1.
168 (number)
–
168 is the natural number following 167 and preceding 169. 168 is a number, a composite number, an abundant number. There are 168 primes less than 1000,168 is the product of the first two perfect numbers. 168 is the order of the group PSL, the second smallest nonabelian simple group, from Hurwitzs automorphisms theorem,168 is the maximum possible number of automorphisms of a genus 3 Riemann surface, this maximum being achieved by the Klein quartic, whose symmetry group is PSL. The Fano plane has 168 symmetries,168 is the sum of four consecutive prime numbers,37 +41 +43 +47
168 (number)
–
There are 168
pips on a double-six set of
dominoes
2.
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
3.
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).
4.
100 (number)
–
100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
100 (number)
–
The
U.S. hundred-dollar bill, Series 2009.
5.
Factorization
–
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.
6.
Divisor
–
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
Divisor
–
The divisors of 10 illustrated with
Cuisenaire rods: 1, 2, 5, and 10
7.
Greek numerals
–
Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek numerals
–
Numeral systems
Greek numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
8.
Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
–
Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
–
Numeral systems
Roman numerals
–
A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
–
An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
9.
Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary number
–
Numeral systems
Binary number
–
Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
10.
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
11.
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
12.
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
13.
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
14.
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
15.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
Duodecimal
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Numeral systems
Duodecimal
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A duodecimal multiplication table
16.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
Hexadecimal
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Numeral systems
Hexadecimal
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Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
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Hexadecimal finger-counting scheme.
17.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
Vigesimal
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Numeral systems
Vigesimal
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The
Maya numerals are a base-20 system.
18.
Base 36
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Base 36
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Numeral systems
Base 36
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34 senary = 22 decimal, in senary finger counting
Base 36
19.
Odd number
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
Odd number
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Rubik's Revenge in solved state
20.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
Composite number
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Overview
21.
Deficient number
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In number theory, a deficient or deficient number is a number n for which the sum of divisors σ<2n, or, equivalently, the sum of proper divisors s<n. The value 2n − σ is called the numbers deficiency, as an example, consider the number 21. Its proper divisors are 1,3 and 7, and their sum is 11, because 11 is less than 21, the number 21 is deficient. Its deficiency is 2 ×21 −32 =10, since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. An infinite number of even and odd deficient numbers exist. All odd numbers with one or two prime factors are deficient. All proper divisors of deficient or perfect numbers are deficient, there exists at least one deficient number in the interval for all sufficiently large n. Closely related to deficient numbers are perfect numbers with σ = 2n, the natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica. Almost perfect number Amicable number Sociable number Sándor, József, Mitrinović, Dragoslav S. Crstici, Borislav, the Prime Glossary, Deficient number Weisstein, Eric W. Deficient Number
Deficient number
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Overview
22.
Square number
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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
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m = 1 2 = 1
23.
31 (number)
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31 is the natural number following 30 and preceding 32. As a Mersenne prime,31 is related to the perfect number 496,31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a triangular number, the lowest prime centered pentagonal number. For the Steiner tree problem,31 is the number of possible Steiner topologies for Steiner trees with 4 terminals, at 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31,31 is a repdigit in base 5, and base 2. The numbers 31,331,3331,33331,333331,3333331, for a time it was thought that every number of the form 3w1 would be prime. Here,31 divides every fifteenth number in 3w1, the atomic number of gallium Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is visible to the naked eye in a modestly dark sky. The New General Catalogue object NGC31, a galaxy in the constellation Phoenix The Saros number of the solar eclipse series which began on -1805 January 31. The duration of Saros series 31 was 1316.2 years, the Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, the jersey number 31 has been retired by several North American sports teams in honor of past playing greats, In Major League Baseball, The San Diego Padres, for Dave Winfield. The Chicago Cubs, for Ferguson Jenkins and Greg Maddux, the Atlanta Braves, also for Maddux. The New York Mets, for Mike Piazza, in the NBA, The Boston Celtics, for Cedric Maxwell. The Indiana Pacers, for Reggie Miller, in the NHL, The Edmonton Oilers, for Grant Fuhr. The New York Islanders, for Billy Smith, in the NFL, The Atlanta Falcons, for William Andrews. The New Orleans Saints, for Jim Taylor, NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company. In ice hockey goaltenders often wear the number 31, in football the number 31 has been retired by Queens Park Rangers F. C.31 from the Prime Pages
31 (number)
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31 is a
centered pentagonal number
24.
Centered hexagonal number
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The nth centered hexagonal number is given by the formula n 3 −3 =3 n +1. Expressing the formula as 1 +6 shows that the centered hexagonal number for n is 1 more than 6 times the th triangular number. The first few centered hexagonal numbers are,1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919. In base 10 one can notice that the hexagonal numbers rightmost digits follow the pattern 1–7–9–7–1, the sum of the first n centered hexagonal numbers is n3. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. In particular, prime centered hexagonal numbers are cuban primes, the difference between 2 and the nth centered hexagonal number is a number of the form 3n2 + 3n −1, while the difference between 2 and the nth centered hexagonal number is a pronic number. Hexagonal number Magic hexagon Star number
Centered hexagonal number
25.
Centered octagonal number
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A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered octagonal numbers are the same as the odd square numbers, thus, the nth centered octagonal number is given by the formula 2 =4 n 2 −4 n +1. The first few centered octagonal numbers are 1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089. Calculating Ramanujans tau function on an octagonal number yields an odd number
Centered octagonal number
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See also [edit]
26.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers
Pell number
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Rational approximations to regular
octagons, with coordinates derived from the Pell numbers.
27.
Markov number
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The first few Markov numbers are 1,2,5,13,29,34,89,169,194,233,433,610,985,1325. Appearing as coordinates of the Markov triples, etc, there are infinitely many Markov numbers and Markov triples. There are two ways to obtain a new Markov triple from an old one. First, one may permute the 3 numbers x, y, z, second, if is a Markov triple then by Vieta jumping so is. Applying this operation twice returns the same triple one started with, joining each normalized Markov triple to the 1,2, or 3 normalized triples one can obtain from this gives a graph starting from as in the diagram. This graph is connected, in other words every Markov triple can be connected to by a sequence of these operations. If we start, as an example, with we get its three neighbors, and in the Markov tree if x is set to 1,5 and 13, respectively. For instance, starting with and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers, starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers. All the Markov numbers on the adjacent to 2s region are odd-indexed Pell numbers. Thus, there are infinitely many Markov triples of the form, likewise, there are infinitely many Markov triples of the form, where Px is the xth Pell number. Aside from the two smallest singular triples and, every Markov triple consists of three distinct integers, odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32. In his 1982 paper, Don Zagier conjectured that the nth Markov number is given by m n =13 e C n + o with C =2.3523414972 …. Moreover, he pointed out that x 2 + y 2 + z 2 =3 x y z +4 /9, the conjecture was proved by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry. The nth Lagrange number can be calculated from the nth Markov number with the formula L n =9 −4 m n 2, the Markov numbers are sums of pairs of squares. If X⋅Y⋅Z =1 then Tr = Tr, so more symmetrically if X, Y, and Z are in SL2 with X⋅Y⋅Z =1, cambridge Tracts in Mathematics and Mathematical Physics. Markov spectrum problem, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Markoff, A. Sur les formes quadratiques binaires indéfinies
Markov number
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The first levels of the Markov number tree
28.
Prime number
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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
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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.
29.
17 (number)
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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)
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No row 17 in
Alitalia planes.
30.
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
31.
37 (number)
–
37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37
37 (number)
–
House number in
Baarle (in its Belgian part)
32.
Asteroid belt
–
The asteroid belt is the circumstellar disc in the Solar System located roughly between the orbits of the planets Mars and Jupiter. It is occupied by numerous irregularly shaped bodies called asteroids or minor planets, the asteroid belt is also termed the main asteroid belt or main belt to distinguish it from other asteroid populations in the Solar System such as near-Earth asteroids and trojan asteroids. About half the mass of the belt is contained in the four largest asteroids, Ceres, Vesta, Pallas, the total mass of the asteroid belt is approximately 4% that of the Moon, or 22% that of Pluto, and roughly twice that of Plutos moon Charon. Ceres, the belts only dwarf planet, is about 950 km in diameter, whereas Vesta, Pallas. The remaining bodies range down to the size of a dust particle, the asteroid material is so thinly distributed that numerous unmanned spacecraft have traversed it without incident. Nonetheless, collisions between large asteroids do occur, and these can form a family whose members have similar orbital characteristics. Individual asteroids within the belt are categorized by their spectra. The asteroid belt formed from the solar nebula as a group of planetesimals. Planetesimals are the precursors of the protoplanets. Between Mars and Jupiter, however, gravitational perturbations from Jupiter imbued the protoplanets with too much energy for them to accrete into a planet. Collisions became too violent, and instead of fusing together, the planetesimals, as a result,99. 9% of the asteroid belts original mass was lost in the first 100 million years of the Solar Systems history. Some fragments eventually found their way into the inner Solar System, Asteroid orbits continue to be appreciably perturbed whenever their period of revolution about the Sun forms an orbital resonance with Jupiter. At these orbital distances, a Kirkwood gap occurs as they are swept into other orbits. Classes of small Solar System bodies in other regions are the objects, the centaurs, the Kuiper belt objects, the scattered disc objects, the sednoids. On 22 January 2014, ESA scientists reported the detection, for the first definitive time, of water vapor on Ceres, the detection was made by using the far-infrared abilities of the Herschel Space Observatory. The finding was unexpected because comets, not asteroids, are considered to sprout jets. According to one of the scientists, The lines are becoming more and more blurred between comets and asteroids. This pattern, now known as the Titius–Bode law, predicted the semi-major axes of the six planets of the provided one allowed for a gap between the orbits of Mars and Jupiter
Asteroid belt
–
By far the largest object within the belt is
Ceres. The total mass of the asteroid belt is significantly less than
Pluto 's, and approximately twice that of Pluto's moon
Charon.
Asteroid belt
–
Sun Jupiter trojans Orbits of
planets
Asteroid belt
–
Giuseppe Piazzi, discoverer of
Ceres, the largest object in the asteroid belt. For several decades after its discovery Ceres was known as a planet, after which it was reclassified as asteroid number 1. In 2006 it was recognized to be a dwarf planet.
Asteroid belt
–
951 Gaspra, the first asteroid imaged by a spacecraft, as viewed during
Galileo ' s 1991 flyby; colors are exaggerated
33.
Asteroid
–
Asteroids are minor planets, especially those of the inner Solar System. The larger ones have also been called planetoids and these terms have historically been applied to any astronomical object orbiting the Sun that did not show the disc of a planet and was not observed to have the characteristics of an active comet. As minor planets in the outer Solar System were discovered and found to have volatile-based surfaces that resemble those of comets, in this article, the term asteroid refers to the minor planets of the inner Solar System including those co-orbital with Jupiter. There are millions of asteroids, many thought to be the remnants of planetesimals. The large majority of known asteroids orbit in the belt between the orbits of Mars and Jupiter, or are co-orbital with Jupiter. However, other orbital families exist with significant populations, including the near-Earth objects, individual asteroids are classified by their characteristic spectra, with the majority falling into three main groups, C-type, M-type, and S-type. These were named after and are identified with carbon-rich, metallic. The size of asteroids varies greatly, some reaching as much as 1000 km across, asteroids are differentiated from comets and meteoroids. In the case of comets, the difference is one of composition, while asteroids are composed of mineral and rock, comets are composed of dust. In addition, asteroids formed closer to the sun, preventing the development of the aforementioned cometary ice, the difference between asteroids and meteoroids is mainly one of size, meteoroids have a diameter of less than one meter, whereas asteroids have a diameter of greater than one meter. Finally, meteoroids can be composed of either cometary or asteroidal materials, only one asteroid,4 Vesta, which has a relatively reflective surface, is normally visible to the naked eye, and this only in very dark skies when it is favorably positioned. Rarely, small asteroids passing close to Earth may be visible to the eye for a short time. As of March 2016, the Minor Planet Center had data on more than 1.3 million objects in the inner and outer Solar System, the United Nations declared June 30 as International Asteroid Day to educate the public about asteroids. The date of International Asteroid Day commemorates the anniversary of the Tunguska asteroid impact over Siberia, the first asteroid to be discovered, Ceres, was found in 1801 by Giuseppe Piazzi, and was originally considered to be a new planet. In the early half of the nineteenth century, the terms asteroid. Asteroid discovery methods have improved over the past two centuries. This task required that hand-drawn sky charts be prepared for all stars in the band down to an agreed-upon limit of faintness. On subsequent nights, the sky would be charted again and any moving object would, hopefully, the expected motion of the missing planet was about 30 seconds of arc per hour, readily discernible by observers
Asteroid
–
253 Mathilde, a
C-type asteroid measuring about 50 kilometres (30 mi) across, covered in craters half that size. Photograph taken in 1997 by the
NEAR Shoemaker probe.
Asteroid
–
2013 EC, shown here in radar images, has a provisional designation
Asteroid
–
⚵
Asteroid
–
243 Ida and its moon Dactyl. Dactyl is the first satellite of an asteroid to be discovered.
34.
Orange (colour)
–
Orange is the colour of carrots, pumpkins and apricots. It is between red and yellow in the spectrum of light, and on the traditional painters colour wheel and it is named after the fruit of the same name. In Asia it is an important symbolic colour of Buddhism and Hinduism, the colour orange is named after the appearance of the ripe orange fruit. The word comes from the Old French orange, from the old term for the fruit, the French word, in turn, comes from the Italian arancia, based on Arabic nāranj, derived from the Sanskrit naranga. The first recorded use of orange as a name in English was in 1512. Prior to this word being introduced to the English-speaking world, saffron already existed in the English language, crog also referred to the saffron colour, so that orange was also referred to as ġeolurēad for reddish orange, or ġeolucrog for yellowish orange. Alternatively, orange things were sometimes described as red such as red deer, red hair, in ancient Egypt artists used an orange mineral pigment called realgar for tomb paintings, as well as other uses. It was also used later by Medieval artists for the colouring of manuscripts, pigments were also made in ancient times from a mineral known as orpiment. Orpiment was an important item of trade in the Roman Empire and was used as a medicine in China although it contains arsenic and is highly toxic and it was also used as a fly poison and to poison arrows. Because of its colour, it was also a favourite with alchemists searching for a way to make gold. Before the late 15th century, the colour orange existed in Europe, portuguese merchants brought the first orange trees to Europe from Asia in the late 15th and early 16th century, along with the Sanskrit naranga, which gradually became orange in English. The House of Orange-Nassau was one of the most influential houses in Europe in the 16th and 17th centuries. It originated in 1163 the tiny Principality of Orange, a state of 108 square miles north of Avignon in southern France. The family of the Prince of Orange eventually adopted the name, the colour came to be associated with Protestantism, due to participation by the House of Orange on the Protestant side in the French Wars of Religion. One member of the house, William I of Orange, organised the Dutch resistance against Spain, another member, William III of Orange, became King of England in 1689, after the downfall of the Catholic James II. Due to William III, orange became an important political colour in Britain, William was a Protestant, and as such he defended the Protestant minority of Ireland against the majority Roman Catholic population. As a result, the Protestants of Ireland were known as Orangemen, Orange eventually became one of the colours of the Irish flag, symbolising the Protestant heritage. When the Dutch settlers of South Africa rebelled against the British in the late 19th century, in the United States, the flag of the City of New York has an orange stripe, to remember the Dutch colonists who founded the city
Orange (colour)
–
The colour orange derives its name from the orange fruit.
Orange (colour)
–
Orange
Orange (colour)
–
Lifeboats in
Arklow Harbour, Ireland. Orange is chosen for lifeboats and lifesaving jackets because of its high visibility.
Orange (colour)
–
A young
Buddhist monk in
Laos
35.
Constellation
–
A constellation is formally defined as a region of the celestial sphere, with boundaries laid down by the International Astronomical Union. The constellation areas mostly had their origins in Western-traditional patterns of stars from which the constellations take their names, in 1922, the International Astronomical Union officially recognized the 88 modern constellations, which cover the entire sky. They began as the 48 classical Greek constellations laid down by Ptolemy in the Almagest, Constellations in the far southern sky are late 16th- and mid 18th-century constructions. 12 of the 88 constellations compose the zodiac signs, though the positions of the constellations only loosely match the dates assigned to them in astrology. The term constellation can refer to the stars within the boundaries of that constellation. Notable groupings of stars that do not form a constellation are called asterisms, when astronomers say something is “in” a given constellation they mean it is within those official boundaries. Any given point in a coordinate system can unambiguously be assigned to a single constellation. Many astronomical naming systems give the constellation in which an object is found along with a designation in order to convey a rough idea in which part of the sky it is located. For example, the Flamsteed designation for bright stars consists of a number, the word constellation seems to come from the Late Latin term cōnstellātiō, which can be translated as set of stars, and came into use in English during the 14th century. It also denotes 88 named groups of stars in the shape of stellar-patterns, the Ancient Greek word for constellation was ἄστρον. Colloquial usage does not draw a distinction between constellation in the sense of an asterism and constellation in the sense of an area surrounding an asterism. The modern system of constellations used in astronomy employs the latter concept, the term circumpolar constellation is used for any constellation that, from a particular latitude on Earth, never sets below the horizon. From the North Pole or South Pole, all constellations south or north of the equator are circumpolar constellations. In the equatorial or temperate latitudes, the term equatorial constellation has sometimes been used for constellations that lie to the opposite the circumpolar constellations. They generally include all constellations that intersect the celestial equator or part of the zodiac, usually the only thing the stars in a constellation have in common is that they appear near each other in the sky when viewed from the Earth. In galactic space, the stars of a constellation usually lie at a variety of distances, since stars also travel on their own orbits through the Milky Way, the star patterns of the constellations change slowly over time. After tens to hundreds of thousands of years, their familiar outlines will become unrecognisable, the terms chosen for the constellation themselves, together with the appearance of a constellation, may reveal where and when its constellation makers lived. The earliest direct evidence for the constellations comes from inscribed stones and it seems that the bulk of the Mesopotamian constellations were created within a relatively short interval from around 1300 to 1000 BC
Constellation
Constellation
Constellation
–
Babylonian tablet recording
Halley's comet in 164 BC.
Constellation
–
Chinese star map with a cylindrical projection (
Su Song)
36.
Taurus (constellation)
–
Taurus is one of the constellations of the zodiac, which means it is crossed by the plane of the ecliptic. Taurus is a large and prominent constellation in the northern hemispheres winter sky and it is one of the oldest constellations, dating back to at least the Early Bronze Age when it marked the location of the Sun during the spring equinox. Its importance to the agricultural calendar influenced various bull figures in the mythologies of Ancient Sumer, Akkad, Assyria, Babylon, Egypt, Greece, a number of features exist that are of interest to astronomers. Taurus hosts two of the nearest open clusters to Earth, the Pleiades and the Hyades, both of which are visible to the naked eye, at first magnitude, the red giant Aldebaran is the brightest star in the constellation. In the northwest part of Taurus is the supernova remnant Messier 1, one of the closest regions of active star formation, the Taurus-Auriga complex, crosses into the northern part of the constellation. The variable star T Tauri is the prototype of a class of pre-main-sequence stars, in September and October, Taurus is visible in the evening along the eastern horizon. The most favorable time to observe Taurus in the sky is during the months of December. By March and April, the constellation will appear to the west during the evening twilight and this constellation forms part of the zodiac, and hence is intersected by the ecliptic. This circle across the sphere forms the apparent path of the Sun as the Earth completes its annual orbit. As the orbital plane of the Moon and the planets lie near the ecliptic, the galactic plane of the Milky Way intersects the northeast corner of the constellation and the galactic anticenter is located near the border between Taurus and Auriga. Taurus is the only constellation crossed by all three of the equator, celestial equator, and ecliptic. A ring-like galactic structure known as the Goulds Belt passes through the Taurus constellation, the recommended three-letter abbreviation for the constellation, as adopted by the International Astronomical Union in 1922, is Tau. The official constellation boundaries, as set by Eugène Delporte in 1930, are defined by a polygon of 26 segments. In the equatorial coordinate system, the right ascension coordinates of these borders lie between 03h 23. 4m and 05h 53. 3m, while the coordinates are between 31. 10° and −1. 35°. Because a small part of the lies to the south of the celestial equator. During November, the Taurid meteor shower appears to radiate from the direction of this constellation. The Beta Taurid meteor shower occurs during the months of June and July in the daytime, between 18 and 29 October, both the Northern Taurids and the Southern Taurids are active, though the latter stream is stronger. However, between November 1 and 10, the two streams equalize, the brightest member of this constellation is Aldebaran, an orange-hued, spectral class K5 III giant star
Taurus (constellation)
–
The constellation Taurus as it can be seen by the naked eye. The constellation lines have been added for clarity.
Taurus (constellation)
–
List of stars in Taurus
Taurus (constellation)
–
Taurus as depicted in the astronomical treatise
Book of Fixed Stars by the Persian astronomer
Abd al-Rahman al-Sufi, c. 964.
Taurus (constellation)
–
Taurus as depicted in
Urania's Mirror, a set of constellation cards published in London c.1825.
37.
Quasar
–
A quasar is an active galactic nucleus of very high luminosity. A quasar consists of a black hole surrounded by an orbiting accretion disk of gas. As gas in the accretion disk falls toward the black hole, quasars emit energy across the electromagnetic spectrum and can be observed at radio, infrared, visible, ultraviolet, and X-ray wavelengths. The most powerful quasars have luminosities exceeding 1041 W, thousands of greater than the luminosity of a large galaxy such as the Milky Way. Quasars are found over a broad range of distances. The peak epoch of quasar activity in the Universe corresponds to redshifts around 2, as of 2011, the most distant known quasar is at redshift z=7.085, light observed from this quasar was emitted when the Universe was only 770 million years old. Because quasars are distant objects, any light which reaches the Earth is redshifted due to the expansion of space. In early optical images, quasars appeared as point sources, indistinguishable from stars, with infrared telescopes and the Hubble Space Telescope, the host galaxies surrounding the quasars have been detected in some cases. These galaxies are normally too dim to be seen against the glare of the quasar, most quasars, with the exception of 3C273 whose average apparent magnitude is 12.9, cannot be seen with small telescopes. The luminosity of some quasars changes rapidly in the optical range, because these changes occur very rapidly they define an upper limit on the volume of a quasar, quasars are not much larger than the Solar System. This implies a high power density. The mechanism of brightness changes probably involves relativistic beaming of astrophysical jets pointed nearly directly toward Earth, the highest redshift quasar known is ULAS J1120+0641, with a redshift of 7.085, which corresponds to a comoving distance of approximately 29 billion light-years from Earth. Since light cannot escape the black holes, the energy is actually generated outside the event horizon by gravitational stresses. Central masses of 105 to 109 solar masses have been measured in quasars by using reverberation mapping. The matter accreting onto the hole is unlikely to fall directly in. Quasars may also be ignited or re-ignited when normal galaxies merge, in fact, it has been suggested that a quasar could form as the Andromeda Galaxy collides with our own Milky Way galaxy in approximately 3–5 billion years. More than 200,000 quasars are known, most from the Sloan Digital Sky Survey, all observed quasar spectra have redshifts between 0.056 and 7.085. Applying Hubbles law to these redshifts, it can be shown that they are between 600 million and 28.85 billion light-years away
Quasar
–
Artist's rendering of
ULAS J1120+0641, a very distant quasar powered by a black hole with a mass two billion times that of the Sun. Credit:
ESO /M. Kornmesser
Quasar
–
A
Hubble picture showing a quasar core
Quasar
–
Quasar QSO-160913+653228 is so distant its light has taken nine billion years to reach us, two thirds of the time that has elapsed since the
Big Bang.
Quasar
–
The
Chandra X-ray image is of the quasar PKS 1127-145, a highly luminous source of X-rays and visible light about 10 billion light years from Earth. An enormous X-ray jet extends at least a million light years from the quasar. Image is 60 arcsec on a side.
RA 11h 30m 7.10s
Dec -14° 49' 27" in Crater. Observation date: May 28, 2000. Instrument: ACIS.
38.
Aries (constellation)
–
Aries is one of the constellations of the zodiac. It is located in the celestial hemisphere between Pisces to the west and Taurus to the east. The name Aries is Latin for ram, and its symbol is and it is one of the 48 constellations described by the 2nd century astronomer Ptolemy, and remains one of the 88 modern constellations. It is a constellation, ranking 39th overall size, with an area of 441 square degrees. Although Aries came to represent specifically the ram whose fleece became the Golden Fleece of Ancient Greek mythology, before that, the stars of Aries formed a farmhand. Different cultures have incorporated the stars of Aries into different constellations including twin inspectors in China, Aries is a relatively dim constellation, possessing only four bright stars, Hamal, Sheratan, Mesarthim, and 41 Arietis. The few deep-sky objects within the constellation are faint and include several pairs of interacting galaxies. Several meteor showers appear to radiate from Aries, including the Daytime Arietids, Aries is now recognized as an official constellation, albeit as a specific region of the sky, by the International Astronomical Union. It was originally defined in ancient texts as a pattern of stars. In the description of the Babylonian zodiac given in the clay tablets known as the MUL. APIN, the MUL. APIN was a comprehensive table of the risings and settings of stars, which likely served as an agricultural calendar. Modern-day Aries was known as MULLÚ. ḪUN. GÁ, The Agrarian Worker or The Hired Man, the earliest identifiable reference to Aries as a distinct constellation comes from the boundary stones that date from 1350 to 1000 BC. On several boundary stones, a zodiacal ram figure is distinct from the characters present. The shift in identification from the constellation as the Agrarian Worker to the Ram likely occurred in later Babylonian tradition because of its association with Dumuzi the Shepherd. By the time the MUL. APIN was created—by 1000 BC—modern Aries was identified with both Dumuzis ram and a hired laborer, the exact timing of this shift is difficult to determine due to the lack of images of Aries or other ram figures. In ancient Egyptian astronomy, Aries was associated with the god Amon-Ra, because it was the location of the vernal equinox, it was called the Indicator of the Reborn Sun. During the times of the year when Aries was prominent, priests would process statues of Amon-Ra to temples, Aries acquired the title of Lord of the Head in Egypt, referring to its symbolic and mythological importance. Aries was not fully accepted as a constellation until classical times, in Hellenistic astrology, the constellation of Aries is associated with the golden ram of Greek mythology that rescued Phrixos and Helle on orders from Hermes, taking Phrixos to the land of Colchis. Phrixos and Helle were the son and daughter of King Athamas, the kings second wife, Ino, was jealous and wished to kill his children
Aries (constellation)
–
Aries and
Musca Borealis as depicted in
Urania's Mirror, a set of constellation cards published in London c.1825
Aries (constellation)
–
List of stars in Aries
Aries (constellation)
–
The constellation Aries as it can be seen with the naked eye
Aries (constellation)
–
NGC 772, with a notated
supernova.
39.
Moon
–
The Moon is an astronomical body that orbits planet Earth, being Earths only permanent natural satellite. It is the fifth-largest natural satellite in the Solar System, following Jupiters satellite Io, the Moon is second-densest satellite among those whose densities are known. The average distance of the Moon from the Earth is 384,400 km, the Moon is thought to have formed about 4.51 billion years ago, not long after Earth. It is the second-brightest regularly visible celestial object in Earths sky, after the Sun and its surface is actually dark, although compared to the night sky it appears very bright, with a reflectance just slightly higher than that of worn asphalt. Its prominence in the sky and its cycle of phases have made the Moon an important cultural influence since ancient times on language, calendars, art. The Moons gravitational influence produces the ocean tides, body tides, and this matching of apparent visual size will not continue in the far future. The Moons linear distance from Earth is currently increasing at a rate of 3.82 ±0.07 centimetres per year, since the Apollo 17 mission in 1972, the Moon has been visited only by uncrewed spacecraft. The usual English proper name for Earths natural satellite is the Moon, the noun moon is derived from moone, which developed from mone, which is derived from Old English mōna, which ultimately stems from Proto-Germanic *mǣnōn, like all Germanic language cognates. Occasionally, the name Luna is used, in literature, especially science fiction, Luna is used to distinguish it from other moons, while in poetry, the name has been used to denote personification of our moon. The principal modern English adjective pertaining to the Moon is lunar, a less common adjective is selenic, derived from the Ancient Greek Selene, from which is derived the prefix seleno-. Both the Greek Selene and the Roman goddess Diana were alternatively called Cynthia, the names Luna, Cynthia, and Selene are reflected in terminology for lunar orbits in words such as apolune, pericynthion, and selenocentric. The name Diana is connected to dies meaning day, several mechanisms have been proposed for the Moons formation 4.51 billion years ago, and some 60 million years after the origin of the Solar System. These hypotheses also cannot account for the angular momentum of the Earth–Moon system. This hypothesis, although not perfect, perhaps best explains the evidence, eighteen months prior to an October 1984 conference on lunar origins, Bill Hartmann, Roger Phillips, and Jeff Taylor challenged fellow lunar scientists, You have eighteen months. Go back to your Apollo data, go back to computer, do whatever you have to. Dont come to our conference unless you have something to say about the Moons birth, at the 1984 conference at Kona, Hawaii, the giant impact hypothesis emerged as the most popular. Afterward there were only two groups, the giant impact camp and the agnostics. Giant impacts are thought to have been common in the early Solar System, computer simulations of a giant impact have produced results that are consistent with the mass of the lunar core and the present angular momentum of the Earth–Moon system
Moon
–
Full moon as seen from Earth's
northern hemisphere
Moon
–
The Moon, tinted reddish, during a
lunar eclipse
Moon
–
Near side of the Moon
Moon
–
Far side of the Moon
40.
Meteorite
–
When the object enters the atmosphere, various factors like friction, pressure, and chemical interactions with the atmospheric gases cause it to heat up and radiate that energy. It then becomes a meteor and forms a fireball, also known as a shooting/falling star, meteorites that survive atmospheric entry and impact vary greatly in size. For geologists, a bolide is a large enough to create a crater. Meteorites that are recovered after being observed as they transit the atmosphere or impact the Earth are called meteorite falls, all others are known as meteorite finds. As of April 2016, there were about 1,140 witnessed falls that have specimens in the worlds collections, there are more than 38,660 well-documented meteorite finds. Modern classification schemes divide meteorites into groups according to their structure, chemical and isotopic composition, meteorites smaller than 2 mm are classified as micrometeorites. Extraterrestrial meteorites are such objects that have impacted other celestial bodies and they have been found on the Moon and Mars. Meteorites are always named for the places they were found, usually a town or geographic feature. In cases where many meteorites were found in one place, the name may be followed by a number or letter, the name designated by the Meteoritical Society is used by scientists, catalogers, and most collectors. Most meteoroids disintegrate when entering the Earths atmosphere, usually, five to ten a year are observed to fall and are subsequently recovered and made known to scientists. Few meteorites are large enough to create large impact craters, instead, they typically arrive at the surface at their terminal velocity and, at most, create a small pit. Large meteoroids may strike the ground with a significant fraction of their escape velocity, the kind of crater will depend on the size, composition, degree of fragmentation, and incoming angle of the impactor. The force of such collisions has the potential to cause widespread destruction, the most frequent hypervelocity cratering events on the Earth are caused by iron meteoroids, which are most easily able to transit the atmosphere intact. In contrast, even relatively large stony or icy bodies like small comets or asteroids, up to millions of tons, are disrupted in the atmosphere, and do not make impact craters. Although such disruption events are uncommon, they can cause a concussion to occur. Very large stony objects, hundreds of meters in diameter or more, weighing tens of millions of tons or more, can reach the surface and cause large craters, such events are generally so energetic that the impactor is completely destroyed, leaving no meteorites. Several phenomena are well documented during witnessed meteorite falls too small to produce hypervelocity craters, various colors have been reported, including yellow, green, and red. Flashes and bursts of light can occur as the object breaks up, explosions, detonations, and rumblings are often heard during meteorite falls, which can be caused by sonic booms as well as shock waves resulting from major fragmentation events
Meteorite
–
The
Hoba meteorite in
Namibia is the largest known intact meteorite, 2.7 metres long and 60 tonnes weight.
Meteorite
–
The 'crater' made by a 61.9 gram
Novato meteorite when it hit a person's roof on October 17, 2012.
Meteorite
–
A
bolide: a very bright meteor of an apparent magnitude of −14 or brighter.
Meteorite
–
NWA 859 iron meteorite showing effects of atmospheric ablation
41.
Oman
–
Oman, officially the Sultanate of Oman, is an Arab country on the southeastern coast of the Arabian Peninsula. The coast is formed by the Arabian Sea on the southeast, the Madha and Musandam exclaves are surrounded by the UAE on their land borders, with the Strait of Hormuz and Gulf of Oman forming Musandams coastal boundaries. From the late 17th century, the Omani Sultanate was an empire, vying with Portugal and Britain for influence in the Persian Gulf. At its peak in the 19th century, Omani influence or control extended across the Strait of Hormuz to modern-day Iran and Pakistan, as its power declined in the 20th century, the sultanate came under the influence of the United Kingdom. Historically, Muscat was the trading port of the Persian Gulf region. Muscat was also among the most important trading ports of the Indian Ocean, the Sultan Qaboos bin Said al Said has been the hereditary leader of the country since 1970. Sultan Qaboos is the current ruler in the Middle East. Oman has modest oil reserves, ranking 25th globally, nevertheless, in 2010 the UNDP ranked Oman as the most improved nation in the world in terms of development during the preceding 40 years. A significant portion of its economy is tourism and trade of fish, dates and this sets it apart from its neighbors solely oil-dependent economies. Oman is categorized as an economy and ranks as the 74th most peaceful country in the world according to the Global Peace Index. Two optically stimulated luminescence age estimates place the Arabian Nubian Complex at 106,000 years old and this supports the proposition that early human populations moved from Africa into Arabia during the Late Pleistocene. Dereaze, located in the city of Ibri, is the oldest known settlement in the area. Archaeological remains have been discovered here from the Stone Age and the Bronze Age, findings have included stone implements, animal bones, shells and fire hearths, with the latter dating back to 7615 BC as the oldest signs of human settlement in the area. Other discoveries include hand-molded pottery bearing distinguishing pre-Bronze Age marks, heavy flint implements, pointed tools, sumerian tablets refer to a country called Magan or Makan, a name believed to refer to Omans ancient copper mines. Mazoon, another used for the region, is derived from the word muzn. The present-day name of the country, Oman, is believed to originate from the Arab tribes who migrated to its territory from the Uman region of Yemen. Many such tribes settled in Oman, making a living by fishing, herding or stock breeding, from the 6th century BC to the arrival of Islam in the 7th century AD, Oman was controlled and/or influenced by three Persian dynasties, the Achaemenids, Parthians and Sassanids. A few scholars believe that in the 6th century BC, the Achaemenids exerted a strong degree of control over the Omani peninsula, Central Oman has its own indigenous so-called Late Iron Age cultural assemblage, the Samad al-Shan
Oman
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A grave at
Al Ayn, Oman, a World Heritage site.
Oman
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Flag
Oman
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The
Sultan's Palace in
Zanzibar, which was once Oman's capital and residence of its Sultans.
Oman
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The coast of
Sur, Oman.
42.
Saskatchewan, Canada
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Saskatchewan is a prairie and boreal province in west-central Canada, the only province without natural borders. It has an area of 651,900 square kilometres, nearly 10 percent of which is water, composed mostly of rivers, reservoirs. As of December 2013, Saskatchewans population was estimated at 1,114,170, residents primarily live in the southern prairie half of the province, while the northern boreal half is mostly forested and sparsely populated. Of the total population, roughly half live in the provinces largest city, Saskatoon, or the provincial capital, other notable cities include Prince Albert, Moose Jaw, Yorkton, Swift Current, North Battleford, and the border city Lloydminster. Saskatchewan is a province with large distances to moderating bodies of waters. As a result, its climate is continental, rendering severe winters throughout the province. Southern areas have very warm or hot summers, Midale and Yellow Grass near the U. S. border are tied for the highest ever recorded temperatures in Canada with 45 °C observed at both locations on July 5,1937. In winter, temperatures below −45 °C are possible even in the south during extreme cold snaps, Saskatchewan has been inhabited for thousands of years by various indigenous groups, and first explored by Europeans in 1690 and settled in 1774. It became a province in 1905, carved out from the vast North-West Territories, in the early 20th century the province became known as a stronghold for Canadian social democracy, North Americas first social-democratic government was elected in 1944. The provinces economy is based on agriculture, mining, and energy, Saskatchewans current premier is Brad Wall and its lieutenant-governor is Vaughn Solomon Schofield. In 1992, the federal and provincial governments signed a land claim agreement with First Nations in Saskatchewan. The First Nations received compensation and were permitted to buy land on the market for the tribes, they have acquired about 3,079 square kilometres. Some First Nations have used their settlement to invest in urban areas and its name derived from the Saskatchewan River. The river was known as kisiskāciwani-sīpiy in the Cree language, as Saskatchewans borders largely follow the geographic coordinates of longitude and latitude, the province is roughly a quadrilateral, or a shape with four sides. However the 49th parallel boundary and the 60th northern border appear curved on globes, additionally, the eastern boundary of the province is partially crooked rather than following a line of longitude, as correction lines were devised by surveyors prior to the homestead program. S. States of Montana and North Dakota, Saskatchewan has the distinction of being the only Canadian province for which no borders correspond to physical geographic features. Along with Alberta, Saskatchewan is one of only two land-locked provinces, the overwhelming majority of Saskatchewans population is located in the southern third of the province, south of the 53rd parallel. Saskatchewan contains two natural regions, the Canadian Shield in the north and the Interior Plains in the south
Saskatchewan, Canada
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Henry Kelsey sees the
buffalo on the western plains.
Saskatchewan, Canada
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Flag
Saskatchewan, Canada
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Cree Pipe Stem Carrier, a painting of a Plains Cree warrior by
Paul Kane.
Saskatchewan, Canada
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The
Battle of Batoche, 1885
43.
Connecticut
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Connecticut is the southernmost state in the New England region of the northeastern United States. Connecticut is also often grouped along with New York and New Jersey as the Tri-State Area and it is bordered by Rhode Island to the east, Massachusetts to the north, New York to the west, and Long Island Sound to the south. Its capital city is Hartford, and its most populous city is Bridgeport, the state is named for the Connecticut River, a major U. S. river that approximately bisects the state. The word Connecticut is derived from various anglicized spellings of an Algonquian word for long tidal river, Connecticut is the third smallest state by area, the 29th most populous, and the fourth most densely populated of the 50 United States. It is known as the Constitution State, the Nutmeg State, the Provisions State, and it was influential in the development of the federal government of the United States. Connecticuts center of population is in Cheshire, New Haven County, Connecticuts first European settlers were Dutch. They established a small, short-lived settlement in present-day Hartford at the confluence of the Park, initially, half of Connecticut was a part of the Dutch colony New Netherland, which included much of the land between the Connecticut and Delaware rivers. The first major settlements were established in the 1630s by England, the Connecticut and New Haven Colonies established documents of Fundamental Orders, considered the first constitutions in North America. In 1662, the three colonies were merged under a charter, making Connecticut a crown colony. This colony was one of the Thirteen Colonies that revolted against British rule in the American Revolution, the Connecticut River, Thames River, and ports along the Long Island Sound have given Connecticut a strong maritime tradition which continues today. The state also has a history of hosting the financial services industry, including insurance companies in Hartford. As of the 2010 Census, Connecticut features the highest per-capita income, Human Development Index, and median household income in the United States. Landmarks and Cities of Connecticut Connecticut is bordered on the south by Long Island Sound, on the west by New York, on the north by Massachusetts, and on the east by Rhode Island. The state capital and third largest city is Hartford, and other cities and towns include Bridgeport, New Haven, Stamford, Waterbury, Norwalk, Danbury, New Britain, Greenwich. Connecticut is slightly larger than the country of Montenegro, there are 169 incorporated towns in Connecticut. The highest peak in Connecticut is Bear Mountain in Salisbury in the northwest corner of the state, the highest point is just east of where Connecticut, Massachusetts, and New York meet, on the southern slope of Mount Frissell, whose peak lies nearby in Massachusetts. At the opposite extreme, many of the towns have areas that are less than 20 feet above sea level. Connecticut has a maritime history and a reputation based on that history—yet the state has no direct oceanfront
Connecticut
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Bear Mountain, highest peak in Connecticut
Connecticut
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Flag
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Lake Mcdonough reservoir as seen from the Tunxis Trail Overlook Spur trail, Barkhamsted
Connecticut
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New Haven
44.
USNS Private Jose F. Valdez (T-AG-169)
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USNS Private Jose F. Valdez, named after World War II Medal of Honor recipient PFC Jose F. Valdez, was a technical research ship in operation during the 1960s. The Galloping Ghost of the Ivory Coast or Grey Ghost of the African Coast, Private Jose F. Valdez, originally Joe P. Martinez, was laid down by Walter Butler Shipbuilders, Inc. The Round Splice was transferred to the War Department 30 August 1946, on 2 September 1950 she was acquired by the United States Navy, designated T-APC-119, and assigned to Military Sea Transportation Service. Manned by a service crew she operated in the Gulf of Mexico. Between then and December she cruised the Mediterranean Sea and in January 1952 began runs to Newfoundland and Greenland which continued until she was ordered inactivated in late 1959. On 22 December she arrived in the James River National Defense Reserve Fleet berthing area and was transferred to the custody of the Maritime Administration, Private Jose F. Valdez was reacquired by the Navy in August 1961. The USNS designation indicates that the ship was manned by civilians, a crew of approximately 55 civilians operated the ship while a detachment of approximately 100 Navy personnel carried out the research operations. The Navy detachment typically included three officers, almost all enlisted men were Communications Technicians, an advantage of the USNS designation is that the ship was not required to return to an American port on a regular basis. Thus the first deployment of Private Jose F. Valdez started in 1961, since the Happy Jose did not regularly return to the USA, the crew was rotated by flying them to a major port city in Africa, such as Cape Town. This occurred on an annual basis, the old crew would be flown back to the USA. Private Jose F. Valdez was typically at sea for about 30 days, in May 1967 tensions were rising in the Middle East between Israel and her Arab neighbors, this resulted in the Six-Day War in June 1967. The National Security Agency decided to deploy a SIGINT collection ship to the area to monitor the situation, most of the technical research ships were too far away, Oxford and Jamestown were in Southeast Asia, Georgetown and Belmont were in South America, and USNS Sgt. Joseph E. Muller was off Cuba, choice of a ship for the operation narrowed between Private Jose F. Valdez, then headed from the eastern Mediterranean to Gibraltar, and USS Liberty in port at Abidjan, Ivory Coast. On 23 May 1967 Liberty was diverted for duty in the eastern Mediterranean, Liberty stopped at Rota on 1 June and departed the next day for the eastern Mediterranean. Eastbound Liberty passed westbound Private Jose F. Valdez on the night of June 5/6, June 7 Contact X removed from Libertys navigation chart. Seven days after arriving Rota, Liberty was attacked by Israeli forces and suffered heavy damages, Private Jose F. Valdez arrived in Bayonne, New Jersey in June 1967. After repair and overhaul, Private Jose F. Valdez departed for her second extended tour in the African region on 18 September 1967 and this was not a new system, it had already been used on Liberty and Oxford. But its major disadvantage is that it could work if the moon was visible
USNS Private Jose F. Valdez (T-AG-169)
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History
USNS Private Jose F. Valdez (T-AG-169)
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Shellback patch from 1964.
45.
United States Navy
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The United States Navy is the naval warfare service branch of the United States Armed Forces and one of the seven uniformed services of the United States. The U. S. Navy is the largest, most capable navy in the world, the U. S. Navy has the worlds largest aircraft carrier fleet, with ten in service, two in the reserve fleet, and three new carriers under construction. The service has 323,792 personnel on duty and 108,515 in the Navy Reserve. It has 274 deployable combat vessels and more than 3,700 operational aircraft as of October 2016, the U. S. Navy traces its origins to the Continental Navy, which was established during the American Revolutionary War and was effectively disbanded as a separate entity shortly thereafter. It played a role in the American Civil War by blockading the Confederacy. It played the role in the World War II defeat of Imperial Japan. The 21st century U. S. Navy maintains a global presence, deploying in strength in such areas as the Western Pacific, the Mediterranean. The Navy is administratively managed by the Department of the Navy, the Department of the Navy is itself a division of the Department of Defense, which is headed by the Secretary of Defense. The Chief of Naval Operations is an admiral and the senior naval officer of the Department of the Navy. The CNO may not be the highest ranking officer in the armed forces if the Chairman or the Vice Chairman of the Joint Chiefs of Staff. The mission of the Navy is to maintain, train and equip combat-ready Naval forces capable of winning wars, deterring aggression, the United States Navy is a seaborne branch of the military of the United States. The Navys three primary areas of responsibility, The preparation of naval forces necessary for the prosecution of war. The development of aircraft, weapons, tactics, technique, organization, U. S. Navy training manuals state that the mission of the U. S. Armed Forces is to prepare and conduct prompt and sustained combat operations in support of the national interest, as part of that establishment, the U. S. Navys functions comprise sea control, power projection and nuclear deterrence, in addition to sealift duties. It follows then as certain as that night succeeds the day, that without a decisive naval force we can do nothing definitive, the Navy was rooted in the colonial seafaring tradition, which produced a large community of sailors, captains, and shipbuilders. In the early stages of the American Revolutionary War, Massachusetts had its own Massachusetts Naval Militia, the establishment of a national navy was an issue of debate among the members of the Second Continental Congress. Supporters argued that a navy would protect shipping, defend the coast, detractors countered that challenging the British Royal Navy, then the worlds preeminent naval power, was a foolish undertaking. Commander in Chief George Washington resolved the debate when he commissioned the ocean-going schooner USS Hannah to interdict British merchant ships, and reported the captures to the Congress
United States Navy
United States Navy
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United States Navy portal
United States Navy
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USS Constellation vs L'Insurgente during the
Quasi-War
United States Navy
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USS Constitution vs HMS Guerriere during the
War of 1812
46.
Technical research ship
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At the time these ships were active, the mission of the ships was covert and discussion of the true mission was prohibited. The mission of the ships was given as conducting research into atmospheric. However, the mission was more or less an open secret. These ships carried a crew of U. S. Navy personnel whose specialty was intercepting wireless electronic communications, in the 1960s those personnel had a U. S. Navy rating of Communications Technician, or CT. Communications could occur only when the moon was visible simultaneously at the ships location, the gyro stabilization of the antenna kept the antenna pointed at the moon while the ship rolled and pitched on the surface of the ocean. These ships were classified as auxiliaries with a hull designation of AGTR. Five of these ships were built with hull numbers of 1–5, the first three ships of this type were converted from World War II-era Liberty ships. The last two ships were converted from Victory ships, the former Liberty ships top speed of 11 knots limited the first three AGTRs to missions of slow steaming on station with a minimum of transits. Victory ships sustained speed of 18 knots enabled Belmont to shadow Mediterranean Sea operations of the Soviet helicopter carrier Moskva in 1969, all of the technical research ships were decommissioned and stricken by 1970. One of these ships received a Presidential Unit Citation for heroism in combat. The USS Liberty was attacked, severely damaged and 34 crew members killed by shelling, napalm bombing and torpedoing from Israeli jet fighter aircraft, the USS Jamestown was awarded a Meritorious Unit Commendation along with the USS Oxford. The citation reads For meritorious service from 1 November 1965 to 30 June 1969 while participating in combat operations in Southeast Asia. Signed E. R. Zumwalt, Admiral, USN, Chief of Naval Operations, for specifications of these ships, see Liberty ship and Victory ship. In contrast to the high freeboard of the AGTR Liberty and Victory hulls, the USS Pueblo, technically still in commission, has been held by North Korea since its attack and capture by on January 23,1968. Two ships were Maritime Commission C1-M-AV1 types, james E. Robinson, was a VC2-S-AP2 type that operated in this role December 1962-April 1964 before being reclassified AK‑274 and resuming cargo operations. USNS Private Jose F. Valdez USNS LT, joseph E. Muller Spy ship USS Parche and USS Jimmy Carter, nuclear submarines modified or designed for intelligence gathering
Technical research ship
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USS Liberty (AGTR-5) Spy ship
Technical research ship
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USS Pueblo (AGER-2)
Technical research ship
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USNS Private Jose F Valdez (T-AG-169)
47.
1960s
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The 1960s was a decade that began on 1 January 1960, and ended on 31 December 1969. The term 1960s also refers to an era more often called the Sixties and this cultural decade is more loosely defined than the actual decade, beginning around 1963 with the Kennedy assassination and ending around 1972 with the Watergate scandal. The decade was also labeled the Swinging Sixties because of the fall or relaxation of social taboos especially relating to racism and sexism that occurred during this time and he charts the rise, success, fall/nightmare and explosion in the London scene of the 1960s. Several Western nations such as the United States, United Kingdom, France, by the end of the 1950s, war-ravaged Europe had largely finished reconstruction and began a tremendous economic boom. World War II had brought about a huge leveling of social classes in which the remnants of the old feudal gentry disappeared, the United States, after sluggish economic growth during the 1950s, also experienced a major 60s boom. Real GDP growth averaged 6% a year during the half of the decade. Thus, the worldwide economic trend in the 1960s was one of prosperity, expansion of the middle class. Kennedys assassination in 1963 was a shock, Liberal reforms were finally passed under Lyndon B. Johnson including civil rights for African Americans and healthcare for the elderly and the poor. Despite his large-scale Great Society programs, Johnson was increasingly reviled by the New Left at home, the heavy-handed American role in the Vietnam War outraged student protestors around the globe. The assassination of Martin Luther King, Jr, in Britain, the Labour Party gained power in 1964. In France, the protests of 1968 led to President Charles de Gaulle temporarily fleeing the country, for some, May 1968 meant the end of traditional collective action and the beginning of a new era to be dominated mainly by the so-called new social movements. Italy formed its first left-of-center government in March 1962 with a coalition of Christian Democrats, Social Democrats, socialists joined the ruling block in December 1963. In Brazil, João Goulart became president after Jânio Quadros resigned, in Africa the 1960s was a period of radical political change as 32 countries gained independence from their European colonial rulers. The Cold War, The Vietnam War 1961 – Substantial American advisory forces first arrive in Vietnam,1962 – By mid-1962, the number of U. S. military advisers in South Vietnam had risen from 900 to 12,000. The resolution gave U. S. President Lyndon B. Johnson authorization, without a declaration of war by Congress. The Johnson administration subsequently cited the resolution as legal authority for its rapid escalation of U. S. military involvement in the Vietnam War. 1966 – After 1966 with the draft in more than 500,000 troops were sent to Vietnam by the Johnson administration. Portuguese Colonial War – the war was fought between Portugals military and the emerging nationalist movements in Portugals African colonies and it was a decisive ideological struggle and armed conflict of the cold war in African and European scenarios
1960s
1960s
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Vietnam War (1955–1975)
1960s
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A child suffering the effects of severe hunger and
malnutrition during the Nigerian blockade of
Biafra 1967–1970.
1960s
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Pictures of Soviet missile silos in
Cuba, taken by United States spy planes on 14 October 1962.
48.
USS Chatham (AK-169)
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USS Chatham was an Alamosa-class cargo ship commissioned by the U. S. Navy for service in World War II. She was responsible for delivering troops, goods and equipment to locations in the war zone, the third Chatham commissioned by the Navy, was launched 13 May 1944 by Froemming Brothers, Inc. Milwaukee, Wisconsin, under a Maritime Commission contract, MC hull 2142, salisbury, acquired by the Navy 20 January 1945, and commissioned at Galveston, Texas,22 February 1945, Lieutenant Commander N. C. Chatham arrived at Pearl Harbor 6 May 1945 to carry cargo to Eniwetok, Saipan, from the US West Coast, she sailed to Baltimore, Maryland, where she was decommissioned 2 April 1946 and returned to the Maritime Commission,4 April 1946. She was sold in 1963 to the Bahamas Line, Panama and she broke in two and sank 15 December 1972, in heavy weather West of San Juan, Puerto Rico, with a load of Gypsum. All but one of her crew were rescued by the USCG buoy tender Sagebrush, the record does not indicate any battle stars for Chatham
USS Chatham (AK-169)
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USS Chatham (AK-169) departing an island port in the Pacific, circa mid-1945.
49.
Cargo ship
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A cargo ship or freighter is any sort of ship or vessel that carries cargo, goods, and materials from one port to another. Thousands of cargo carriers ply the worlds seas and oceans each year, cargo ships are usually specially designed for the task, often being equipped with cranes and other mechanisms to load and unload, and come in all sizes. Today, they are almost always built by welded steel, cargo ships/freighters can be divided into five groups, according to the type of cargo they carry. Tankers carry petroleum products or other liquid cargo, dry bulk carriers carry coal, grain, ore and other similar products in loose form. Multi-purpose vessels, as the name suggests, carry different classes of cargo – e. g. liquid, a Reefer ship is specifically designed and used for shipping perishable commodities which require temperature-controlled, mostly fruits, meat, fish, vegetables, dairy products and other foodstuffs. Specialized types of cargo vessels include ships and bulk carriers. Cargo ships fall into two categories that reflect the services they offer to industry, liner and tramp services. Those on a published schedule and fixed tariff rates are cargo liners. Tramp ships do not have fixed schedules, users charter them to haul loads. Generally, the shipping companies and private individuals operate tramp ships. Cargo liners run on fixed schedules published by the shipping companies, each trip a liner takes is called a voyage. However, some cargo liners may carry passengers also, a cargo liner that carries 12 or more passengers is called a combination or passenger-cum-cargo line. The desire to trade routes over longer distances, and throughout more seasons of the year. Before the middle of the 19th century, the incidence of piracy resulted in most cargo ships being armed, sometimes heavily, as in the case of the Manila galleons. They were also escorted by warships. Piracy is still common in some waters, particularly in the Malacca Straits. In 2004, the governments of three nations agreed to provide better protection for the ships passing through the Straits. The waters off Somalia and Nigeria are also prone to piracy, while smaller vessels are also in danger along parts of the South American, Southeast Asian coasts, the words cargo and freight have become interchangeable in casual usage
Cargo ship
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The
Colombo Express, one of the largest container ships in the world (when she was built in 2005), owned and operated by
Hapag-Lloyd of
Germany
Cargo ship
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Loading of a general cargo vessel in 1959
Cargo ship
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A Delmas container ship unloading at the Zanzibar port in Tanzania
50.
World War II
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World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945, although related conflicts began earlier. It involved the vast majority of the worlds countries—including all of the great powers—eventually forming two opposing alliances, the Allies and the Axis. It was the most widespread war in history, and directly involved more than 100 million people from over 30 countries. Marked by mass deaths of civilians, including the Holocaust and the bombing of industrial and population centres. These made World War II the deadliest conflict in human history, from late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, and formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Poland, Finland, Romania and the Baltic states. In December 1941, Japan attacked the United States and European colonies in the Pacific Ocean, and quickly conquered much of the Western Pacific. The Axis advance halted in 1942 when Japan lost the critical Battle of Midway, near Hawaii, in 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained all of its territorial losses and invaded Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in South Central China and Burma, while the Allies crippled the Japanese Navy, thus ended the war in Asia, cementing the total victory of the Allies. World War II altered the political alignment and social structure of the world, the United Nations was established to foster international co-operation and prevent future conflicts. The victorious great powers—the United States, the Soviet Union, China, the United Kingdom, the Soviet Union and the United States emerged as rival superpowers, setting the stage for the Cold War, which lasted for the next 46 years. Meanwhile, the influence of European great powers waned, while the decolonisation of Asia, most countries whose industries had been damaged moved towards economic recovery. Political integration, especially in Europe, emerged as an effort to end pre-war enmities, the start of the war in Europe is generally held to be 1 September 1939, beginning with the German invasion of Poland, Britain and France declared war on Germany two days later. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or even the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred simultaneously and this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935. The British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the forces of Mongolia and the Soviet Union from May to September 1939, the exact date of the wars end is also not universally agreed upon. It was generally accepted at the time that the war ended with the armistice of 14 August 1945, rather than the formal surrender of Japan
World War II
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Clockwise from top left: Chinese forces in the
Battle of Wanjialing, Australian
25-pounder guns during the
First Battle of El Alamein, German
Stuka dive bombers on the
Eastern Front in December 1943, a U.S. naval force in the
Lingayen Gulf,
Wilhelm Keitel signing the
German Instrument of Surrender, Soviet troops in the
Battle of Stalingrad
World War II
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The
League of Nations assembly, held in
Geneva,
Switzerland, 1930
World War II
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Adolf Hitler at a German
National Socialist political rally in
Weimar, October 1930
World War II
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Italian soldiers recruited in 1935, on their way to fight the
Second Italo-Abyssinian War