1.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
2.
Addition
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Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two together, making a total of five apples. This observation is equivalent to the mathematical expression 3 +2 =5 i. e.3 add 2 is equal to 5, besides counting fruits, addition can also represent combining other physical objects. In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others, in algebra, addition is studied more abstractly. It is commutative, meaning that order does not matter, and it is associative, repeated addition of 1 is the same as counting, addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication, performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers, the most basic task,1 +1, can be performed by infants as young as five months and even some members of other animal species. In primary education, students are taught to add numbers in the system, starting with single digits. Mechanical aids range from the ancient abacus to the modern computer, Addition is written using the plus sign + between the terms, that is, in infix notation. The result is expressed with an equals sign, for example, 3½ =3 + ½ =3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead, the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example, ∑ k =15 k 2 =12 +22 +32 +42 +52 =55. The numbers or the objects to be added in addition are collectively referred to as the terms, the addends or the summands. This is to be distinguished from factors, which are multiplied, some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an addend at all, today, due to the commutative property of addition, augend is rarely used, and both terms are generally called addends. All of the above terminology derives from Latin, using the gerundive suffix -nd results in addend, thing to be added. Likewise from augere to increase, one gets augend, thing to be increased, sum and summand derive from the Latin noun summa the highest, the top and associated verb summare
3.
Subtraction
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Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign, for example, in the picture on the right, there are 5 −2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. It is anticommutative, meaning that changing the order changes the sign of the answer and it is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 does not change a number, subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers, general binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks, subtraction of very small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the system, starting with single digits. Subtraction is written using the minus sign − between the terms, that is, in infix notation, the result is expressed with an equals sign. This is most common in accounting, formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. All of this terminology derives from Latin, subtraction is an English word derived from the Latin verb subtrahere, which is in turn a compound of sub from under and trahere to pull, thus to subtract is to draw from below, take away. Using the gerundive suffix -nd results in subtrahend, thing to be subtracted, likewise from minuere to reduce or diminish, one gets minuend, thing to be diminished. Imagine a line segment of length b with the left end labeled a, starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition, a + b = c, from c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction, c − b = a, now, a line segment labeled with the numbers 1,2, and 3. From position 3, it takes no steps to the left to stay at 3 and it takes 2 steps to the left to get to position 1, so 3 −2 =1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3, to represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number, from 3, it takes 3 steps to the left to get to 0, so 3 −3 =0. But 3 −4 is still invalid since it leaves the line
4.
Multiplication
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Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
5.
Division (mathematics)
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Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture on the right, the 20 apples are divided into groups of five apples, Division can also be thought of as the process of evaluating a fraction, and fractional notation is commonly used to represent division. Division is the inverse of multiplication, if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the numbers and most other contexts, because if b =0, then a cannot be deduced from b and c. In some contexts, division by zero can be defined although to a limited extent, in division, the dividend is divided by the divisor to get a quotient. In the above example,20 is the dividend, five is the divisor, in some cases, the divisor may not be contained fully by the dividend, for example,10 ÷3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. Besides dividing apples, division can be applied to other physical, Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces and fields. Division is the most mentally difficult of the four operations of arithmetic. Teaching the objective concept of dividing integers introduces students to the arithmetic of fractions, unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder, to complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. When students advance to algebra, the theory of division intuited from arithmetic naturally extends to algebraic division of variables, polynomials. Division is often shown in algebra and science by placing the dividend over the divisor with a line, also called a fraction bar. For example, a divided by b is written a b This can be read out loud as a divided by b, a fraction is a division expression where both dividend and divisor are integers, and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus, common in arithmetic, in this manner, ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the operation itself. In some non-English-speaking cultures, a divided by b is written a, b and this notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz
6.
Summation
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In mathematics, summation is the addition of a sequence of numbers, the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a sum, prefix sum. The numbers to be summed may be integers, rational numbers, real numbers, besides numbers, other types of values can be added as well, vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of elements, summation always produces a well-defined sum. The summation of a sequence of values is called a series. A value of such a series may often be defined by means of a limit, another notion involving limits of finite sums is integration. The summation of the sequence is an expression whose value is the sum of each of the members of the sequence, in the example,1 +2 +4 +2 =9. Addition is also commutative, so permuting the terms of a sequence does not change its sum. There is no notation for the summation of such explicit sequences. If, however, the terms of the sequence are given by a pattern, possibly of variable length. For the summation of the sequence of integers from 1 to 100. In this case, the reader can guess the pattern. However, for more complicated patterns, one needs to be precise about the used to find successive terms. Using this sigma notation the above summation is written as, ∑ i =1100 i, the value of this summation is 5050. It can be found without performing 99 additions, since it can be shown that ∑ i =1 n i = n 2 for all natural numbers n, more generally, formulae exist for many summations of terms following a regular pattern. By contrast, summation as discussed in this article is called definite summation, when it is necessary to clarify that numbers are added with their signs, the term algebraic sum is used. Mathematical notation uses a symbol that compactly represents summation of many terms, the summation symbol, ∑. The i = m under the symbol means that the index i starts out equal to m
7.
Product (mathematics)
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In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance,6 is the product of 2 and 3, the order in which real or complex numbers are multiplied has no bearing on the product, this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, matrix multiplication, for example, and multiplication in other algebras is in general non-commutative. There are many different kinds of products in mathematics, besides being able to multiply just numbers, polynomials or matricies, an overview of these different kinds of products is given here. Placing several stones into a pattern with r rows and s columns gives r ⋅ s = ∑ i =1 s r = ∑ j =1 r s stones. Integers allow positive and negative numbers, the product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, the product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏. The product of a sequence consisting of one number is just that number itself. The product of no factors at all is known as the empty product, commutative rings have a product operation. Under the Fourier transform, convolution becomes point-wise function multiplication, others have very different names but convey essentially the same idea. A brief overview of these is given here, by the very definition of a vector space, one can form the product of any scalar with any vector, giving a map R × V → V. A scalar product is a map, ⋅, V × V → R with the following conditions. From the scalar product, one can define a norm by letting ∥ v ∥, = v ⋅ v, now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to U, then one can get g ∘ f = g = g j k f i j v i b U k. Or in matrix form, g ∘ f = G F v, in which the i-row, j-column element of F, denoted by Fij, is fji, the composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication. To see this, let r = dim, s = dim, let U = be a basis of U, V = be a basis of V and W = be a basis of W. Then B ⋅ A = M W U ∈ R s × t is the matrix representing g ∘ f, U → W, in other words, the matrix product is the description in coordinates of the composition of linear functions. For inifinite-dimensional vector spaces, one also has the, Tensor product of Hilbert spaces Topological tensor product, the tensor product, outer product and Kronecker product all convey the same general idea
8.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
9.
Hippolytus of Rome
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Hippolytus of Rome was the most important 3rd-century theologian in the Christian Church in Rome, where he was probably born. He came into conflict with the popes of his time and seems to have headed a group as a rival to the Bishop of Rome. He opposed the Roman bishops who softened the penitential system to accommodate the number of new pagan converts. However, he was probably reconciled to the Church when he died as a martyr. Starting in the 4th century AD, various legends arose about him and he has also been confused with another martyr of the same name. As a presbyter of the church at Rome under Pope Zephyrinus, Hippolytus was distinguished for his learning and it was at this time that Origen of Alexandria, then a young man, heard him preach. He accused Pope Zephyrinus of modalism, the heresy which held that the names Father, Hippolytus championed the Logos doctrine of the Greek apologists, most notably Justin Martyr, which distinguished the Father from the Logos. An ethical conservative, he was scandalized when Pope Callixtus I extended absolution to Christians who had committed grave sins, Hippolytus himself advocated an excessive rigorism. At this time, he seems to have allowed himself to be elected as a rival Bishop of Rome, G. Salmon suggests that Hippolytus was the leader of the Greek-speaking Christians of Rome. Allen Brent sees the development of Roman house-churches into something akin to Greek philosophical schools gathered around a compelling teacher, under the persecution at the time of Emperor Maximinus Thrax, Hippolytus and Pontian were exiled together in 235 AD to Sardinia, likely dying in the mines. It is quite probable that, before his death there, he was reconciled to the party at Rome, for, under Pope Fabian, his body. The facts of his life as well as his writing were soon forgotten in the West, perhaps by reason of his criticism of the bishops of Rome and because he wrote in Greek. In the Passionals of the 7th and 8th centuries he is represented as a converted by Saint Lawrence. He was also confused with a martyr of the name who was buried in Portus, of which city he was believed to have been a bishop. According to Prudentius account, Hippolytus was dragged to death by horses, a striking parallel to the story of the mythological Hippolytus. He described the tomb of the saint and states that he saw there a picture representing Hippolytus’ execution. He also confirms August 13 as the date on which a Hippolytus was celebrated but this refers to the convert of Lawrence. The latter account led to Hippolytus being considered the saint of horses
10.
Iamblichus
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Iamblichus, also known as Iamblichus Chalcidensis, or Iamblichus of Apamea, was a Syrian Neoplatonist philosopher who determined the direction taken by later Neoplatonic philosophy. Iamblichus was the representative of Syrian Neoplatonism, though his influence spread over much of the ancient world. The events of his life and his religious beliefs are not entirely known, according to the Suda, and his biographer Eunapius, he was born at Chalcis in Syria. He was the son of a rich and illustrious family, and he initially studied under Anatolius of Laodicea, and later went on to study under Porphyry, a pupil of Plotinus, the founder of Neoplatonism. He disagreed with Porphyry over the practice of theurgy, Iamblichus responds to Porphyrys criticisms of theurgy in a book attributed to him, around 304, he returned to Syria to found his own school at Apameia, a city famous for its Neoplatonic philosophers. Here he designed a curriculum for studying Plato and Aristotle, still, for Iamblichus, Pythagoras was the supreme authority. He is known to have written the Collection of Pythagorean Doctrines, only the first four books, and fragments of the fifth, survive. Scholars noted that the Exhortation to Philosophy of Iamblichus was composed in Apamea in the early 4th c, Iamblichus was said to have been a man of great culture and learning. He was also renowned for his charity and self-denial, many students gathered around him, and he lived with them in genial friendship. According to Fabricius, he died during the reign of Constantine, only a fraction of Iamblichus books have survived. For our knowledge of his system, we are indebted partly to the fragments of writings preserved by Stobaeus and others. The notes of his successors, especially Proclus, as well as his five extant books, besides these, Proclus seems to have ascribed to him the authorship of the celebrated treatise Theurgia, or On the Egyptian Mysteries. However, the differences between this book and Iamblichus other works in style and in points of doctrine have led some to question whether Iamblichus was the actual author. As a speculative theory, Neoplatonism had received its highest development from Plotinus, unlike Plotinus who broke from Platonic tradition and asserted an undescended soul, Iamblichus re-affirmed the souls embodiment in matter believing matter to be as divine as the rest of the cosmos. It is most likely on account that lamblichus was venerated. Iamblichus was highly praised by those who followed his thought, by his contemporaries, Iamblichus was accredited with miraculous powers. During the revival of interest in his philosophy in the 15th and 16th centuries, at the head of his system, Iamblichus placed the transcendent incommunicable One, the monad, whose first principle is intellect, nous. Immediately after the absolute One, lamblichus introduced a second superexistent One to stand between it and the many as the producer of intellect, or soul, psyche and these three entities, the psyche, and the nous split into the intelligible and the intellective, form a triad
11.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
12.
Avicenna
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Avicenna or Ibn Sīnā was a Persian polymath who is regarded as one of the most significant thinkers and writers of the Islamic Golden Age. Of the 450 works he is known to have written, around 240 have survived, in 1973, Avicennas Canon Of Medicine was reprinted in New York. Besides philosophy and medicine, Avicennas corpus includes writings on astronomy, alchemy, geography and geology, psychology, Islamic theology, logic, mathematics, physics and poetry. Avicenna is a Latin corruption of the Arabic patronym Ibn Sīnā, meaning Son of Sina, however, Avicenna was not the son, but the great-great-grandson of a man named Sina. His full name was Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn al-Ḥasan ibn ʿAlī ibn Sīnā, Ibn Sina created an extensive corpus of works during what is commonly known as the Islamic Golden Age, in which the translations of Greco-Roman, Persian, and Indian texts were studied extensively. Under the Samanids, Bukhara rivaled Baghdad as a capital of the Islamic world. The study of the Quran and the Hadith thrived in such a scholarly atmosphere, philosophy, Fiqh and theology were further developed, most noticeably by Avicenna and his opponents. Al-Razi and Al-Farabi had provided methodology and knowledge in medicine and philosophy, Avicenna had access to the great libraries of Balkh, Khwarezm, Gorgan, Rey, Isfahan and Hamadan. Various texts show that he debated philosophical points with the greatest scholars of the time, aruzi Samarqandi describes how before Avicenna left Khwarezm he had met Al-Biruni, Abu Nasr Iraqi, Abu Sahl Masihi and Abu al-Khayr Khammar. Avicenna was born c. 980 in Afshana, a village near Bukhara, the capital of the Samanids, a Persian dynasty in Central Asia and Greater Khorasan. His mother, named Setareh, was from Bukhara, his father, Abdullah, was a respected Ismaili scholar from Balkh and his father worked in the government of Samanid in the village Kharmasain, a Sunni regional power. After five years, his brother, Mahmoud, was born. Avicenna first began to learn the Quran and literature in such a way that when he was ten years old he had learned all of them. According to his autobiography, Avicenna had memorised the entire Quran by the age of 10 and he learned Indian arithmetic from an Indian greengrocer, ءMahmoud Massahi and he began to learn more from a wandering scholar who gained a livelihood by curing the sick and teaching the young. He also studied Fiqh under the Sunni Hanafi scholar Ismail al-Zahid, Avicenna was taught some extent of philosophy books such as Introduction s Porphyry, Euclids Elements, Ptolemys Almagest by an unpopular philosopher, Abu Abdullah Nateli, who claimed philosophizing. As a teenager, he was troubled by the Metaphysics of Aristotle. For the next year and a half, he studied philosophy, in such moments of baffled inquiry, he would leave his books, perform the requisite ablutions, then go to the mosque, and continue in prayer till light broke on his difficulties. Deep into the night, he would continue his studies, and even in his dreams problems would pursue him and work out their solution
13.
Buckminster Fuller
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Richard Buckminster Bucky Fuller was an American architect, systems theorist, author, designer, and inventor. Fuller published more than 30 books, coining or popularizing terms such as Spaceship Earth, ephemeralization and he also developed numerous inventions, mainly architectural designs, and popularized the widely known geodesic dome. Carbon molecules known as fullerenes were later named by scientists for their structural and mathematical resemblance to geodesic spheres, Fuller was the second World President of Mensa from 1974 to 1983. Fuller was born on July 12,1895, in Milton, Massachusetts, the son of Richard Buckminster Fuller and Caroline Wolcott Andrews and he spent much of his youth on Bear Island, in Penobscot Bay off the coast of Maine. He often made items from materials he found in the woods and he experimented with designing a new apparatus for human propulsion of small boats. Later in life, Fuller took exception to the term invention, Fuller earned a machinists certification, and knew how to use the press brake, stretch press, and other tools and equipment used in the sheet metal trade. Fuller attended Milton Academy in Massachusetts, and after that studying at Harvard College. He was expelled from Harvard twice, first for spending all his money partying with a vaudeville troupe, by his own appraisal, he was a non-conforming misfit in the fraternity environment. Between his sessions at Harvard, Fuller worked in Canada as a mechanic in a textile mill, and later as a laborer in the meat-packing industry. He also served in the U. S. Navy in World War I, as a radio operator, as an editor of a publication. After discharge, he worked again in the packing industry. In 1917, he married Anne Hewlett, Buckminster Fuller recalled 1927 as a pivotal year of his life. His daughter Alexandra had died in 1922 of complications from polio, Fuller dwelled on her death, suspecting that it was connected with the Fullers damp and drafty living conditions. This provided motivation for Fullers involvement in Stockade Building Systems, a business which aimed to provide affordable, in 1927, at age 32, Fuller lost his job as president of Stockade. The Fuller family had no savings, and the birth of their daughter Allegra in 1927 added to the financial challenges, Fuller drank heavily and reflected upon the solution to his familys struggles on long walks around Chicago. During the autumn of 1927, Fuller contemplated suicide, so that his family could benefit from an insurance payment. Fuller said that he had experienced a profound incident which would provide direction and he felt as though he was suspended several feet above the ground enclosed in a white sphere of light. A voice spoke directly to Fuller, and declared, From now on you need never await temporal attestation to your thought and you do not have the right to eliminate yourself
14.
Signal processing
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According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the digitalization or digital refinement of techniques can be found in the digital control systems of the 1940s and 1950s. Feature extraction, such as understanding and speech recognition. Quality improvement, such as reduction, image enhancement. Including audio compression, image compression, and video compression and this involves linear electronic circuits as well as non-linear ones. The former are, for instance, passive filters, active filters, additive mixers, integrators, non-linear circuits include compandors, multiplicators, voltage-controlled filters, voltage-controlled oscillators and phase-locked loops. Continuous-time signal processing is for signals that vary with the change of continuous domain, the methods of signal processing include time domain, frequency domain, and complex frequency domain. This technology was a predecessor of digital processing, and is still used in advanced processing of gigahertz signals. Digital signal processing is the processing of digitized discrete-time sampled signals, processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors. Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, other typical operations supported by the hardware are circular buffers and look-up tables. Examples of algorithms are the Fast Fourier transform, finite impulse response filter, Infinite impulse response filter, nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatio-temporal domains. Nonlinear systems can produce complex behaviors including bifurcations, chaos, harmonics
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Checksum
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A checksum is a small-sized datum derived from a block of digital data for the purpose of detecting errors which may have been introduced during its transmission or storage. It is usually applied to an installation file after it is received from the download server, by themselves, checksums are often used to verify data integrity but are not relied upon to verify data authenticity. The actual procedure which yields the checksum from an input is called a checksum function or checksum algorithm. Depending on its goals, a good checksum algorithm will usually output a significantly different value. Checksum functions are related to functions, fingerprints, randomization functions. However, each of those concepts has different applications and therefore different design goals, for instance a function returning the start of a string can provide a hash appropriate for some applications but will never be a suitable checksum. Checksums are used as cryptographic primitives in larger authentication algorithms, for cryptographic systems with these two specific design goals, see HMAC. Check digits and parity bits are special cases of checksums, appropriate for small blocks of data, some error-correcting codes are based on special checksums which not only detect common errors but also allow the original data to be recovered in certain cases. The simplest checksum algorithm is the so-called longitudinal parity check, which breaks the data into words with a number n of bits. The result is appended to the message as an extra word, with this checksum, any transmission error which flips a single bit of the message, or an odd number of bits, will be detected as an incorrect checksum. However, an error which affects two bits will not be detected if those bits lie at the position in two distinct words. Also swapping of two or more words will not be detected, if the affected bits are independently chosen at random, the probability of a two-bit error being undetected is 1/n. A variant of the algorithm is to add all the words as unsigned binary numbers, discarding any overflow bits. To validate a message, the receiver adds all the words in the manner, including the checksum, if the result is not a word full of zeros. This variant too detects any single-bit error, but the sum is used in SAE J1708. This feature generally increases the cost of computing the checksum, a message that is m bits long can be viewed as a corner of the m-dimensional hypercube. The effect of an algorithm that yields an n-bit checksum is to map each m-bit message to a corner of a larger hypercube. The 2m+n corners of this hypercube represent all possible received messages, the valid received messages comprise a smaller set, with only 2m corners
16.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
17.
The Mathematical Association of America
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The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, the organization publishes mathematics journals and books, including the American Mathematical Monthly, the most widely read mathematics journal in the world according to records on JSTOR. The MAA sponsors the annual summer MathFest and cosponsors with the American Mathematical Society the Joint Mathematics Meeting, on occasion the Society for Industrial and Applied Mathematics joins in these meetings. Twenty-nine regional sections also hold regular meetings, the association publishes multiple journals, The American Mathematical Monthly is expository, aimed at a broad audience from undergraduate students to research mathematicians. Mathematics Magazine is expository, aimed at teachers of undergraduate mathematics, the College Mathematics Journal is expository, aimed at teachers of undergraduate mathematics, especially at the freshman-sophomore level. Math Horizons is expository, aimed at undergraduate students, MAA FOCUS is the association member newsletter. The Association publishes an online resource, Mathematical Sciences Digital Library, the service launched in 2001 with the online-only Journal of Online Mathematics and its Applications and a set of classroom tools, Digital Classroom Resources. These were followed in 2004 by Convergence, a history magazine, and in 2005 by MAA Reviews, an online book review service, and Classroom Capsules and Notes. Ultimately, six high school students are chosen to represent the U. S. at the International Mathematics Olympiad. Allendoerfer Award, Trevor Evans Award, Lester R. Ford Award, George Pólya Award, Merten M. Hasse Award, Henry L. Alder Award, a detailed history of the first fifty years of the MAA appears in May. A report on activities prior to World War II appears in Bennet, further details of its history can be found in Case. In addition numerous regional sections of the MAA have published accounts of their local history, the MAA has for a long time followed a strict policy of inclusiveness and non-discrimination. In previous periods it was subject to the problems of discrimination that were widespread across the United States. M. Holloway came to the meeting and were able to attend the scientific sessions, however, the organizer for the closing banquet refused to honor the reservations of these four mathematicians. Lorch and his colleagues wrote to the bodies of the AMS. Bylaws were not changed, but non-discriminatory policies were established and have been observed since then. The Associations first woman president was Dorothy Lewis Bernstein, the Carriage House that belonged to the residents at 1529 18th Street, N. W. dates to around 1900. It is older than the 5-story townhouse where the MAA Headquarters is currently located, charles Evans Hughes occupied the house while he was Secretary of State and a Supreme Court Justice
18.
Alexander Roberts
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Alexander Roberts DD was a Scottish biblical scholar. Born at Marykirk, Kincardineshire, on 12 May 1826, he was the son of Alexander Roberts and he was educated at the grammar school and Kings College, Aberdeen, where he graduated M. A. in March 1847, being the Simpson Greek prizeman. Roberts was a minister in Scotland and London. In 1864, then a minister at Carlton Hill, London and he was also minister at St. Johns Wood, and was a member of the New Testament revision company. In 1872, he succeeded John Campbell Shairp in the chair of humanity at the University of St. Andrews and he died at St. Andrews, Mitcham Park, Surrey, on 8 March 1901. He was returned to St Andrews for burial and lies in the south-east corner of the churchyard of St Andrews Cathedral, Roberts married on 2 December 1852 Mary Anne Speid, and had fourteen children, of whom four sons and eight daughters survived him. Roberts Discussions on the Gospels was published in 1862, one of a series of works in which he maintained that Greek was the speech of Jesus. He also translated the Works of Sulpitius Severus in the Select Library of Nicene and Post-Nicene Fathers and this article incorporates text from a publication now in the public domain, Gordon, Alexander. Dictionary of National Biography,1912 supplement, Works by or about Alexander Roberts at Internet Archive Works by Alexander Roberts at LibriVox
19.
James Donaldson (classical scholar)
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Sir James Donaldson FRSE LLD, was a Scottish classical scholar, and educational and theological writer. He was born in Aberdeen on 26 April 1831 and he was educated at Aberdeen Grammar School, Marischal College, Aberdeen, New College, London, and Berlin University. In 1854 he was appointed Rector of the Stirling High School where he remained for two years, before leaving for the Royal High School of Edinburgh, of which he was appointed Rector in 1866. He was elected a Fellow of the Royal Society of Edinburgh in 1867 and he became in 1881 Professor of Humanity in the University of Aberdeen, and in 1890 Principal of the University of St Andrews by the Universities Act. He died on 9 March 1915. and is buried with his wife in the churchyard of St Andrews Cathedral and he also has a memorial in the Church of St John the Evangelist, Edinburgh. Thurston, H. T. Colby, F. M. eds
20.
Brady Haran
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Haran is also the co-host of the Hello Internet podcast along with fellow Youtuber CGP Grey. Brady Haran studied journalism for a year before being hired by The Adelaide Advertiser, in 2002, he moved from Australia to Nottingham, United Kingdom. In Nottingham, he worked for the BBC, began to work with film, in 2007, Haran worked as a filmmaker-in-residence for Nottingham Science City, as part of an agreement between the BBC and The University of Nottingham. Haran then left the BBC to work full-time making YouTube videos, following Test Tube, Haran decided to create new YouTube channels. In his first 5 years as an independent filmmaker he made over 1500 videos, in 2012, he was the producer, editor, and interviewer behind 12 YouTube channels such as The Periodic Table of Videos, Sixty Symbols and Numberphile. Working with Poliakoff, Harans videos explaining chemistry and science for non-technical persons received positive recognition, together, they have made over 500 short videos that cover the elements and other chemistry-related topics. Their YouTube channel has had more than 120 million views and their Gold Bullion Vault, shot in the vaults of The Bank of England, was released 7 December 2012, and received more than two million hits in the next two months. Also, Haran and Poliakoff authored an article in the Nature Chemistry journal, Haran frequently collaborates with professionals and experts, who often appear in his videos to discuss subjects relevant to their work. Most notably his series Periodic Videos features chemist Sir Martyn Poliakoff, séquin, scientists Brian Butterworth, Ed Copeland, Laurence Eaves, and Clifford Stoll, and scientific writers and popularizers Alex Bellos, Steve Mould, Matt Parker, Tom Scott, and Simon Singh. In January 2014, Haran launched the podcast Hello Internet along with co-host CGP Grey, the podcast peaked as the No.1 iTunes podcast in United Kingdom, United States, Germany, Canada, and Australia. It was selected as one of Apples best new podcasts of 2014, Grey reported a podcast listenership of approximately a quarter million downloads per episode as of September 2015. The podcast features discussions pertaining to their lives as professional content creators for YouTube, as well as their interests, typical topics include new gadgets, technology etiquette, workplace efficiency, wristwatches, plane accidents, vexillology, and the differences between Harans and Greys personal mannerisms. As a result of their conversations, Haran has been credited for coining the term freebooting to refer to the unauthorized rehosting of online media, the podcast has an official flag called Nail & Gear which was chosen by the listeners. Test tube, behind the scenes in the world of science, teaching chem eng – Martyn Poliakoff and Brady Haran on Nottingham Unis periodic table for the YouTube generation. How to measure the impact of chemistry on the small screen