1.
Papyrus Oxyrhynchus 29
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Papyrus Oxyrhynchus 29 is a fragment of the second book of the Elements of Euclid in Greek. It was discovered by Grenfell and Hunt in 1897 in Oxyrhynchus, the fragment was originally dated to the end of the third century or the beginning of the fourth century, although more recent scholarship suggests a date of 75–125 CE. It is housed in the library of the University of Pennsylvania, the text was published by Grenfell and Hunt in 1898. The manuscript was written on papyrus in sloping irregular uncial letters, with no iota adscript, the fragment measures 85 by 152 mm. The fragment provides a statement of the 5th proposition of Book 2 of the Elements, together with a diagram. No part of the proof is provided, Oxyrhynchus Papyri Papyrus Oxyrhynchus 28 Papyrus Oxyrhynchus 30 This article incorporates text from a publication now in the public domain, B. P. Grenfell, A. S. Hunt
2.
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
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Theorems
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In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
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Conjecture
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In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found. Conjectures such as the Riemann hypothesis or Fermats Last Theorem have shaped much of history as new areas of mathematics are developed in order to prove them. In number theory, Fermats Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics, two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23,1852 when Francis Guthrie, while trying to color the map of counties of England, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. The four color theorem was proven in 1976 by Kenneth Appel and it was the first major theorem to be proved using a computer. Appel and Hakens approach started by showing that there is a set of 1,936 maps. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. The Hauptvermutung of geometric topology is the conjecture that any two triangulations of a triangulable space have a refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze and this conjecture is now known to be false. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion, the manifold version is true in dimensions m ≤3
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Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
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Formal proof
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A formal proof or derivation is a finite sequence of sentences, each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system, the notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof, the theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. Formal proofs often are constructed with the help of computers in interactive theorem proving, significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, while the problem of finding proofs is usually computationally intractable and/or only semi-decidable, a formal language is a set of finite sequences of symbols. Such a language can be defined without reference to any meanings of any of its expressions, it can exist before any interpretation is assigned to it – that is, Formal proofs are expressed in some formal language. A formal grammar is a description of the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the language which constitute well formed formulas. However, it does not describe their semantics, a formal system consists of a formal language together with a deductive apparatus. The deductive apparatus may consist of a set of rules or a set of axioms. A formal system is used to derive one expression from one or more other expressions, an interpretation of a formal system is the assignment of meanings to the symbols, and values to the sentences of a formal system. The study of interpretations is called formal semantics, giving an interpretation is synonymous with constructing a model. Proof Mathematical proof Proof theory Axiomatic system A Special Issue on Formal Proof, notices of the American Mathematical Society. 2πix. com, Logic Part of a series of articles covering mathematics and logic
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Philosophy of mathematics
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The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics, the logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably, the latter, however, may be used to refer to several other areas of study. Another refers to the philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Recurrent themes include, What is the role of Mankind in developing mathematics, what are the sources of mathematical subject matter. What is the status of mathematical entities. What does it mean to refer to a mathematical object, what is the character of a mathematical proposition. What is the relation between logic and mathematics, what is the role of hermeneutics in mathematics. What kinds of play a role in mathematics. What are the objectives of mathematical inquiry, what gives mathematics its hold on experience. What are the human traits behind mathematics, what is the source and nature of mathematical truth. What is the relationship between the world of mathematics and the material universe. The origin of mathematics is subject to argument, whether the birth of mathematics was a random happening or induced by necessity duly contingent upon other subjects, say for example physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics, there are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Greek philosophy on mathematics was strongly influenced by their study of geometry, for example, at one time, the Greeks held the opinion that 1 was not a number, but rather a unit of arbitrary length. A number was defined as a multitude, therefore,3, for example, represented a certain multitude of units, and was thus not truly a number. At another point, an argument was made that 2 was not a number. These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the root of two
8.
History of logic
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The history of logic deals with the study of the development of the science of valid inference. Formal logics developed in ancient times in China, India, Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic, christian and Islamic philosophers such as Boethius and William of Ockham further developed Aristotles logic in the Middle Ages, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the century saw largely decline and neglect. Empirical methods ruled the day, as evidenced by Sir Francis Bacons Novum Organon of 1620, valid reasoning has been employed in all periods of human history. However, logic studies the principles of reasoning, inference. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, the ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid. Ancient Babylon was also skilled in mathematics, while the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers seem aware of geometrys methods, fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. The proofs of Euclid of Alexandria are a paradigm of Greek geometry, the three basic principles of geometry are as follows, Certain propositions must be accepted as true without demonstration, such a proposition is known as an axiom of geometry. Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry, the proof must be formal, that is, the derivation of the proposition must be independent of the particular subject matter in question. Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi and this is part of a protracted debate about truth and falsity. Thales was said to have had a sacrifice in celebration of discovering Thales Theorem just as Pythagoras had the Pythagorean Theorem, Indian and Babylonian mathematicians knew his theorem for special cases before he proved it. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon, before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c.54 years older Thales. The systematic study of proof seems to have begun with the school of Pythagoras in the sixth century BC. Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize rather than matter. He is known for his obscure sayings and this logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. But other people fail to notice what they do when awake, in contrast to Heraclitus, Parmenides held that all is one and nothing changes
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Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
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Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
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Thales
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Thales of Miletus was a pre-Socratic Greek/Phoenician philosopher, mathematician and astronomer from Miletus in Asia Minor. He was one of the Seven Sages of Greece, Thales is recognized for breaking from the use of mythology to explain the world and the universe, and instead explaining natural objects and phenomena by theories and hypothesis, i. e. science. Aristotle reported Thales hypothesis that the principle of nature and the nature of matter was a single material substance. In mathematics, Thales used geometry to calculate the heights of pyramids and he is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to Thales theorem. He is the first known individual to whom a mathematical discovery has been attributed, the ancient source, Apollodorus of Athens, writing during the 2nd century BCE, thought Thales was born about the year 625 BCE. The dates of Thales life are not exactly known, but are roughly established by a few events mentioned in the sources. According to Herodotus Thales predicted the eclipse of May 28,585 BC. Diogenes Laërtius quotes the chronicle of Apollodorus of Athens as saying that Thales died at the age of 78 during the 58th Olympiad and attributes his death to heat stroke while watching the games. Plutarch had earlier told this version, Solon visited Thales and asked him why he remained single, nevertheless, several years later, anxious for family, he adopted his nephew Cybisthus. Thales involved himself in many activities, taking the role of an innovator, some say that he left no writings, others say that he wrote On the Solstice and On the Equinox. Diogenes Laërtius quotes two letters from Thales, one to Pherecydes of Syros, offering to review his book on religion, Thales identifies the Milesians as Athenian colonists. He was aware of the existence of the lodestone, and was the first to be connected to knowledge of this in history, according to Aristotle, Thales thought lodestones had souls, because iron is attracted to them. According to Hieronymus, historically quoted by Diogenes Laertius, Thales found the height of pyramids by comparison between the lengths of the shadows cast by a person and by the pyramids, several anecdotes suggest Thales was not just a philosopher, but also a businessman. A story, with different versions, recounts how Thales achieved riches from an olive harvest by prediction of the weather, in one version, he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Thales’ political life had mainly to do with the involvement of the Ionians in the defense of Anatolia against the power of the Persians. In neighbouring Lydia, a king had come to power, Croesus and he had conquered most of the states of coastal Anatolia, including the cities of the Ionians. The story is told in Herodotus, the war endured for five years, but in the sixth an eclipse of the Sun spontaneously halted a battle in progress. It seems that Thales had predicted this solar eclipse, the Seven Sages were most likely already in existence, as Croesus was also heavily influenced by Solon of Athens, another sage
12.
Hippocrates of Chios
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Hippocrates of Chios was an ancient Greek mathematician, geometer, and astronomer, who lived c.470 – c.410 BCE. He was born on the isle of Chios, where he originally was a merchant, after some misadventures he went to Athens, possibly for litigation. There he grew into a leading mathematician, on Chios, Hippocrates may have been a pupil of the mathematician and astronomer Oenopides of Chios. The reductio ad absurdum argument has been traced to him, only a single, and famous, fragment of Hippocrates Elements is existent, embedded in the work of Simplicius. In this fragment the area is calculated of some so-called Hippocratic lunes — see Lune of Hippocrates. This was part of a programme to achieve the quadrature of the circle. The strategy apparently was to divide a circle into a number of crescent-shaped parts, if it were possible to calculate the area of each of those parts, then the area of the circle as a whole would be known too. Only much later was it proven that this approach had no chance of success, the number π is the ratio of the circumference to the diameter of a circle, and also the ratio of the area to the square of the radius. In the century after Hippocrates at least four other mathematicians wrote their own Elements, steadily improving terminology, in this way Hippocrates pioneering work laid the foundation for Euclids Elements that was to remain the standard geometry textbook for many centuries. Hippocrates is believed to have originated the use of letters to refer to the points and figures in a proposition, e. g. triangle ABC for a triangle with vertices at points A, B. Two other contributions by Hippocrates in the field of mathematics are noteworthy and he found a way to tackle the problem of duplication of the cube, that is, the problem of how to construct a cube root. Like the quadrature of the circle this was another of the three great mathematical problems of Antiquity. Hippocrates also invented the technique of reduction, that is, to transform specific mathematical problems into a general problem that is more easy to solve. The solution to the general problem then automatically gives a solution to the original problem. In the field of astronomy Hippocrates tried to explain the phenomena of comets, ivor Bulmer-Thomas, Hippocrates of Chios, in, Dictionary of Scientific Biography, Charles Coulston Gillispie, ed. pp. 410–418. Björnbo, Hippokrates, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, G. Wissowa, oConnor, John J. Robertson, Edmund F. Hippocrates of Chios, MacTutor History of Mathematics archive, University of St Andrews. The Quadrature of the Circle and Hippocrates Lunes at Convergence
13.
Aristotle
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Aristotle was an ancient Greek philosopher and scientist born in the city of Stagira, Chalkidice, on the northern periphery of Classical Greece. His father, Nicomachus, died when Aristotle was a child, at seventeen or eighteen years of age, he joined Platos Academy in Athens and remained there until the age of thirty-seven. Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, teaching Alexander the Great gave Aristotle many opportunities and an abundance of supplies. He established a library in the Lyceum which aided in the production of many of his hundreds of books and he believed all peoples concepts and all of their knowledge was ultimately based on perception. Aristotles views on natural sciences represent the groundwork underlying many of his works, Aristotles views on physical science profoundly shaped medieval scholarship. Their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, some of Aristotles zoological observations, such as on the hectocotyl arm of the octopus, were not confirmed or refuted until the 19th century. His works contain the earliest known study of logic, which was incorporated in the late 19th century into modern formal logic. Aristotle was well known among medieval Muslim intellectuals and revered as The First Teacher and his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotles philosophy continue to be the object of academic study today. Though Aristotle wrote many elegant treatises and dialogues – Cicero described his style as a river of gold – it is thought that only around a third of his original output has survived. Aristotle, whose means the best purpose, was born in 384 BC in Stagira, Chalcidice. His father Nicomachus was the physician to King Amyntas of Macedon. Aristotle was orphaned at a young age, although there is little information on Aristotles childhood, he probably spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy. At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Platos Academy and he remained there for nearly twenty years before leaving Athens in 348/47 BC. Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor, there, he traveled with Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island. Aristotle married Pythias, either Hermiass adoptive daughter or niece and she bore him a daughter, whom they also named Pythias. Soon after Hermias death, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander in 343 BC, Aristotle was appointed as the head of the royal academy of Macedon. During that time he gave not only to Alexander
14.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
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Euclid's Elements
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
16.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
17.
Mathematics in medieval Islam
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Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries, the study of algebra, the name of which is derived from the Arabic word meaning completion or reunion of broken parts, flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi deals with ways to solve for the roots of first. He also introduces the method of reduction, and unlike Diophantus, Al-Khwarizmis algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the work of Diophantus, which was syncopated. The transition to symbolic algebra, where symbols are used, can be seen in the work of Ibn al-Banna al-Marrakushi. It is important to understand just how significant this new idea was and it was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a theory which allowed rational numbers, irrational numbers, geometrical magnitudes. It gave mathematics a whole new development path so much broader in concept to that which had existed before, another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. Several other mathematicians during this time expanded on the algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation, omar Khayyam found the general geometric solution of a cubic equation. Omar Khayyám wrote the Treatise on Demonstration of Problems of Algebra containing the solution of cubic or third-order equations. Khayyám obtained the solutions of equations by finding the intersection points of two conic sections. This method had used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī developed an approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. His surviving works give no indication of how he discovered his formulae for the maxima of these curves, various conjectures have been proposed to account for his discovery of them. The earliest implicit traces of mathematical induction can be found in Euclids proof that the number of primes is infinite, the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique
18.
Arithmetic
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Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
19.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
20.
Iraqi people
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For The Iraqis political party, refer to The Iraqis The Iraqi people are the citizens of the modern country of Iraq. Arabs have had a presence in Mesopotamia since the Sasanian Empire. Arabic was spoken by the majority in the Kingdom of Araba in the first and second centuries, Arabs were common in Mesopotamia at the time of the Seleucid Empire. The first Arab kingdom outside of Arabia was established in Iraqs Al-Hirah in the third century, Kurds who are Iraqi citizens live in the Zagros Mountains of northeast Iraq to the east of the upper Tigris. Arabic and Kurdish are Iraqs national languages, in ancient and medieval times Mesopotamia was the political and cultural centre of many great empires, such as the Akkadian Empire, Assyria, and Babylonia. The ancient Mesopotamian civilization of Sumer is the oldest known civilization in the world, Iraq remained an important centre of civilization for millennia, up until the Abbasid Caliphate, which was the most advanced empire of the medieval world. One study found that Haplogroup J-M172 originated in northern Iraq, Iraqi mitochondrial DNA haplogroup distribution is similar to that of Iran, Turkey, Georgia, and Armenia, whereas it substantially differs from that observed in Yemen. Iraqi Y-chromosome DNA haplogroup distribution is similar to that of Kuwait, Lebanon, Turkey, no significant differences in Y-DNA variation were observed among Iraqi Arabs, Assyrians, or Kurds in the study because of very small sample sizes for Kurds and Assyrians. Modern genetic studies indicate that Iraqi Arabs and Iraqi Kurds are distantly related, the Assyrian Christian population is fairly related to other Iraqis, and also to Jordanians, yet due to religious endogamy have a distinct genetic profile that distinguishes their population. The Assyrians are a homogeneous group of people, believed to originate from the land of old Assyria in northern Iraq they are Christians and are bona fide descendants of their namesakes. Most North-West and Central Iraqis who today speak Arabic are originally of Assyrian roots, studies have reported that most Irish and Britons have ancestry to Neolithic farmers who left modern day Iraq, Jordan and Syria 10,000 years ago. Genetic researchers say they have found compelling evidence that, on average, in another study, scientists analysed DNA from the 8, 000-year-old remains of early farmers found at an ancient graveyard in Germany. They compared the signatures to those of modern populations and found similarities with the DNA of people living in todays Turkey. Iraqis have historically been a multilingual, multiethnic people, conversant in several languages, Iraqi identity transcends ethnic and language boundaries, and is more associated with geography, the Tigris–Euphrates alluvial plain and its environs. The shared identity and culture of Iraqi people is most commonly seen in the Iraqi cuisine, nowadays, the demonym Iraqi includes all ethnic minorities in the country, such as the Assyrians, Kurds and Iraqi Turkmen. Iraqis trace their ancestry back to the ancient people of the land, and are proud of their ancient Mesopotamian roots and legacy, Iraqs national languages are Arabic and the Kurdish languages. Arabic is spoken as a first language by around 79 percent of Iraqi people, the two main regional dialects of Arabic spoken by the Iraqi people are Mesopotamian Arabic and North Mesopotamian Arabic. The two main dialects of Kurdish spoken by Kurdish Iraqis are Central Kurdish and Northern Kurdish, in addition to Arabic, most Assyrians and Mandaeans speak Neo-Aramaic languages
21.
Irrational number
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In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
22.
Mathematical induction
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Mathematical induction is a mathematical proof technique used to prove a given statement about any well-ordered set. Most commonly, it is used to establish statements for the set of all natural numbers, mathematical induction is a form of direct proof, usually done in two steps. When trying to prove a statement for a set of natural numbers. The second step, known as the step, is to prove that, if the statement is assumed to be true for any one natural number. Having proved these two steps, the rule of inference establishes the statement to be true for all natural numbers, in common terminology, using the stated approach is referred to as using the Principle of mathematical induction. Mathematical induction in this sense is closely related to recursion. Mathematical induction, in form, is the foundation of all correctness proofs for computer programs. Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning, mathematical induction is an inference rule used in proofs. In mathematics, proofs including those using mathematical induction are examples of deductive reasoning, in 370 BC, Platos Parmenides may have contained an early example of an implicit inductive proof. The earliest implicit traces of mathematical induction may be found in Euclids proof that the number of primes is infinite, none of these ancient mathematicians, however, explicitly stated the inductive hypothesis. Another similar case was that of Francesco Maurolico in his Arithmeticorum libri duo, the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. Another Frenchman, Fermat, made use of a related principle. The inductive hypothesis was also employed by the Swiss Jakob Bernoulli, the modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole, Augustus de Morgan, Charles Sanders Peirce, Giuseppe Peano, and Richard Dedekind. The simplest and most common form of mathematical induction infers that a statement involving a number n holds for all values of n. The proof consists of two steps, The basis, prove that the statement holds for the first natural number n, usually, n =0 or n =1, rarely, n = –1. The inductive step, prove that, if the statement holds for some number n. The hypothesis in the step that the statement holds for some n is called the induction hypothesis. To perform the step, one assumes the induction hypothesis
23.
Binomial theorem
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In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. For example,4 = x 4 +4 x 3 y +6 x 2 y 2 +4 x y 3 + y 4, the coefficient a in the term of a xb yc is known as the binomial coefficient or. These coefficients for varying n and b can be arranged to form Pascals triangle and these numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set. Special cases of the theorem were known from ancient times. Greek mathematician Euclid mentioned the case of the binomial theorem for exponent 2. There is evidence that the theorem for cubes was known by the 6th century in India. Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to the ancient Hindus. The earliest known reference to this problem is the Chandaḥśāstra by the Hindu lyricist Pingala. The commentator Halayudha from the 10th century A. D. explains this method using what is now known as Pascals triangle. By the 6th century A. D. the Hindu mathematicians probably knew how to express this as a quotient n. k. the binomial theorem as such can be found in the work of 11th-century Persian mathematician Al-Karaji, who described the triangular pattern of the binomial coefficients. He also provided a proof of both the binomial theorem and Pascals triangle, using a primitive form of mathematical induction. The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, the binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, in 1544, Michael Stifel introduced the term binomial coefficient and showed how to use them to express n in terms of n −1, via Pascals triangle. Blaise Pascal studied the eponymous triangle comprehensively in the treatise Traité du triangle arithmétique, however, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin. Isaac Newton is generally credited with the binomial theorem, valid for any rational exponent. This formula is also referred to as the formula or the binomial identity. Using summation notation, it can be written as n = ∑ k =0 n x n − k y k = ∑ k =0 n x k y n − k. A simple variant of the formula is obtained by substituting 1 for y
24.
Pascal's triangle
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In mathematics, Pascals triangle is a triangular array of the binomial coefficients. In the Western world, it is named after French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, the rows of Pascals triangle are conventionally enumerated starting with row n =0 at the top. The entries in each row are numbered from the beginning with k =0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the manner, In row 0. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the number in the first row is 1. The entry in the nth row and kth column of Pascals triangle is denoted, for example, the unique nonzero entry in the topmost row is =1. With this notation, the construction of the previous paragraph may be written as follows, = +, for any integer n. This recurrence for the coefficients is known as Pascals rule. Pascals triangle has higher dimensional generalizations, the three-dimensional version is called Pascals pyramid or Pascals tetrahedron, while the general versions are called Pascals simplices. The pattern of numbers that forms Pascals triangle was known well before Pascals time, centuries before, discussion of the numbers had arisen in the context of Indian studies of combinatorics and of binomial numbers and Greeks study of figurate numbers. From later commentary, it appears that the coefficients and the additive formula for generating them. Halayudha also explained obscure references to Meru-prastaara, the Staircase of Mount Meru, in approximately 850, the Jain mathematician Mahāvīra gave a different formula for the binomial coefficients, using multiplication, equivalent to the modern formula = n. r. At around the time, it was discussed in Persia by the Persian mathematician. It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám, thus the triangle is referred to as the Khayyam triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem, Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. Pascals triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian, in the 13th century, Yang Hui presented the triangle and hence it is still called Yang Huis triangle in China. In the west, the binomial coefficients were calculated by Gersonides in the early 14th century, petrus Apianus published the full triangle on the frontispiece of his book on business calculations in 1527
25.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
26.
Parallel postulate
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In geometry, the parallel postulate, also called Euclids fifth postulate because it is the fifth postulate in Euclids Elements, is a distinctive axiom in Euclidean geometry. This postulate does not specifically talk about parallel lines, it is only a postulate related to parallelism, Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. Euclidean geometry is the study of geometry that satisfies all of Euclids axioms, a geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclids fifth postulate is known as absolute geometry and this axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the axioms which give Euclidean geometry, each of these can be used to prove the other. These equivalent statements include, There is at most one line that can be parallel to another given one through an external point. The sum of the angles in every triangle is 180°, There exists a triangle whose angles add up to 180°. The sum of the angles is the same for every triangle, There exists a pair of similar, but not congruent, triangles. If three angles of a quadrilateral are right angles, then the angle is also a right angle. There exists a quadrilateral in which all angles are right angles, that is, There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the line are also parallel to each other. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the two sides. There is no limit to the area of a triangle. The summit angles of the Saccheri quadrilateral are 90°, if a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. In the list above, it is taken to refer to non-intersecting lines. Note that the two definitions are not equivalent, because in hyperbolic geometry the second definition holds only for ultraparallel lines. For two thousand years, many attempts were made to prove the parallel postulate using Euclids first four postulates, the main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order the postulates were listed in the Elements is significant, many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found
27.
Data structure
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In computer science, a data structure is a particular way of organizing data in a computer so that it can be used efficiently. Data structures can implement one or more abstract data types, which specify the operations that can be performed on a data structure. In comparison, a structure is a concrete implementation of the specification provided by an ADT. Different kinds of structures are suited to different kinds of applications. For example, relational databases commonly use B-tree indexes for data retrieval, Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms, some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Data structures can be used to organize the storage and retrieval of stored in both main memory and secondary memory. Many data structures use both principles, sometimes combined in non-trivial ways, the implementation of a data structure usually requires writing a set of procedures that create and manipulate instances of that structure. The efficiency of a data structure cannot be analyzed separately from those operations, there are numerous types of data structures, generally built upon simpler primitive data types, An array is a number of elements in a specific order, typically all of the same type. Elements are accessed using an index to specify which element is required. Typical implementations allocate contiguous memory words for the elements of arrays, arrays may be fixed-length or resizable. A linked list is a collection of data elements of any type, called nodes, where each node has itself a value. The principal advantage of a linked list over an array, is that values can always be efficiently inserted and removed without relocating the rest of the list, certain other operations, such as random access to a certain element, are however slower on lists than on arrays. A record is a data structure. A record is a value that contains other values, typically in fixed number and sequence, the elements of records are usually called fields or members. A union is a structure that specifies which of a number of permitted primitive types may be stored in its instances. Contrast with a record, which could be defined to contain a float, enough space is allocated to contain the widest member datatype. A tagged union contains an additional field indicating its current type, a class is a data structure that contains data fields, like a record, as well as various methods which operate on the contents of the record
28.
Axiomatic set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
29.
Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
30.
Formal language
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In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it. The alphabet of a language is the set of symbols, letters. The strings formed from this alphabet are called words, and the words belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is defined by means of a formal grammar such as a regular grammar or context-free grammar. The field of language theory studies primarily the purely syntactical aspects of such languages—that is. Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages. The first formal language is thought to be the one used by Gottlob Frege in his Begriffsschrift, literally meaning concept writing, axel Thues early semi-Thue system, which can be used for rewriting strings, was influential on formal grammars. The elements of an alphabet are called its letters, alphabets may be infinite, however, most definitions in formal language theory specify finite alphabets, and most results only apply to them. A word over an alphabet can be any sequence of letters. The set of all words over an alphabet Σ is usually denoted by Σ*, the length of a word is the number of letters it is composed of. For any alphabet there is one word of length 0, the empty word. By concatenation one can combine two words to form a new word, whose length is the sum of the lengths of the original words, the result of concatenating a word with the empty word is the original word. A formal language L over an alphabet Σ is a subset of Σ*, that is, sometimes the sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of well-formed expressions. In computer science and mathematics, which do not usually deal with natural languages, in practice, there are many languages that can be described by rules, such as regular languages or context-free languages. The notion of a formal grammar may be closer to the concept of a language. By an abuse of the definition, a formal language is often thought of as being equipped with a formal grammar that describes it. The following rules describe a formal language L over the alphabet Σ =, Every nonempty string that does not contain + or =, a string containing = is in L if and only if there is exactly one =, and it separates two valid strings of L. A string containing + but not = is in L if, no string is in L other than those implied by the previous rules
31.
Immanuel Kant
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Immanuel Kant was a German philosopher who is considered a central figure in modern philosophy. Kant took himself to have effected a Copernican revolution in philosophy and his beliefs continue to have a major influence on contemporary philosophy, especially the fields of metaphysics, epistemology, ethics, political theory, and aesthetics. Politically, Kant was one of the earliest exponents of the idea that peace could be secured through universal democracy. He believed that this will be the outcome of universal history. Kant wanted to put an end to an era of futile and speculative theories of human experience, Kant argued that our experiences are structured by necessary features of our minds. In his view, the shapes and structures experience so that, on an abstract level. Among other things, Kant believed that the concepts of space and time are integral to all human experience, as are our concepts of cause, Kant published other important works on ethics, religion, law, aesthetics, astronomy, and history. These included the Critique of Practical Reason, the Metaphysics of Morals, which dealt with ethics, and the Critique of Judgment, Immanuel Kant was born in 1724 in Königsberg, Prussia. His mother, Anna Regina Reuter, was born in Königsberg to a father from Nuremberg. His father, Johann Georg Kant, was a German harness maker from Memel, Immanuel Kant believed that his paternal grandfather Hans Kant was of Scottish origin. Kant was the fourth of nine children, baptized Emanuel, he changed his name to Immanuel after learning Hebrew. Young Kant was a solid, albeit unspectacular, student and he was brought up in a Pietist household that stressed religious devotion, humility, and a literal interpretation of the Bible. His education was strict, punitive and disciplinary, and focused on Latin and religious instruction over mathematics, despite his religious upbringing and maintaining a belief in God, Kant was skeptical of religion in later life, various commentators have labelled him agnostic. Common myths about Kants personal mannerisms are listed, explained, and refuted in Goldthwaits introduction to his translation of Observations on the Feeling of the Beautiful and Sublime. It is often held that Kant lived a strict and disciplined life. He never married, but seemed to have a social life — he was a popular teacher. He had a circle of friends whom he met, among them Joseph Green. A common myth is that Kant never traveled more than 16 kilometres from Königsberg his whole life, in fact, between 1750 and 1754 he worked as a tutor in Judtschen and in Groß-Arnsdorf
32.
Mathematical beauty
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Mathematical beauty describes the notion that some mathematicians may derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful, Mathematicians describe mathematics as an art form or, at a minimum, as a creative activity. Comparisons are often made music and poetry. The true spirit of delight, the exaltation, the sense of being more than Man, Paul Erdős expressed his views on the ineffability of mathematics when he said, Why are numbers beautiful. Its like asking why is Beethovens Ninth Symphony beautiful, if you dont see why, someone cant tell you. If they arent beautiful, nothing is, Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean, A proof that uses a minimum of additional assumptions or previous results, a proof that is unusually succinct. A proof that derives a result in a surprising way A proof that is based on new, a method of proof that can be easily generalized to solve a family of similar problems. In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—the first proof that is found may not be the best. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, another theorem that has been proved in many different ways is the theorem of quadratic reciprocity—Carl Friedrich Gauss alone published eight different proofs of this theorem. Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated and these results are often described as deep. While it is difficult to find agreement on whether a result is deep. One is Eulers identity, e i π +1 =0 and this is a special case of Eulers formula, which the physicist Richard Feynman called our jewel and the most remarkable formula in mathematics. Other examples of deep results include unexpected insights into mathematical structures, for example, Gausss Theorema Egregium is a deep theorem which relates a local phenomenon to a global phenomenon in a surprising way. In particular, the area of a triangle on a surface is proportional to the excess of the triangle. Another example is the theorem of calculus. The opposite of deep is trivial, sometimes, however, a statement of a theorem can be original enough to be considered deep, even though its proof is fairly obvious. In his A Mathematicians Apology, Hardy suggests that a proof or result possesses inevitability, unexpectedness
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Direct proof
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In order to directly prove a conditional statement of the form If p, then q, it suffices to consider the situations in which the statement p is true. Logical deduction is employed to reason from assumptions to conclusion, the type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. Common proof rules used are modus ponens and universal instantiation, in contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example, instead of showing directly p ⇒ q, one proves its contrapositive ~q ⇒ ~p, since p ⇒ q and ~q ⇒ ~p are equivalent by the principle of transposition, p ⇒ q is indirectly proved. Proof methods that are not direct include proof by contradiction, including proof by infinite descent, direct proof methods include proof by exhaustion and proof by induction. A direct proof is the simplest form of proof there is, the word ‘proof’ comes from the Latin word probare, which means “to test”. The earliest use of proofs was prominent in legal proceedings, a person with authority, such as a nobleman, was said to have probity, which means that the evidence was by his relative authority, which outweighed empirical testimony. In days gone by, mathematics and proof was often intertwined with practical questions – with populations like the Egyptians and this led to a natural curiosity with regards to geometry and trigonometry – particularly triangles and rectangles. Another shape which is crucial in the history of proof is the circle. This meant that ancient geometry discussed circles, the earliest form of mathematics was phenomenological. For example, if someone could draw a picture, or give a convincing description. On occasion, analogical arguments took place, or even by “invoking the gods”, the idea that mathematical statements could be proven had not been developed yet, so these were the earliest forms of the concept of proof, despite not being actual proof at all. Proof as we know it came about one specific question, “what is a proof. ”Traditionally. Naturally, one would assume that the best way to prove the truth of something like this would be to draw up a comparison with something old that has already been proven as true, thus was created the concept of deriving a new result from an old result. Consider two even integers x and y, since they are even, they can be written as x =2 a y =2 b respectively for integers a and b. Observe that we have four right-angled triangles and a square packed into a large square, each of the triangles has sides a and b and hypotenuse c. The area of a square is defined as the square of the length of its sides - in this case,2, however, the area of the large square can also be expressed as the sum of the areas of its components. In this case, that would be the sum of the areas of the four triangles and we know that the area of the large square is equal to 2
34.
Even and odd numbers
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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
35.
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
36.
Material conditional
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The material conditional is a logical connective that is often symbolized by a forward arrow →. The material conditional is used to form statements of the form
37.
Square root of 2
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The square root of 2, or the th power of 2, written in mathematics as √2 or 2 1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational, the rational approximation of the square root of two,665, 857/470,832, derived from the fourth step in the Babylonian algorithm starting with a0 =1, is too large by approx. 1. 6×10−12, its square is 2. 0000000000045… The rational approximation 99/70 is frequently used, despite having a denominator of only 70, it differs from the correct value by less than 1/10,000. The numerical value for the root of two, truncated to 65 decimal places, is,1. 41421356237309504880168872420969807856967187537694807317667973799….41421296 ¯. That is,1 +13 +13 ×4 −13 ×4 ×34 =577408 =1.4142156862745098039 ¯. This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as a secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras number or Pythagoras constant, for example by Conway & Guy, there are a number of algorithms for approximating √2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots. It goes as follows, First, pick a guess, a0 >0, then, using that guess, iterate through the following recursive computation, a n +1 = a n +2 a n 2 = a n 2 +1 a n. The more iterations through the algorithm, the approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits, starting with a0 =1 the next approximations are 3/2 =1.5 17/12 =1.416. The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanadas team in 1997, in February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010, for a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely, such computations aim to check empirically whether such numbers are normal
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Contraposition
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In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped, for instance, the proposition All bats are mammals can be restated as the conditional If something is a bat, then it is a mammal. Now, the law says that statement is identical to the contrapositive If something is not a mammal, then it is not a bat. The contrapositive can be compared with three other relationships between conditional statements, Inversion, ¬ P → ¬ Q If something is not a bat, then it is not a mammal. Unlike the contrapositive, the truth value is not at all dependent on whether or not the original proposition was true. The inverse here is not true. Conversion, Q → P If something is a mammal, then it is a bat, the converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. Negation, ¬ There exists a bat that is not a mammal, If the negation is true, the original proposition is false. Here, of course, the negation is false, note that if P → Q is true and we are given that Q is false, ¬ Q, it can logically be concluded that P must be false, ¬ P. This is often called the law of contrapositive, or the modus tollens rule of inference, according to this diagram, if something is in A, it must be in B as well. So we can all of A is in B as, A → B It is also clear that anything that is not within B cannot be within A. This statement, ¬ B → ¬ A is the contrapositive, therefore, we can say that →. Practically speaking, this may make much easier when trying to prove something. Alternatively, we can try to prove ¬ B → ¬ A by checking all girls without brown hair to see if they are all outside the US. This means that if we find at least one girl without brown hair within the US, we will have disproved ¬ B → ¬ A, to conclude, for any statement where A implies B, then not B always implies not A. Proving or disproving either one of these statements automatically proves or disproves the other. A proposition Q is implicated by a proposition P when the relationship holds, This states that, if P, then Q, or, if Socrates is a man. In a conditional such as this, P is the antecedent, one statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa
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Contrapositive
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In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped, for instance, the proposition All bats are mammals can be restated as the conditional If something is a bat, then it is a mammal. Now, the law says that statement is identical to the contrapositive If something is not a mammal, then it is not a bat. The contrapositive can be compared with three other relationships between conditional statements, Inversion, ¬ P → ¬ Q If something is not a bat, then it is not a mammal. Unlike the contrapositive, the truth value is not at all dependent on whether or not the original proposition was true. The inverse here is not true. Conversion, Q → P If something is a mammal, then it is a bat, the converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. Negation, ¬ There exists a bat that is not a mammal, If the negation is true, the original proposition is false. Here, of course, the negation is false, note that if P → Q is true and we are given that Q is false, ¬ Q, it can logically be concluded that P must be false, ¬ P. This is often called the law of contrapositive, or the modus tollens rule of inference, according to this diagram, if something is in A, it must be in B as well. So we can all of A is in B as, A → B It is also clear that anything that is not within B cannot be within A. This statement, ¬ B → ¬ A is the contrapositive, therefore, we can say that →. Practically speaking, this may make much easier when trying to prove something. Alternatively, we can try to prove ¬ B → ¬ A by checking all girls without brown hair to see if they are all outside the US. This means that if we find at least one girl without brown hair within the US, we will have disproved ¬ B → ¬ A, to conclude, for any statement where A implies B, then not B always implies not A. Proving or disproving either one of these statements automatically proves or disproves the other. A proposition Q is implicated by a proposition P when the relationship holds, This states that, if P, then Q, or, if Socrates is a man. In a conditional such as this, P is the antecedent, one statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa
40.
Coprime
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In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
41.
Joseph Liouville
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Joseph Liouville was a French mathematician. Liouville graduated from the École Polytechnique in 1827, after some years as an assistant at various institutions including the École Centrale Paris, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the Collège de France in 1850, besides his academic achievements, he was very talented in organisational matters. Liouville founded the Journal de Mathématiques Pures et Appliquées which retains its high reputation up to today and he was the first to read, and to recognize, the importance of the unpublished work of Évariste Galois which appeared in his journal in 1846. Liouville was also involved in politics for some time, and he became a member of the Constituting Assembly in 1848, however, after his defeat in the legislative elections in 1849, he turned away from politics. Liouville worked in a number of different fields in mathematics, including number theory, complex analysis, differential geometry and topology and he is remembered particularly for Liouvilles theorem, a nowadays rather basic result in complex analysis. In number theory, he was the first to prove the existence of numbers by a construction using continued fractions. In Hamiltonian dynamics, Liouville also introduced the notion of variables as a description of completely integrable systems. The modern formulation of this is called the Liouville-Arnold theorem. In 1851, he was elected a member of the Royal Swedish Academy of Sciences. The crater Liouville on the Moon is named after him, so is the Liouville function, an important function in number theory. List of things named after Joseph Liouville Liouvilles theorem OConnor, John J. Robertson, Edmund F. Joseph Liouville, MacTutor History of Mathematics archive, icelandic Mathematical Society, Reykjavik, pp. 149–160
42.
Transcendental number
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In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with integer coefficients. The best-known transcendental numbers are π and e, though only a few classes of transcendental numbers are known, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the numbers are countable while the sets of real. All real transcendental numbers are irrational, since all numbers are algebraic. Another irrational number that is not transcendental is the ratio, φ or ϕ. The name transcendental comes from the root trans meaning across and length of numbers, euler was probably the first person to define transcendental numbers in the modern sense. Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of πs transcendence. In other words, the nth digit of this number is 1 only if n is one of the numbers 1. Liouville showed that number is what we now call a Liouville number. Liouville showed that all Liouville numbers are transcendental, the first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. In 1874, Georg Cantor proved that the numbers are countable. He also gave a new method for constructing transcendental numbers, in 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers. Cantors work established the ubiquity of transcendental numbers, in 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that ea is transcendental when a is algebraic, then, since eiπ = −1 is algebraic, iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem, the transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem and this work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms. The set of numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a number of zeroes
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Four color theorem
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Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers, according to an article by the math historian Kenneth May, “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. Martin Gardner wrote an account of what was known at the time about the four color theorem in his September 1960 Mathematical Games column in Scientific American magazine. In 1975 Gardner revisited the topic by publishing a map said to be a counter-example in his infamous April fools hoax column of April 1975, the four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer, Appel and Hakens approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exist because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. To dispel remaining doubt about the Appel–Haken proof, a proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour. Additionally, in 2005, the theorem was proved by Georges Gonthier with general purpose theorem proving software, the intuitive statement of the four color theorem, i. e. First, all corners, points that belong to three or more countries, must be ignored. In addition, bizarre maps can require more than four colors, second, for the purpose of the theorem, every country has to be a connected region, or contiguous. In the real world, this is not true, because all the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map, In this map and this map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. A similar construction also applies if a color is used for all bodies of water. For maps in which more than one country may have multiple disconnected regions, a simpler statement of the theorem uses graph theory
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Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω
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Collatz conjecture
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The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz. The conjecture can be summarized as follows, If n is even, divide it by 2 to get n /2. If n is odd, multiply it by 3 and add 1 to obtain 3n +1, the conjecture is that no matter what number you start with, you will always eventually reach 1. Paul Erdős said about the Collatz conjecture, Mathematics may not be ready for such problems and he also offered $500 for its solution. Jeffrey Lagarias in 2010 claimed that only on known information about this problem. Consider the following operation on a positive integer, If the number is even. If the number is odd, triple it and add one, in modular arithmetic notation, define the function f as follows, f = { n /2 if n ≡03 n +1 if n ≡1. Now, form a sequence by performing this operation repeatedly, beginning with any positive integer, in notation, a i = { n for i =0 f for i >0. The Collatz conjecture is, This process will eventually reach the number 1 and that smallest i such that ai =1 is called the total stopping time of n. The conjecture asserts that every n has a well-defined total stopping time, If, for some n, such an i doesnt exist, we say that n has infinite total stopping time and the conjecture is false. If the conjecture is false, it can only be there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence might enter a cycle that excludes 1. No such sequence has been found, for instance, starting with n =12, one gets the sequence 12,6,3,10,5,16,8,4,2,1. N =19, for example, takes longer to reach 1,19,58,29,88,44,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1. The sequence for n =27, listed and graphed below, takes 111 steps, any counterexample to the Collatz conjecture would have to consist either of an infinite divergent trajectory or a cycle different from the trivial cycle. Thus, if one could prove that neither of these types of counterexample could exist, such a strong result is not known, but certain types of cycles have been ruled out. The type of a cycle may be defined with reference to the definition of the Collatz map. A cycle is a sequence where f = a 1, f = a 2, for this shortcut definition, the only known cycle is