1.
15 (number)
–
15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name, in spoken English, the numbers 15 and 50 are often confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed,15 /fɪfˈtiːn/ vs 50 /ˈfɪfti/, however, in dates such as 1500 or when contrasting numbers in the teens, the stress generally shifts to the first syllable,15 /ˈfɪftiːn/. In a 24-hour clock, the hour is in conventional language called three or three oclock. A composite number, its divisors being 1,3 and 5. A repdigit in binary and quaternary, in hexadecimal, as well as all higher bases,15 is represented as F. the 4th discrete semiprime and the first member of the discrete semiprime family. It is thus the first odd discrete semiprime, the number proceeding 15,14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21, the smallest number that can be factorized using Shors quantum algorithm. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet, the aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members. 15 is the composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree, fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33. 15 and 16 form a Ruth-Aaron pair under the definition in which repeated prime factors are counted as often as they occur. There are 15 solutions to Známs problem of length 7, if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems. Group 15 of the table are sometimes known as the pnictogens. 15 Madadgar is designated as a number in Pakistan, for mobile phones, similar to the international GSM emergency number 112, if 112 is used in Pakistan. 112 can be used in an emergency if the phone is locked. The Hanbali Sunni madhab states that the age of fifteen of a solar or lunar calendar is when ones taklif begins and is the stage whereby one has his deeds recorded. In the Hebrew numbering system, the number 15 is not written according to the method, with the letters that represent 10 and 5
2.
17 (number)
–
17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution
3.
9 (number)
–
9 is the natural number following 8 and preceding 10. In the NATO phonetic alphabet, the digit 9 is called Niner, five-digit produce PLU codes that begin with 9 are organic. Common terminal digit in psychological pricing, Nine is a number that appears often in Indian Culture and mythology. Nine influencers are attested in Indian astrology, in the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements, Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. Navaratri is a festival dedicated to the nine forms of Durga. Navaratna, meaning 9 jewels may also refer to Navaratnas - accomplished courtiers, Navratan - a kind of dish, according to Yoga, the human body has nine doors - two eyes, two ears, the mouth, two nostrils, and the openings for defecation and procreation. In Indian aesthetics, there are nine kinds of Rasa, Nine is considered a good number in Chinese culture because it sounds the same as the word long-lasting. Nine is strongly associated with the Chinese dragon, a symbol of magic, there are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales –81 yang and 36 yin, all three numbers are multiples of 9 as well as having the same digital root of 9. The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City, the name of the area called Kowloon in Hong Kong literally means, nine dragons. The nine-dotted line delimits certain island claims by China in the South China Sea, the nine-rank system was a civil service nomination system used during certain Chinese dynasties. 9 Points of the Heart / Heart Master Channels in Traditional Chinese Medicine, the nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. The Ennead is a group of nine Egyptian deities, who, in versions of the Osiris myth. The Nine Worthies are nine historical, or semi-legendary figures who, in Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil. The nine Muses in Greek mythology are Calliope, Clio, Erato, Euterpe, Melpomene, Polyhymnia, Terpsichore, Thalia and it takes nine days to fall from heaven to earth, and nine more to fall from earth to Tartarus—a place of torment in the underworld. Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo, according to Georges Ifrah, the origin of the 9 integers can be attributed to ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0. In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot, the Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, as time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller
4.
10 (number)
–
10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十
5.
12 (number)
–
12 is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a highly composite number, divisible by 2,3,4. It is central to systems of counting, including the Western calendar and units of time. The word twelve is the largest number with a name in English. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance and it derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels Book of John. It has cognates in every Germanic language, whose Proto-Germanic ancestor has been reconstructed as *twaliƀi, from *twa and suffix *-lif- or *-liƀ- of uncertain meaning. It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19, every other Indo-European language instead uses a form of two+ten, such as the Latin duōdecim. The usual ordinal form is twelfth but dozenth or duodecimal is also used in some contexts, similarly, a group of twelve things is usually a dozen but may also be referred to as a duodecad. The adjective referring to a group of twelve is duodecuple, as with eleven, the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan, with the implicit meaning that two is left after having already counted to ten. The Lithuanian suffix is also considered to share a similar development, the suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten. While, as mentioned above,12 has its own name in Germanic languages such as English and German, it is a number in many other languages, e. g. Italian dodici. In Germany, according to an old tradition, the numbers 0 through 12 were spelt out, the Duden now calls this tradition outdated and no longer valid, but many writers still follow it. Another system spells out all numbers written in one or two words, Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1,2,3,4,6 and 12. Twelve is also a composite number, the next one being twenty-four. Twelve is also a highly composite number, the next one being sixty. It is the first composite number of the form p2q, a square-prime,12 has an aliquot sum of 16. Accordingly,12 is the first abundant number and demonstrates an 8-member aliquot sequence,12 is the 3rd composite number in the 3-aliquot tree, the only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant, Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
Numeral system
–
A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
17.
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
18.
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
19.
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
20.
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
21.
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
22.
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
23.
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
24.
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
25.
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
26.
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
27.
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
28.
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
29.
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
30.
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
31.
Hebrew (language)
–
Hebrew is a language native to Israel, spoken by over 9 million people worldwide, of whom over 5 million are in Israel. Historically, it is regarded as the language of the Israelites and their ancestors, the earliest examples of written Paleo-Hebrew date from the 10th century BCE. Hebrew belongs to the West Semitic branch of the Afroasiatic language family, Hebrew is the only living Canaanite language left, and the only truly successful example of a revived dead language. Hebrew had ceased to be a spoken language somewhere between 200 and 400 CE, declining since the aftermath of the Bar Kokhba revolt. Aramaic and to a lesser extent Greek were already in use as international languages, especially among elites and it survived into the medieval period as the language of Jewish liturgy, rabbinic literature, intra-Jewish commerce, and poetry. Then, in the 19th century, it was revived as a spoken and literary language, and, according to Ethnologue, had become, as of 1998, the language of 5 million people worldwide. After Israel, the United States has the second largest Hebrew-speaking population, with 220,000 fluent speakers, Modern Hebrew is one of the two official languages of the State of Israel, while premodern Hebrew is used for prayer or study in Jewish communities around the world today. Ancient Hebrew is also the tongue of the Samaritans, while modern Hebrew or Arabic is their vernacular. For this reason, Hebrew has been referred to by Jews as Leshon Hakodesh, the modern word Hebrew is derived from the word Ivri, one of several names for the Israelite people. It is traditionally understood to be a based on the name of Abrahams ancestor, Eber. This name is based upon the root ʕ-b-r meaning to cross over. Interpretations of the term ʕibrim link it to this verb, cross over, in the Bible, the Hebrew language is called Yәhudit because Judah was the surviving kingdom at the time of the quotation. In Isaiah 19,18 it is called the Language of Canaan, Hebrew belongs to the Canaanite group of languages. In turn, the Canaanite languages are a branch of the Northwest Semitic family of languages, according to Avraham ben-Yosef, Hebrew flourished as a spoken language in the Kingdoms of Israel and Judah during about 1200 to 586 BCE. Scholars debate the degree to which Hebrew was a vernacular in ancient times following the Babylonian exile. In July 2008 Israeli archaeologist Yossi Garfinkel discovered a ceramic shard at Khirbet Qeiyafa which he claimed may be the earliest Hebrew writing yet discovered, dating around 3000 years ago. The Gezer calendar also dates back to the 10th century BCE at the beginning of the Monarchic Period, classified as Archaic Biblical Hebrew, the calendar presents a list of seasons and related agricultural activities. The Gezer calendar is written in an old Semitic script, akin to the Phoenician one that through the Greeks, the Gezer calendar is written without any vowels, and it does not use consonants to imply vowels even in the places where later Hebrew spelling requires it
32.
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
33.
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
34.
Square number
–
In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
35.
4 (number)
–
4 is a number, numeral, and glyph. It is the number following 3 and preceding 5. Four is the only cardinal numeral in the English language that has the number of letters as its number value. Four is the smallest composite number, its divisors being 1 and 2. Four is also a composite number. The next highly composite number is 6, Four is the second square number, the second centered triangular number. 4 is the smallest squared prime and the even number in this form. It has a sum of 3 which is itself prime. The aliquot sequence of 4 has 4 members and is accordingly the first member of the 3-aliquot tree, a number is a multiple of 4 if its last two digits are a multiple of 4. For example,1092 is a multiple of 4 because 92 =4 ×23, only one number has an aliquot sum of 4 and that is squared prime 9. Four is the smallest composite number that is equal to the sum of its prime factors, however, it is the only composite number n for which. It is also a Motzkin number, in bases 6 and 12,4 is a 1-automorphic number. In addition,2 +2 =2 ×2 =22 =4, continuing the pattern in Knuths up-arrow notation,2 ↑↑2 =2 ↑↑↑2 =4, and so on, for any number of up arrows. A four-sided plane figure is a quadrilateral which include kites, rhombi, a circle divided by 4 makes right angles and four quadrants. Because of it, four is the number of plane. Four cardinal directions, four seasons, duodecimal system, and vigesimal system are based on four, a solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces of a polyhedron. The regular tetrahedron is the simplest Platonic solid, a tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only regular polyhedron
36.
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
37.
2 (number)
–
2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set
38.
8 (number)
–
8 is the natural number following 7 and preceding 9. 8 is, a number, its proper divisors being 1,2. It is twice 4 or four times 2, a power of two, being 23, and is the first number of the form p3, p being an integer greater than 1. The first number which is neither prime nor semiprime, the base of the octal number system, which is mostly used with computers. In octal, one digit represents 3 bits, in modern computers, a byte is a grouping of eight bits, also called an octet. A Fibonacci number, being 3 plus 5, the next Fibonacci number is 13. 8 is the only positive Fibonacci number, aside from 1, the order of the smallest non-abelian group all of whose subgroups are normal. The dimension of the octonions and is the highest possible dimension of a division algebra. The first number to be the sum of two numbers other than itself, the discrete biprime 10, and the square number 49. It has a sum of 7 in the 4 member aliquot sequence being the first member of 7-aliquot tree. All powers of 2, have a sum of one less than themselves. A number is divisible by 8 if its last 3 digits,8 and 9 form a Ruth–Aaron pair under the second definition in which repeated prime factors are counted as often as they occur. There are a total of eight convex deltahedra, a polygon with eight sides is an octagon. Figurate numbers representing octagons are called octagonal numbers, a polyhedron with eight faces is an octahedron. A cuboctahedron has as faces six equal squares and eight regular triangles. Sphenic numbers always have exactly eight divisors, the number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example, if O is the limit of the inclusions of real orthogonal groups O ↪ O ↪ … ↪ O ↪ …. Clifford algebras also display a periodicity of 8, for example, the algebra Cl is isomorphic to the algebra of 16 by 16 matrices with entries in Cl
39.
Power of two
–
In mathematics, a power of two means a number of the form 2n where n is an integer, i. e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to values, so we have 1,2. Because two is the base of the numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100…000 or 0. 00…001, just like a power of ten in the decimal system, a word, interpreted as an unsigned integer, can represent values from 0 to 2n −1 inclusively. Corresponding signed integer values can be positive, negative and zero, either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this show up frequently in computer software. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees at any time. Powers of two are used to measure computer memory. A byte is now considered eight bits (an octet, resulting in the possibility of 256 values, the prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024. However, in general, the term kilo has been used in the International System of Units to mean 1,000, binary prefixes have been standardized, such as kibi meaning 1,024. Nearly all processor registers have sizes that are powers of two,32 or 64 being most common, powers of two occur in a range of other places as well. For many disk drives, at least one of the size, number of sectors per track. The logical block size is almost always a power of two. Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of two or three powers of two, or powers of two minus one. For example,640 =512 +128 =128 ×5, put another way, they have fairly regular bit patterns. A prime number that is one less than a power of two is called a Mersenne prime, for example, the prime number 31 is a Mersenne prime because it is 1 less than 32. Similarly, a number that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational, the numbers that can be represented as sums of consecutive positive integers are called polite numbers, they are exactly the numbers that are not powers of two
40.
Parity (mathematics)
–
Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
41.
Tetration
–
In mathematics, tetration is the next hyperoperation after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra-, Tetration is used for the notation of very large numbers. The notation n a means a a ⋅ ⋅ a, the application of exponentiation n −1 times. Shown here are the first four hyperoperations, with tetration as the fourth, multiplication a × n = a + a + ⋯ + a ⏟ n n copies of a combined by addition. Exponentiation a n = a × a × ⋯ × a ⏟ n n copies of a combined by multiplication, Tetration n a = a a ⋅ ⋅ a ⏟ n n copies of a combined by exponentiation, right-to-left. The above example is read as the nth tetration of a, each operation is defined by iterating the previous one. Tetration is not an elementary function, analogously, tetration can be thought of as a chained power involving n numbers a. The parameter a may be called the base-parameter in the following, computer programmers refer to this choice as right-associative. When a and 10 are coprime, we can compute the last m decimal digits of n a using Eulers theorem, there are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale, the term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory, has gained dominance. It was also popularized in Rudy Ruckers Infinity and the Mind, the term superexponentiation was published by Bromer in his paper Superexponentiation in 1987. It was used earlier by Ed Nelson in his book Predicative Arithmetic, the term hyperpower is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence, when considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is misleading, since it is referring to tetration. The term power tower is used, in the form the power tower of order n for a a ⋅ ⋅ a ⏟ n. This is a misnomer, however, because tetration cannot be expressed with iterated power functions, the term snap is occasionally used in informal contexts, in the form a snap n for a a ⋅ ⋅ a ⏟ n. This term is not yet accepted, although it is used within select communities. It is believed to be a reference to Jounce, the derivative of position in physics
42.
Computer science
–
Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems and its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory, are highly abstract, other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623, in 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner. He may be considered the first computer scientist and information theorist, for, among other reasons and he started developing this machine in 1834, and in less than two years, he had sketched out many of the salient features of the modern computer. A crucial step was the adoption of a card system derived from the Jacquard loom making it infinitely programmable. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information, when the machine was finished, some hailed it as Babbages dream come true. During the 1940s, as new and more powerful computing machines were developed, as it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as an academic discipline in the 1950s. The worlds first computer science program, the Cambridge Diploma in Computer Science. The first computer science program in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights and it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM704 and later the IBM709 computers, still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. During the late 1950s, the science discipline was very much in its developmental stages. Time has seen significant improvements in the usability and effectiveness of computing technology, modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base
43.
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
44.
Sulfur
–
Sulfur or sulphur is a chemical element with symbol S and atomic number 16. It is abundant, multivalent, and nonmetallic, under normal conditions, sulfur atoms form cyclic octatomic molecules with chemical formula S8. Elemental sulfur is a yellow crystalline solid at room temperature. Chemically, sulfur reacts with all elements except for gold, platinum, iridium, tellurium, though sometimes found in pure, native form, sulfur usually occurs as sulfide and sulfate minerals. Being abundant in native form, sulfur was known in ancient times, being mentioned for its uses in ancient India, ancient Greece, China, in the Bible, sulfur is called brimstone. Today, almost all elemental sulfur is produced as a byproduct of removing sulfur-containing contaminants from natural gas, the greatest commercial use of the element is the production of sulfuric acid for sulfate and phosphate fertilizers, and other chemical processes. The element sulfur is used in matches, insecticides, and fungicides, many sulfur compounds are odoriferous, and the smells of odorized natural gas, skunk scent, grapefruit, and garlic are due to organosulfur compounds. Hydrogen sulfide gives the characteristic odor to rotting eggs and other biological processes, sulfur is an essential element for all life, but almost always in the form of organosulfur compounds or metal sulfides. Three amino acids and two vitamins are organosulfur compounds, many cofactors also contain sulfur including glutathione and thioredoxin and iron–sulfur proteins. Disulfides, S–S bonds, confer mechanical strength and insolubility of the keratin, found in outer skin, hair. Sulfur is one of the chemical elements needed for biochemical functioning and is an elemental macronutrient for all organisms. Sulfur is derived from the Latin word sulpur, which was Hellenized to sulphur, the spelling sulfur appears toward the end of the Classical period. In 12th-century Anglo-French, it was sulfre, in the 14th century the Latin ph was restored, for sulphre, the parallel f~ph spellings continued in Britain until the 19th century, when the word was standardized as sulphur. Sulfur was the form chosen in the United States, whereas Canada uses both, the IUPAC adopted the spelling sulfur in 1990, as did the Nomenclature Committee of the Royal Society of Chemistry in 1992, restoring the spelling sulfur to Britain. Sulfur forms polyatomic molecules with different chemical formulas, the best-known allotrope being octasulfur, cyclo-S8. The point group of cyclo-S8 is D4d and its dipole moment is 0 D. Octasulfur is a soft, bright-yellow solid that is odorless and it melts at 115.21 °C, boils at 444.6 °C and sublimes easily. At 95.2 °C, below its melting temperature, cyclo-octasulfur changes from α-octasulfur to the β-polymorph, the structure of the S8 ring is virtually unchanged by this phase change, which affects the intermolecular interactions. At higher temperatures, the viscosity decreases as depolymerization occurs, molten sulfur assumes a dark red color above 200 °C
45.
Group (periodic table)
–
In chemistry, a group is a column of elements in the periodic table of the chemical elements. There are 18 numbered groups in the table, but the f-block columns are not numbered. There are three systems of group numbering, the modern numbering group 1 to group 18 is recommended by the International Union of Pure and Applied Chemistry. It replaces two older naming schemes that were mutually confusing, also, groups may be identified by their topmost element or have a specific name. For example, group 16 is variously described as oxygen group, two earlier group number systems exist, CAS and old IUPAC. Both use numerals and letters A and B, both systems agree on the numbers. The numbers indicate approximately the highest oxidation number of the elements in that group, the number proceeds in a linearly increasing fashion for the most part, once on the left of the table, and once on the right, with some irregularities in the transition metals. However, the two use the letters differently. For example, potassium has one valence electron, therefore, it is located in group 1. Calcium is in group 2, for it contains two valence electrons, the old IUPAC system was frequently used in Europe while the CAS is most common in America. The new IUPAC scheme was developed to both systems as they confusingly used the same names to mean different things. The new system simply numbers the groups increasingly from left to right on the periodic table. The IUPAC proposal was first circulated in 1985 for public comments, in history, several sets of group names have been used, Scerri, E. R. The periodic table, its story and its significance
46.
Periodic table
–
The periodic table is a tabular arrangement of the chemical elements, ordered by their atomic number, electron configurations, and recurring chemical properties. This ordering shows periodic trends, such as elements with similar behaviour in the same column and it also shows four rectangular blocks with some approximately similar chemical properties. In general, within one row the elements are metals on the left, the rows of the table are called periods, the columns are called groups. Six groups have names as well as numbers, for example, group 17 elements are the halogens, and group 18, the noble gases. The periodic table can be used to derive relationships between the properties of the elements, and predict the properties of new elements yet to be discovered or synthesized, the periodic table provides a useful framework for analyzing chemical behaviour, and is widely used in chemistry and other sciences. The Russian chemist Dmitri Mendeleev published the first widely recognized periodic table in 1869 and he developed his table to illustrate periodic trends in the properties of the then-known elements. Mendeleev also predicted some properties of elements that would be expected to fill gaps in this table. Most of his predictions were proved correct when the elements in question were subsequently discovered, Mendeleevs periodic table has since been expanded and refined with the discovery or synthesis of further new elements and the development of new theoretical models to explain chemical behaviour. The first 94 elements exist naturally, although some are only in trace amounts and were synthesized in laboratories before being found in nature. Elements with atomic numbers from 95 to 118 have only been synthesized in laboratories or nuclear reactors, synthesis of elements having higher atomic numbers is being pursued. Numerous synthetic radionuclides of naturally occurring elements have also produced in laboratories. Each chemical element has an atomic number representing the number of protons in its nucleus. Most elements have differing numbers of neutrons among different atoms, with variants being referred to as isotopes. Isotopes are never separated in the table, they are always grouped together under a single element. Elements with no stable isotopes have the masses of their most stable isotopes. In the standard periodic table, the elements are listed in order of increasing atomic number, a new row is started when a new electron shell has its first electron. Columns are determined by the configuration of the atom, elements with the same number of electrons in a particular subshell fall into the same columns. Thus, it is easy to predict the chemical properties of an element if one knows the properties of the elements around it
47.
Chalcogens
–
The chalcogens are the chemical elements in group 16 of the periodic table. This group is known as the oxygen family. It consists of the oxygen, sulfur, selenium, tellurium. The chemically uncharacterized synthetic element livermorium is predicted to be a chalcogen as well, the word chalcogen is derived from a combination of the Greek word khalkόs principally meaning copper, and the Latinised Greek word genēs, meaning born or produced. Sulfur has been known since antiquity, and oxygen was recognized as an element in the 18th century, selenium, tellurium and polonium were discovered in the 19th century, and livermorium in 2000. All of the chalcogens have six valence electrons, leaving them two electrons short of an outer shell. Their most common oxidation states are −2, +2, +4 and they have relatively low atomic radii, especially the lighter ones. Lighter chalcogens are typically nontoxic in their form, and are often critical to life. All of the chalcogens have some role in functions, either as a nutrient or a toxin. The lighter chalcogens, such as oxygen and sulfur, are rarely toxic, selenium is an important nutrient but is also commonly toxic. Tellurium often has unpleasant effects, and polonium is always extremely harmful, sulfur has more than 20 allotropes, oxygen has nine, selenium has at least five, polonium has two, and only one crystal structure of tellurium has so far been discovered. There are numerous organic chalcogen compounds, not counting oxygen, organic sulfur compounds are generally the most common, followed by organic selenium compounds and organic tellurium compounds. This trend also occurs with chalcogen pnictides and compounds containing chalcogens, oxygen is generally extracted from air and sulfur is extracted from oil and natural gas. Selenium and tellurium are produced as byproducts of copper refining, polonium and livermorium are most available in particle accelerators. The primary use of oxygen is in steelmaking. Sulfur is mostly converted into sulfuric acid, which is used in the chemical industry. Seleniums most common application is glassmaking, tellurium compounds are mostly used in optical disks, electronic devices, and solar cells. Some of poloniums applications are due to its radioactivity, all of the solid, stable chalcogens are soft and do not conduct heat well
48.
Driver's license
–
The laws relating to the licensing of drivers vary between jurisdictions. In some jurisdictions, a license is issued after the recipient has passed a driving test, while in others, different categories of license often exist for different types of motor vehicles, particularly large trucks and passenger vehicles. The difficulty of the driving test varies considerably between jurisdictions, as do such as age and the required level of practice. Up until the start of the 20th century, European authorities issued licenses to drive motor vehicles similarly ad hoc, the first mandatory license requirement for driving was introduced in the Motor Car Act 1903 in the United Kingdom. Every car owner had to register his automobile with his local government authority, the minimum qualifying age was set at 17. The license gave its holder freedom of the road with a maximum 20 mph speed limit, compulsory testing was introduced in 1934, with the passing of the Road Traffic Act. Prussia, then a state within the German Empire, introduced compulsory licensing on September 29,1903, a test on mechanical aptitude had to be passed and the Dampfkesselüberwachungsverein was charged with conducting these tests. In 1910, the German imperial government mandated the licensing of drivers on a scale, establishing a system of tests. As automobile-related fatalities soared in North America, public outcry provoked legislators to begin studying the French, on August 1,1910, North Americas first licensing law for motor vehicles went into effect in the US state of New York, though it initially applied only to professional chauffeurs. In July 1913, the state of New Jersey became the first to all drivers to pass a mandatory examination before receiving a license. Many countries, including Australia, New Zealand, Canada, the United Kingdom, as many people have drivers licenses, they are often accepted as de facto proof of identity. Most identity cards and drivers licenses are credit card size—the ID-1 size, many European countries require drivers to produce their license on demand when driving. Some European countries require adults to carry proof of identity at all times, in the United Kingdom drivers are not required to carry their licence. A driver may be required by a constable or vehicle examiner to produce their licence, but may provide it in a police station within seven days. In Spain, Sweden and Finland, the license number is the same as the citizens ID number. In Bulgaria, Italy, Poland, Romania and Spain, a vehicle registration card. In Hong Kong a driving license carries the number as the holders ID card. Upon inspection both must be presented, plans to make the newly phased in Smart ID contain driving license information have been shelved
49.
United States
–
Forty-eight of the fifty states and the federal district are contiguous and located in North America between Canada and Mexico. The state of Alaska is in the northwest corner of North America, bordered by Canada to the east, the state of Hawaii is an archipelago in the mid-Pacific Ocean. The U. S. territories are scattered about the Pacific Ocean, the geography, climate and wildlife of the country are extremely diverse. At 3.8 million square miles and with over 324 million people, the United States is the worlds third- or fourth-largest country by area, third-largest by land area. It is one of the worlds most ethnically diverse and multicultural nations, paleo-Indians migrated from Asia to the North American mainland at least 15,000 years ago. European colonization began in the 16th century, the United States emerged from 13 British colonies along the East Coast. Numerous disputes between Great Britain and the following the Seven Years War led to the American Revolution. On July 4,1776, during the course of the American Revolutionary War, the war ended in 1783 with recognition of the independence of the United States by Great Britain, representing the first successful war of independence against a European power. The current constitution was adopted in 1788, after the Articles of Confederation, the first ten amendments, collectively named the Bill of Rights, were ratified in 1791 and designed to guarantee many fundamental civil liberties. During the second half of the 19th century, the American Civil War led to the end of slavery in the country. By the end of century, the United States extended into the Pacific Ocean. The Spanish–American War and World War I confirmed the status as a global military power. The end of the Cold War and the dissolution of the Soviet Union in 1991 left the United States as the sole superpower. The U. S. is a member of the United Nations, World Bank, International Monetary Fund, Organization of American States. The United States is a developed country, with the worlds largest economy by nominal GDP. It ranks highly in several measures of performance, including average wage, human development, per capita GDP. While the U. S. economy is considered post-industrial, characterized by the dominance of services and knowledge economy, the United States is a prominent political and cultural force internationally, and a leader in scientific research and technological innovations. In 1507, the German cartographer Martin Waldseemüller produced a map on which he named the lands of the Western Hemisphere America after the Italian explorer and cartographer Amerigo Vespucci
50.
Australia
–
Australia, officially the Commonwealth of Australia, is a country comprising the mainland of the Australian continent, the island of Tasmania and numerous smaller islands. It is the worlds sixth-largest country by total area, the neighbouring countries are Papua New Guinea, Indonesia and East Timor to the north, the Solomon Islands and Vanuatu to the north-east, and New Zealand to the south-east. Australias capital is Canberra, and its largest urban area is Sydney, for about 50,000 years before the first British settlement in the late 18th century, Australia was inhabited by indigenous Australians, who spoke languages classifiable into roughly 250 groups. The population grew steadily in subsequent decades, and by the 1850s most of the continent had been explored, on 1 January 1901, the six colonies federated, forming the Commonwealth of Australia. Australia has since maintained a liberal democratic political system that functions as a federal parliamentary constitutional monarchy comprising six states. The population of 24 million is highly urbanised and heavily concentrated on the eastern seaboard, Australia has the worlds 13th-largest economy and ninth-highest per capita income. With the second-highest human development index globally, the country highly in quality of life, health, education, economic freedom. The name Australia is derived from the Latin Terra Australis a name used for putative lands in the southern hemisphere since ancient times, the Dutch adjectival form Australische was used in a Dutch book in Batavia in 1638, to refer to the newly discovered lands to the south. On 12 December 1817, Macquarie recommended to the Colonial Office that it be formally adopted, in 1824, the Admiralty agreed that the continent should be known officially as Australia. The first official published use of the term Australia came with the 1830 publication of The Australia Directory and these first inhabitants may have been ancestors of modern Indigenous Australians. The Torres Strait Islanders, ethnically Melanesian, were originally horticulturists, the northern coasts and waters of Australia were visited sporadically by fishermen from Maritime Southeast Asia. The first recorded European sighting of the Australian mainland, and the first recorded European landfall on the Australian continent, are attributed to the Dutch. The first ship and crew to chart the Australian coast and meet with Aboriginal people was the Duyfken captained by Dutch navigator, Willem Janszoon. He sighted the coast of Cape York Peninsula in early 1606, the Dutch charted the whole of the western and northern coastlines and named the island continent New Holland during the 17th century, but made no attempt at settlement. William Dampier, an English explorer and privateer, landed on the north-west coast of New Holland in 1688, in 1770, James Cook sailed along and mapped the east coast, which he named New South Wales and claimed for Great Britain. The first settlement led to the foundation of Sydney, and the exploration, a British settlement was established in Van Diemens Land, now known as Tasmania, in 1803, and it became a separate colony in 1825. The United Kingdom formally claimed the part of Western Australia in 1828. Separate colonies were carved from parts of New South Wales, South Australia in 1836, Victoria in 1851, the Northern Territory was founded in 1911 when it was excised from South Australia
51.
Iceland
–
Iceland is a Nordic island country in the North Atlantic Ocean. It has a population of 332,529 and an area of 103,000 km2, the capital and largest city is Reykjavík. Reykjavík and the areas in the southwest of the country are home to over two-thirds of the population. Iceland is volcanically and geologically active, the interior consists of a plateau characterised by sand and lava fields, mountains and glaciers, while many glacial rivers flow to the sea through the lowlands. Iceland is warmed by the Gulf Stream and has a climate, despite a high latitude just outside the Arctic Circle. Its high latitude and marine influence still keeps summers chilly, with most of the archipelago having a tundra climate. According to the ancient manuscript Landnámabók, the settlement of Iceland began in the year 874 AD when the Norwegian chieftain Ingólfr Arnarson became the first permanent settler on the island. In the following centuries, Norwegians, and to a lesser extent other Scandinavians, emigrated to Iceland, the island was governed as an independent commonwealth under the Althing, one of the worlds oldest functioning legislative assemblies. Following a period of strife, Iceland acceded to Norwegian rule in the 13th century. The establishment of the Kalmar Union in 1397 united the kingdoms of Norway, Denmark, Iceland thus followed Norways integration to that Union and came under Danish rule after Swedens secession from that union in 1523. In the wake of the French revolution and the Napoleonic wars, Icelands struggle for independence took form and culminated in independence in 1918, until the 20th century, Iceland relied largely on subsistence fishing and agriculture, and was among the poorest in Europe. Industrialisation of the fisheries and Marshall Plan aid following World War II brought prosperity, in 1994, it became a part of the European Economic Area, which further diversified the economy into sectors such as finance, biotechnology, and manufacturing. Iceland has an economy with relatively low taxes compared to other OECD countries. It maintains a Nordic social welfare system that provides health care. Iceland ranks high in economic, political and social stability and equality, in 2013, it was ranked as the 13th most-developed country in the world by the United Nations Human Development Index. Iceland runs almost completely on renewable energy, some bankers were jailed, and the economy has made a significant recovery, in large part due to a surge in tourism. Icelandic culture is founded upon the nations Scandinavian heritage, most Icelanders are descendants of Germanic and Gaelic settlers. Icelandic, a North Germanic language, is descended from Old Norse and is related to Faroese
52.
New Zealand
–
New Zealand /njuːˈziːlənd/ is an island nation in the southwestern Pacific Ocean. The country geographically comprises two main landmasses—the North Island, or Te Ika-a-Māui, and the South Island, or Te Waipounamu—and around 600 smaller islands. New Zealand is situated some 1,500 kilometres east of Australia across the Tasman Sea and roughly 1,000 kilometres south of the Pacific island areas of New Caledonia, Fiji, because of its remoteness, it was one of the last lands to be settled by humans. During its long period of isolation, New Zealand developed a distinct biodiversity of animal, fungal, the countrys varied topography and its sharp mountain peaks, such as the Southern Alps, owe much to the tectonic uplift of land and volcanic eruptions. New Zealands capital city is Wellington, while its most populous city is Auckland, sometime between 1250 and 1300 CE, Polynesians settled in the islands that later were named New Zealand and developed a distinctive Māori culture. In 1642, Dutch explorer Abel Tasman became the first European to sight New Zealand, in 1840, representatives of Britain and Māori chiefs signed the Treaty of Waitangi, which declared British sovereignty over the islands. In 1841, New Zealand became a colony within the British Empire, today, the majority of New Zealands population of 4.7 million is of European descent, the indigenous Māori are the largest minority, followed by Asians and Pacific Islanders. Reflecting this, New Zealands culture is derived from Māori and early British settlers. The official languages are English, Māori and New Zealand Sign Language, New Zealand is a developed country and ranks highly in international comparisons of national performance, such as health, education, economic freedom and quality of life. Since the 1980s, New Zealand has transformed from an agrarian, Queen Elizabeth II is the countrys head of state and is represented by a governor-general. In addition, New Zealand is organised into 11 regional councils and 67 territorial authorities for local government purposes, the Realm of New Zealand also includes Tokelau, the Cook Islands and Niue, and the Ross Dependency, which is New Zealands territorial claim in Antarctica. New Zealand is a member of the United Nations, Commonwealth of Nations, ANZUS, Organisation for Economic Co-operation and Development, Pacific Islands Forum, and Asia-Pacific Economic Cooperation. Dutch explorer Abel Tasman sighted New Zealand in 1642 and called it Staten Landt, in 1645, Dutch cartographers renamed the land Nova Zeelandia after the Dutch province of Zeeland. British explorer James Cook subsequently anglicised the name to New Zealand, Aotearoa is the current Māori name for New Zealand. It is unknown whether Māori had a name for the country before the arrival of Europeans. Māori had several names for the two main islands, including Te Ika-a-Māui for the North Island and Te Waipounamu or Te Waka o Aoraki for the South Island. Early European maps labelled the islands North, Middle and South, in 1830, maps began to use North and South to distinguish the two largest islands and by 1907, this was the accepted norm. The New Zealand Geographic Board discovered in 2009 that the names of the North Island and South Island had never been formalised and this set the names as North Island or Te Ika-a-Māui, and South Island or Te Waipounamu
53.
Norway
–
The Antarctic Peter I Island and the sub-Antarctic Bouvet Island are dependent territories and thus not considered part of the Kingdom. Norway also lays claim to a section of Antarctica known as Queen Maud Land, until 1814, the kingdom included the Faroe Islands, Greenland, and Iceland. It also included Isle of Man until 1266, Shetland and Orkney until 1468, Norway has a total area of 385,252 square kilometres and a population of 5,258,317. The country shares a long border with Sweden. Norway is bordered by Finland and Russia to the north-east, Norway has an extensive coastline, facing the North Atlantic Ocean and the Barents Sea. King Harald V of the Dano-German House of Glücksburg is the current King of Norway, erna Solberg became Prime Minister in 2013, replacing Jens Stoltenberg. A constitutional monarchy, Norway divides state power between the Parliament, the Cabinet and the Supreme Court, as determined by the 1814 Constitution, the kingdom is established as a merger of several petty kingdoms. By the traditional count from the year 872, the kingdom has existed continuously for 1,144 years, Norway has both administrative and political subdivisions on two levels, counties and municipalities. The Sámi people have an amount of self-determination and influence over traditional territories through the Sámi Parliament. Norway maintains close ties with the European Union and the United States, the country maintains a combination of market economy and a Nordic welfare model with universal health care and a comprehensive social security system. Norway has extensive reserves of petroleum, natural gas, minerals, lumber, seafood, the petroleum industry accounts for around a quarter of the countrys gross domestic product. On a per-capita basis, Norway is the worlds largest producer of oil, the country has the fourth-highest per capita income in the world on the World Bank and IMF lists. On the CIAs GDP per capita list which includes territories and some regions, from 2001 to 2006, and then again from 2009 to 2017, Norway had the highest Human Development Index ranking in the world. It also has the highest inequality-adjusted ranking, Norway ranks first on the World Happiness Report, the OECD Better Life Index, the Index of Public Integrity and the Democracy Index. Norway has two names, Noreg in Nynorsk and Norge in Bokmål. The name Norway comes from the Old English word Norðrveg mentioned in 880, meaning way or way leading to the north. In contrasting with suðrvegar southern way for Germany, and austrvegr eastern way for the Baltic, the Anglo-Saxon of Britain also referred to the kingdom of Norway in 880 as Norðmanna land. This was the area of Harald Fairhair, the first king of Norway, and because of him
54.
South Africa
–
South Africa, officially the Republic of South Africa, is the southernmost country in Africa. South Africa is the 25th-largest country in the world by land area and it is the southernmost country on the mainland of the Old World or the Eastern Hemisphere. About 80 percent of South Africans are of Sub-Saharan African ancestry, divided among a variety of ethnic groups speaking different Bantu languages, the remaining population consists of Africas largest communities of European, Asian, and multiracial ancestry. South Africa is a multiethnic society encompassing a variety of cultures, languages. Its pluralistic makeup is reflected in the recognition of 11 official languages. The country is one of the few in Africa never to have had a coup détat, however, the vast majority of black South Africans were not enfranchised until 1994. During the 20th century, the black majority sought to recover its rights from the dominant white minority, with this struggle playing a role in the countrys recent history. The National Party imposed apartheid in 1948, institutionalising previous racial segregation, since 1994, all ethnic and linguistic groups have held political representation in the countrys democracy, which comprises a parliamentary republic and nine provinces. South Africa is often referred to as the Rainbow Nation to describe the multicultural diversity. The World Bank classifies South Africa as an economy. Its economy is the second-largest in Africa, and the 34th-largest in the world, in terms of purchasing power parity, South Africa has the seventh-highest per capita income in Africa. However, poverty and inequality remain widespread, with about a quarter of the population unemployed, nevertheless, South Africa has been identified as a middle power in international affairs, and maintains significant regional influence. The name South Africa is derived from the geographic location at the southern tip of Africa. Upon formation the country was named the Union of South Africa in English, since 1961 the long form name in English has been the Republic of South Africa. In Dutch the country was named Republiek van Zuid-Afrika, replaced in 1983 by the Afrikaans Republiek van Suid-Afrika, since 1994 the Republic has had an official name in each of its 11 official languages. Mzansi, derived from the Xhosa noun umzantsi meaning south, is a name for South Africa. South Africa contains some of the oldest archaeological and human fossil sites in the world, extensive fossil remains have been recovered from a series of caves in Gauteng Province. The area is a UNESCO World Heritage site and has termed the Cradle of Humankind
55.
Isle of Man
–
The Isle of Man, also known simply as Mann, is a self-governing crown dependency in the Irish Sea between England and Northern Ireland. The head of state is Queen Elizabeth II, who holds the title of Lord of Mann, the Lord of Mann is represented by a Lieutenant Governor. Foreign relations and defence are the responsibility of the British Government, the island has been inhabited since before 6500 BC. Gaelic cultural influence began in the 5th century and the Manx language, in 627, Edwin of Northumbria conquered the Isle of Man along with most of Mercia. In the 9th century, Norsemen established the Kingdom of the Isles, magnus III, King of Norway, was also known as King of Mann and the Isles between 1099 and 1103. In 1266, the became part of Scotland under the Treaty of Perth. After a period of alternating rule by the kings of Scotland and England, the short form often used in English, Mann, is derived from the Manx Mannin, though sometimes the name is written as Man. The earliest recorded Manx form of the name is Manu or Mana, the Old Irish form of the name is Manau or Mano. Old Welsh records named it as Manaw, also reflected in Manaw Gododdin, the name for an ancient district in north Britain along the lower Firth of Forth. The oldest known reference to the calls it Mona, in Latin, in the 1st century CE, Pliny the Elder records it as Monapia or Monabia. Later Latin references have Mevania or Mænavia, and Eubonia or Eumonia by Irish writers and it is found in the Sagas of Icelanders as Mön. The name is cognate with the Welsh name of the island of Anglesey, Ynys Môn, usually derived from a Celtic word for mountain. The name was at least secondarily associated with that of Manannán mac Lir in Irish mythology, later, a Manannán is recorded as the first king of Mann in a Manx poem. The island was cut off from the islands around 8000 BC. The first residents were hunter gatherers and fishermen, examples of their tools are kept at the Manx Museum. There were also the local Ronaldsway and Bann cultures, during the Bronze Age, burial mounds became smaller. Bodies were put in stone-lined graves with ornamental containers, the Bronze Age burial mounds created long-lasting markers around the countryside. The ancient Romans knew of the island and called it Insula Manavia although it is whether they conquered the island
56.
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
57.
Happy Birthday Sweet Sixteen
–
Happy Birthday Sweet Sixteen is a pop song released in 1961 by Neil Sedaka. Sedaka wrote the music and performed the song, while the lyrics were written by Howard Greenfield, both of these songs exhibiting similarity to Happy Birthday Sweet Sixteen were penned by the team of Carole King and Gerry Goffin. The song reached #6 on the Billboard Hot 100 chart and no 3 on the UK Singles Chart and this was one of several Sedaka recordings that employed the services of drummer Gary Chester. A Bobby Vee version was released on the album 30 Big Hits from the 60s in 1964, the Swedish dansband Flamingokvintetten in 1968 recorded a cover of the song that reached #1 in the Swedish charts Tio i topp and Kvällstoppen. They also recorded a Swedish version of the song, named Hon är sexton år i dag reaching Svensktoppen the same year. A version called Happy Birthday, Twenty-One with a lyric by Ian Whitcomb. In 2009, Damian McGinty of Celtic Thunder included the song on their album Take Me Home, in the Roseanne episode Pretty in Black, Roseanne, Dan and several of their friends sing the first chorus of this song to a mortified Darlene at her sweet sixteen party. A version of the song by a performer is also played over the closing credits of the episode
58.
You're Sixteen
–
Youre Sixteen is a song written by the Sherman Brothers. It was first performed by American rockabilly singer Johnny Burnette, whose version peaked at eight on the U. S. Billboard Hot 100 in December 1960. The original 1960 version of Youre Sixteen by Johnny Burnette is featured prominently on the 1973 motion picture soundtrack of the film American Graffiti, in January 1974, the song, taken from the album Ringo, hit number one. The latter performance reunited Ringo Starr with his former Beatles bandmate Paul McCartney, McCartney is credited on the liner notes of the album Ringo as having played the solo on a kazoo. But reviewer Michael Verity has quoted the producer, Richard Perry. “In fact, the solo on ‘You’re Sixteen, ’ which sounds like a kazoo or something, was Paul singing very spontaneously as we played that back, so he’s singing the solo on that. ”In any case. Harry Nilsson sang backing vocals on Starrs version, nicky Hopkins is heard playing the piano, including his going up and down the scale in the instrumental fade of the song. Also, the Ringo Starr version was in 6/8 time, compared to the original, in addition, in Ringos version, the melody and the chords were different, in the Bridge Section, which led to a minor key, while the original version used all major keys. The 1978 video of Starrs version features Carrie Fisher as Starrs love interest, a reference to Youre Sixteen is made in the title of Cold Chisels EP Youre Thirteen, Youre Beautiful, and Youre Mine. Footnotes Citations Lyrics of this song at MetroLyrics
59.
Sweet Sixteen (Hilary Duff song)
–
Metamorphosis is the second studio album by American singer Hilary Duff. The album was released on August 26,2003 by Hollywood Records as a follow up to her holiday album, according to Duff, the album incorporates elements of pop and rock music, and it represents changes that are specific to her life and that everyone experiences. Duff worked with producers on the album such as The Matrix who worked with Latin Pop Singer, Myra. Others who collaborated on the album include Chico Bennett, Matthew Gerrard, John Shanks, Metamorphosis coincided with other high-profile projects in which she was involved in other media. The album debuted at two on the US Billboard 200, selling 203,000 copies in its first week. In following week the album rose to one on the chart. The album became the eighth best selling album of 2003 in the US, according to Nielsen SoundScan, by late 2005, the album had sold five million copies worldwide. For Hollywood Records, Metamorphosis was the labels first high seller in several years, three singles were released from the album. The lead single, So Yesterday, was released in July 2003 to mixed reception, the song failed to make much of an impact in the United States, peaking in the top fifty. Internationally, it peaked within the top ten of charts in Australia, Canada, France. The third and final single from the album, Little Voice, was released in Australia. Duff further promoted the album with the Metamorphosis Tour, which began in November 2003, Hilary Duff had always wanted to follow in the footsteps of her elder sister, Haylie. Duff watched her sister rehearsing in 2001, after which she told her mother that she wanted to be involved in singing, during the same time period, she attended a Radio Disney concert where she met Andre Recke, whose client Myra was performing. According to Duff, watching the pop musicians preparing and warming up backstage at the concert made her think, Recke said he thought Duff, who was popular with preteens at the time because of her role in the popular Disney Channel original series Lizzie McGuire, had something special. Sometimes you just have that feeling, that, Wow, shes a star, after becoming determined to start a music career, Duff resumed her vocal lessons—which she had started before her acting career began—and became one of Reckes clients. Ive always had a big drive, she said, when I felt like something looked fun or I wanted to accomplish something, I really just go for it. I didnt really know what it was gonna be like, but I knew I wanted to try it, Duff recorded several songs for Disney soundtrack and compilation albums, and a Christmas album, Santa Claus Lane, in 2002. According to Duff, her mother, Recke and Duff herself worked hard to get music that she related to and was age-appropriate for her
60.
U16 Girls
–
U16 Girls is the second single released from Indie band Travis debut album, Good Feeling. The single was released on 1 April 1997 in the United Kingdom, as the age of consent in Scotland is 16, Healy named the song U16 Girls. The singles artwork depicts a surfboard which was designed by The Stone Roses guitarist John Squire, the surfboard is decorated with lyrics from Beach Boys songs. The single charted at number forty on the UK Singles Chart
61.
The Crests
–
The Crests were an American doo-wop group, formed by bass vocalist J. T. Carter in the mid-1950s. Their most popular song,16 Candles, rose to #2 on the Billboard Hot 100 chart in 1959, selling one million copies. The interracial group had three African American members, one Puerto Rican, and one Italian American, founded by J. T. Carter, the group included Talmadge Gough, Harold Torres and Patricia Van Dross. Carter selected vocalist Johnny Mastrangelo as lead vocalist, maestros vocal style became instantly recognizable, and a jukebox favorite of national teen audiences. His quality vocals, great song selections and recordings, with dance-easy beats and they also charted with Sweetest One in 1957. In the late 1950s, The Crests performed on several national teen dance shows, including American Bandstand. After recording two singles for Joyce Records, Van Dross left The Crests in 1958, Maestro left for a solo career in 1961. He later joined The Del Satins as their lead singer and they merged with The Rhythm Method in March 1968 to become Johnny Maestro & the Brooklyn Bridge. In 1969 they had a #3 hit with The Worst That Could Happen, in 1962, The Crests recorded a new single, Little Miracles, with Tony Middleton singing lead. It was their first single not to chart in the Top 100, james Ancrum then took over the lead, recording Guilty and several other songs. Gough quit the group after the single, and was replaced by Gary Lewis, subsequently, the group failed to find success throughout the decade. By the late 1960s, Torres was gone, the group continued until 1978 as a trio of Carter, Ancrum and Lewis, when the group split. Carter went on to sing with Charlie Thomas Drifters, Carter reformed the group in 1980, auditioning over 200 singers, finally settling on lead Bill Damon, Greg Sereck, Dennis Ray and New York drummer Jon Ihle. The group continued well into the 1990s, Carter sold the trademark The Crests to Thomas Marasciullo in the late 1990s. Mara now continues the group without Carter, in 2016, Carter continued his legendary career performing as J. T. Carters Crests, based on provisions of the Truth in Music Act. Carters new group include Ken Boulden, formerly of Solid Gold, J. T. Carters Crests held their first concert in Upland, California on July 16,2016 to a sold out crowd and received standing ovations. Carter has had a new level of recognition in recent times, in 2013 Carter also appeared on the 1st Annual Palisades Park Reunion concert with Cousin Brucie aka Bruce Morrow, broadcast live on SiriusXM satellite radio. Other performers included Neil Sedaka, Lesley Gore, Bobby Lewis, in January 2014 Carter interviewed Joe Franklin on the Bloomberg Radio network
62.
Obstetrics and gynaecology
–
Obstetrics and Gynecology is the medical specialty that deals with obstetrics and gynecology. The postgraduate training program for both aspects is unified and this combined training prepares the practicing OB/GYN to be adept at the care of female reproductive organs health and at the management of pregnancy. After completing medical school, one must complete a 4 year residency program to be eligible to sit for boards, for the ERAS match in 2017, there will be 238 participating programs accepting applicants. In all, this adds up to 11-14 years of education, the first 7-9 years are general medical training. Experienced OBGYN professionals can seek certifications in sub-specialty areas, including maternal and fetal medicine, the other subspecialties are recognized as informal concentrations of practice. Additionally, physicians of other specialties may become trained in Advanced Life Support in Obstetrics, the salary of an obstetrician varies by country. In the United States, as of 2015, the salary is $222,400. Theriogenology Andrology Urology Neonatal infection Llewellyn-Jones, Derek, Fundamentals of Obstetrics and Gynecology, a provincial surgeon and his obstetric practice, Thomas W. Jones of Henley-in-Arden, 1764-1846. Reprint of Revised Edition Chicago, Alice B
63.
Belgium
–
Belgium, officially the Kingdom of Belgium, is a sovereign state in Western Europe bordered by France, the Netherlands, Germany, Luxembourg, and the North Sea. It is a small, densely populated country which covers an area of 30,528 square kilometres and has a population of about 11 million people. Additionally, there is a group of German-speakers who live in the East Cantons located around the High Fens area. Historically, Belgium, the Netherlands and Luxembourg were known as the Low Countries, the region was called Belgica in Latin, after the Roman province of Gallia Belgica. From the end of the Middle Ages until the 17th century, today, Belgium is a federal constitutional monarchy with a parliamentary system of governance. It is divided into three regions and three communities, that exist next to each other and its two largest regions are the Dutch-speaking region of Flanders in the north and the French-speaking southern region of Wallonia. The Brussels-Capital Region is a bilingual enclave within the Flemish Region. A German-speaking Community exists in eastern Wallonia, Belgiums linguistic diversity and related political conflicts are reflected in its political history and complex system of governance, made up of six different governments. Upon its independence, declared in 1830, Belgium participated in the Industrial Revolution and, during the course of the 20th century, possessed a number of colonies in Africa. This continuing antagonism has led to several far-reaching reforms, resulting in a transition from a unitary to a federal arrangement during the period from 1970 to 1993. Belgium is also a member of the Eurozone, NATO, OECD and WTO. Its capital, Brussels, hosts several of the EUs official seats as well as the headquarters of major international organizations such as NATO. Belgium is also a part of the Schengen Area, Belgium is a developed country, with an advanced high-income economy and is categorized as very high in the Human Development Index. A gradual immigration by Germanic Frankish tribes during the 5th century brought the area under the rule of the Merovingian kings, a gradual shift of power during the 8th century led the kingdom of the Franks to evolve into the Carolingian Empire. Many of these fiefdoms were united in the Burgundian Netherlands of the 14th and 15th centuries, the Eighty Years War divided the Low Countries into the northern United Provinces and the Southern Netherlands. The latter were ruled successively by the Spanish and the Austrian Habsburgs and this was the theatre of most Franco-Spanish and Franco-Austrian wars during the 17th and 18th centuries. The reunification of the Low Countries as the United Kingdom of the Netherlands occurred at the dissolution of the First French Empire in 1815, although the franchise was initially restricted, universal suffrage for men was introduced after the general strike of 1893 and for women in 1949. The main political parties of the 19th century were the Catholic Party, French was originally the single official language adopted by the nobility and the bourgeoisie
64.
Italy
–
Italy, officially the Italian Republic, is a unitary parliamentary republic in Europe. Located in the heart of the Mediterranean Sea, Italy shares open land borders with France, Switzerland, Austria, Slovenia, San Marino, Italy covers an area of 301,338 km2 and has a largely temperate seasonal and Mediterranean climate. Due to its shape, it is referred to in Italy as lo Stivale. With 61 million inhabitants, it is the fourth most populous EU member state, the Italic tribe known as the Latins formed the Roman Kingdom, which eventually became a republic that conquered and assimilated other nearby civilisations. The legacy of the Roman Empire is widespread and can be observed in the distribution of civilian law, republican governments, Christianity. The Renaissance began in Italy and spread to the rest of Europe, bringing a renewed interest in humanism, science, exploration, Italian culture flourished at this time, producing famous scholars, artists and polymaths such as Leonardo da Vinci, Galileo, Michelangelo and Machiavelli. The weakened sovereigns soon fell victim to conquest by European powers such as France, Spain and Austria. Despite being one of the victors in World War I, Italy entered a period of economic crisis and social turmoil. The subsequent participation in World War II on the Axis side ended in defeat, economic destruction. Today, Italy has the third largest economy in the Eurozone and it has a very high level of human development and is ranked sixth in the world for life expectancy. The country plays a prominent role in regional and global economic, military, cultural and diplomatic affairs, as a reflection of its cultural wealth, Italy is home to 51 World Heritage Sites, the most in the world, and is the fifth most visited country. The assumptions on the etymology of the name Italia are very numerous, according to one of the more common explanations, the term Italia, from Latin, Italia, was borrowed through Greek from the Oscan Víteliú, meaning land of young cattle. The bull was a symbol of the southern Italic tribes and was often depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. The name Italia originally applied only to a part of what is now Southern Italy – according to Antiochus of Syracuse, but by his time Oenotria and Italy had become synonymous, and the name also applied to most of Lucania as well. The Greeks gradually came to apply the name Italia to a larger region, excavations throughout Italy revealed a Neanderthal presence dating back to the Palaeolithic period, some 200,000 years ago, modern Humans arrived about 40,000 years ago. Other ancient Italian peoples of undetermined language families but of possible origins include the Rhaetian people and Cammuni. Also the Phoenicians established colonies on the coasts of Sardinia and Sicily, the Roman legacy has deeply influenced the Western civilisation, shaping most of the modern world
65.
Age of consent
–
The distinguishing aspect of the age of consent laws is that the person below the minimum age is regarded as the victim, and their sex partner as the offender. The term age of consent rarely appears in legal statutes, generally, a law will instead establish the age below which it is illegal to engage in sexual activity with that person. It has sometimes used with other meanings, such as the age at which a person becomes competent to consent to marriage. It should not be confused with the age of majority, age of responsibility, the voting age, the drinking age. Age of consent laws vary widely from jurisdiction to jurisdiction, though most jurisdictions set the age of consent in the range 14 to 18. Charges resulting from a breach of laws may range from a misdemeanor such as corruption of a minor. These factors all make age of consent an often confusing subject, in traditional societies, the age of consent for a sexual union was a matter for the family to decide, or a tribal custom. In most cases, this coincided with signs of puberty, menstruation for a woman, the ancient Greek poet Hesiod in Works and Days suggests that a man should marry around the age of thirty, and that he should take a wife who is five years past puberty. Reliable data for ages at marriage is scarce, in England, for example, the only reliable data in the early modern period comes from property records made after death. Modern historians have shown reluctance to accept evidence of young ages of marriage. There were authorities that said that consent could take place earlier, marriage would then be valid as long as neither of the two parties annulled the marital agreement before reaching puberty, or if they had already consummated the marriage. Judges sometimes honored marriages based on mutual consent at ages younger than 7, in contrast to established canon, the phrase within age was later interpreted by jurist Sir Edward Coke as meaning the age of marriage, which at the time was 12 years of age. The American colonies followed the English tradition, and the law was more of a guide, for example, Mary Hathaway was only 9 when she was married to William Williams. In the 16th century, a number of Italian and German states set the minimum age for sexual intercourse for girls. Towards the end of the 18th century, other European countries also began to enact similar laws, the first French Constitution of 1791 established the minimum age at 11 years. Portugal, Spain, Denmark and the Swiss cantons initially set the age at 10–12 years. In Christian societies, sex outside marriage was forbidden, older children were often punished themselves for being complicit in sexual interaction with an adult. Until the late 18th century, there was understanding of childhood as a concept
66.
Scotland
–
Scotland is a country that is part of the United Kingdom and covers the northern third of the island of Great Britain. It shares a border with England to the south, and is surrounded by the Atlantic Ocean, with the North Sea to the east. In addition to the mainland, the country is made up of more than 790 islands, including the Northern Isles, the Kingdom of Scotland emerged as an independent sovereign state in the Early Middle Ages and continued to exist until 1707. By inheritance in 1603, James VI, King of Scots, became King of England and King of Ireland, Scotland subsequently entered into a political union with the Kingdom of England on 1 May 1707 to create the new Kingdom of Great Britain. The union also created a new Parliament of Great Britain, which succeeded both the Parliament of Scotland and the Parliament of England. Within Scotland, the monarchy of the United Kingdom has continued to use a variety of styles, titles, the legal system within Scotland has also remained separate from those of England and Wales and Northern Ireland, Scotland constitutes a distinct jurisdiction in both public and private law. Glasgow, Scotlands largest city, was one of the worlds leading industrial cities. Other major urban areas are Aberdeen and Dundee, Scottish waters consist of a large sector of the North Atlantic and the North Sea, containing the largest oil reserves in the European Union. This has given Aberdeen, the third-largest city in Scotland, the title of Europes oil capital, following a referendum in 1997, a Scottish Parliament was re-established, in the form of a devolved unicameral legislature comprising 129 members, having authority over many areas of domestic policy. Scotland is represented in the UK Parliament by 59 MPs and in the European Parliament by 6 MEPs, Scotland is also a member nation of the British–Irish Council, and the British–Irish Parliamentary Assembly. Scotland comes from Scoti, the Latin name for the Gaels, the Late Latin word Scotia was initially used to refer to Ireland. By the 11th century at the latest, Scotia was being used to refer to Scotland north of the River Forth, alongside Albania or Albany, the use of the words Scots and Scotland to encompass all of what is now Scotland became common in the Late Middle Ages. Repeated glaciations, which covered the land mass of modern Scotland. It is believed the first post-glacial groups of hunter-gatherers arrived in Scotland around 12,800 years ago, the groups of settlers began building the first known permanent houses on Scottish soil around 9,500 years ago, and the first villages around 6,000 years ago. The well-preserved village of Skara Brae on the mainland of Orkney dates from this period and it contains the remains of an early Bronze Age ruler laid out on white quartz pebbles and birch bark. It was also discovered for the first time that early Bronze Age people placed flowers in their graves, in the winter of 1850, a severe storm hit Scotland, causing widespread damage and over 200 deaths. In the Bay of Skaill, the storm stripped the earth from a large irregular knoll, when the storm cleared, local villagers found the outline of a village, consisting of a number of small houses without roofs. William Watt of Skaill, the laird, began an amateur excavation of the site, but after uncovering four houses
67.
Germany
–
Germany, officially the Federal Republic of Germany, is a federal parliamentary republic in central-western Europe. It includes 16 constituent states, covers an area of 357,021 square kilometres, with about 82 million inhabitants, Germany is the most populous member state of the European Union. After the United States, it is the second most popular destination in the world. Germanys capital and largest metropolis is Berlin, while its largest conurbation is the Ruhr, other major cities include Hamburg, Munich, Cologne, Frankfurt, Stuttgart, Düsseldorf and Leipzig. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity, a region named Germania was documented before 100 AD. During the Migration Period the Germanic tribes expanded southward, beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation, in 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the German Revolution of 1918–1919, the Empire was replaced by the parliamentary Weimar Republic, the establishment of the national socialist dictatorship in 1933 led to World War II and the Holocaust. After a period of Allied occupation, two German states were founded, the Federal Republic of Germany and the German Democratic Republic, in 1990, the country was reunified. In the 21st century, Germany is a power and has the worlds fourth-largest economy by nominal GDP. As a global leader in industrial and technological sectors, it is both the worlds third-largest exporter and importer of goods. Germany is a country with a very high standard of living sustained by a skilled. It upholds a social security and universal health system, environmental protection. Germany was a member of the European Economic Community in 1957. It is part of the Schengen Area, and became a co-founder of the Eurozone in 1999, Germany is a member of the United Nations, NATO, the G8, the G20, and the OECD. The national military expenditure is the 9th highest in the world, the English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine. This in turn descends from Proto-Germanic *þiudiskaz popular, derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- people, the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a mine in Schöningen where three 380, 000-year-old wooden javelins were unearthed
68.
Switzerland
–
Switzerland, officially the Swiss Confederation, is a federal republic in Europe. It consists of 26 cantons, and the city of Bern is the seat of the federal authorities. The country is situated in western-Central Europe, and is bordered by Italy to the south, France to the west, Germany to the north, and Austria and Liechtenstein to the east. Switzerland is a country geographically divided between the Alps, the Swiss Plateau and the Jura, spanning an area of 41,285 km2. The establishment of the Old Swiss Confederacy dates to the medieval period, resulting from a series of military successes against Austria. Swiss independence from the Holy Roman Empire was formally recognized in the Peace of Westphalia in 1648. The country has a history of armed neutrality going back to the Reformation, it has not been in a state of war internationally since 1815, nevertheless, it pursues an active foreign policy and is frequently involved in peace-building processes around the world. In addition to being the birthplace of the Red Cross, Switzerland is home to international organisations. On the European level, it is a member of the European Free Trade Association. However, it participates in the Schengen Area and the European Single Market through bilateral treaties, spanning the intersection of Germanic and Romance Europe, Switzerland comprises four main linguistic and cultural regions, German, French, Italian and Romansh. Due to its diversity, Switzerland is known by a variety of native names, Schweiz, Suisse, Svizzera. On coins and stamps, Latin is used instead of the four living languages, Switzerland is one of the most developed countries in the world, with the highest nominal wealth per adult and the eighth-highest per capita gross domestic product according to the IMF. Zürich and Geneva have each been ranked among the top cities in the world in terms of quality of life, with the former ranked second globally, according to Mercer. The English name Switzerland is a compound containing Switzer, a term for the Swiss. The English adjective Swiss is a loan from French Suisse, also in use since the 16th century. The name Switzer is from the Alemannic Schwiizer, in origin an inhabitant of Schwyz and its associated territory, the Swiss began to adopt the name for themselves after the Swabian War of 1499, used alongside the term for Confederates, Eidgenossen, used since the 14th century. The data code for Switzerland, CH, is derived from Latin Confoederatio Helvetica. The toponym Schwyz itself was first attested in 972, as Old High German Suittes, ultimately related to swedan ‘to burn’
69.
Austria
–
Austria, officially the Republic of Austria, is a federal republic and a landlocked country of over 8.7 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Hungary and Slovakia to the east, Slovenia and Italy to the south, the territory of Austria covers 83,879 km2. The terrain is mountainous, lying within the Alps, only 32% of the country is below 500 m. The majority of the population speaks local Bavarian dialects of German as their native language, other local official languages are Hungarian, Burgenland Croatian, and Slovene. The origins of modern-day Austria date back to the time of the Habsburg dynasty, from the time of the Reformation, many northern German princes, resenting the authority of the Emperor, used Protestantism as a flag of rebellion. Following Napoleons defeat, Prussia emerged as Austrias chief competitor for rule of a greater Germany, Austrias defeat by Prussia at the Battle of Königgrätz, during the Austro-Prussian War of 1866, cleared the way for Prussia to assert control over the rest of Germany. In 1867, the empire was reformed into Austria-Hungary, Austria was thus the first to go to war in the July Crisis, which would ultimately escalate into World War I. The First Austrian Republic was established in 1919, in 1938 Nazi Germany annexed Austria in the Anschluss. This lasted until the end of World War II in 1945, after which Germany was occupied by the Allies, in 1955, the Austrian State Treaty re-established Austria as a sovereign state, ending the occupation. In the same year, the Austrian Parliament created the Declaration of Neutrality which declared that the Second Austrian Republic would become permanently neutral, today, Austria is a parliamentary representative democracy comprising nine federal states. The capital and largest city, with a population exceeding 1.7 million, is Vienna, other major urban areas of Austria include Graz, Linz, Salzburg and Innsbruck. Austria is one of the richest countries in the world, with a nominal per capita GDP of $43,724, the country has developed a high standard of living and in 2014 was ranked 21st in the world for its Human Development Index. Austria has been a member of the United Nations since 1955, joined the European Union in 1995, Austria also signed the Schengen Agreement in 1995, and adopted the euro currency in 1999. The German name for Austria, Österreich, meant eastern realm in Old High German, and is cognate with the word Ostarrîchi and this word is probably a translation of Medieval Latin Marchia orientalis into a local dialect. Austria was a prefecture of Bavaria created in 976, the word Austria is a Latinisation of the German name and was first recorded in the 12th century. Accordingly, Norig would essentially mean the same as Ostarrîchi and Österreich, the Celtic name was eventually Latinised to Noricum after the Romans conquered the area that encloses most of modern-day Austria, around 15 BC. Noricum later became a Roman province in the mid-first century AD, heers hypothesis is not accepted by linguists. Settled in ancient times, the Central European land that is now Austria was occupied in pre-Roman times by various Celtic tribes, the Celtic kingdom of Noricum was later claimed by the Roman Empire and made a province
70.
Portugal
–
Portugal, officially the Portuguese Republic, is a country on the Iberian Peninsula in Southwestern Europe. It is the westernmost country of mainland Europe, to the west and south it is bordered by the Atlantic Ocean and to the east and north by Spain. The Portugal–Spain border is 1,214 kilometres long and considered the longest uninterrupted border within the European Union, the republic also includes the Atlantic archipelagos of the Azores and Madeira, both autonomous regions with their own regional governments. The territory of modern Portugal has been settled, invaded. The Pre-Celts, Celts, Carthaginians and the Romans were followed by the invasions of the Visigothic, in 711 the Iberian Peninsula was invaded by the Moors, making Portugal part of Muslim Al Andalus. Portugal was born as result of the Christian Reconquista, and in 1139, Afonso Henriques was proclaimed King of Portugal, in the 15th and 16th centuries, Portugal established the first global empire, becoming one of the worlds major economic, political and military powers. Portugal monopolized the trade during this time, and the Portuguese Empire expanded with military campaigns led in Asia. After the 1910 revolution deposed the monarchy, the democratic but unstable Portuguese First Republic was established, democracy was restored after the Portuguese Colonial War and the Carnation Revolution in 1974. Shortly after, independence was granted to almost all its overseas territories, Portugal has left a profound cultural and architectural influence across the globe and a legacy of over 250 million Portuguese speakers today. Portugal is a country with a high-income advanced economy and a high living standard. It is the 5th most peaceful country in the world, maintaining a unitary semi-presidential republican form of government and it has the 18th highest Social Progress in the world, putting it ahead of other Western European countries like France, Spain and Italy. Portugal is a pioneer when it comes to drug decriminalization, as the nation decriminalized the possession of all drugs for use in 2001. The early history of Portugal is shared with the rest of the Iberian Peninsula located in South Western Europe, the name of Portugal derives from the joined Romano-Celtic name Portus Cale. Other influences include some 5th-century vestiges of Alan settlements, which were found in Alenquer, Coimbra, the region of present-day Portugal was inhabited by Neanderthals and then by Homo sapiens, who roamed the border-less region of the northern Iberian peninsula. These were subsistence societies that, although they did not establish prosperous settlements, neolithic Portugal experimented with domestication of herding animals, the raising of some cereal crops and fluvial or marine fishing. Chief among these tribes were the Calaicians or Gallaeci of Northern Portugal, the Lusitanians of central Portugal, the Celtici of Alentejo, a few small, semi-permanent, commercial coastal settlements were also founded in the Algarve region by Phoenicians-Carthaginians. Romans first invaded the Iberian Peninsula in 219 BC, during the last days of Julius Caesar, almost the entire peninsula had been annexed to the Roman Republic. The Carthaginians, Romes adversary in the Punic Wars, were expelled from their coastal colonies and it suffered a severe setback in 150 BC, when a rebellion began in the north
71.
British Armed Forces
–
They also promote Britains wider interests, support international peacekeeping efforts, and provide humanitarian aid. Repeatedly emerging victorious from conflicts has allowed Britain to establish itself as one of the leading military. The Commander-in-chief of the British Armed Forces is the British monarch, Queen Elizabeth II, the UK Parliament approves the continued existence of the armed forces by passing an Armed Forces Act at least once every five years, as required by the Bill of Rights 1689. The armed forces are managed by the Defence Council of the Ministry of Defence, with the Acts of Union 1707, the armed forces of England and Scotland were merged into the armed forces of the Kingdom of Great Britain. Britain feared that Russian expansionism in the region would eventually threaten the Empire in India and this ultimately led to British involvement in the Crimean War against the Russian Empire. The beginning of the twentieth century served to reduce tensions between Britain and the Russian Empire, partly due to the emergence of a unified German Empire. Allied victory resulted in the defeat of the Central Powers, the end of the German Empire, the Treaty of Versailles, once again tensions accumulated in European relations, and following Germanys invasion of Poland in September 1939, the Second World War began. The conflict was the most widespread in British history, with British Empire and Commonwealth troops fighting in campaigns from Europe and North Africa, to the Middle East, approximately 390,000 British Empire and Commonwealth troops lost their lives. Allied victory resulted in the defeat of the Axis powers and the establishment of the United Nations, reflecting Britains new role in the world and the escalation of the Cold War, the country became a founding member of the NATO military alliance in 1949. By the mid-1970s, the forces had reconfigured to focus on the responsibilities allocated to them by NATO. While NATO obligations took increased prominence, Britain nonetheless found itself engaged in a number of low-intensity conflicts, however the Dhofar Rebellion and The Troubles emerged as the primary operational concerns of the armed forces. Perhaps the most important conflict during the Cold War, at least in the context of British defence policy, was the Falklands War. Since the end of the Cold War, an international role for the armed forces has been pursued, with re-structuring to deliver a greater focus on expeditionary warfare. In addition to the campaign, the British Army has trained and supplied allies on the ground. Figures released by the Ministry of Defence on 31 March 2016 show that 7,185 British Armed Forces personnel have lost their lives in medal earning theatres since the end of the Second World War. As Sovereign and head of state, Queen Elizabeth II is Head of the Armed Forces, the Queen, however, remains the ultimate authority of the military, with officers and personnel swearing allegiance to the monarch. It has been claimed that this includes the power to prevent unconstitutional use of the armed forces, responsibility for the management of the forces is delegated to a number of committees, the Defence Council, Chiefs of Staff Committee, Defence Management Board and three single-service boards. The Defence Council, composed of representatives of the services
72.
United Kingdom
–
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom or Britain, is a sovereign country in western Europe. Lying off the north-western coast of the European mainland, the United Kingdom includes the island of Great Britain, Northern Ireland is the only part of the United Kingdom that shares a land border with another sovereign state—the Republic of Ireland. The Irish Sea lies between Great Britain and Ireland, with an area of 242,500 square kilometres, the United Kingdom is the 78th-largest sovereign state in the world and the 11th-largest in Europe. It is also the 21st-most populous country, with an estimated 65.1 million inhabitants, together, this makes it the fourth-most densely populated country in the European Union. The United Kingdom is a monarchy with a parliamentary system of governance. The monarch is Queen Elizabeth II, who has reigned since 6 February 1952, other major urban areas in the United Kingdom include the regions of Birmingham, Leeds, Glasgow, Liverpool and Manchester. The United Kingdom consists of four countries—England, Scotland, Wales, the last three have devolved administrations, each with varying powers, based in their capitals, Edinburgh, Cardiff and Belfast, respectively. The relationships among the countries of the UK have changed over time, Wales was annexed by the Kingdom of England under the Laws in Wales Acts 1535 and 1542. A treaty between England and Scotland resulted in 1707 in a unified Kingdom of Great Britain, which merged in 1801 with the Kingdom of Ireland to form the United Kingdom of Great Britain and Ireland. Five-sixths of Ireland seceded from the UK in 1922, leaving the present formulation of the United Kingdom of Great Britain, there are fourteen British Overseas Territories. These are the remnants of the British Empire which, at its height in the 1920s, British influence can be observed in the language, culture and legal systems of many of its former colonies. The United Kingdom is a country and has the worlds fifth-largest economy by nominal GDP. The UK is considered to have an economy and is categorised as very high in the Human Development Index. It was the worlds first industrialised country and the worlds foremost power during the 19th, the UK remains a great power with considerable economic, cultural, military, scientific and political influence internationally. It is a nuclear weapons state and its military expenditure ranks fourth or fifth in the world. The UK has been a permanent member of the United Nations Security Council since its first session in 1946 and it has been a leading member state of the EU and its predecessor, the European Economic Community, since 1973. However, on 23 June 2016, a referendum on the UKs membership of the EU resulted in a decision to leave. The Acts of Union 1800 united the Kingdom of Great Britain, Scotland, Wales and Northern Ireland have devolved self-government
73.
Argentina
–
Argentina, officially the Argentine Republic, is a federal republic in the southern half of South America. With a mainland area of 2,780,400 km2, Argentina is the eighth-largest country in the world, the second largest in Latin America, and the largest Spanish-speaking one. The country is subdivided into provinces and one autonomous city, Buenos Aires. The provinces and the capital have their own constitutions, but exist under a federal system, Argentina claims sovereignty over part of Antarctica, the Falkland Islands, and South Georgia and the South Sandwich Islands. The earliest recorded presence in the area of modern-day Argentina dates back to the Paleolithic period. The country has its roots in Spanish colonization of the region during the 16th century, Argentina rose as the successor state of the Viceroyalty of the Río de la Plata, a Spanish overseas viceroyalty founded in 1776. The country thereafter enjoyed relative peace and stability, with waves of European immigration radically reshaping its cultural. The almost-unparalleled increase in prosperity led to Argentina becoming the seventh wealthiest developed nation in the world by the early 20th century, Argentina retains its historic status as a middle power in international affairs, and is a prominent regional power in the Southern Cone and Latin America. Argentina has the second largest economy in South America, the third-largest in Latin America and is a member of the G-15 and it is the country with the second highest Human Development Index in Latin America with a rating of very high. Because of its stability, market size and growing high-tech sector, the description of the country by the word Argentina has to be found on a Venice map in 1536. In English the name Argentina probably comes from the Spanish language, however the naming itself is not Spanish, Argentina means in Italian of silver, silver coloured, probably borrowed from the Old French adjective argentine of silver > silver coloured already mentioned in the 12th century. The French word argentine is the form of argentin and derives of argent silver with the suffix -in. The Italian naming Argentina for the country implies Argentina Terra land of silver or Argentina costa coast of silver, in Italian, the adjective or the proper noun is often used in an autonomous way as a substantive and replaces it and it is said lArgentina. The name Argentina was probably first given by the Venitian and Genoese navigators, in Spanish and Portuguese, the words for silver are respectively plata and prata and of silver is said plateado and prateado. Argentina was first associated with the silver mountains legend, widespread among the first European explorers of the La Plata Basin. The first written use of the name in Spanish can be traced to La Argentina, a 1602 poem by Martín del Barco Centenera describing the region, the 1826 constitution included the first use of the name Argentine Republic in legal documents. The name Argentine Confederation was also used and was formalized in the Argentine Constitution of 1853. In 1860 a presidential decree settled the name as Argentine Republic