1.
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
2.
19 (number)
–
19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
19 (number)
–
A 19x19
Go board
19 (number)
–
19 is a
centered triangular number
3.
24 (number)
–
24 is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta, and for 10−24 yocto and these numbers are the largest and smallest number to receive an SI prefix to date. In a 24-hour clock, the hour is in conventional language called twelve or twelve oclock. 24 is the factorial of 4 and a number, being the first number of the form 23q. It follows that 24 is the number of ways to order 4 distinct items and it is the smallest number with exactly eight divisors,1,2,3,4,6,8,12, and 24. It is a composite number, having more divisors than any smaller number. 24 is a number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. Subtracting 1 from any of its divisors yields a number,24 is the largest number with this property. 24 has a sum of 36 and the aliquot sequence. It is therefore the lowest abundant number whose aliquot sum is itself abundant, the aliquot sum of only one number,529 =232, is 24. There are 10 solutions to the equation φ =24, namely 35,39,45,52,56,70,72,78,84 and 90 and this is more than any integer below 24, making 24 a highly totient number. 24 is the sum of the prime twins 11 and 13, the product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two numbers, one of which is a multiple of four, and there must be a multiple of three. The tesseract has 24 two-dimensional faces,24 is the only nontrivial solution to the cannonball problem, that is,12 +22 +32 + … +242 is a perfect square. In 24 dimensions there are 24 even positive definite unimodular lattices, the Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S and the Mathieu group M24. The modular discriminant Δ is proportional to the 24th power of the Dedekind eta function η, Δ = 12η24, the Barnes-Wall lattice contains 24 lattices. 24 is the number whose divisors — namely 1,2,3,4,6,8,12,24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group × = is isomorphic to the additive group 3 and this fact plays a role in monstrous moonshine
24 (number)
–
Astronomical clock in Prague
4.
25 (number)
–
25 is the natural number following 24 and preceding 26. It is a number, being 52 =5 ×5. It is one of two numbers whose square and higher powers of the number also ends in the same last two digits, e. g.252 =625, the other is 76. It is the smallest square that is also a sum of two squares,25 =32 +42, hence it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the odd natural numbers 1,3,5,7 and 9. 25 is an octagonal number, a centered square number. 25 percent is equal to 1/4,25 has an aliquot sum of 6 and number 6 is the first number to have an aliquot sequence that does not culminate in 0 through a prime. 25 is the sum of three integers,95,119, and 143. Twenty-five is the second member of the 6-aliquot tree. It is the smallest base 10 Friedman number as it can be expressed by its own digits,52 and it is also a Cullen number. 25 is the smallest pseudoprime satisfying the congruence 7n =7 mod n.25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate. Within base 10 one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00,25,50 or 75. 25 and 49 are the perfect squares in the following list,13,25,37,49,511,613,715,817,919,1021,1123,1225,1327,1429. The formula in this list can be described as 10nZ + where n depends on the number of digits in Z, in base 30,25 is a 1-automorphic number, and in base 10 a 2-automorphic number. The percent DNA overlap of a half-sibling, grandparent, grandchild, aunt, uncle, niece, nephew, identical twin cousin, in Ezekiels vision of a new temple, The number twenty-five is of cardinal importance in Ezekiels Temple Vision. In The Book of Revelations New International Version, Surrounding the throne were twenty-four other thrones and they were dressed in white and had crowns of gold on their heads. In Islam, there are 25 prophets mentioned in the Quran, the size of the full roster on a Major League Baseball team for most of the season, except for regular-season games on or after September 1, when teams expand their roster to 40 players. The size of the roster on a Nippon Professional Baseball team for a particular game
25 (number)
–
25 is a square
5.
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.
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.
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.
9.
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.
10.
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)
11.
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.
12.
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.
13.
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.
14.
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
15.
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.
16.
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
17.
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
18.
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
19.
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
20.
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
21.
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
22.
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
23.
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
24.
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.
25.
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.
26.
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
27.
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, …)
28.
Gaussian prime
–
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with addition and multiplication of complex numbers, form an integral domain. This integral domain is a case of a commutative ring of quadratic integers. It does not have an ordering that respects arithmetic. Formally, the Gaussian integers are the set Z =, where i 2 = −1, note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice. The norm of a Gaussian integer is the square of its value as a complex number. It is the natural number defined as N = a 2 + b 2 = ¯ =, the norm is multiplicative, since the absolute value of complex numbers is multiplicative, i. e. one has N = N N. The latter can also be verified by a straightforward check, the units of Z are precisely those elements with norm 1, i. e. the set. The Gaussian integers form a principal ideal domain with units, for x ∈ Z, the four numbers ±x, ±ix are called the associates of x. As for every principal ideal domain, Z is also a unique factorization domain and it follows that a Gaussian integer is prime if and only if it is irreducible. The prime elements of Z are also known as Gaussian primes, an associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes, the positive integer Gaussian primes are the prime numbers that are congruent to 3 modulo 4. One should not refer to only these numbers as the Gaussian primes, which refers to all the Gaussian primes, many of which do not lie in Z. In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, for example,5 = · and 13 = ·. If p =2, we have 2 = = i2, the ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q consisting of the complex numbers whose real and imaginary part are both rational. It is easy to see graphically that every number is no farther than a distance of 22 from some Gaussian integer. Put another way, every number has a maximal distance of 22 N units to some multiple of z, where z is any Gaussian integer, this turns Z into a Euclidean domain. The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his monograph on quartic reciprocity
Gaussian prime
–
Gaussian integers as
lattice points in the
complex plane
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.
Motzkin number
–
In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords between n points on a circle. The Motzkin numbers are named after Theodore Motzkin, and have diverse applications in geometry, combinatorics. The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle, the following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle. Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers, a Motzkin prime is a Motzkin number that is prime. Guibert, Pergola & Pinzani showed that vexillary involutions are enumerated by Motzkin numbers
Motzkin number
–
Contents
31.
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
32.
Composite number
–
A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
Composite number
–
Overview
33.
1 (number)
–
1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few
1 (number)
–
The 24-hour tower clock in
Venice, using J as a symbol for 1.
34.
3 (number)
–
3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
3 (number)
–
The
Shield of the Trinity is a diagram of the Christian doctrine of the Trinity
35.
7 (number)
–
7 is the natural number following 6 and preceding 8. Seven, the prime number, is not only a Mersenne prime. It is also a Newman–Shanks–Williams prime, a Woodall prime, a prime, a lucky prime, a happy number, a safe prime. Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers, Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree. N =7 is the first natural number for which the statement does not hold, Two nilpotent endomorphisms from Cn with the same minimal polynomial. 7 is the only number D for which the equation 2n − D = x2 has more than two solutions for n and x natural, in particular, the equation 2n −7 = x2 is known as the Ramanujan–Nagell equation. 7 is the dimension, besides the familiar 3, in which a vector cross product can be defined. 7 is the lowest dimension of an exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere. 999,999 divided by 7 is exactly 142,857, for example, 1/7 =0.142857142857. and 2/7 =0.285714285714. In fact, if one sorts the digits in the number 142857 in ascending order,124578, the remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example,628 ÷7 =89 5/7, here 5 is the remainder, so in this case,628 ÷7 =89.714285. Another example,5238 ÷7 =748 2/7, hence the remainder is 2, in this case,5238 ÷7 =748.285714. A seven-sided shape is a heptagon, the regular n-gons for n ≤6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons are called heptagonal numbers, Seven is also a centered hexagonal number. Seven is the first integer reciprocal with infinitely repeating sexagesimal representation, There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers. There are seven types of catastrophes. When rolling two standard six-sided dice, seven has a 6 in 36 probability of being rolled, the greatest of any number, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved, in quaternary,7 is the smallest prime with a composite sum of digits
7 (number)
–
Seven Days of Creation - 1765 book
7 (number)
–
Graph of the probability distribution of the sum of 2 six-sided dice
7 (number)
–
The
Seven Lucky Gods in
Japanese mythology
36.
Positive integers
–
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
Positive integers
–
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.
Positive integers
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
37.
Squaring the square
–
Squaring the square is the problem of tiling an integral square using only other integral squares. The name was coined in an analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set, the most studied restriction is that the squaring be perfect, meaning that the sizes of the smaller squares are all different. A related problem is squaring the plane, which can be even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a square is its number of constituent squares. A perfect squared square is a such that each of the smaller squares has a different size. It is first recorded as being studied by R. L. Brooks, smith, A. H. Stone and W. T. Tutte at Cambridge University. The first perfect squared square, a one of side 4205. Martin Gardner published an article written by W. T. Tutte about the early history of squaring the square in his mathematical games column in November 1958. A simple squared square is one where no subset of the forms a rectangle or square. In 1978, A. J. W. Duijvestijn discovered a perfect squared square of side 112 with the smallest number of squares using a computer search. His tiling uses 21 squares, and has proved to be minimal. This squared square forms the logo of the Trinity Mathematical Society, Duijvestijn also found 2 simple perfect squared squares of sides 110 but each comprising 22 squares. Gambini proved that these 3 are the smallest perfect squared squares in terms of side length, the perfect compound squared square with the fewest squares was discovered by T. H. Willcocks in 1946 and has 24 squares, however, it was not until 1982 that Duijvestijn, Pasquale Joseph Federico, in other words, the greatest common divisor of all the smaller side lengths should be 1. The Mrs. Perkinss quilt problem is to find a Mrs. Perkinss quilt with the fewest pieces for a given n × n square. A cute number means a positive integer n such that some square admits a dissection into n squares of no more than two different sizes, without other restrictions and it can be shown that aside from 2,3, and 5, every positive integer is cute. In 1975, Solomon Golomb raised the question whether the plane can be tiled by squares, one of each integer edge-length
Squaring the square
–
Periodic
Squaring the square
–
The first perfect squared square discovered, a compound one of side 4205 and order 55. Each number denotes the side length of its square.
Squaring the square
38.
Padovan sequence
–
The Padovan sequence is the sequence of integers P defined by the initial values P = P = P =1, and the recurrence relation P = P + P. The first few values of P are 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265. The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom, Hans van der Laan, Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996 and he also writes about it in one of his books, Math Hysteria, Fun Games With Mathematics. The above definition is the one given by Ian Stewart and by MathWorld, other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets. This is a property of recurrence relations, the Perrin sequence can be obtained from the Padovan sequence by the following formula, P e r r i n = P + P. e. The Padovan sequence also satisfies the identity P2 − P P = P. The Padovan sequence is related to sums of binomial coefficients by the following identity, P = ∑2 m + n = k = ∑ m = ⌈ k /3 ⌉ ⌊ k /2 ⌋. For example, for k =12, the values for the pair with 2m + n =12 which give non-zero binomial coefficients are, and, and, + + =1 +10 +1 =12 = P. The Padovan sequence numbers can be written in terms of powers of the roots of the equation x 3 − x −1 =0 and this equation has 3 roots, one real root p and two complex conjugate roots q and r. Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r, P = a p n + b q n + c r n where a, b and c are constants. Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n, and P − a p n tends to zero. For all n ≥0, P is the integer closest to p n −1 s, the ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the ratio does to the Fibonacci sequence. P is the number of ways of writing n +2 as a sum in which each term is either 2 or 3. This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as, ∑ n =0 ∞ P α n = α2 α3 − α −1. A Padovan prime is P that is prime, the first few Padovan primes are 2,3,5,7,37,151,3329,23833. Also, if you count the number of As, Bs and Cs in each string, then for the nth string, you have P As, P Bs, the count of BB pairs, AA pairs and CC pairs are also Padovan numbers
Padovan sequence
–
Spiral of equilateral triangles with side lengths which follow the Padovan sequence.
39.
Atomic number
–
The atomic number or proton number of a chemical element is the number of protons found in the nucleus of an atom of that element. It is identical to the number of the nucleus. The atomic number identifies a chemical element. In an uncharged atom, the number is also equal to the number of electrons. The atomic number Z, should not be confused with the mass number A and this number of neutrons, N, completes the weight, A = Z + N. Atoms with the atomic number Z but different neutron numbers N. Historically, it was these atomic weights of elements that were the quantities measurable by chemists in the 19th century. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge, loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, and so they can be numbered in order. Dmitri Mendeleev claimed that he arranged his first periodic tables in order of atomic weight, however, in consideration of the elements observed chemical properties, he changed the order slightly and placed tellurium ahead of iodine. This placement is consistent with the practice of ordering the elements by proton number, Z. A simple numbering based on periodic table position was never entirely satisfactory and this central charge would thus be approximately half the atomic weight. This proved eventually to be the case, the experimental position improved dramatically after research by Henry Moseley in 1913. To do this, Moseley measured the wavelengths of the innermost photon transitions produced by the elements from aluminum to gold used as a series of movable anodic targets inside an x-ray tube. The square root of the frequency of these photons increased from one target to the next in an arithmetic progression and this led to the conclusion that the atomic number does closely correspond to the calculated electric charge of the nucleus, i. e. the element number Z. Among other things, Moseley demonstrated that the series must have 15 members—no fewer. After Moseleys death in 1915, the numbers of all known elements from hydrogen to uranium were examined by his method. There were seven elements which were not found and therefore identified as still undiscovered, from 1918 to 1947, all seven of these missing elements were discovered. By this time the first four transuranium elements had also been discovered, in 1915 the reason for nuclear charge being quantized in units of Z, which were now recognized to be the same as the element number, was not understood
Atomic number
–
An explanation of the superscripts and subscripts seen in atomic number notation. Atomic number is the number of protons, and therefore also the total positive charge, in the atomic nucleus.
Atomic number
–
Russian chemist Dmitri Mendeleev created a periodic table of the elements that ordered them numerically by atomic weight, yet occasionally used chemical properties in contradiction to weight.
Atomic number
–
Niels Bohr's 1913
Bohr model of the atom required van den Broek's atomic number of nuclear charges, and Bohr believed that Moseley's work contributed greatly to the acceptance of the model.
Atomic number
–
Henry Moseley helped develop the concept of atomic number by showing experimentally (1913) that Van den Broek's 1911 hypothesis combined with the
Bohr model nearly correctly predicted atomic X-ray emissions.
40.
Scandium
–
Scandium is a chemical element with symbol Sc and atomic number 21. A silvery-white metallic d-block element, it has historically been classified as a rare earth element, together with yttrium. It was discovered in 1879 by spectral analysis of the minerals euxenite and gadolinite from Scandinavia, Scandium is present in most of the deposits of rare earth and uranium compounds, but it is extracted from these ores in only a few mines worldwide. Because of the low availability and the difficulties in the preparation of metallic scandium, the positive effects of scandium on aluminium alloys were discovered in the 1970s, and its use in such alloys remains its only major application. The global trade of scandium oxide is about 10 tonnes per year, the properties of scandium compounds are intermediate between those of aluminium and yttrium. A diagonal relationship exists between the behavior of magnesium and scandium, just as there is between beryllium and aluminium, in the chemical compounds of the elements in group 3, the predominant oxidation state is +3. Scandium is a metal with a silvery appearance. It develops a slightly yellowish or pinkish cast when oxidized by air and it is susceptible to weathering and dissolves slowly in most dilute acids. It does not react with a 1,1 mixture of acid and 48% hydrofluoric acid. Scandium turnings ignite in air with a brilliant yellow flame to form scandium oxide, in nature, scandium is found exclusively as the isotope 45Sc, which has a nuclear spin of 7/2. Thirteen radioisotopes have been characterized with the most stable being 46Sc, which has a half-life of 83.8 days, 47Sc,3.35 days, the positron emitter 44Sc,4 h, and 48Sc,43.7 hours. All of the radioactive isotopes have half-lives less than 4 hours. This element also has five meta states, with the most stable being 44mSc, the isotopes of scandium range from 36Sc to 60Sc. The primary decay mode at masses lower than the stable isotope, 45Sc, is electron capture. The primary decay products at atomic weights below 45Sc are calcium isotopes, in Earths crust, scandium is not rare. Estimates vary from 18 to 25 ppm, which is comparable to the abundance of cobalt, Scandium is only the 50th most common element on Earth, but it is the 23rd most common element in the Sun. However, scandium is distributed sparsely and occurs in trace amounts in many minerals, rare minerals from Scandinavia and Madagascar such as thortveitite, euxenite, and gadolinite are the only known concentrated sources of this element. Thortveitite can contain up to 45% of scandium in the form of scandium oxide, the stable form of scandium is created in supernovas via the r-process
Scandium
–
Scandium, 21 Sc
Scandium
–
Parts of the
MiG-29 are made from Al-Sc alloy.
41.
Age of majority
–
The age of majority is the threshold of adulthood as recognized or declared in law. Most countries set the age of majority at 18, the word majority here refers to having greater years and being of full age as opposed to minority, the state of being a minor. The law in a jurisdiction may not actually use the term age of majority. The term typically refers to a collection of laws bestowing the status of adulthood, the age of majority does not necessarily correspond to the mental or physical maturity of an individual. Age of majority can be confused with the concept of the age of license. As a legal term of art, license means permission, thus, an age of license is an age at which one has legal permission from government to do something. The age of majority, on the hand, is legal recognition that one has grown into an adult. Many ages of license are correlated to the age of majority, one need not have attained the age of majority to have permission to exercise certain rights and responsibilities. Some ages of license are actually higher than the age of majority, for example, the age of license to purchase alcoholic beverages is 21 in all U. S. states. Another example is the age, which prior to the 1970s was 21. In the Republic of Ireland the age of majority is 18, also, in Portugal the age of majority is 18, but one must be at least 25 years of age to run for public office. A child who is legally emancipated by a court of competent jurisdiction automatically attains to their maturity upon the signing of the court order, only emancipation confers the status of maturity before a person has actually reached the age of majority. In almost all places, minors who are married are automatically emancipated, some places also do the same for minors who are in the armed forces or who have a certain degree or diploma. In the United States, all states have some form of emancipation of minors, judaism,13 years of age for males and 12 years of age for females, such persons are considered adults Roman Catholic Church,18 years of age
Age of majority
42.
Coming of age
–
Coming of age is a young persons transition from being a child to being an adult. The certain age at which this takes place changes in society. It can be a simple legal convention or can be part of a ritual or spiritual event, in the past, and in some societies today, such a change is associated with the age of sexual maturity, especially menarche and spermarche. In others, it is associated with an age of religious responsibility, particularly in western societies, modern legal conventions which stipulate points in late adolescence or early adulthood are the focus of the transition. In either case, many cultures retain ceremonies to confirm the coming of age, Coming of age is often a topic of fiction, in the form of a coming-of-age story. In written literature, a novel which deals with the psychological and moral growth often associated with coming of age is called a bildungsroman. Similar stories that are told in film are called coming-of-age films, turning 15, the age of maturity, as the Bahai faith terms it, is a time when a child is considered spiritually mature. Declared Bahais that have reached the age of maturity are expected to begin observing certain Bahai laws, such as obligatory prayer, theravada boys, typically just under the age of 20 years, undergo a Shinbyu ceremony, where they are initiated into the Temple as Novice Monks. They will typically stay in the monastery for between 3 days and 3 years, most commonly for one 3-month rainy season retreat, held annually from late July to early October. After living the monastic life for some time, the boy, now considered to have come of age. Confucianism had a ceremony called Guan Li for young men, at around the age of 20, they would receive their style name. In many Western Christian churches, a young person celebrates his/her Coming of Age with the Sacrament of Confirmation and this is usually done by the Bishop laying his hands upon the foreheads of the young person, and marking them with the seal of the Holy Spirit. In some denominations during this sacrament the child adopts a name which is added onto their Christian name. In Christian denominations that practice Believers Baptism, the ritual can be carried out after the age of accountability has arrived. Some traditions withhold the rite of Holy Communion from those not yet at the age of accountability, mid-teens in the United States, early teens in Ireland and Britain, has in some areas been abandoned in favour of restoring the traditional order of the three sacraments of initiation. These individuals are seen, according to some Christians, as existing in a perpetual state of innocence. The Church of Jesus Christ of Latter-day Saints sets the age for baptism at 8 years of age and calls this the age of accountability. All persons younger than 8 are considered innocent and incapable of sinning, the LDS Church considers mentally challenged individuals whose mental age is under 8 to be in a perpetual state of innocence, similar to other Christian churches
Coming of age
–
Confirmation (
Roman Catholic)
Coming of age
–
Quinceañera (
Latin America)
Coming of age
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Jugendweihe (
East Germany)
Coming of age
–
Coming of Age Day (
Japan)
43.
Legal drinking age
–
The legal drinking age is the age at which a person can legally consume or purchase alcoholic beverages. These laws cover a range of issues and behaviors, addressing when. The minimum age alcohol can be legally consumed can be different from the age when it can be purchased in some countries and these laws vary among different countries and many laws have exemptions or special circumstances. Most laws apply only to drinking alcohol in public places, with consumption in the home being mostly unregulated. Some countries also have different age limits for different types of alcoholic drinks, some Islamic nations prohibit Muslims, or both Muslims and non-Muslims, from drinking alcohol at any age, due to the Quraan forbidding the consumption of wine. In other countries, it is not illegal for minors to drink alcohol, in some cases, it is illegal to sell or give alcohol to minors. The following list indicates the age of the person for whom it is legal to consume, austria, Belgium, Denmark, Germany, Georgia, Luxembourg, Moldova, Morocco, and Western Sahara have the lowest set drinking ages. ARA Industry Association for Responsible Alcohol Use Association established in 1989 of major manufacturers of alcohol beverages in South Africa, IARD International Alliance for Responsible Drinking
Legal drinking age
–
Illegal
44.
Hawaii
–
Hawaii is the 50th and most recent state to have joined the United States of America, having received statehood on August 21,1959. Hawaii is the only U. S. state located in Oceania and it is the northernmost island group in Polynesia, occupying most of an archipelago in the central Pacific Ocean. Hawaii is the only U. S. state not located in the Americas, the state encompasses nearly the entire volcanic Hawaiian archipelago, which comprises hundreds of islands spread over 1,500 miles. At the southeastern end of the archipelago, the eight main islands are—in order from northwest to southeast, Niʻihau, Kauaʻi, Oʻahu, Molokaʻi, Lānaʻi, Kahoʻolawe, Maui, and the Island of Hawaiʻi. The last is the largest island in the group, it is called the Big Island or Hawaiʻi Island to avoid confusion with the state or archipelago. The archipelago is physiographically and ethnologically part of the Polynesian subregion of Oceania, Hawaii has over a million permanent residents, along with many visitors and U. S. military personnel. Its capital is Honolulu on the island of Oʻahu, Hawaii is the 8th-smallest and the 11th-least populous, but the 13th-most densely populated of the fifty U. S. states. It is the state with an Asian plurality. The states coastline is about 750 miles long, the fourth longest in the U. S. after the coastlines of Alaska, Florida, the state of Hawaii derives its name from the name of its largest island, Hawaiʻi. A common Hawaiian explanation of the name of Hawaiʻi is that was named for Hawaiʻiloa and he is said to have discovered the islands when they were first settled. The Hawaiian language word Hawaiʻi is very similar to Proto-Polynesian *Sawaiki, cognates of Hawaiʻi are found in other Polynesian languages, including Māori, Rarotongan and Samoan. According to linguists Pukui and Elbert, lsewhere in Polynesia, Hawaiʻi or a cognate is the name of the underworld or of the home, but in Hawaii. A somewhat divisive political issue arose in 1978 when the Constitution of the State of Hawaii added Hawaiian as an official state language. The title of the constitution is The Constitution of the State of Hawaii. Article XV, Section 1 of the Constitution uses The State of Hawaii, diacritics were not used because the document, drafted in 1949, predates the use of the okina and the kahakō in modern Hawaiian orthography. The exact spelling of the name in the Hawaiian language is Hawaiʻi. In the Hawaii Admission Act that granted Hawaiian statehood, the government recognized Hawaii as the official state name. Official government publications, department and office titles, and the Seal of Hawaii use the spelling with no symbols for glottal stops or vowel length
Hawaii
–
Hawaii from space, January 26, 2014
Hawaii
–
Flag
Hawaii
–
Nā Pali coast, Kaua ʻ i
Hawaii
–
The main islands and undersea terrain of Hawaii
45.
New York (state)
–
New York is a state in the northeastern United States, and is the 27th-most extensive, fourth-most populous, and seventh-most densely populated U. S. state. New York is bordered by New Jersey and Pennsylvania to the south and Connecticut, Massachusetts, and Vermont to the east. With an estimated population of 8.55 million in 2015, New York City is the most populous city in the United States, the New York Metropolitan Area is one of the most populous urban agglomerations in the world. New York City makes up over 40% of the population of New York State, two-thirds of the states population lives in the New York City Metropolitan Area, and nearly 40% lives on Long Island. Both the state and New York City were named for the 17th-century Duke of York, the next four most populous cities in the state are Buffalo, Rochester, Yonkers, and Syracuse, while the state capital is Albany. New York has a diverse geography and these more mountainous regions are bisected by two major river valleys—the north-south Hudson River Valley and the east-west Mohawk River Valley, which forms the core of the Erie Canal. Western New York is considered part of the Great Lakes Region and straddles Lake Ontario, between the two lakes lies Niagara Falls. The central part of the state is dominated by the Finger Lakes, New York had been inhabited by tribes of Algonquian and Iroquoian-speaking Native Americans for several hundred years by the time the earliest Europeans came to New York. The first Europeans to arrive were French colonists and Jesuit missionaries who arrived southward from settlements at Montreal for trade, the British annexed the colony from the Dutch in 1664. The borders of the British colony, the Province of New York, were similar to those of the present-day state, New York is home to the Statue of Liberty, a symbol of the United States and its ideals of freedom, democracy, and opportunity. In the 21st century, New York has emerged as a node of creativity and entrepreneurship, social tolerance. On April 17,1524 Verrazanno entered New York Bay, by way of the now called the Narrows into the northern bay which he named Santa Margherita. Verrazzano described it as a vast coastline with a delta in which every kind of ship could pass and he adds. This vast sheet of water swarmed with native boats and he landed on the tip of Manhattan and possibly on the furthest point of Long Island. Verrazannos stay was interrupted by a storm which pushed him north towards Marthas Vineyard, in 1540 French traders from New France built a chateau on Castle Island, within present-day Albany, due to flooding, it was abandoned the next year. In 1614, the Dutch under the command of Hendrick Corstiaensen, rebuilt the French chateau, Fort Nassau was the first Dutch settlement in North America, and was located along the Hudson River, also within present-day Albany. The small fort served as a trading post and warehouse, located on the Hudson River flood plain, the rudimentary fort was washed away by flooding in 1617, and abandoned for good after Fort Orange was built nearby in 1623. Henry Hudsons 1609 voyage marked the beginning of European involvement with the area, sailing for the Dutch East India Company and looking for a passage to Asia, he entered the Upper New York Bay on September 11 of that year
New York (state)
–
British general
John Burgoyne surrenders at
Saratoga in 1777.
New York (state)
–
Flag
New York (state)
–
1800 map of New York from
Low's Encyclopaedia
New York (state)
–
The
Erie Canal at
Lockport, New York in 1839
46.
Cigarettes
–
A cigarette, or cigaret, is a small cylinder of finely cut tobacco leaves rolled in thin paper for smoking. Most modern manufactured cigarettes are filtered, and also include reconstituted tobacco, the term cigarette, as commonly used, refers to a tobacco cigarette, but can apply to similar devices other substances, such as cannabis. A cigarette is distinguished from a cigar by its size, use of processed leaf, and paper wrapping. Cigars are typically composed entirely of whole-leaf tobacco, rates of cigarette smoking vary widely throughout the world and have changed considerably since cigarettes were first widely used in the mid-19th century. While rates of smoking have over time leveled off or declined in the developed world, Cigarettes carry serious health risks, which are more prevalent than with other tobacco products. Nicotine, the psychoactive chemical in tobacco and therefore cigarettes, is very addictive. About half of cigarette smokers die of tobacco-related disease and lose on average 14 years of life, cigarette use by pregnant women has also been shown to cause birth defects, including low birth weight, fetal abnormalities, and premature birth. Second-hand smoke from cigarettes has been shown to be injurious to bystanders, Cigarettes produce an aerosol containing over 4,000 chemical compounds, including nicotine, carbon monoxide, acrolein, and other harmful substances. Over 50 of these are carcinogenic, the earliest forms of cigarettes were similar to their predecessor, the cigar. Cigarettes appear to have had antecedents in Mexico and Central America around the 9th century in the form of reeds, the Maya, and later the Aztecs, smoked tobacco and other psychoactive drugs in religious rituals and frequently depicted priests and deities smoking on pottery and temple engravings. The cigarette and the cigar were the most common methods of smoking in the Caribbean, Mexico, and Central and South America until recent times. The North American, Central American, and South American cigarette used various plant wrappers, when it was back to Spain, maize wrappers were introduced. The resulting product was called papelate and is documented in Goyas paintings La Cometa, La Merienda en el Manzanares, by 1830, the cigarette had crossed into France, where it received the name cigarette, and in 1845, the French state tobacco monopoly began manufacturing them. The first patented cigarette machine was by Juan Nepomuceno Adorno of Mexico in 1847 and this was helped by the development of tobaccos suitable for cigarette use, and by the development of the Egyptian cigarette export industry. Cigarettes may have initially used in a manner similar to pipes, cigars. As cigarette tobacco became milder and more acidic, inhaling may have perceived as more agreeable. However, Moltke noticed in the 1830s that Ottomans inhaled the Turkish tobacco, the widespread smoking of cigarettes in the Western world is largely a 20th-century phenomenon. At the start of the 20th century, the per capita consumption in the USA was 54 cigarettes
Cigarettes
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Unlit filtered cigarette
Cigarettes
–
A reproduction of a carving from the temple at
Palenque, Mexico, depicting a Mayan priest smoking from a smoking tube
Cigarettes
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Francisco Goya 's La Cometa, depicting a (foreground left) man smoking an early quasicigarette
Cigarettes
–
Tabak-Trafik in Vienna: Since 1 January 2007, all
cigarette machines in Austria must attempt to verify a customer's age by requiring the insertion of a debit card or mobile phone verification.
47.
Voting age
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A voting age is a minimum age established by law that a person must attain to be eligible to vote in a public election. Typically, the age is set at 18 years, however, ages as low as 16, most countries have set a minimum voting age, often set in their constitution. When the right to vote was first accorded in democracies the voting age was set at 21 or higher. In the 1970s the voting age was reduced to 18 in many countries, debate is currently under way in many places on proposals to reduce the voting age to or below 16. In May 2009, Danish Member of Parliament Mogens Jensen presented an initiative to the Parliamentary Assembly of the Council of Europe in Strasbourg to lower the age in Europe to 16. Before the Second World War, almost all countries had voting ages of 21 or higher, czechoslovakia was early to act, reducing its age to 18 in 1946, and by 1968 a total of 17 states had made the same reduction. A large number of countries, particularly in Western Europe, reduced their voting ages to 18 during the 1970s, the USA, Canada, Australia, France and others followed soon afterwards. By the end of the 20th century,18 had become by far the most common age at which citizens acquired the right to vote, however, a few countries maintained voting ages of 20 years or higher. Eighteen-year-old men could be drafted to go to war, so people felt they should be able to vote at the age of 18. Consideration of a reduction to 18 continued into the late 20th, reductions were seen in India, Switzerland, Austria and Morocco during this time. Japan is due to make the change to 18 in 2016, a dispute is continuing in the Maldives. Around the year 2000 a number of countries began to consider whether the voting age ought to be reduced further, the earliest moves came during the 1990s, when the voting age for municipal elections in some States of Germany was lowered to 16. Lower Saxony was the first state to such a reduction, in 1995. During the 2000s several proposals for a voting age were put forward in U. S. states, including California, Florida and Alaska. A national reduction was proposed in 2005 in Canada and a reduction in New South Wales, Australia. In 2007 Austria became the first member of the European Union to adopt a voting age of 16 for most purposes, the voting age had been reduced in Austria from 19 to 18 at all levels in 1992. At that time an age of 16 was proposed by the Green Party. The voting age for elections in some states was lowered to 16 shortly after 2000
Voting age
–
Demonstration in favor of lowering the voting age by members of
NYRA Berkeley, California, 2004.
48.
R-rated
–
The Motion Picture Association of America film rating system is used in the United States and its territories to rate a films suitability for certain audiences based on its content. The MPAA rating system is a scheme that is not enforced by law, films can be exhibited without a rating. Non-members of MPAA may also submit films for rating, other media, such as television programs and video games, is rated by other entities such as the ESRB and the TV Parental Guidelines. The MPAA rating system is one of various motion picture rating systems that are used to help decide what films are appropriate for their children. It is administered by the Classification & Ratings Administration, an independent division of the MPAA, if a film has not yet been assigned a final rating, the label This Film Is Not Yet Rated is used in trailers and television commercials. The MPAA also rates film trailers, print advertising, posters, green, yellow, or red title cards displayed before the start of a trailer indicate the trailers rating. Green, When the trailer accompanies another rated feature, the wording on the title card states The following preview has been approved to accompany this feature. For trailers hosted on the internet, the wording has been altered to The following preview has been approved for appropriate audiences. Yellow, A yellow title card exists solely for trailers hosted on the internet, the MPAA defines age-appropriate internet users as visitors to sites either frequented mainly by adults or accessible only between 9,00 p. m. and 4,00 a. m. The yellow card is reserved for trailers previewing films rated PG-13 or stronger, for trailers hosted on the internet, the wording is tweaked to The following restricted preview has been approved for appropriate audiences. The red title card is reserved for trailers previewing R and NC-17 rated films, in 1989, Tennessee enacted a law whereby patrons of an R-rated motion picture were not admitted unless they were at least 18 years old or accompanied by a parent or adult guardian. He revised the Code to include the SMA advisory as a stopgap measure, Rated X, Persons Under 16 Not Admitted This content classification system originally was to have three ratings, with the intention of allowing parents to take their children to any film they chose. However, the National Association of Theater Owners urged the creation of an adults-only category, the X rating was not an MPAA trademark and would not receive the MPAA seal, any producer not submitting a film for MPAA rating could self-apply the X rating. In 1970 the ages for R and X were raised from 16 to 17, in 1972, GP was revised to PG. In late 1989 and early 1990, two critically acclaimed art films featuring strong adult content, Henry, Portrait of a Serial Killer and The Cook, the Thief, His Wife & Her Lover, were released. Neither film was approved for an MPAA rating, thus limiting their commercial distribution, in September 1990, the MPAA introduced the rating NC-17. Henry & June – previously to be assigned an X rating – was the first film to receive the NC-17 rating instead, in 1996, the minimum age for NC-17 films was raised to 18, by rewording it to No One 17 and Under Admitted. For example, some films explanations may read Strong Brutal Violence, Pervasive Language, Some Strong Sexual Content, around the late 1990s, the MPAA began applying rating explanations for PG, PG-13, and NC-17 films as well
R-rated
–
PG – Parental Guidance Suggested Some material may not be suitable for children. Parents urged to give "parental guidance". May contain some material parents might not like for their young children.
49.
Gambling
–
Gambling is the wagering of money or something of value on an event with an uncertain outcome with the primary intent of winning money and/or material goods. Gambling thus requires three elements be present, consideration, chance and prize, the term gaming in this context typically refers to instances in which the activity has been specifically permitted by law. However, this distinction is not universally observed in the English-speaking world, for instance, in the United Kingdom, the regulator of gambling activities is called the Gambling Commission. Gambling is also an international commercial activity, with the legal gambling market totaling an estimated $335 billion in 2009. In other forms, gambling can be conducted with materials which have a value, many popular games played in modern casinos originate from Europe and China. Games such as craps, baccarat, roulette, and blackjack originate from different areas of Europe, a version of keno, an ancient Chinese lottery game, is played in casinos around the world. In addition, pai gow poker, a hybrid between pai gow and poker is also played, many jurisdictions, local as well as national, either ban gambling or heavily control it by licensing the vendors. Such regulation generally leads to gambling tourism and illegal gambling in the areas where it is not allowed, there is generally legislation requiring that the odds in gaming devices are statistically random, to prevent manufacturers from making some high-payoff results impossible. Since these high-payoffs have very low probability, a bias can quite easily be missed unless the odds are checked carefully. Most jurisdictions that allow gambling require participants to be above a certain age, in some jurisdictions, the gambling age differs depending on the type of gambling. For example, in many American states one must be over 21 to enter a casino, E. g. Nonetheless, both insurance and gambling contracts are typically considered aleatory contracts under most legal systems, though they are subject to different types of regulation. Under common law, particularly English Law, a contract may not give a casino bona fide purchaser status. For case law on recovery of gambling losses where the loser had stolen the funds see Rights of owner of money as against one who won it in gambling transaction from thief. This was a plot point in a Perry Mason novel, The Case of the Singing Skirt. Religious perspectives on gambling have been mixed, ancient Hindu poems like the Gamblers Lament and the Mahabharata testify to the popularity of gambling among ancient Indians. However, the text Arthashastra recommends taxation and control of gambling, ancient Jewish authorities frowned on gambling, even disqualifying professional gamblers from testifying in court. For these social and religious reasons, most legal jurisdictions limit gambling, in at least one case, the same bishop opposing a casino has sold land to be used for its construction. Although different interpretations of law exist in the Muslim world
Gambling
–
Caravaggio,
The Cardsharps, c. 1594
Gambling
–
Gamblers in the
Ship of Fools, 1494
Gambling
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Bag with 65 Inlaid Gambling Sticks, Tsimshian (Native American), 19th century,
Brooklyn Museum
Gambling
–
The
Caesars Palace main fountain. The statue is a copy of the ancient
Winged Victory of Samothrace.
50.
Casino
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A casino is a facility which houses and accommodates certain types of gambling activities. The industry that deals in casinos is called the gaming industry, casinos are most commonly built near or combined with hotels, restaurants, retail shopping, cruise ships or other tourist attractions. There is much debate whether or not the social and economic consequences of casino gambling outweigh the initial revenue that may be generated. Some casinos are also known for hosting live entertainment events, such as comedy, concerts. The term casino is a confusing linguistic false friend for translators, Casino is of Italian origin, the root casa originally meant a small country villa, summerhouse, or social club. In modern-day Italian, the term designates a bordello, while the gambling house is spelled casinò with an accent. Not all casinos were used for gaming, the Copenhagen Casino was a theatre, known for the mass public meetings often held in its hall during the 1848 Revolution, which made Denmark a constitutional monarchy. Until 1937, it was a well-known Danish theatre, the Hanko Casino in Hanko, Finland—one of that towns most conspicuous landmarks—was never used for gambling. Rather, it was a hall for the Russian nobility which frequented this spa resort in the late 19th century and is now used as a restaurant. In military and non-military usage in German and Spanish, a casino or kasino is an officers mess, in Italian—the source-language of the word—a casino is either a brothel, a mess, or a noisy environment, while a gaming house is called a casinò. The precise origin of gambling is unknown and it is generally believed that gambling in some form or another has been seen in almost every society in history. From the Ancient Greeks and Romans to Napoleons France and Elizabethan England and it was closed in 1774 as the city government felt it was impoverishing the local gentry. In American history, early gambling establishments were known as saloons, the creation and importance of saloons was greatly influenced by four major cities, New Orleans, St. Louis, Chicago and San Francisco. It was in the saloons that travelers could find people to talk to, drink with, during the early 20th century in America, gambling became outlawed and banned by state legislation and social reformers of the time. However, in 1931, gambling was legalized throughout the state of Nevada, Americas first legalized casinos were set up in those places. In 1976 New Jersey allowed gambling in Atlantic City, now Americas second largest gambling city, most jurisdictions worldwide have a minimum gambling age. Customers gamble by playing games of chance, in cases with an element of skill, such as craps, roulette, baccarat, blackjack. Most games played have mathematically determined odds that ensure the house has at all times an overall advantage over the players and this can be expressed more precisely by the notion of expected value, which is uniformly negative
Casino
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Slot machines in Atlantic City. Slot machines are a standard attraction of casinos
Casino
–
The
Marina Bay Sands
Casino
–
The Venetian Macao
Casino
–
A sign at the
Thousand Islands Casino