1.
21 (number)
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21 is the natural number following 20 and preceding 22. In a 24-hour clock, the twenty-first hour is in conventional language called nine or nine oclock,21 is, the fifth discrete semiprime and the second in the family. With 22 it forms the second discrete semiprime pair, a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes. A composite number, its divisors being 1,3 and 7. The sum of the first six numbers, making it a triangular number. The sum of the sum of the divisors of the first 5 positive integers, the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number. The smallest natural number that is not close to a power of 2, 2n,21 has an aliquot sum of 11 though it is the second composite number found in the 11-aliquot tree with the abundant square prime 18 being the first such member. Twenty-one is the first number to be the sum of three numbers 18,51,91. 21 appears in the Padovan sequence, preceded by the terms 9,12,16, in several countries 21 is the age of majority. In most US states,21 is the drinking age, however, in Puerto Rico and U. S. Virgin Island, the drinking age is 18. In Hawaii and New York,21 is the age that one person may purchase cigarettes. In some countries it is the voting age, in the United States,21 is the age at which one can purchase multiple tickets to an R-rated film without providing Identifications. It is also the age to one under the age of 17 as their parent or adult guardian for an R-rated movie. In some states,21 is the age, persons may gamble or enter casinos. In 2011, Adele named her second studio album 21, because of her age at the time, the Milwaukee Braves, for Hall of Famer Warren Spahn, the number continues to be honored by the team in its current home of Atlanta. The Pittsburgh Pirates, for Hall of Famer Roberto Clemente, following his death in a crash while attempting to deliver humanitarian aid to victims of an earthquake in Nicaragua. In the NBA, The Atlanta Hawks, for Hall of Famer Dominique Wilkins, the Boston Celtics, for Hall of Famer Bill Sharman. The Detroit Pistons, for Hall of Famer Dave Bing, the Sacramento Kings, for Vlade Divac
2.
19 (number)
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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
3.
20 (number)
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20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants
4.
24 (number)
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24 is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta, and for 10−24 yocto and these numbers are the largest and smallest number to receive an SI prefix to date. In a 24-hour clock, the hour is in conventional language called twelve or twelve oclock. 24 is the factorial of 4 and a number, being the first number of the form 23q. It follows that 24 is the number of ways to order 4 distinct items and it is the smallest number with exactly eight divisors,1,2,3,4,6,8,12, and 24. It is a composite number, having more divisors than any smaller number. 24 is a number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. Subtracting 1 from any of its divisors yields a number,24 is the largest number with this property. 24 has a sum of 36 and the aliquot sequence. It is therefore the lowest abundant number whose aliquot sum is itself abundant, the aliquot sum of only one number,529 =232, is 24. There are 10 solutions to the equation φ =24, namely 35,39,45,52,56,70,72,78,84 and 90 and this is more than any integer below 24, making 24 a highly totient number. 24 is the sum of the prime twins 11 and 13, the product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two numbers, one of which is a multiple of four, and there must be a multiple of three. The tesseract has 24 two-dimensional faces,24 is the only nontrivial solution to the cannonball problem, that is,12 +22 +32 + … +242 is a perfect square. In 24 dimensions there are 24 even positive definite unimodular lattices, the Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S and the Mathieu group M24. The modular discriminant Δ is proportional to the 24th power of the Dedekind eta function η, Δ = 12η24, the Barnes-Wall lattice contains 24 lattices. 24 is the number whose divisors — namely 1,2,3,4,6,8,12,24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group × = is isomorphic to the additive group 3 and this fact plays a role in monstrous moonshine
5.
25 (number)
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25 is the natural number following 24 and preceding 26. It is a number, being 52 =5 ×5. It is one of two numbers whose square and higher powers of the number also ends in the same last two digits, e. g.252 =625, the other is 76. It is the smallest square that is also a sum of two squares,25 =32 +42, hence it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the odd natural numbers 1,3,5,7 and 9. 25 is an octagonal number, a centered square number. 25 percent is equal to 1/4,25 has an aliquot sum of 6 and number 6 is the first number to have an aliquot sequence that does not culminate in 0 through a prime. 25 is the sum of three integers,95,119, and 143. Twenty-five is the second member of the 6-aliquot tree. It is the smallest base 10 Friedman number as it can be expressed by its own digits,52 and it is also a Cullen number. 25 is the smallest pseudoprime satisfying the congruence 7n =7 mod n.25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate. Within base 10 one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00,25,50 or 75. 25 and 49 are the perfect squares in the following list,13,25,37,49,511,613,715,817,919,1021,1123,1225,1327,1429. The formula in this list can be described as 10nZ + where n depends on the number of digits in Z, in base 30,25 is a 1-automorphic number, and in base 10 a 2-automorphic number. The percent DNA overlap of a half-sibling, grandparent, grandchild, aunt, uncle, niece, nephew, identical twin cousin, in Ezekiels vision of a new temple, The number twenty-five is of cardinal importance in Ezekiels Temple Vision. In The Book of Revelations New International Version, Surrounding the throne were twenty-four other thrones and they were dressed in white and had crowns of gold on their heads. In Islam, there are 25 prophets mentioned in the Quran, the size of the full roster on a Major League Baseball team for most of the season, except for regular-season games on or after September 1, when teams expand their roster to 40 players. The size of the roster on a Nippon Professional Baseball team for a particular game
6.
30 (number)
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30 is the natural number following 29 and preceding 31. Thirty is the sum of the first four squares, which makes it a square pyramidal number and it is a primorial and is the smallest Giuga number. 30 is the smallest sphenic number, and the smallest of the form 2 ×3 × r,30 has an aliquot sum of 42, the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises Thirty has but one number for which it is the aliquot sum, adding up some subsets of its divisors gives 30, hence 30 is a semiperfect number. 30 is the largest number such that all smaller than itself. A polygon with thirty sides is called a triacontagon, the icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a graph with 30 vertices. The atomic number of zinc is 30 Messier object M30, a magnitude 8, the duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series began on June 19,1803 BC. The duration of Saros series 30 was 1316.2 years, Thirty is, Used to indicate the end of a newspaper story, a copy editors typographical notation. S. Judas Iscariot betrayed Jesus for 30 pieces of silver, one of the rallying-cries of the 1960s student/youth protest movement was the slogan, Dont trust anyone over thirty. In Franz Kafkas novel The Trial Joseph wakes up on the morning of his birthday to find himself under arrest for an unspecified crime. After making many attempts to find the nature of the crime or the name of his accuser. The number of uprights that formed the Sarsen Circle at Stonehenge, western Christianitys most prolific 20th century essayist, F. W. Also in that essay Boreham writes It was said of Keats, in tennis, the number 30 represents the second point gained in a game. Under NCAA rules for basketball, the offensive team has 30 seconds to attempt a shot. As of 2012, three of the four major leagues in the United States and Canada have 30 teams each. The California Angels baseball team retired the number in honor of its most notable wearer, Nolan Ryan, the San Francisco Giants extended the same honor to Orlando Cepeda
7.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
8.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
9.
40 (number)
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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
10.
60 (number)
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60 is the natural number following 59 and preceding 61. Being three times 20, it is called three score in older literature. It is a number, with divisors 1,2,3,4,5,6,10,12,15,20,30. Because it is the sum of its divisors, it is a unitary perfect number. Being ten times a number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6 and it is the smallest number with exactly 12 divisors. It is the sum of a pair of twin primes and the sum of four consecutive primes and it is adjacent to two primes. It is the smallest number that is the sum of two odd primes in six ways, the smallest non-solvable group has order 60. There are four Archimedean solids with 60 vertices, the icosahedron, the rhombicosidodecahedron, the snub dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs, there are also two Archimedean solids with 60 edges, the snub cube and the icosidodecahedron. The skeleton of the forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares, in geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, a number system with base 60 is called sexagesimal. It is the smallest positive integer that is written only the smallest. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule and this ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 is an isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz
11.
80 (number)
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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
12.
90 (number)
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90 is the natural number preceded by 89 and followed by 91. In English speech, the numbers 90 and 19 are often confused, when carefully enunciated, they differ in which syllable is stressed,19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in such as 1999, and when contrasting numbers in the teens and when counting, such as 17,18,19. 90 is, a perfect number because it is the sum of its unitary divisors. A semiperfect number because it is equal to the sum of a subset of its divisors, a Perrin number, preceded in the sequence by 39,51,68. Palindromic and a repdigit in bases 14,17,29, a Harshad number since 90 is divisible by the sum of its base 10 digits. In normal space, the angles of a rectangle measure 90 degrees each. Also, in a triangle, the angle opposing the hypotenuse measures 90 degrees. Thus, an angle measuring 90 degrees is called a right angle, ninety is, the atomic number of thorium, an actinide. As an atomic weight,90 identifies an isotope of strontium, the latitude in degrees of the North and the South geographical poles. NFL, New York Jets Dennis Byrds #90 is retired +90 is the code for international direct dial phone calls to Turkey,90 is the code for the French département Belfort
13.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
14.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
15.
Divisor
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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
16.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
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Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
18.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
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Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
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Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
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Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
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Senary
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
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Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
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Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
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Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
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Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
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Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
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Deficient number
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In number theory, a deficient or deficient number is a number n for which the sum of divisors σ<2n, or, equivalently, the sum of proper divisors s<n. The value 2n − σ is called the numbers deficiency, as an example, consider the number 21. Its proper divisors are 1,3 and 7, and their sum is 11, because 11 is less than 21, the number 21 is deficient. Its deficiency is 2 ×21 −32 =10, since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. An infinite number of even and odd deficient numbers exist. All odd numbers with one or two prime factors are deficient. All proper divisors of deficient or perfect numbers are deficient, there exists at least one deficient number in the interval for all sufficiently large n. Closely related to deficient numbers are perfect numbers with σ = 2n, the natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica. Almost perfect number Amicable number Sociable number Sándor, József, Mitrinović, Dragoslav S. Crstici, Borislav, the Prime Glossary, Deficient number Weisstein, Eric W. Deficient Number
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Composite number
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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
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Semiprime
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In mathematics, a semiprime is a natural number that is the product of two prime numbers. The semiprimes less than 100 are 4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94, and 95. Semiprimes that are not perfect squares are called discrete, or distinct, by definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its factors are 1,2,13. The total number of prime factors Ω for a n is two, by definition. A semiprime is either a square of a prime or square-free, the square of any prime number is a semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a number is a semiprime without knowing the two factors. A composite n non-divisible by primes ≤ n 3 is semiprime, various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits. These are considered novelties, since their construction method might prove vulnerable to factorization, for a semiprime n = pq the value of Eulers totient function is particularly simple when p and q are distinct, φ = = p q − +1 = n − +1. If otherwise p and q are the same, φ = φ = p = p2 − p = n − p and these methods rely on the fact that finding two large primes and multiplying them together is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of specific large semiprimes, the most recent such challenge closed in 2007. In practical cryptography, it is not sufficient to choose just any semiprime, the factors p and q of n should both be very large, around the same order of magnitude as the square root of n, this makes trial division and Pollards rho algorithm impractical. At the same time they should not be too close together, or else the number can be quickly factored by Fermats factorization method. The number may also be chosen so that none of p −1, p +1, q −1, or q +1 are smooth numbers, protecting against Pollards p −1 algorithm or Williams p +1 algorithm. However, these checks cannot take future algorithms or secret algorithms into account, in 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. It consisted of 1679 binary digits intended to be interpreted as a 23×73 bitmap image, the number 1679 = 23×73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns. Chens theorem Weisstein, Eric W. Semiprime
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1 (number)
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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
32.
2 (number)
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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
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Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
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Line segment
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In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
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Lazy caterer's sequence
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The lazy caterers sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a circle that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a point inside the circle. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines, for generalizations to higher dimensions, the analogue of this sequence in 3 dimensions is the cake number. The maximum number p of pieces that can be created with a number of cuts n. Using binomial coefficients, the formula can be expressed as p =1 + = + +. This sequence, starting with n =0, results in 1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211. Each number equals 1 plus a triangular number, to obtain the maximum number of pieces, the nth cut line should cross all the other previous cut lines inside the circle, but not cross any intersection of previous cut lines. Thus, the nth line itself is cut in n −1 places, each segment divides one piece of the -cut pancake into 2 parts, adding exactly n to the number of pieces. The new line cant have any more segments since it can cross each previous line once. Thus, the number of pieces after n cuts is f = n + f. This recurrence relation can be solved, if ƒ is expanded one term the relation becomes f = n + + f. Expansion of the term ƒ can continue until the last term is reduced to ƒ, thus, since f =1, because there is one piece before any cuts are made, this can be rewritten as f =1 +. This can be simplified, using the formula for the sum of an arithmetic progression, using Eulers formula to solve plane separation problems, The College Mathematics Journal, Mathematical Association of America,22, 125–130, doi,10. 2307/2686448, JSTOR2686448. Steiner, J. Einige Gesetze über die Theilung der Ebene und des Raumes, on the division of the plane by lines, American Mathematical Monthly, Mathematical Association of America,85, 647–656, doi,10. 2307/2320333, JSTOR2320333. Weisstein, Eric W. Circle Division by Lines