Crying Out Loud, in 2017. The bands music has been described as a mix between The Stone Roses and Primal Scream with the swagger of Oasis, the band was formerly called Saracuse and started recording at Bedrock Studios in Leicester where Chris Edwards worked as an engineer. The original band members all hailed from Countesthorpe and Blaby, the first demo EP was produced by Scott Gilbert and handed to the band late on 24 December 1998. Three songs were recorded live, Whats Going On, Life of Luxury and their first public appearance was at the Vipers Rugby Club to celebrate Edwards 18th birthday with family and friends. The band was signed to BMG by London DJ and producer Sam Young and changed their name to Kasabian, after Linda Kasabian. In an interview with Ukula, bassist Chris Edwards explained how former guitarist Chris Karloff picked the name, Reading up on Charles Manson, the Kasabian name stuck with Karloff. He just thought the word was cool, it took about a minute after the rest of us heard it. So it was decided, says Edwards, Kasabian is a common Armenian surname, from the dialectal Armenian ղասաբ ġasab taken from Arabic butcher and the patronymic ending յան -yan. Sam Young had been working at small independent label Concept Music when the bands manager dropped the Saracuse demo on his desk, EMI were interested in the band as well. The band and Sam Young fell out and their eponymous debut album was released in the UK on 13 September 2004, receiving good sales and generally positive reviews. During the recording, the band lived in a farmhouse near Rutland Water to avoid being disturbed, Kasabian featured at Glastonbury Festival 2005 on the Other Stage. During this period, various drummers played with Kasabian, including current keyboard player Ben Kealey, DJ Dan Ralph Martin, brothers Mitch and Ryan Glovers and some others. While recording in Bristol, the band met Ian Matthews, who plays on Processed Beats, Butcher Blues, Beneficial Herbs and possibly some other songs on the debut album and he was asked to tour with them in 2004 and became a permanent member in April 2005
Kasabian
–
Kasabian performing at
The O2 Arena in 2011
44.
55 Cadillac
–
55 Cadillac is the fourth studio album by American musician Andrew W. K. released on September 7,2009 through Ecstatic Peace. On vinyl format and W. K. s own record label,55 Cadillac is an instrumental solo piano album featuring new age spontaneous solo piano improvisations and was recorded in Cleveland, Ohio. Writing about the release in The Guardian, Andrew W. K. No one telling me what to play, or how to play it, and no masterplans, high-concept visions, worldwide goals with roll-out schedules. No style consultants or acting coaches, no more meetings with sponsors or computerized yelling
55 Cadillac
–
55 Cadillac
45.
55 Days at Peking
–
The film was released by Allied Artists. The screenplay was written by Philip Yordan, Bernard Gordon, Ben Barzman, the music score was composed by Dimitri Tiomkin, while the theme song So Little Time was composed by Tiomkin with lyrics by Paul Francis Webster. 55 Days in Peking is a dramatization of the siege of the foreign compounds in Peking during the Boxer Rebellion. It is based on the book by Noel Gerson, in addition to directing, Nicholas Ray plays the minor role as the head of the American diplomatic mission in China. This film is also the first known appearance of future martial arts film star Yuen Siu Tien, Japanese film director Juzo Itami, credited in the film as Ichizo Itami, appears as Col. Goro Shiba. In the early years of the 20th century, Peking is a city with the Chinese. The Boxers, who oppose Christianity, are agitating against the foreigners, the head of the U. S military garrison is Maj. Matt Lewis, USMC, an experienced China hand who knows local conditions well, the political situation is tense with the Boxers having the tacit approval of the Dowager Empress. Fed up with foreign encroachment, the Dowager Empress Tzu-Hsi uses the Boxer secret societies to attack foreigners within China. This leads to the siege and subsequent defense of the compounds, from June 20 to August 14,1900. The foreign embassies in Peking are being held in a grip of terror as the Boxers set about killing Christians in an anti-Christian nationalistic fever, Lewis heads a contingent of multinational soldiers and American Marines defending the compound. When the Boxers attack, Lewis, working with the officer from the British Embassy. Inside the besieged compound, the British ambassador gathers the beleaguered ambassadors into a defensive formation, included in the group of high-level dignitaries is the sultry Russian Baroness Natalie Ivanoff, who begins a romantic liaison with Lewis. Eventually, the forces of the Eight-Nation Alliance arrive to put down the rebellion and they relieve the siege of the foreign ligations compound following the Battle of Peking, foreshadowing the demise of the Qing Dynasty, rulers of China for the previous two and one-half centuries. The historical events which this film concerns were, and remain, the conflicts between Chinese, Japanese, and European nationalism are addressed. Most of the starring Chinese roles, including the Empress Dowager and her Prime Minister, are played by white performers, the Japanese in the foreign legation are played by Asian actors, but they have relatively minor roles. The film opens with cacophonous displays of nationalism inside the Foreign Legation quarter, with each nation raising its own flag, the camera pans to two old, Pekingese men eating a meal in a crowded Chinese street, Old Pekingese Man 1, What is this terrible noise. Old Pekingese Man 2, Different nations saying the same thing at the same time, opposing this aggressive stance is Gen. Jung-Lu
55 Days at Peking
–
Original cinema poster by
Howard Terpning
46.
1955
–
January 2 – José Antonio Remón Cantera, president of Panama, is assassinated at a race track in Panama City. January 3 – José Ramón Guizado becomes president of Panama, January 7 – Marian Anderson is the first African-American singer to perform at the Metropolitan Opera in New York City. January 17 – USS Nautilus, the first nuclear-powered submarine, puts to sea for the first time, from Groton, January 18–January 20 – Battle of Yijiangshan Islands, The Chinese Communist Peoples Liberation Army seizes the islands from the Republic of China. January 22 – In the United States, The Pentagon announces a plan to develop ballistic missiles armed with nuclear weapons. January 23 – Sutton Coldfield rail crash kills 17 near Birmingham, January 25 – Presidium of the Supreme Soviet of the Soviet Union announces the end of the war between the USSR and Germany, which began during World War II in 1941. January 28 – United States Congress authorizes President Dwight D. Eisenhower to use force to protect Formosa from the Peoples Republic of China, february 4 – Baghdad Pact, Military treaty signed between Iraq and Turkey. February 9 – Apartheid in South Africa,60,000 non-white residents of the Sophiatown suburb of Johannesburg are forcibly evicted, february 10 – The United States Seventh Fleet helps the Republic of China evacuate Chinese Nationalist army and residents from the Tachen Islands to Taiwan. February 12 – U. S. President Dwight D. Eisenhower sends the first U. S. advisors to South Vietnam, february 14 – WFLA-TV signs on the air in Tampa/St. February 16 – Nearly 100 die in a fire at a home for the elderly in Yokohama, february 19 – Southeast Asia Treaty Organization, established at a meeting in Bangkok. February 22 – In Chicagos Democratic primary, Mayor Martin H. Kennelly loses to the head of the Cook County Democratic Party, March – A young Jim Henson builds the first version of Kermit the Frog. March 2 Claudette Colvin, a fifteen-year-old African-American girl, refuses to give up her seat on a bus in Montgomery, Alabama and she is carried off the bus backwards while being kicked and handcuffed and harassed on the way to the police station. She becomes a plaintiff in Browder v. Gayle which rules bus segregation to be unconstitutional, March 5 WBBJ-TV signs on the air in the Jackson, Tennessee, with WDXI as its initial call-letters, to expanded American commercial television in mostly-rural areas. Elvis Presley makes his debut on Louisiana Hayride carried by KSLA-TV Shreveport. It is also the first time that a musical is presented in its entirety on TV almost exactly as it was performed on stage. This program gains the largest viewership of a TV special up to this time, March 19 – KXTV signs on the air in Sacramento, California. March 20 – The movie adaptation of Evan Hunters novel Blackboard Jungle premieres in the United States, teenagers jump from their seats to dance to the song. April 1 – EOKA A starts a terrorist campaign against British rule in the Crown colony of Cyprus, april 5 Winston Churchill resigns as Prime Minister of the United Kingdom due to ill-health at the age of 80. Richard J. Daley defeats Robert Merrian to become Mayor of Chicago by a vote of 708,222 to 581,555, april 6 – Anthony Eden becomes Prime Minister of the United Kingdom
1955
–
January 7:
Marian Anderson at the
Met
1955
–
January 22:
ICBM
1955
–
April 15:
McDonald's
1955
–
September 18: Britain annexes
Rockall
47.
Gazeta 55
–
Gazeta 55 is an Albanian language newspaper published in Tirana, Albania and is politically unaffiliated daily with nine reporters on staff. The tabloids owner is Fahri Balliu, an Albanian businessman, Gazeta 55 was first published on 18 October 1997. At the beginning of the 2000s Gazeta 55 had a circulation of 4,500 copies, the number 55 was chosen because Article 55 of the Constitution of communist Albania, dealt with Agitation and Propaganda related crimes. The newspaper is organized in three sections, including the magazine, News, Includes International, National, Tirana, Politics, Business, Technology, Science, Health, Sports, Education. Opinion, Includes Editorials, Op-Eds and Letters to the Editor, features, Includes Arts, Movies, Theater, and a Sigurimi file Gazeta 55 has had a web presence since 2008. Accessing articles requires no registration, and the newspaper is available in PDF. The online editor is Leonard Quku, list of newspapers in Albania News in Albania Official website
Gazeta 55
–
Gazeta 55 Newspaper 55
48.
Agitation and Propaganda against the State
–
Constitution 55 or Agitation and Propaganda against the State was a criminal offence in Communist Albania. This law was used from 1945 until 1991 and was into the Constitution of the Peoples Socialist Republic of Albania, the term was interchangeably used with counterrevolutionary agitation. The latter one was in use after the Albanian Resistance of World War II and was phased out by the end of the 1990s in favor of the former one. Article 55 was defined, The creation of any type of organization of a fascist, anti-democratic, religious, fascist, anti-democratic, religious, war-mongering, and anti-socialist activities and propaganda, as well as the incitement of national and racial hatred are prohibited. The penalty was from six months to 7 years to max life in imprisonment, the article 55 was considered by the critics of the Albanian System as a grave violation of the freedom of speech. It was among the legal instruments for the prosecution of the Albanian dissidents, some other being the punitive psychiatry. This article was the most common tool in fighting Albanian dissidents, the first time the law was used was in 1951 at the bombing of Soviet Embassy. Gazeta 55 uses number 55 because Article 55 of the Constitution of communist Albania, dealt with Agitation and Propaganda related crimes
Agitation and Propaganda against the State
–
Execution in Communist Albania
49.
Code for international direct dial
–
Country calling codes or country dial in codes are telephone dialing prefixes for the member countries of the International Telecommunication Union. They are defined by the ITU-T in standards E.123, the prefixes enable international direct dialing, and are also referred to as international subscriber dialing codes. Country codes are a component of the telephone numbering plan. Country codes are dialed before the telephone number. For example, the call prefix in all countries belonging to the North American Numbering Plan is 011. On GSM networks, the prefix may automatically be inserted when the user prefixes a dialed number with the plus sign, Country calling codes are prefix codes, hence, they can be organized as a tree. In each row of the table below, the country codes given in the left-most column share the same first digit, while there is a general geographic grouping to the zones, some exceptions exist for political and historical reasons. Thus, the geographical indicators below are approximations only, countries within NANP administered areas are assigned area codes as if they were all within one country. The codes below in format +1 XXX represent area code XXX within the +1 NANP zone – not a country code. Small countries, such as Iceland, were assigned three-digit codes, since the 1980s, all new assignments have been three-digit regardless of countries’ populations.164 assigned country codes as of 15 November 2016. List of ITU-T Recommendation E.164 Dialling Procedures as of 15 December 2011, complement to Recommendation ITU-T E.164 - List of Recommendation ITU-T E.164 Assigned Country Codes. Telephone and Internet Country Codes in 10 Languages
Code for international direct dial
–
Worldwide distribution of country calling codes colored by first digit