92-466: In mathematics , Hilbert–Schmidt may refer to a Hilbert–Schmidt operator ; a Hilbert–Schmidt integral operator ; the Hilbert–Schmidt theorem . Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Hilbert–Schmidt . If an internal link led you here, you may wish to change
184-460: A corruption of Greek mathematical terms. Euclid is best known for his thirteen-book treatise, the Elements ( ‹See Tfd› Greek : Στοιχεῖα ; Stoicheia ), considered his magnum opus . Much of its content originates from earlier mathematicians, including Eudoxus , Hippocrates of Chios , Thales and Theaetetus , while other theorems are mentioned by Plato and Aristotle. It
276-477: A mathematical tradition there. The city was founded by Alexander the Great in 331 BC, and the rule of Ptolemy I from 306 BC onwards gave it a stability which was relatively unique amid the chaotic wars over dividing Alexander's empire . Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education. Euclid
368-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
460-516: A "reservoir of results". Despite this, Sialaros furthers that "the remarkably tight structure of the Elements reveals authorial control beyond the limits of a mere editor". The Elements does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry (books 1–6), basic number theory (books 7–10) and solid geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) do not exactly fit this scheme. The heart of
552-439: A copy thereof, and is sometimes synonymous with 'geometry'. As with many ancient Greek mathematicians , the details of Euclid's life are mostly unknown. He is accepted as the author of four mostly extant treatises—the Elements , Optics , Data , Phaenomena —but besides this, there is nothing known for certain of him. The traditional narrative mainly follows the 5th century AD account by Proclus in his Commentary on
644-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
736-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
828-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of
920-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it
1012-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
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#17327755545411104-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
1196-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
1288-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating
1380-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to
1472-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
1564-460: Is flat " and "a field is always a ring ". Euclid Euclid ( / ˈ j uː k l ɪ d / ; ‹See Tfd› Greek : Εὐκλείδης ; fl. 300 BC) was an ancient Greek mathematician active as a geometer and logician . Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated
1656-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
1748-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
1840-428: Is difficult to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics. The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps" and the historian Serafina Cuomo described it as
1932-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
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#17327755545412024-628: Is in Apollonius' prefatory letter to the Conics (early 2nd century BC): "The third book of the Conics contains many astonishing theorems that are useful for both the syntheses and the determinations of number of solutions of solid loci . Most of these, and the finest of them, are novel. And when we discovered them we realized that Euclid had not made the synthesis of the locus on three and four lines but only an accidental fragment of it, and even that
2116-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
2208-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
2300-465: Is no royal road to geometry". This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus . Both accounts were written in the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature. Any firm dating of Euclid's activity c. 300 BC is called into question by a lack of contemporary references. The earliest original reference to Euclid
2392-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
2484-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
2576-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
2668-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
2760-600: Is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum ; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria. In the Elements , Euclid deduced the theorems from a small set of axioms . He also wrote works on perspective , conic sections , spherical geometry , number theory , and mathematical rigour . In addition to
2852-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
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2944-457: Is speculated to have been among the Musaeum's first scholars. Euclid's date of death is unknown; it has been speculated that he died c. 270 BC . Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Euclid of Megara , a pupil of Socrates included in dialogues of Plato with whom he was historically conflated. Valerius Maximus ,
3036-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
3128-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
3220-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
3312-540: Is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it a natural fit. As a whole, the authorial voice remains general and impersonal. Book 1 of the Elements is foundational for the entire text. It begins with a series of 20 definitions for basic geometric concepts such as lines , angles and various regular polygons . Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions. These assumptions are intended to provide
3404-459: Is unknown; some scholars estimate around 330 or 325 BC, but others refrain from speculating. It is presumed that he was of Greek descent, but his birthplace is unknown. Proclus held that Euclid followed the Platonic tradition , but there is no definitive confirmation for this. It is unlikely he was a contemporary of Plato, so it is often presumed that he was educated by Plato's disciples at
3496-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
3588-483: The Elements in works whose dates are firmly known are not until the 2nd century AD, by Galen and Alexander of Aphrodisias ; by this time it was a standard school text. Some ancient Greek mathematicians mention Euclid by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of Elements "). In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from
3680-621: The Elements was published in 1570 by Henry Billingsley and John Dee . The mathematician Oliver Byrne published a well-known version of the Elements in 1847 entitled The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners , which included colored diagrams intended to increase its pedagogical effect. David Hilbert authored
3772-657: The Elements , Euclid wrote a central early text in the optics field, Optics , and lesser-known works including Data and Phaenomena . Euclid's authorship of On Divisions of Figures and Catoptrics has been questioned. He is thought to have written many lost works . The English name 'Euclid' is the anglicized version of the Ancient Greek name Eukleídes ( Εὐκλείδης ). It is derived from ' eu- ' ( εὖ ; 'well') and 'klês' ( -κλῆς ; 'fame'), meaning "renowned, glorious". In English, by metonymy , 'Euclid' can mean his most well-known work, Euclid's Elements , or
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3864-412: The Elements , at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements , with definitions and proved propositions. Four other works are credibly attributed to Euclid, but have been lost. Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity. Many commentators cite him as one of
3956-544: The Elements , book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes. The final three books (11–13) primarily discuss solid geometry . By introducing a list of 37 definitions, Book 11 contextualizes the next two. Although its foundational character resembles Book 1, unlike the latter it features no axiomatic system or postulates. The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37). In addition to
4048-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
4140-694: The Platonic Academy in Athens. Historian Thomas Heath supported this theory, noting that most capable geometers lived in Athens, including many of those whose work Euclid built on; historian Michalis Sialaros considers this a mere conjecture. In any event, the contents of Euclid's work demonstrate familiarity with the Platonic geometry tradition. In his Collection , Pappus mentions that Apollonius studied with Euclid's students in Alexandria , and this has been taken to imply that Euclid worked and founded
4232-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
4324-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
4416-493: The area of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48). The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate". Book 2 is traditionally understood as concerning " geometric algebra ", though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since
4508-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
4600-501: The pentagon . Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion". Book 6 utilizes the "theory of ratios " in the context of plane geometry. It is built almost entirely of its first proposition: "Triangles and parallelograms which are under the same height are to one another as their bases". From Book 7 onwards, the mathematician Benno Artmann [ de ] notes that "Euclid starts afresh. Nothing from
4692-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
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#17327755545414784-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
4876-625: The 1st century AD Roman compiler of anecdotes, mistakenly substituted Euclid's name for Eudoxus (4th century BC) as the mathematician to whom Plato sent those asking how to double the cube . Perhaps on the basis of this mention of a mathematical Euclid roughly a century early, Euclid became mixed up with Euclid of Megara in medieval Byzantine sources (now lost), eventually leading Euclid the mathematician to be ascribed details of both men's biographies and described as Megarensis ( lit. ' of Megara ' ). The Byzantine scholar Theodore Metochites ( c. 1300 ) explicitly conflated
4968-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
5060-495: The First Book of Euclid's Elements , as well as a few anecdotes from Pappus of Alexandria in the early 4th century. According to Proclus, Euclid lived shortly after several of Plato 's ( d. 347 BC) followers and before the mathematician Archimedes ( c. 287 – c. 212 BC); specifically, Proclus placed Euclid during the rule of Ptolemy I ( r. 305/304–282 BC). Euclid's birthdate
5152-577: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
5244-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
5336-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
5428-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is
5520-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
5612-556: The description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
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#17327755545415704-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
5796-490: The fictionalization was done to strengthen the connection between a revered mathematician and the Arab world. There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man". The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading his Elements , which Euclid replied with "there
5888-409: The field until the early 19th century. His system, now referred to as Euclidean geometry , involved innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus , Hippocrates of Chios , Thales and Theaetetus . With Archimedes and Apollonius of Perga , Euclid is generally considered among the greatest mathematicians of antiquity, and one of
5980-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",
6072-411: The foundations of even nascent algebra occurred many centuries later. The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes. It focuses on the area of rectangles and squares (see Quadrature ), and leads up to a geometric precursor of the law of cosines . Book 3 focuses on circles, while the 4th discusses regular polygons , especially
6164-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
6256-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
6348-501: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Hilbert–Schmidt&oldid=932875336 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for
6440-492: The logical basis for every subsequent theorem, i.e. serve as an axiomatic system . The common notions exclusively concern the comparison of magnitudes . While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions of plane geometry and triangle congruence (1–26); parallel lines (27–34);
6532-611: The lunar crater Euclides , and the minor planet 4354 Euclides . The Elements is often considered after the Bible as the most frequently translated, published, and studied book in the Western World 's history. With Aristotle's Metaphysics , the Elements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds. The first English edition of
6624-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
6716-475: The most influential figures in the history of mathematics . The geometrical system established by the Elements long dominated the field; however, today that system is often referred to as ' Euclidean geometry ' to distinguish it from other non-Euclidean geometries discovered in the early 19th century. Among Euclid's many namesakes are the European Space Agency 's (ESA) Euclid spacecraft,
6808-530: The most influential in the history of mathematics . Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara . It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato 's students and before Archimedes. There
6900-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
6992-409: The needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves
7084-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains
7176-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
7268-533: The preceding books is used". Number theory is covered by books 7 to 10, the former beginning with a set of 22 definitions for parity , prime numbers and other arithmetic-related concepts. Book 7 includes the Euclidean algorithm , a method for finding the greatest common divisor of two numbers. The 8th book discusses geometric progressions , while book 9 includes the proposition, now called Euclid's theorem , that there are infinitely many prime numbers . Of
7360-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
7452-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
7544-506: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
7636-546: The text is the theorems scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" ( ‹See Tfd› Greek : ὅρος or ὁρισμός ), "postulate" ( αἴτημα ), or a "common notion" ( κοινὴ ἔννοια ); only the first book includes postulates—later known as axioms —and common notions. The second group consists of propositions, presented alongside mathematical proofs and diagrams. It
7728-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
7820-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
7912-769: The two Euclids, as did printer Erhard Ratdolt 's 1482 editio princeps of Campanus of Novara 's Latin translation of the Elements . After the mathematician Bartolomeo Zamberti [ fr ; de ] appended most of the extant biographical fragments about either Euclid to the preface of his 1505 translation of the Elements , subsequent publications passed on this identification. Later Renaissance scholars, particularly Peter Ramus , reevaluated this claim, proving it false via issues in chronology and contradiction in early sources. Medieval Arabic sources give vast amounts of information concerning Euclid's life, but are completely unverifiable. Most scholars consider them of dubious authenticity; Heath in particular contends that
8004-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
8096-406: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
8188-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
8280-556: Was not felicitously done." The Elements is speculated to have been at least partly in circulation by the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted; however, Archimedes employs an older variant of the theory of proportions than the one found in the Elements . The oldest physical copies of material included in the Elements , dating from roughly 100 AD, can be found on papyrus fragments unearthed in an ancient rubbish heap from Oxyrhynchus , Roman Egypt . The oldest extant direct citations to
8372-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
8464-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
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