1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit

3.
Geometric series
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In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r

4.
Limit of a sequence
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In mathematics, the limit of a sequence is the value that the terms of a sequence tend to. If such a limit exists, the sequence is called convergent, a sequence which does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests, limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes, leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series, Newton dealt with series in his works on Analysis with infinite series, Method of fluxions and infinite series and Tractatus de Quadratura Curvarum. In the latter work, Newton considers the binomial expansion of n which he then linearizes by taking limits, at the end of the century, Lagrange in his Théorie des fonctions analytiques opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series for the first time rigorously investigated under which conditions a series converged to a limit, the modern definition of a limit was given by Bernhard Bolzano and by Karl Weierstrass in the 1870s. In the real numbers, a number L is the limit of the if the numbers in the sequence become closer and closer to L. If x n = c for some constant c, then x n → c, if x n =1 n, then x n →0. If x n =1 / n when n is even, given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence 0.3,0.33,0.333,0.3333, note that the decimal representation 0.3333. is the limit of the previous sequence, defined by 0.3333. ≜ lim n → ∞ ∑ i =1 n 310 i, finding the limit of a sequence is not always obvious. Two examples are lim n → ∞ n and the Arithmetic–geometric mean, the squeeze theorem is often useful in such cases. In other words, for measure of closeness ϵ, the sequences terms are eventually that close to the limit. The sequence is said to converge to or tend to the limit x, symbolically, this is, ∀ ϵ >0 ∃ N ∈ R ∀ n ∈ N. If a sequence converges to some limit, then it is convergent, limits of sequences behave well with respect to the usual arithmetic operations. For any continuous function f, if x n → x then f → f, in fact, any real-valued function f is continuous if and only if it preserves the limits of sequences. Some other important properties of limits of sequences include the following

5.
Zeno's paradoxes
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It is usually assumed, based on Platos Parmenides, that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides view. Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point, some of Zenos nine surviving paradoxes are essentially equivalent to one another. Aristotle offered a refutation of some of them, three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below. Zenos arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction and they are also credited as a source of the dialectic method used by Socrates. Some mathematicians and historians, such as Carl Boyer, hold that Zenos paradoxes are simply mathematical problems, some philosophers, however, say that Zenos paradoxes and their variations remain relevant metaphysical problems. The origins of the paradoxes are somewhat unclear, Diogenes Laertius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zenos teacher Parmenides was the first to introduce the Achilles and the tortoise paradox. But in a passage, Laertius attributes the origin of the paradox to Zeno. In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. – as recounted by Aristotle, Physics VI,9, 239b15 In the paradox of Achilles, Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at constant speed, then after some finite time, Achilles will have run 100 meters. During this time, the tortoise has run a shorter distance. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go, therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. That which is in locomotion must arrive at the stage before it arrives at the goal. – as recounted by Aristotle, Physics VI,9. Before he can get there, he must get halfway there, before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth, before an eighth, one-sixteenth, the resulting sequence can be represented as, This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a problem in that it contains no first distance to run, for any possible first distance could be divided in half. Hence, the trip cannot even begin, the paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. An alternative conclusion, proposed by Henri Bergson, is that motion is not actually divisible and this argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts

6.
Eye of Horus
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The Eye of Horus is an ancient Egyptian symbol of protection, royal power and good health. The eye is personified in the goddess Wadjet, the Eye of Horus is similar to the Eye of Ra, which belongs to a different god, Ra, but represents many of the same concepts. Wadjet was one of the earliest of Egyptian deities who later associated with other goddesses such as Bast, Sekhmet, Mut. She was the deity of Lower Egypt and the major Delta shrine the per-nu was under her protection. Hathor is also depicted with this eye, funerary amulets were often made in the shape of the Eye of Horus. The Wadjet or Eye of Horus is the element of seven gold, faience, carnelian. The Wedjat was intended to protect the pharaoh in the afterlife, Ancient Egyptian and Middle-Eastern sailors would frequently paint the symbol on the bow of their vessel to ensure safe sea travel. Horus was the ancient Egyptian sky god who was depicted as a falcon. His right eye was associated with the sun god, Ra, the eye symbol represents the marking around the eye of the falcon, including the teardrop marking sometimes found below the eye. The mirror image, or left eye, sometimes represented the moon, in one myth, when Set and Horus were fighting for the throne after Osiriss death, Set gouged out Horuss left eye. The majority of the eye was restored by either Hathor or Thoth, when Horuss eye was recovered, he offered it to his father, Osiris, in hopes of restoring his life. Hence, the eye of Horus was often used to sacrifice, healing, restoration. There are seven different hieroglyphs used to represent the eye, most commonly ir. t in Egyptian, in Egyptian myth the eye was not the passive organ of sight but more an agent of action, protection or wrath. The Eye of Horus was represented as a hieroglyph, designated D10 in Gardiners sign list and it is represented in the Unicode character block for Egyptian hieroglyphs as U+13080. In Ancient Egyptian most fractions were written as the sum of two or more unit fractions, with scribes possessing tables of answers, thus instead of 3⁄4, one would write 1⁄2 + 1⁄4. Studies from the 1970s to this day in Egyptian mathematics have clearly shown this theory was fallacious, the evolution of the symbols used in mathematics, although similar to the different parts of the Eye of Horus, is now known to be distinct. Wadjet eye tatoos associated with Hathor depicted on 3, 000-year-old mummy

7.
0.999...
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In mathematics, the repeating decimal 0. 999… denotes a real number that can be shown to be the number one. In other words, the symbols 0. 999… and 1 represent the same number, more generally, every nonzero terminating decimal has an equal twin representation with infinitely many trailing 9s. The terminating decimal representation is preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases or in any representation of the real numbers. The equality of 0. 999… and 1 is closely related to the absence of nonzero infinitesimals in the number system. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals, the equality 0. 999… =1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of studies in mathematics education. Algebraic proofs showing that 0. 999… represents the number 1 use concepts such as fractions, long division, however, these proofs are not rigorous as they do not include a careful analytic definition of 0. 999…. One reason that infinite decimals are an extension of finite decimals is to represent fractions. Using long division, a division of integers like 1⁄9 becomes a recurring decimal,0. 111…. This decimal yields a quick proof for 0. 999… =1, If 0. 999… is to be consistent, it must equal 9⁄9 =1. 0.333 … =390.888 … =890.999 … =99 =1 When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 ×0. 999… equals 9. 999…, which is 9 greater than the original number, in introductory arithmetic, such proofs help explain why 0. 999… =1 but 0. 333… <0.34. In introductory algebra, the proofs help explain why the method of converting between fractions and repeating decimals works. Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can demonstrate that the decimals 0. 999… and 1. 000… both represent the same real number, it is built into the definition. Since the question of 0. 999… does not affect the development of mathematics

8.
Dotted note
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In Western musical notation, a dotted note is a note with a small dot written after it. In modern practice the first dot increases the duration of the note by half of its original value. A dotted note is equivalent to writing the basic note tied to a note of half the value, or with more than one dot, a rhythm using longer notes alternating with shorter notes is sometimes called a dotted rhythm. Historical examples of performance styles using dotted rhythm include notes inégales. The precise performance of dotted rhythms can be a complex issue, even in notation that includes dots, their performed values may be longer than the dot mathematically indicates, a practice known as over-dotting. If the note to be dotted is on a space, the dot goes on the space, while if the note is on a line. Theoretically, any note value can be dotted, as can rests of any value, if the rest is in its normal position, dots are always placed in third staff space from the bottom. The use of a dot for augmentation of a note dates back at least to the 10th century, although the amount of augmentation is disputed. More than one dot may be added, each dot adds half of the duration added by the previous dot, a double-dotted note is a note with two small dots written after it. Its duration is 1 3⁄4 times its basic note value, the double-dotted note is used less frequently than the dotted note. Typically, as in the example below, it is followed by a note whose duration is one-quarter the length of the note value. Before the mid 18th century, double dots were not used, until then, in some circumstances, single dots could mean double dots. Example 2 is a fragment of the movement of Joseph Haydns string quartet, Op.74. Haydns theme was adapted for piano by a composer, the adapted version can be heard here. In a French overture, notes written as dotted notes are often interpreted to mean double-dotted notes, a triple-dotted note is a note with three dots written after it, its duration is 1 7⁄8 times its basic note value. Use of a note value is not common in the Baroque and Classical periods. An example of the use of double- and triple-dotted notes is the Prelude in G major for piano, Op.28, the piece, in common time, contains running semiquavers in the left hand. Several times during the piece Chopin asks for the hand to play a triple-dotted minim

9.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today