1.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
2.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
3.
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
100 (number)
–
The
U.S. hundred-dollar bill, Series 2009.
4.
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
Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.
5.
Prime number
–
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime number
–
The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
6.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek numerals
–
Numeral systems
Greek numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
7.
Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
–
Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
–
Numeral systems
Roman numerals
–
A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
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An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
8.
Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
9.
Ternary numeral system
–
The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
Ternary numeral system
–
Numeral systems
10.
Quaternary numeral system
–
Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
Quaternary numeral system
–
Numeral systems
11.
Quinary
–
Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
Quinary
–
Numeral systems
12.
Senary
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Senary
–
Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting
Senary
13.
Octal
–
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
Octal
–
Numeral systems
14.
Duodecimal
–
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
Duodecimal
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Numeral systems
Duodecimal
–
A duodecimal multiplication table
15.
Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
Hexadecimal
–
Numeral systems
Hexadecimal
–
Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
–
Hexadecimal finger-counting scheme.
16.
Vigesimal
–
The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
Vigesimal
–
Numeral systems
Vigesimal
–
The
Maya numerals are a base-20 system.
17.
Base 36
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Base 36
–
Numeral systems
Base 36
–
34 senary = 22 decimal, in senary finger counting
Base 36
18.
Natural number
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Natural number
–
The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
19.
Eisenstein prime
–
In mathematics, an Eisenstein prime is an Eisenstein integer z = a + b ω that is irreducible in the ring-theoretic sense, its only Eisenstein divisors are the units, a + bω itself and its associates. The associates and the conjugate of any Eisenstein prime are also prime. It follows that the absolute value squared of every Eisenstein prime is a prime or the square of a natural prime. The first few Eisenstein primes that equal a natural prime 3n −1 are,2,5,11,17,23,29,41,47,53,59,71,83,89,101. Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes, some non-real Eisenstein primes are 2 + ω,3 + ω,4 + ω,5 + 2ω,6 + ω,7 + ω,7 + 3ω. Up to conjugacy and unit multiples, the primes listed above, as of March 2017, the largest known Eisenstein prime is the seventh largest known prime 10223 ×231172165 +1, discovered by Péter Szabolcs and PrimeGrid. All larger known primes are Mersenne primes, discovered by GIMPS, real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes are congruent to 1 mod 3, thus no Mersenne prime is an Eisenstein prime
Eisenstein prime
–
Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3 n − 1. All others have an absolute value squared equal to a natural prime.
20.
MIT AI Lab
–
Housed within the Stata Center, CSAIL is the largest on-campus laboratory as measured by research scope and membership. CSAILs research activities are organized around a number of research groups. Research at MIT in the field of artificial intelligence began in 1959, on July 1,1963, Project MAC was launched with a $2 million grant from the Defense Advanced Research Projects Agency. Project MACs original director was Robert Fano of MITs Research Laboratory of Electronics, the program manager responsible for the DARPA grant was J. C. R. Licklider, who had previously been at MIT conducting research in RLE, Project MAC would become famous for groundbreaking research in operating systems, artificial intelligence, and the theory of computation. Its contemporaries included Project Genie at Berkeley, the Stanford Artificial Intelligence Laboratory, an AI Group including Marvin Minsky, John McCarthy and a talented community of computer programmers was incorporated into the newly formed Project MAC. It was interested principally in the problems of vision, mechanical motion and manipulation, and language, in the 1960s - 1970s the AI Group shared a computer room with a computer for which they built a time-sharing operating system called ITS. These founders envisioned the creation of a utility whose computational power would be as reliable as an electric utility. To this end, Corbató brought the first computer time-sharing system, CTSS, with him from the MIT Computation Center, in 1966, Scientific American featured Project MAC in the September thematic issue devoted to computer science, that was later published in book form. At the time, the system was described as having approximately 100 TTY terminals, mostly on campus, only 30 users could be logged in at the same time. In the late 1960s, Minskys artificial intelligence group was seeking more space, talented programmers such as Richard Stallman, who used TECO to write EMACS, flourished in the AI Lab during this time. Two professors, Hal Abelson and Gerald Jay Sussman, chose to remain neutral – their group was referred to variously as Switzerland, the AI Lab led to the invention of Lisp machines and their attempted commercialization by two companies in the 1980s, Symbolics and Lisp Machines Inc. This divided the AI lab into camps and resulted in a hiring away of many employees, the experience was influential on Stallmans later work on the GNU project. Nobody had envisioned that the AI labs hacker group would be wiped out, on the fortieth anniversary of Project MACs establishment, July 1,2003, LCS re-merged with the AI Lab to form the MIT Computer Science and Artificial Intelligence Laboratory, or CSAIL. This merger created the largest laboratory on the MIT campus and was regarded as a reuniting of the elements of Project MAC. The IMARA group sponsors a variety of programs which bridge the Global Digital Divide. Its aim is to find and implement long-term, sustainable solutions which increase the availability of educational technology and resources to domestic. Rus, 2012– A Marriage of Convenience, The Founding of the MIT Artificial Intelligence Laboratory, Chious et al. - includes important information on the Incompatible Timesharing System Weizenbaum
MIT AI Lab
MIT AI Lab
–
MIT Computer Science and Artificial Intelligence Laboratory
21.
Square number
–
In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
Square number
–
m = 1 2 = 1
22.
Cube (arithmetic)
–
In arithmetic and algebra, the cube of a number n is its third power, the result of the number multiplied by itself twice, n3 = n × n × n. It is also the number multiplied by its square, n3 = n × n2 and this is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n and it determines the side of the cube of a given volume. It is also n raised to the one-third power, both cube and cube root are odd functions,3 = −. The cube of a number or any other mathematical expression is denoted by a superscript 3, a cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The perfect cubes up to 603 are, Geometrically speaking, an integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger. For example,27 small cubes can be arranged into one larger one with the appearance of a Rubiks Cube, the difference between the cubes of consecutive integers can be expressed as follows, n3 −3 = 3n +1. There is no minimum perfect cube, since the cube of an integer is negative. For example, −4 × −4 × −4 = −64, unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25,75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6, some cube numbers are also square numbers, for example,64 is a square number and a cube number. This happens if and only if the number is a perfect sixth power, the last digits of each 3rd power are, It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1,8 or 9. That is their values modulo 9 may be only −1,1 and 0, every positive integer can be written as the sum of nine positive cubes. The equation x3 + y3 = z3 has no solutions in integers. In fact, it has none in Eisenstein integers, both of these statements are also true for the equation x3 + y3 = 3z3. The sum of the first n cubes is the nth triangle number squared,13 +23 + ⋯ + n 3 =2 =2. Proofs Charles Wheatstone gives a simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. Indeed, he begins by giving the identity n 3 = + + + ⋯ + ⏟ n consecutive odd numbers, kanim provides a purely visual proof, Benjamin & Orrison provide two additional proofs, and Nelsen gives seven geometric proofs
Cube (arithmetic)
–
y = x 3 for values of 0 ≤ x ≤ 25.
23.
Square root of 2
–
The square root of 2, or the th power of 2, written in mathematics as √2 or 2 1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational, the rational approximation of the square root of two,665, 857/470,832, derived from the fourth step in the Babylonian algorithm starting with a0 =1, is too large by approx. 1. 6×10−12, its square is 2. 0000000000045… The rational approximation 99/70 is frequently used, despite having a denominator of only 70, it differs from the correct value by less than 1/10,000. The numerical value for the root of two, truncated to 65 decimal places, is,1. 41421356237309504880168872420969807856967187537694807317667973799….41421296 ¯. That is,1 +13 +13 ×4 −13 ×4 ×34 =577408 =1.4142156862745098039 ¯. This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as a secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras number or Pythagoras constant, for example by Conway & Guy, there are a number of algorithms for approximating √2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots. It goes as follows, First, pick a guess, a0 >0, then, using that guess, iterate through the following recursive computation, a n +1 = a n +2 a n 2 = a n 2 +1 a n. The more iterations through the algorithm, the approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits, starting with a0 =1 the next approximations are 3/2 =1.5 17/12 =1.416. The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanadas team in 1997, in February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010, for a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely, such computations aim to check empirically whether such numbers are normal
Square root of 2
–
Babylonian clay tablet YBC 7289 with annotations. Besides showing the square root of 2 in
sexagesimal (1 24 51 10), the tablet also gives an example where one side of the square is 30 and the diagonal then is 42 25 35. The sexagesimal digit 30 can also stand for 1/2, in which case 42 25 35 is approximately 0.7071065.
Square root of 2
–
The square root of 2 is equal to the length of the
hypotenuse of a
right triangle with legs of length 1.
24.
Pi
–
The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
Pi
–
The constant π is represented in this
mosaic outside the Mathematics Building at the
Technical University of Berlin.
Pi
–
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
–
Archimedes developed the polygonal approach to approximating π.
Pi
–
Isaac Newton used
infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
25.
Radian
–
The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
Radian
–
A chart to convert between degrees and radians
Radian
–
An arc of a
circle with the same length as the
radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to an angle of 2
π radians.
26.
Degree (angle)
–
A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
Degree (angle)
–
One degree (shown in red) and eighty nine (shown in blue)
27.
Binary numeral system
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary numeral system
–
Numeral systems
Binary numeral system
–
Gottfried Leibniz
Binary numeral system
–
George Boole
28.
Factorial
–
In mathematics, the factorial of a non-negative integer n, denoted by n. is the product of all positive integers less than or equal to n. =5 ×4 ×3 ×2 ×1 =120, the value of 0. is 1, according to the convention for an empty product. The factorial operation is encountered in areas of mathematics, notably in combinatorics, algebra. Its most basic occurrence is the fact there are n. ways to arrange n distinct objects into a sequence. This fact was known at least as early as the 12th century, fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial, Now the nature of these methods is such, the factorial function is formally defined by the product n. = ∏ k =1 n k, or by the relation n. = {1 if n =0. The factorial function can also be defined by using the rule as n. All of the above definitions incorporate the instance 0, =1, in the first case by the convention that the product of no numbers at all is 1. This is convenient because, There is exactly one permutation of zero objects, = n. ×, valid for n >0, extends to n =0. It allows for the expression of many formulae, such as the function, as a power series. It makes many identities in combinatorics valid for all applicable sizes, the number of ways to choose 0 elements from the empty set is =0. More generally, the number of ways to choose n elements among a set of n is = n. n, the factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica, although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. There are n. different ways of arranging n distinct objects into a sequence, often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations from a set with n elements, one can obtain such a combination by choosing a k-permutation, successively selecting and removing an element of the set, k times, for a total of n k _ = n ⋯ possibilities. This however produces the k-combinations in an order that one wishes to ignore, since each k-combination is obtained in k. different ways. This number is known as the coefficient, because it is also the coefficient of Xk in n
Factorial
–
Amplitude and phase of factorial of complex argument
Factorial
–
Plot of the natural logarithm of the factorial
29.
Wolfgang Amadeus Mozart
–
Wolfgang Amadeus Mozart, baptised as Johannes Chrysostomus Wolfgangus Theophilus Mozart, was a prolific and influential composer of the Classical era. Born in Salzburg, he showed prodigious ability from his earliest childhood, already competent on keyboard and violin, he composed from the age of five and performed before European royalty. At 17, Mozart was engaged as a musician at the Salzburg court, while visiting Vienna in 1781, he was dismissed from his Salzburg position. He chose to stay in the capital, where he achieved fame, during his final years in Vienna, he composed many of his best-known symphonies, concertos, and operas, and portions of the Requiem, which was largely unfinished at the time of his death. The circumstances of his death have been much mythologized. He was survived by his wife Constanze and two sons and he composed more than 600 works, many acknowledged as pinnacles of symphonic, concertante, chamber, operatic, and choral music. He is among the most enduringly popular of classical composers, Ludwig van Beethoven composed his own early works in the shadow of Mozart, and Joseph Haydn wrote, posterity will not see such a talent again in 100 years. Wolfgang Amadeus Mozart was born on 27 January 1756 to Leopold Mozart and Anna Maria, née Pertl and this was the capital of the Archbishopric of Salzburg, an ecclesiastic principality in what is now Austria, then part of the Holy Roman Empire. He was the youngest of seven children, five of whom died in infancy and his elder sister was Maria Anna Mozart, nicknamed Nannerl. Mozart was baptized the day after his birth, at St. Ruperts Cathedral in Salzburg, the baptismal record gives his name in Latinized form, as Joannes Chrysostomus Wolfgangus Theophilus Mozart. He generally called himself Wolfgang Amadè Mozart as an adult, Leopold Mozart, a native of Augsburg, Germany, was a minor composer and an experienced teacher. In 1743, he was appointed as fourth violinist in the establishment of Count Leopold Anton von Firmian. Four years later, he married Anna Maria in Salzburg, Leopold became the orchestras deputy Kapellmeister in 1763. During the year of his sons birth, Leopold published a textbook, Versuch einer gründlichen Violinschule. When Nannerl was 7, she began lessons with her father. Years later, after her brothers death, she reminisced, He often spent much time at the clavier, picking out thirds, which he was ever striking, and his pleasure showed that it sounded good. In the fourth year of his age his father, for a game as it were, began to teach him a few minuets and he could play it faultlessly and with the greatest delicacy, and keeping exactly in time. At the age of five, he was composing little pieces
Wolfgang Amadeus Mozart
–
Mozart c. 1780, detail from portrait by
Johann Nepomuk della Croce
Wolfgang Amadeus Mozart
–
Anonymous portrait of the child Mozart, possibly by
Pietro Antonio Lorenzoni; painted in 1763 on commission from Leopold Mozart
Wolfgang Amadeus Mozart
–
Mozart's birthplace at Getreidegasse 9, Salzburg
Wolfgang Amadeus Mozart
–
The Mozart family on tour: Leopold, Wolfgang, and Nannerl. Watercolor by
Carmontelle, ca. 1763
30.
Orchestra
–
The term orchestra derives from the Greek ὀρχήστρα, the name for the area in front of a stage in ancient Greek theatre reserved for the Greek chorus. A full-size orchestra may sometimes be called an orchestra or philharmonic orchestra. The actual number of employed in a given performance may vary from seventy to over one hundred musicians, depending on the work being played. The term chamber orchestra usually refers to smaller-sized ensembles of about fifty musicians or fewer, the typical orchestra grew in size throughout the 18th and 19th centuries, reaching a peak with the large orchestras called for in the works of Richard Wagner, and later, Gustav Mahler. Orchestras are usually led by a conductor who directs the performance with movements of the hands and arms, the conductor unifies the orchestra, sets the tempo and shapes the sound of the ensemble. The first violin, commonly called the concertmaster, also plays an important role in leading the musicians, the typical symphony orchestra consists of four groups of related musical instruments called the woodwinds, brass, percussion, and strings. The orchestra, depending on the size, contains almost all of the instruments in each group. Chamber orchestra usually refers to smaller-sized ensembles, a chamber orchestra might employ as many as fifty musicians. The term concert orchestra may also be used, as in the BBC Concert Orchestra, the so-called standard complement of doubled winds and brass in the orchestra from the first half of the 19th century is generally attributed to the forces called for by Beethoven. The composers instrumentation almost always included paired flutes, oboes, clarinets, bassoons, horns, the exceptions to this are his Symphony No. 4, Violin Concerto, and Piano Concerto No,4, which each specify a single flute. Beethoven carefully calculated the expansion of this particular timbral palette in Symphonies 3,5,6, the third horn in the Eroica Symphony arrives to provide not only some harmonic flexibility, but also the effect of choral brass in the Trio movement. Piccolo, contrabassoon, and trombones add to the finale of his Symphony No.5. A piccolo and a pair of trombones help deliver the effect of storm and sunshine in the Sixth, for several decades after his death, symphonic instrumentation was faithful to Beethovens well-established model, with few exceptions. Apart from the core orchestral complement, various instruments are called for occasionally. These include the guitar, heckelphone, flugelhorn, cornet, harpsichord. Saxophones, for example, appear in some 19th- through 21st-century scores.6 and 9 and William Waltons Belshazzars Feast, and many other works as a member of the orchestral ensemble. The euphonium is featured in a few late Romantic and 20th-century works, usually playing parts marked tenor tuba, including Gustav Holsts The Planets, cornets appear in Pyotr Ilyich Tchaikovskys ballet Swan Lake, Claude Debussys La Mer, and several orchestral works by Hector Berlioz
Orchestra
–
The Jalisco Philharmonic Orchestra.
Orchestra
–
Dublin Philharmonic Orchestra
Orchestra
–
Göttinger Symphonie Orchester
Orchestra
–
A modern orchestra concert hall: Philharmony in
Szczecin, Poland
31.
Plutonium
–
Plutonium is a transuranic radioactive chemical element with symbol Pu and atomic number 94. It is a metal of silvery-gray appearance that tarnishes when exposed to air. The element normally exhibits six allotropes and four oxidation states and it reacts with carbon, halogens, nitrogen, silicon and hydrogen. When exposed to moist air, it forms oxides and hydrides that can expand the sample up to 70% in volume and it is radioactive and can accumulate in bones, which makes the handling of plutonium dangerous. Plutonium was first produced and isolated on December 14,1940 by Dr. Glenn T. Seaborg, Joseph W. Kennedy, Edwin M. McMillan, wahl by deuteron bombardment of uranium-238 in the 60-inch cyclotron at the University of California, Berkeley. They first synthesized neptunium-238 which subsequently beta-decayed to form a new element with atomic number 94. Uranium had been named after the planet Uranus and neptunium after the planet Neptune, and so element 94 was named after Pluto, wartime secrecy prevented them from announcing the discovery until 1948. Plutonium is the heaviest element to occur in nature as trace quantities arising similarly from the capture of natural uranium-238. Both plutonium-239 and plutonium-241 are fissile, meaning that they can sustain a chain reaction, leading to applications in nuclear weapons. Plutonium-240 exhibits a high rate of fission, raising the neutron flux of any sample containing it. The presence of plutonium-240 limits a plutonium samples usability for weapons or its quality as reactor fuel, plutonium-238 has a half-life of 88 years and emits alpha particles. It is a source in radioisotope thermoelectric generators, which are used to power some spacecraft. Plutonium isotopes are expensive and inconvenient to separate, so particular isotopes are usually manufactured in specialized reactors, producing plutonium in useful quantities for the first time was a major part of the Manhattan Project during World War II that developed the first atomic bombs. The Fat Man bombs used in the Trinity nuclear test in July 1945, Human radiation experiments studying plutonium were conducted without informed consent, and several criticality accidents, some lethal, occurred after the war. Disposal of plutonium waste from power plants and dismantled nuclear weapons built during the Cold War is a nuclear-proliferation. Other sources of plutonium in the environment are fallout from numerous above-ground nuclear tests, Plutonium, like most metals, has a bright silvery appearance at first, much like nickel, but it oxidizes very quickly to a dull gray, although yellow and olive green are also reported. At room temperature plutonium is in its α form and this, the most common structural form of the element, is about as hard and brittle as gray cast iron unless it is alloyed with other metals to make it soft and ductile. Unlike most metals, it is not a conductor of heat or electricity
Plutonium
–
Plutonium, 94 Pu
Plutonium
–
A ring of
weapons-grade 99.96% pure electrorefined plutonium, enough for one
bomb core. The ring weighs 5.3 kg, is ca. 11 cm in diameter and its shape helps with
criticality safety.
Plutonium
–
Various oxidation states of plutonium in solution
Plutonium
–
Plutonium
pyrophoricity can cause it to look like a glowing ember under certain conditions.
32.
Pu-239
–
Plutonium-239 is an isotope of plutonium. Plutonium-239 is the fissile isotope used for the production of nuclear weapons. Plutonium-239 is also one of the three main isotopes demonstrated usable as fuel in thermal spectrum reactors, along with uranium-235. Plutonium-239 has a half-life of 24,110 years, plutonium-239 can also absorb neutrons and fission along with the uranium-235 in a reactor. Of all the common nuclear fuels, Pu-239 has the smallest critical mass, a spherical untamped critical mass is about 11 kg,10.2 cm in diameter. Using appropriate triggers, neutron reflectors, implosion geometry and tampers and this optimization usually requires a large nuclear development organization supported by a sovereign nation. The fission of one atom of Pu-239 generates 207.1 MeV =3.318 × 10−11 J, i. e.19.98 TJ/mol =83.61 TJ/kg, or about 2322719 kilowatt hours/kg. Pu-239 is normally created in nuclear reactors by transmutation of atoms of one of the isotopes of uranium present in the fuel rods. Occasionally, when an atom of U-238 is exposed to radiation, its nucleus will capture a neutron. This happens more easily with lower kinetic energy, the U-239 then rapidly undergoes two beta decays, becoming Pu-239.5 m i n β −93239 N p →2. Only if the fuel has been exposed for a few days in the reactor, Pu-239 has a higher probability for fission than U-235 and a larger number of neutrons produced per fission event, so it has a smaller critical mass. In practice, however, reactor-bred plutonium will invariably contain an amount of Pu-240 due to the tendency of Pu-239 to absorb an additional neutron during production. Pu-240 has a rate of spontaneous fission events, making it an undesirable contaminant. It is because of this limitation that plutonium-based weapons must be implosion-type, moreover, Pu-239 and Pu-240 cannot be chemically distinguished, so expensive and difficult isotope separation would be necessary to separate them. Weapons-grade plutonium is defined as containing no more than 7% Pu-240, Pu-240 exposed to alpha particles will incite a nuclear fission. A reactor running on unenriched or moderately enriched uranium contains a great deal of U-238, however, most commercial nuclear power reactor designs require the entire reactor to shut down, often for weeks, in order to change the fuel elements. They therefore produce plutonium in a mix of isotopes that is not well-suited to weapon construction, in practice, their construction and operation is sufficiently difficult that they are generally only used to produce plutonium. Breeder reactors are generally fast reactors, since fast neutrons are more efficient at plutonium production
Pu-239
–
Full table
33.
Florida
–
Florida /ˈflɒrᵻdə/ is a state located in the southeastern region of the United States. It is bordered to the west by the Gulf of Mexico, to the north by Alabama and Georgia, to the east by the Atlantic Ocean, Florida is the 22nd-most extensive, the 3rd-most populous, and the 8th-most densely populated of the U. S. states. Jacksonville is the most populous municipality in the state and is the largest city by area in the contiguous United States, the Miami metropolitan area is Floridas most populous urban area. The city of Tallahassee is the state capital, much of the state is at or near sea level and is characterized by sedimentary soil. The climate varies from subtropical in the north to tropical in the south, the American alligator, American crocodile, Florida panther, and manatee can be found in the Everglades National Park. It was a location of the Seminole Wars against the Native Americans. Today, Florida is distinctive for its large Cuban expatriate community and high population growth, the states economy relies mainly on tourism, agriculture, and transportation, which developed in the late 19th century. Florida is also renowned for amusement parks, orange crops, the Kennedy Space Center, Florida has attracted many writers such as Marjorie Kinnan Rawlings, Ernest Hemingway and Tennessee Williams, and continues to attract celebrities and athletes. It is internationally known for golf, tennis, auto racing, by the 16th century, the earliest time for which there is a historical record, major Native American groups included the Apalachee, the Timucua, the Ais, the Tocobaga, the Calusa and the Tequesta. Florida was the first part of the continental United States to be visited and settled by Europeans, the earliest known European explorers came with the Spanish conquistador Juan Ponce de León. Ponce de León spotted and landed on the peninsula on April 2,1513 and he named the region La Florida. The story that he was searching for the Fountain of Youth is a myth, in May 1539, Conquistador Hernando de Soto skirted the coast of Florida, searching for a deep harbor to land. He described seeing a wall of red mangroves spread mile after mile, some reaching as high as 70 feet. Very soon, many smokes appeared along the whole coast, billowing against the sky, the Spanish introduced Christianity, cattle, horses, sheep, the Spanish language, and more to Florida. Both the Spanish and French established settlements in Florida, with varying degrees of success, in 1559, Don Tristán de Luna y Arellano established a settlement at present-day Pensacola, making it the first attempted settlement in Florida, but it was abandoned by 1561. Spain maintained tenuous control over the region by converting the tribes to Christianity. The area of Spanish Florida diminished with the establishment of English settlements to the north, the English attacked St. Augustine, burning the city and its cathedral to the ground several times. Florida attracted numerous Africans and African-Americans from adjacent British colonies who sought freedom from slavery, in 1738, Governor Manuel de Montiano established Fort Gracia Real de Santa Teresa de Mose near St
Florida
–
St. Augustine is the oldest city in the U.S., established in 1565 by Spain.
Florida
–
Flag
Florida
–
Aerial view of
Castillo De San Marcos (Florida).
Florida
–
The five flags of Florida from the right,
Spain (1565–1763), the
Kingdom of Great Britain, Spain (1784–1821), the
Confederacy, and the United States. France (flag not shown) also controlled part of Florida.
34.
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
United States
–
Native Americans meeting with Europeans, 1764
United States
–
Flag
United States
–
The signing of the
Mayflower Compact, 1620.
United States
–
The
Declaration of Independence: the
Committee of Five presenting their draft to the
Second Continental Congress in 1776
35.
Saint Petersburg Lyceum 239
–
Presidential Physics and Mathematics Lyceum №239, is a public high school in Saint Petersburg, Russia that specializes in mathematics and physics. The school opened in 1918 and it became a city school in 1961. The school is noted for its academic programs. It is the alma mater of numerous winners of International Mathematical Olympiads, the lyceum has been named the best school in Russia in both 2015 and 2016. The school was founded in 1918, originally, it was located in the Lobanov-Rostovsky house, also known as house with lions at the corner of Saint Isaacs Square and Admiralteysky Prospect. It was one of handful of schools to remain open during Siege of Leningrad. In 1961 the school was granted status of school with specialization in physics and mathematics. In 1964 the school moved to the building on Kazansky Street 48/1, which was occupied by school of working youth. Finally, in 1975 the school relocated to its current location, in 1994, the school won the George Soros grant. The US Mathematical society voted the school as one of top ten schools of former Soviet Union, famous for his research on Big Data with Michael Stonebraker. Official web site of Lyceum 239 History of physico-mathematical lyceum #239
Saint Petersburg Lyceum 239
–
Lyceum 239
Saint Petersburg Lyceum 239
–
Saint Petersburg Lyceum 239, January 2008
Saint Petersburg Lyceum 239
–
Annenschule gymnasium in 1912.
36.
Saint-Petersburg
–
Saint Petersburg is Russias second-largest city after Moscow, with five million inhabitants in 2012, and an important Russian port on the Baltic Sea. It is politically incorporated as a federal subject, situated on the Neva River, at the head of the Gulf of Finland on the Baltic Sea, it was founded by Tsar Peter the Great on May 271703. In 1914, the name was changed from Saint Petersburg to Petrograd, in 1924 to Leningrad, between 1713 and 1728 and 1732–1918, Saint Petersburg was the capital of imperial Russia. In 1918, the government bodies moved to Moscow. Saint Petersburg is one of the cities of Russia, as well as its cultural capital. The Historic Centre of Saint Petersburg and Related Groups of Monuments constitute a UNESCO World Heritage Site, Saint Petersburg is home to The Hermitage, one of the largest art museums in the world. A large number of consulates, international corporations, banks. Swedish colonists built Nyenskans, a fortress, at the mouth of the Neva River in 1611, in a then called Ingermanland. A small town called Nyen grew up around it, Peter the Great was interested in seafaring and maritime affairs, and he intended to have Russia gain a seaport in order to be able to trade with other maritime nations. He needed a better seaport than Arkhangelsk, which was on the White Sea to the north, on May 1703121703, during the Great Northern War, Peter the Great captured Nyenskans, and soon replaced the fortress. On May 271703, closer to the estuary 5 km inland from the gulf), on Zayachy Island, he laid down the Peter and Paul Fortress, which became the first brick and stone building of the new city. The city was built by conscripted peasants from all over Russia, tens of thousands of serfs died building the city. Later, the city became the centre of the Saint Petersburg Governorate, Peter moved the capital from Moscow to Saint Petersburg in 1712,9 years before the Treaty of Nystad of 1721 ended the war, he referred to Saint Petersburg as the capital as early as 1704. During its first few years, the city developed around Trinity Square on the bank of the Neva, near the Peter. However, Saint Petersburg soon started to be built out according to a plan, by 1716 the Swiss Italian Domenico Trezzini had elaborated a project whereby the city centre would be located on Vasilyevsky Island and shaped by a rectangular grid of canals. The project was not completed, but is evident in the layout of the streets, in 1716, Peter the Great appointed French Jean-Baptiste Alexandre Le Blond as the chief architect of Saint Petersburg. In 1724 the Academy of Sciences, University and Academic Gymnasium were established in Saint Petersburg by Peter the Great, in 1725, Peter died at the age of fifty-two. His endeavours to modernize Russia had met opposition from the Russian nobility—resulting in several attempts on his life
Saint-Petersburg
–
Top left to bottom right:
Peter and Paul Fortress on
Zayachy Island,
Smolny Cathedral,
Moyka river with the
General Staff Building,
Trinity Cathedral,
Bronze Horseman on
Senate Square, and the
Winter Palace.
Saint-Petersburg
–
The
Bronze Horseman, monument to Peter the Great
Saint-Petersburg
–
Palace Square backed by the General Staff arch and building, as the main square of the Russian Empire it was the setting of many events of historic significance
Saint-Petersburg
–
Map of Saint Petersburg, 1903
37.
Russia
–
Russia, also officially the Russian Federation, is a country in Eurasia. The European western part of the country is more populated and urbanised than the eastern. Russias capital Moscow is one of the largest cities in the world, other urban centers include Saint Petersburg, Novosibirsk, Yekaterinburg, Nizhny Novgorod. Extending across the entirety of Northern Asia and much of Eastern Europe, Russia spans eleven time zones and incorporates a range of environments. It shares maritime borders with Japan by the Sea of Okhotsk, the East Slavs emerged as a recognizable group in Europe between the 3rd and 8th centuries AD. Founded and ruled by a Varangian warrior elite and their descendants, in 988 it adopted Orthodox Christianity from the Byzantine Empire, beginning the synthesis of Byzantine and Slavic cultures that defined Russian culture for the next millennium. Rus ultimately disintegrated into a number of states, most of the Rus lands were overrun by the Mongol invasion. The Soviet Union played a role in the Allied victory in World War II. The Soviet era saw some of the most significant technological achievements of the 20th century, including the worlds first human-made satellite and the launching of the first humans in space. By the end of 1990, the Soviet Union had the second largest economy, largest standing military in the world. It is governed as a federal semi-presidential republic, the Russian economy ranks as the twelfth largest by nominal GDP and sixth largest by purchasing power parity in 2015. Russias extensive mineral and energy resources are the largest such reserves in the world, making it one of the producers of oil. The country is one of the five recognized nuclear weapons states and possesses the largest stockpile of weapons of mass destruction, Russia is a great power as well as a regional power and has been characterised as a potential superpower. The name Russia is derived from Rus, a state populated mostly by the East Slavs. However, this name became more prominent in the later history, and the country typically was called by its inhabitants Русская Земля. In order to distinguish this state from other states derived from it, it is denoted as Kievan Rus by modern historiography, an old Latin version of the name Rus was Ruthenia, mostly applied to the western and southern regions of Rus that were adjacent to Catholic Europe. The current name of the country, Россия, comes from the Byzantine Greek designation of the Kievan Rus, the standard way to refer to citizens of Russia is Russians in English and rossiyane in Russian. There are two Russian words which are translated into English as Russians
Russia
–
Kievan Rus' in the 11th century
Russia
–
Flag
Russia
–
The
Baptism of Kievans, by
Klavdy Lebedev
Russia
–
Sergius of Radonezh blessing
Dmitry Donskoy in
Trinity Sergius Lavra, before the
Battle of Kulikovo, depicted in a painting by
Ernst Lissner
38.
The Simpsons
–
The Simpsons is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a depiction of working-class life epitomized by the Simpson family, which consists of Homer, Marge, Bart, Lisa. The show is set in the town of Springfield and parodies American culture, society, television. The family was conceived by Groening shortly before a solicitation for a series of animated shorts with producer James L. Brooks, Groening created a dysfunctional family and named the characters after members of his own family, substituting Bart for his own name. The shorts became a part of The Tracey Ullman Show on April 19,1987. After a three-season run, the sketch was developed into a prime time show and became an early hit for Fox. Since its debut on December 17,1989,615 episodes of The Simpsons have been broadcast and its 28th season began on September 25,2016. It is the longest-running American sitcom and the longest-running American animated program, the Simpsons Movie, a feature-length film, was released in theaters worldwide on July 27,2007, and grossed over $527 million. On May 4,2015, the series was renewed for seasons 27 and 28, on November 4,2016, the series was renewed for seasons 29 and 30, consisting of 22 episodes each. The Simpsons received widespread critical acclaim throughout its first nine or ten seasons, Time named it the 20th centurys best television series, and Erik Adams of The A. V. Club named it televisions crowning achievement regardless of format, on January 14,2000, the Simpson family was awarded a star on the Hollywood Walk of Fame. It has won dozens of awards since it debuted as a series, including 31 Primetime Emmy Awards,30 Annie Awards, Homers exclamatory catchphrase Doh. has been adopted into the English language, while The Simpsons has influenced many adult-oriented animated sitcoms. Despite this, the show has also criticized for what many perceive as a decline in quality over the years. The Simpsons are a family who live in a fictional Middle American town of Springfield, Homer, the father, works as a safety inspector at the Springfield Nuclear Power Plant, a position at odds with his careless, buffoonish personality. He is married to Marge Simpson, a stereotypical American housewife, although the family is dysfunctional, many episodes examine their relationships and bonds with each other and they are often shown to care about one another. The family owns a dog, Santas Little Helper, and a cat, Snowball V, renamed Snowball II in I, both pets have had starring roles in several episodes. The show includes an array of supporting characters, co-workers, teachers, family friends, extended relatives, townspeople. The creators originally intended many of these characters as jokes or for fulfilling needed functions in the town
The Simpsons
–
James L. Brooks (pictured) asked
Matt Groening to create a series of animated shorts for
The Tracey Ullman Show
The Simpsons
The Simpsons
–
Matt Groening, creator
The Simpsons
–
Part of the writing staff of The Simpsons in 1992. Back row, left to right: Mike Mendel, Colin ABV Lewis (partial), Jeff Goldstein,
Al Jean (partial),
Conan O'Brien,
Bill Oakley,
Josh Weinstein,
Mike Reiss, Ken Tsumura,
George Meyer,
John Swartzwelder,
Jon Vitti (partial), CJ Gibson and
David M. Stern. Front row, left to right: Dee Capelli,
Lona Williams, and unknown.
39.
Homer Simpson
–
Homer Jay Simpson is a fictional character and the main protagonist of the American animated television series The Simpsons as the patriarch of the eponymous family. He is voiced by Dan Castellaneta and first appeared on television, along with the rest of his family, Homer was created and designed by cartoonist Matt Groening while he was waiting in the lobby of James L. Brooks office. Groening had been called to pitch a series of shorts based on his comic strip Life in Hell and he named the character after his father, Homer Groening. After appearing for three seasons on The Tracey Ullman Show, the Simpson family got their own series on Fox that debuted December 17,1989, Homer and his wife Marge have three children, Bart, Lisa, and Maggie. As the familys provider, he works at the Springfield Nuclear Power Plant as a plant operator. Despite the suburban blue-collar routine of his life, he has had a number of remarkable experiences. He has appeared in other media relating to The Simpsons – including video games, The Simpsons Movie, The Simpsons Ride, commercials and comic books – and inspired an entire line of merchandise. His signature catchphrase, the annoyed grunt Doh. has been included in The New Oxford Dictionary of English since 1998, Homer is one of the most influential characters in the history of television. The British newspaper The Sunday Times described him as the greatest comic creation of time, for voicing Homer, Castellaneta has won four Primetime Emmy Awards for Outstanding Voice-Over Performance and a special-achievement Annie Award. In 2000, Homer and his family were awarded a star on the Hollywood Walk of Fame, Homer is the bumbling husband of Marge and father of Bart, Lisa and Maggie Simpson. He is the son of Mona and Abraham Grampa Simpson, Homer held over 188 different jobs in the first 400 episodes of The Simpsons. In most episodes, he works as the Nuclear Safety Inspector at the Springfield Nuclear Power Plant, a position he has held since Homers Odyssey, the third episode of the series. At the plant, Homer is often ignored and completely forgotten by his boss Mr. Burns, Matt Groening has stated that he decided to have Homer work at the power plant because of the potential for Homer to wreak havoc. Each of his other jobs has lasted only one episode, in the first half of the series, the writers developed an explanation about how he got fired from the plant and was then rehired in every episode. In later episodes, he began a new job on impulse. The Simpsons uses a floating timeline in which the characters never age, and, as such. Nevertheless, in episodes, events in Homers life have been linked to specific time periods. However, the episode That 90s Show contradicted much of this backstory, portraying Homer and Marge as a childless couple in the early 1990s
Homer Simpson
–
Homer's design has been revised several times over the course of the series. Left to right: Homer as he appeared in "
Good Night " (1987), "
Bathtime " (1989), and "
Bart the Genius " (1990).
Homer Simpson
–
Homer Simpson
Homer Simpson
–
—
Dan Castellaneta
Homer Simpson
–
The first sketch of Homer strangling Bart, drawn in 1988.
40.
King-Size Homer
–
King-Size Homer is the seventh episode of The Simpsons seventh season. It originally aired on the Fox network in the United States on November 5,1995, in the episode, Homer despises the nuclear plants new exercise program, and decides to gain 61 pounds in order to claim a disability and work at home. The episode was written by Dan Greaney and directed by Jim Reardon, joan Kenley makes her second of three guest appearances on The Simpsons in the episode as the voice of the telephone lady. It features cultural references to the worlds heaviest twins, the 1993 film Whats Eating Gilbert Grape, since airing, the episode has received positive reviews from fans and television critics, and Empire named it the best episode of the series. It acquired a Nielsen rating of 10.0, and was the third highest rated show on the Fox network that week, Mr. Burns organizes a morning calisthenics program at the nuclear power plant, much to the dismay of Homer. After learning that someone who is disabled can work from home, Homer soon discovers that any employee that weighs 300 pounds or more qualifies as disabled, and so he decides to gain the 61 pounds needed to reach 300. He begins eating excessively, despite Marge and Lisas repeated warnings that he could seriously endanger his health, with Barts help, Homer soon reaches his goal, and Mr. Burns installs a stay-at-home work terminal in the Simpson house. Homer continues to neglect his responsibilities as a safety inspector by simply typing yes every time the system prompts him, looking for shortcuts, he leaves his terminal with a drinking bird to press the Y key to indicate yes on the keyboard and goes out. Returning home, he discovers that the bird has fallen over, Mr. Burns gives Homer a medal and pays for him to undergo liposuction. King-Size Homer was written by Dan Greaney, and directed by Jim Reardon and it was the first episode Greaney wrote for The Simpsons. Prior to this episode he was working as a lawyer and was contemplating moving to Ukraine to work for a start-up company and he said that this episode saved him from doing so. Greaney pitched some ideas to the staff, but none of them were satisfactory. Oakley had Greaney come to Hollywood to write it, and when Greaney showed the first draft to the staff, they liked it, the writers wanted the title of the episode to make Homer sound proud about his weight, so they decided to name it King-Size Homer. Greaney really enjoyed working on the episode because Homer is constantly happy and goal oriented in it, animator David Silverman designed the obese Homer for the episode. There was a discussion about what Homer would wear when he became fat, the writers were also discussing about how they were going to treat Homers obesity. They did not want Homer to come off as a hog, action figurines based on obese Homer were made for the World of Springfield series shortly after the episode had aired. Homer has a dream in which he is standing at the foot of a mountain with 300 pounds as the goal at the top, a pig wearing a tuxedo appears next to him in the dream, and motivates Homer to reach the top of the mountain. This scene was inspired by the cover of the Sweetness and Light issue of the National Lampoon magazine
King-Size Homer
–
Then-show runner
Bill Oakley came up with the idea for the episode, and assigned
Dan Greaney to write it.
King-Size Homer
–
"King-Size Homer"
King-Size Homer
–
Homer believes he can get a cup of
Tab by pressing the
tab key
41.
Book of Mormon
–
It was first published in March 1830 by Joseph Smith as The Book of Mormon, An Account Written by the Hand of Mormon upon Plates Taken from the Plates of Nephi. According to Smiths account and the narrative, the Book of Mormon was originally written in otherwise unknown characters referred to as reformed Egyptian engraved on golden plates. Critics claim that it was fabricated by Smith, drawing on material, the pivotal event of the book is an appearance of Jesus Christ in the Americas shortly after his resurrection. The Book of Mormon is divided into books, titled after the individuals named as primary authors and, in most versions, divided into chapters. It is written in English very similar to the Early Modern English linguistic style of the King James Version of the Bible, as of 2011, more than 150 million copies of the Book of Mormon have been published. The writings were said to describe a people whom God had led from Jerusalem to the Western hemisphere 600 years before Jesus birth. According to the narrative, Moroni was the last prophet among these people and had buried the record, which God had promised to bring forth in the latter days. e. Smiths description of these events recounts that he was allowed to take the plates on September 22,1827, exactly four years from that date, accounts vary of the way in which Smith dictated the Book of Mormon. Smith himself implied that he read the plates directly using spectacles prepared for the purpose of translating, other accounts variously state that he used one or more seer stones placed in a top hat. Both the special spectacles and the stone were at times referred to as the Urim and Thummim. During the translating process itself, Smith sometimes separated himself from his scribe with a blanket between them, additionally, the plates were not always present during the translating process and, when present, they were always covered up. Smiths first published description of the said that the plates had the appearance of gold. They were described by Martin Harris, one of Smiths early scribes, Smith called the engraved writing on the plates reformed Egyptian. A portion of the text on the plates was also sealed according to his account, in addition to Smiths account regarding the plates, eleven others stated that they saw the golden plates and, in some cases, handled them. Their written testimonies are known as the Testimony of Three Witnesses and these statements have been published in most editions of the Book of Mormon. Smith enlisted his neighbor Martin Harris as a scribe during his work on the text. In 1828, Harris, prompted by his wife Lucy Harris, Smith reluctantly acceded to Harriss requests. Lucy Harris is thought to have stolen the first 116 pages, after the loss, Smith recorded that he had lost the ability to translate, and that Moroni had taken back the plates to be returned only after Smith repented
Book of Mormon
–
A page from the original manuscript of the Book of Mormon, covering 1 Nephi 4:38 - 5:14
Book of Mormon
–
A depiction of Joseph Smith dictating the Book of Mormon by peering at a seer stone in a hat.
Book of Mormon
–
Books of the Book of Mormon
42.
On-Line Encyclopedia of Integer Sequences
–
The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
On-Line Encyclopedia of Integer Sequences
–
On-Line Encyclopedia of Integer Sequences
43.
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
12 (number)
–
12 stars are featured on the
Flag of Europe
44.
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
15 (number)
–
Fifteen total individuals in the
mollusk species
Donax variabilis comprise the entire coloration and patterning in their
phenotypes.
15 (number)
–
The 15 perfect matchings of K 6
45.
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
17 (number)
–
No row 17 in
Alitalia planes.
46.
19 (number)
–
19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
19 (number)
–
A 19x19
Go board
19 (number)
–
19 is a
centered triangular number
47.
20 (number)
–
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants
20 (number)
–
An
icosahedron has 20
faces
48.
21 (number)
–
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
21 (number)
–
Number 21 on the road bicycle of
Ellen van Dijk at the
Ronde van Drenthe.
21 (number)
–
Building called "21" in
Zlín,
Czech Republic.
21 (number)
–
Detail of the building entrance
49.
24 (number)
–
24 is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta, and for 10−24 yocto and these numbers are the largest and smallest number to receive an SI prefix to date. In a 24-hour clock, the hour is in conventional language called twelve or twelve oclock. 24 is the factorial of 4 and a number, being the first number of the form 23q. It follows that 24 is the number of ways to order 4 distinct items and it is the smallest number with exactly eight divisors,1,2,3,4,6,8,12, and 24. It is a composite number, having more divisors than any smaller number. 24 is a number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. Subtracting 1 from any of its divisors yields a number,24 is the largest number with this property. 24 has a sum of 36 and the aliquot sequence. It is therefore the lowest abundant number whose aliquot sum is itself abundant, the aliquot sum of only one number,529 =232, is 24. There are 10 solutions to the equation φ =24, namely 35,39,45,52,56,70,72,78,84 and 90 and this is more than any integer below 24, making 24 a highly totient number. 24 is the sum of the prime twins 11 and 13, the product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two numbers, one of which is a multiple of four, and there must be a multiple of three. The tesseract has 24 two-dimensional faces,24 is the only nontrivial solution to the cannonball problem, that is,12 +22 +32 + … +242 is a perfect square. In 24 dimensions there are 24 even positive definite unimodular lattices, the Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S and the Mathieu group M24. The modular discriminant Δ is proportional to the 24th power of the Dedekind eta function η, Δ = 12η24, the Barnes-Wall lattice contains 24 lattices. 24 is the number whose divisors — namely 1,2,3,4,6,8,12,24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group × = is isomorphic to the additive group 3 and this fact plays a role in monstrous moonshine
24 (number)
–
Astronomical clock in Prague