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数学 Matematika Математика Matemática Matematiko Matemàtiques Μαθηματικά Matemáticas 수학 Mathematik Matematika Matematik 数学 Matematica Mathematics Matematyka Mathématiques رياضيات Математика Matematika Wiskunde Matamaitic
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La matemática​ (del latín mathematĭca, y este del griego μαθηματικά, transliterado como mathēmatiká, derivado de μάθημα, tr. máthēma. 'conocimiento') es una ciencia formal que surgió del estudio de las figuras geométricas y la aritmética con números. No existe una definición generalmente aceptada de las matemáticas; hoy en día se suelen describir como una ciencia que utiliza la lógica para examinar las propiedades y los patrones de las estructuras abstractas creadas por las definiciones lógicas. Is éard is matamaitic ann (ón tSean-Ghréigis μάθημα; máthēma: 'eolas, staidéar, foghlaim ') ná réimse eolais a chuimsíonn topaicí amhail; uimhreacha (Teoiric na matamaitice agus na huimhirtheoirice), foirmlí agus struchtúir ghaolmhara (ailgéabar), cruthanna agus na spásanna ina bhfuil siad (geoiméadracht), agus cainníochtaí agus a n-athruithe (calcalas agus anailís). Baineann an chuid is mó den ghníomhaíocht mhatamaiticiúil le húsáid a bhaint as glanréasúnaíocht theibí chun airíonna réad teibí a fháil amach nó a chruthú, arb éard atá i gceist leo ná teibithe ó nádúr, nó sa mhatamaitic nua-aimseartha, aonáin a bhfuil airíonna áirithe acu, ar a dtugtar aicsímí. Is éard atá i gcruthúnas matamaiticiúil ná sraith d'feidhmeanna ar roinnt rialacha déaduchtacha do thorthaí atá ar eolas cheana féin La matemàtica (encara que, per a referir-se a l'estudi i ciència, s'acostuma a utilitzar el plural matemàtiques) és aquella ciència que estudia patrons en les estructures de cossos abstractes i en les relacions que s'estableixen entre ells (del mot derivat del grec μάθημα, máthēma: ciència, coneixement, aprenentatge; μαθηματικός, mathēmatikós). Wiskunde (minder gebruikelijk: mathematiek, mathematica of mathesis) is een formele wetenschap die onder andere getallen, patronen en abstracte structuren bestudeert. De wiskunde komt voort uit het rekenen en de meetkunde, maar omvat veel meer dan dat. La matematica (dal greco: μάθημα (máthema), traducibile con i termini "scienza", "conoscenza" o "apprendimento"; μαθηματικός (mathematikós) significa "incline ad apprendere") è la disciplina che studia le quantità, i numeri, lo spazio, le strutture e i calcoli. La potenza e la generalità dei risultati della matematica le ha reso l'appellativo di regina delle scienze: ogni disciplina scientifica o tecnica, dalla fisica all'ingegneria, dall'economia all'informatica, fa largo uso degli strumenti di analisi, di calcolo e di modellazione offerti dalla matematica. 数学(すうがく、 英:Mathematics)とは、数・量・図形などに関する学問である。数学は形式科学に分類され、自然科学とは区別される。 Die Mathematik (bundesdeutsches Hochdeutsch: [matemaˈtiːk], [matemaˈtik]; österreichisches Hochdeutsch: [mateˈmaːtik]; altgriechisch μαθηματικὴ τέχνη mathēmatikē téchnē ‚die Kunst des Lernens‘) ist eine Formalwissenschaft, die aus der Untersuchung von geometrischen Figuren und dem Rechnen mit Zahlen entstand. Für Mathematik gibt es keine allgemein anerkannte Definition; heute wird sie üblicherweise als eine Wissenschaft beschrieben, die durch logische Definitionen selbstgeschaffene abstrakte Strukturen mittels der Logik auf ihre Eigenschaften und Muster untersucht. Τα μαθηματικά είναι μια επιστήμη που μελετά θέματα που αφορούν την ποσότητα (αριθμούς), τη (γεωμετρικά σχήματα), το χώρο, τη μεταβολή, τις σχέσεις όλων των μετρήσιμων αντικειμένων της πραγματικότητας και της φαντασίας μας, καθώς επίσης, σύμφωνα με ορισμένους ερευνητές, και μερικά άλλα που δεν είναι γενικώς δεκτά ότι πρέπει να περιλαμβάνονται στον . Matematika (grezierazko μάθημα, "máthema" –euskeraz, zientzia, ezagutza, ikaskuntza–, eta μαθηματικóς, "mathematikós" –zientzia zalea–) zenbaketa, neurketa eta geometriako kontzeptuetatik eratorritako kontzeptu abstraktoak aztertzen dituen zientzia zehatza da, logikan oinarritzen dena. Matematika (z řeckého μαθηματικός (mathématikos) = milující poznání; μάθημα (mathéma) = věda, vědění, poznání) je věda zabývající se z formálního hlediska kvantitou, strukturou, prostorem a změnou. Matematika je též popisována jako disciplína, jež se zabývá vytvářením abstraktních entit a vyhledáváním zákonitých vztahů mezi nimi. Matematiko (de la greka μαθημα [matema] - scienco, lernado) estas ekzakta logika dedukta scienco, kiu studas aksiomajn abstraktajn strukturojn (laŭ kvanto, formo, ) uzante logikan formalan lingvon. La specifaj strukturoj de matematiko plejofte originas de natursciencoj, plej multe de fiziko, sed matematikistoj difinas ankaŭ aliajn konceptojn por pure internaj bezonoj de la scienco. Matematiko jam penetris tra la tuta moderna vivo: modeligi precizajn instrumentojn, evoluigi novajn teknologiaĵojn kaj komputilojn, konstrui domojn; eĉ baki kukon bezonas aplikon de nocioj de nombroj, geometrio, mezuro kaj spaco. Matematiko estas iasence la fundamenta scienco. 数学,是研究數量、结构以及空间等概念及其变化的一門学科,屬於形式科學的一種。數學利用抽象化和邏輯推理,從計數、計算、量度、對物體形狀及運動的觀察發展而成。數學家們拓展這些概念,以公式化新的猜想,以及從選定的公理及定義出發,嚴謹地推導出一些定理。 基礎數學的知識與運用是生活中不可或缺的一環。對數學基本概念的完善,早在古埃及、及历史上的古代數學文本便可觀見,而在古希臘那裡有更為嚴謹的處理。從那時開始,數學的發展便持續不斷地小幅進展,至16世紀的文藝復興時期,因为新的科學發現和數學革新兩者的交互,致使數學的加速发展,直至今日。数学并成为許多國家及地區的教育中的一部分。 数学在许多领域都有应用,包括科學、工程、醫學、經濟學和金融學等。數學對這些領域的應用通常被稱為應用數學,有時亦會激起新的數學發現,並導致全新學科的發展,例如物理学的实质性发展中建立的某些理论激发数学家对于某些问题的不同角度的思考。數學家也研究純粹數學,就是數學本身的实质性內容,而不以任何實際應用為目標。許多研究雖然以純粹數學開始,但其过程中也發現許多可用之处。 Matematyka (z łac. mathematicus, od gr. μαθηματικός mathēmatikós, od μαθηματ-, μαθημα mathēmat-, mathēma, „nauka, lekcja, poznanie”, od μανθάνειν manthánein, „uczyć się, dowiedzieć”; prawd. spokr. z goc. mundon, „baczyć, uważać”) – nauka dostarczająca narzędzi do otrzymywania ścisłych wniosków z przyjętych założeń, zatem dotycząca prawidłowości rozumowania. Ponieważ ścisłe założenia mogą dotyczyć najróżniejszych dziedzin myśli ludzkiej, a muszą być czynione w naukach ścisłych, technice, a nawet w naukach humanistycznych, zakres matematyki jest szeroki i stale się powiększa. الرياضيات هي مجموعة من المعارف المجردة الناتجة عن الاستنتاجات المنطقية المطبقة على مختلف الكائنات الرياضية مثل المجموعات، والأعداد، والأشكال والبنيات والتحويلات. وتهتم الرياضيات أيضًا بدراسة مواضيع مثل الكمية والبنية والفضاء والتغير. ولا يوجد حتى الآن تعريف عام متفق عليه للمصطلح. Matematik (från grekiska: Μαθηματικά) är en abstrakt och generell vetenskap om problemlösning och metodutveckling – abstrakt därför att den frigjort sig från problemens konkreta ursprung och generell därför att den är tillämpbar i ett stort antal områden.Alternativt kan man även benämna den som en vetenskap om kvantitativa relationer och rumsliga strukturer i den verkliga världen. Exempel på matematiska begrepp är tal, data, struktur, kvantiteter, rum och deras förhållanden. Antingen som abstrakta koncept (ren matematik) eller tillämpningar i vetenskapliga discipliner såsom fysik och teknik (tillämpad matematik). Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets divers tels que les ensembles mathématiques, les nombres, les formes, les structures, les transformations, etc. ; ainsi qu'aux relations et opérations mathématiques qui existent entre ces objets. Elles sont aussi le domaine de recherche développant ces connaissances, ainsi que la discipline qui les enseigne. Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Matematika (dari bahasa Yunani Kuno μάθημα (máthēma), berarti "pengetahuan, pemikiran, pengkajian, pembelajaran"), adalah bidang ilmu, yang mencakup studi tentang topik-topik seperti bilangan (aritmetika dan teori bilangan), rumus dan struktur terkait (aljabar), bangun dan ruang tempat mereka berada (geometri), dan besaran serta perubahannya (kalkulus dan analisis). Tidak ada kesepakatan umum tentang ruang lingkup yang tepat atau status epistemologisnya. 수학(數學, 영어: mathematics, 줄여서 math)은 수, 양, 구조, 공간, 변화, 논리 등의 개념을 다루는 학문이다. 널리 받아들여지는 명확한 정의는 없으나 현대 수학은 일반적으로 엄밀한 논리에 근거하여 을 탐구하며, 이는 규칙의 발견과 문제의 제시 및 해결의 과정으로 이루어진다. 수학은 그 발전 과정에 있어서 철학, 과학과 깊은 연관을 맺고 있으며, 다만 엄밀한 논리와 특유의 추상성, 보편성에 의해 다른 학문들과 구별된다. 특히 수학은 과학의 여느 분야들과는 달리 자연계에서 관측되지 않는 개념들에 대해서까지 이론을 추상화시키는 특징을 보이는데, 수학자들은 그러한 개념들에 대한 추측을 제시하고 적절하게 선택된 정의와 공리로부터 엄밀한 연역을 거쳐 그 진위를 파악한다. 수학의 개념들은 기원전 600년 경에 활동하며 최초의 수학자로도 여겨지는 탈레스의 기록은 물론, 다른 고대 문명들에서도 찾아볼 수 있으며 인류의 문명과 함께 발전해 왔다. 오늘날 수학은 자연과학, 사회과학, 공학, 의학 등 거의 모든 학문에서도 핵심적인 역할을 하며 다양한 방식으로 응용된다. A matemática (dos termos gregos, μάθημα, transliterado máthēma, 'ciência', conhecimento' ou 'aprendizagem; e μαθηματικός, transliterado mathēmatikós, 'inclinado a aprender') é a ciência do raciocínio lógico e abstrato, que estuda quantidades (teoria dos números), espaço e medidas (geometria), estruturas, variações e estatística. Não há, porém, uma definição consensual por parte da comunidade científica. O trabalho matemático consiste em procurar e relacionar padrões, de modo a formular conjecturas cuja veracidade ou falsidade é provada por meio de deduções rigorosas, a partir de axiomas e definições. A matemática desenvolveu-se principalmente na Mesopotâmia, no Egito, na Grécia, na Índia e no Oriente Médio. Após a Renascença, o desenvolvimento da matemática intensificou-se na Europa, quand Матема́тика (грец. μάθημα — наука, знання, вивчення) — наука, яка первісно виникла як один з напрямків пошуку істини (у грецькій філософії) у сфері просторових відношень (землеміряння — геометрії) і обчислень (арифметики), для практичних потреб людини рахувати, обчислювати, вимірювати, досліджувати форми та рух фізичних тіл. Пізніше розвинулась у доволі складну і багатогранну науку про абстрактні кількісні та якісні співвідношення, форми і структури. Загальноприйнятого визначення математики немає. Початково її використовували для підрахунку, вимірювання, а також для вивчення форм і руху фізичних об'єктів шляхом дедуктивних розмірковувань і абстракцій. Математики формулюють нові висновки та намагаються з'ясувати їхню правдивість, виходячи із вдало вибраних аксіом і визначень. Матема́тика (др.-греч. μᾰθημᾰτικά < μάθημα «изучение; наука») — точная (формальная) наука, первоначально исследовавшая количественные отношения и пространственные формы. В более современном понимании, это наука об отношениях между объектами, о которых ничего не известно, кроме описывающих их некоторых свойств, — именно тех, которые в качестве аксиом положены в основание той или иной математической теории.
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Gottfried Wilhelm von Leibniz Isaac Newton
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Isaac Newton and Gottfried Wilhelm Leibniz developed infinitesimal calculus.
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Mathematics
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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A proof consists 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 as true starting points of the theory under consideration. Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. The fundamental truths of mathematics are independent from any scientific experimentation, although mathematics is extensively used for modeling phenomena. 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 mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later. A fitting example is the problem of integer factorization, which goes back to Euclid, but which had no practical application before its use in the RSA cryptosystem (for the security of computer networks). Historically, 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 essentially 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 areas of the subject. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep 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. This gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. This can be seen, for example, in the contemporary Mathematics Subject Classification, which lists more than 60 first-level areas of mathematics. Matematika (z řeckého μαθηματικός (mathématikos) = milující poznání; μάθημα (mathéma) = věda, vědění, poznání) je věda zabývající se z formálního hlediska kvantitou, strukturou, prostorem a změnou. Matematika je též popisována jako disciplína, jež se zabývá vytvářením abstraktních entit a vyhledáváním zákonitých vztahů mezi nimi. Matematika je založena a budována jako exaktní věda. Její exaktnost (podobně jako jiných exaktních věd) tkví v tom že, jak matematické objekty, tak i operace nad nimi jsou exaktně vytyčeny (tj. s nulovou vnitřní vágností), tedy tak, že každý v matematice (v dané exaktní vědě) vzdělaný člověk naprosto přesně (bez jakýchkoli pochyb) ví, co znamenají. To je podstata exaktnosti této disciplíny. V rámci matematiky existuje ale ještě jinak chápaná exaktnost, a to exaktnost použitých metod a jejich výsledků: Příkladem může být exaktní a neexaktní řešení: Některé aplikace jsou řešitelné pouze opuštěním přísného a omezujícího požadavku exaktnosti výsledku. Například proto, že neexistuje matematická funkce, která by byla (exaktním) řešením dané diferenciální rovnice. Může ale existovat posloupnost funkcí, která s libovolnou přesností (nikoli však exaktně), řešením té rovnice je. Dosazením exaktního výsledku (řešení) do výchozího vztahu (rovnice) dostáváme identitu. Neexaktní výsledek se od exaktního liší o „chybu “, takže po jeho dosazení identitu nedostaneme. Charakteristickou vlastností matematiky je její důraz na absolutní přesnost metod a nezpochybnitelnost výsledků. Tyto vlastnosti, které matematiku odlišují od všech ostatních vědních disciplín, mají původ již v antickém Řecku. Nejstarším dochovaným příkladem tohoto přístupu je kniha řeckého matematika Euklida Základy pocházející z 4. století př. n. l. Široké veřejnosti je známa tzv. elementární matematika, která se zabývá operováním s čísly, řešením praktických úloh, jednoduchých rovnic a popisem základních geometrických objektů. Ve fyzice, informatice, chemii, ekonomii a dalších oborech se často využívají výsledky aplikované matematiky, která je také těmito obory zpětně ovlivňována. Tzv. čistá matematika se zabývá pouze vysoce abstraktními pojmy, jejichž definování není přímo motivováno praktickým užitkem v reálném světě. Některé obory čisté matematiky se nacházejí na pomezí s logikou či filozofií. Матема́тика (грец. μάθημα — наука, знання, вивчення) — наука, яка первісно виникла як один з напрямків пошуку істини (у грецькій філософії) у сфері просторових відношень (землеміряння — геометрії) і обчислень (арифметики), для практичних потреб людини рахувати, обчислювати, вимірювати, досліджувати форми та рух фізичних тіл. Пізніше розвинулась у доволі складну і багатогранну науку про абстрактні кількісні та якісні співвідношення, форми і структури. Загальноприйнятого визначення математики немає. Початково її використовували для підрахунку, вимірювання, а також для вивчення форм і руху фізичних об'єктів шляхом дедуктивних розмірковувань і абстракцій. Математики формулюють нові висновки та намагаються з'ясувати їхню правдивість, виходячи із вдало вибраних аксіом і визначень. 数学(すうがく、 英:Mathematics)とは、数・量・図形などに関する学問である。数学は形式科学に分類され、自然科学とは区別される。 Wiskunde (minder gebruikelijk: mathematiek, mathematica of mathesis) is een formele wetenschap die onder andere getallen, patronen en abstracte structuren bestudeert. De wiskunde komt voort uit het rekenen en de meetkunde, maar omvat veel meer dan dat. Wiskundige structuren worden met strikte logische redeneringen opgebouwd. Wiskundige beweringen waarvan de juistheid is aangetoond heten stellingen; zij doen uitspraken over gedefinieerde objecten en formuleren verbanden daartussen. De formele redenering die aantoont dat een stelling waar is, noemt men een wiskundig bewijs. Bij het opstellen van een bewijs wordt uitgegaan van een (klein) aantal uitgangspunten (axioma's) en van definities. Wiskunde wordt niet alleen zelfstandig bestudeerd, de opgedane kennis wordt toegepast in allerlei dagelijkse situaties en in andere wetenschappen. Men spreekt dan van toegepaste wiskunde tegenover zuivere wiskunde. De scheidslijn is echter niet heel duidelijk en wat begon als zuivere wiskunde blijkt later toch regelmatig een toepassing te vinden. In toepassingen wordt gerekend op basis van reeds bewezen stellingen. Dat kan eenvoudig zijn, bijvoorbeeld de stelling van Pythagoras in de meetkunde om een afstand te bepalen. Maar soms is het probleem in het dagelijks leven zo uitgebreid, dat er een (super)computer voor nodig is om binnen een redelijke tijd een oplossing te vinden. Een voorbeeld daarvan is de weersverwachting in de meteorologie: de atmosfeer wordt toegepast wiskundig gemodelleerd met behulp van differentiaalvergelijkingen. Meetwaarden, afkomstig van meetpunten liefst over de hele aardbol op verschillende hoogten, bepalen na bewerking een begintoestand vanwaaruit de toekomstige druk, wind en luchtvochtigheid wordt berekend. Het doorrekenen van de differentiaalvergelijkingen die uit deze meetgegevens volgen, kost enorm veel computertijd. Matematika (grezierazko μάθημα, "máthema" –euskeraz, zientzia, ezagutza, ikaskuntza–, eta μαθηματικóς, "mathematikós" –zientzia zalea–) zenbaketa, neurketa eta geometriako kontzeptuetatik eratorritako kontzeptu abstraktoak aztertzen dituen zientzia zehatza da, logikan oinarritzen dena. Matematik (från grekiska: Μαθηματικά) är en abstrakt och generell vetenskap om problemlösning och metodutveckling – abstrakt därför att den frigjort sig från problemens konkreta ursprung och generell därför att den är tillämpbar i ett stort antal områden.Alternativt kan man även benämna den som en vetenskap om kvantitativa relationer och rumsliga strukturer i den verkliga världen. Exempel på matematiska begrepp är tal, data, struktur, kvantiteter, rum och deras förhållanden. Antingen som abstrakta koncept (ren matematik) eller tillämpningar i vetenskapliga discipliner såsom fysik och teknik (tillämpad matematik). Medan naturvetenskapen studerar entiteter i tid och rum är det inte uppenbart att samma är sant för de objekt som studeras i matematik. Vidare skiljer sig metoderna för undersökning åt: naturvetenskapen tenderar att använda metoder av induktion och matematiken metoder av deduktion. Av bland annat nämnda skäl väcker matematik ontologiska och kunskapsteoretiska frågor skilda från vetenskapsteorin. Dessa frågor behandlas i matematikfilosofi. Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets divers tels que les ensembles mathématiques, les nombres, les formes, les structures, les transformations, etc. ; ainsi qu'aux relations et opérations mathématiques qui existent entre ces objets. Elles sont aussi le domaine de recherche développant ces connaissances, ainsi que la discipline qui les enseigne. Elles possèdent plusieurs branches telles que : l'arithmétique, l'algèbre, l'analyse, la géométrie, la logique mathématique, les probabilités, etc. Il existe également une certaine séparation entre les mathématiques pures et les mathématiques appliquées. Les mathématiques se distinguent des autres sciences par un rapport particulier au réel car l'observation et l'expérience ne s'y portent pas sur des objets physiques ; les mathématiques ne sont pas une science empirique. Elles sont de nature entièrement intellectuelle, fondées sur des axiomes déclarés vrais ou sur des postulats provisoirement admis. Ces axiomes en constituent les fondements et ne dépendent donc d'aucune autre proposition. Un énoncé mathématique – dénommé généralement, après être validé, théorème, proposition, lemme, fait, scholie ou corollaire – est considéré comme valide lorsque le discours formel qui établit sa vérité respecte une certaine structure rationnelle appelée démonstration, ou raisonnement logicodéductif. Un énoncé qui n'a pas encore fait l'objet d'une démonstration mais qui est néanmoins considéré plausible est appelé conjecture. Bien que les résultats mathématiques soient des vérités purement formelles, ils trouvent des applications dans les autres sciences et dans différents domaines de la technique. C'est ainsi qu'Eugene Wigner déclare que la « déraisonnable efficacité des mathématiques dans les sciences de la nature est une chose presque mystérieuse ». Matematyka (z łac. mathematicus, od gr. μαθηματικός mathēmatikós, od μαθηματ-, μαθημα mathēmat-, mathēma, „nauka, lekcja, poznanie”, od μανθάνειν manthánein, „uczyć się, dowiedzieć”; prawd. spokr. z goc. mundon, „baczyć, uważać”) – nauka dostarczająca narzędzi do otrzymywania ścisłych wniosków z przyjętych założeń, zatem dotycząca prawidłowości rozumowania. Ponieważ ścisłe założenia mogą dotyczyć najróżniejszych dziedzin myśli ludzkiej, a muszą być czynione w naukach ścisłych, technice, a nawet w naukach humanistycznych, zakres matematyki jest szeroki i stale się powiększa. Wiele dziedzin nauki i technologii w pewnym momencie zaczyna definiować swoje pojęcia z dostatecznie dużą precyzją, aby można było stosować do nich metody matematyczne, co często zapoczątkowuje kolejny dział matematyki teoretycznej lub stosowanej. Tak stało się np. z mechaniką klasyczną, mechaniką statystyczną, ekonomią (ekonometria), lingwistyką (lingwistyka matematyczna), teorią gier, a nawet niektórymi działami politologii. Obecnie standardem w naukach eksperymentalnych jest potwierdzanie istnienia obserwowanych zależności za pomocą metod statystyki, będącej działem matematyki. Pomaga to odróżnić rzeczywiste zależności od przypadkowej zbieżności. Leonardo da Vinci stwierdził w Traktacie o malarstwie: „Żadne ludzkie badania nie mogą być nazywane prawdziwą nauką, jeśli nie mogą być zademonstrowane matematycznie”. Matematyka teoretyczna, nazywana czasami matematyką czystą, jest często rozwijana bez wyraźnego związku z konkretnymi zastosowaniami. W tej odmianie jest ona przez niektórych matematyków uważana za formę sztuki. Jednak niektóre działy matematyki teoretycznej znalazły swoje praktyczne zastosowanie, kiedy okazało się, że potrzebuje ich nowoczesna fizyka lub informatyka. Szkolne rozumienie matematyki jako nauki wyłącznie o liczbach i pojęciach geometrycznych zdezaktualizowało się już w XIX wieku wraz z postępami algebry i teorii mnogości. 数学,是研究數量、结构以及空间等概念及其变化的一門学科,屬於形式科學的一種。數學利用抽象化和邏輯推理,從計數、計算、量度、對物體形狀及運動的觀察發展而成。數學家們拓展這些概念,以公式化新的猜想,以及從選定的公理及定義出發,嚴謹地推導出一些定理。 基礎數學的知識與運用是生活中不可或缺的一環。對數學基本概念的完善,早在古埃及、及历史上的古代數學文本便可觀見,而在古希臘那裡有更為嚴謹的處理。從那時開始,數學的發展便持續不斷地小幅進展,至16世紀的文藝復興時期,因为新的科學發現和數學革新兩者的交互,致使數學的加速发展,直至今日。数学并成为許多國家及地區的教育中的一部分。 数学在许多领域都有应用,包括科學、工程、醫學、經濟學和金融學等。數學對這些領域的應用通常被稱為應用數學,有時亦會激起新的數學發現,並導致全新學科的發展,例如物理学的实质性发展中建立的某些理论激发数学家对于某些问题的不同角度的思考。數學家也研究純粹數學,就是數學本身的实质性內容,而不以任何實際應用為目標。許多研究雖然以純粹數學開始,但其过程中也發現許多可用之处。 Матема́тика (др.-греч. μᾰθημᾰτικά < μάθημα «изучение; наука») — точная (формальная) наука, первоначально исследовавшая количественные отношения и пространственные формы. В более современном понимании, это наука об отношениях между объектами, о которых ничего не известно, кроме описывающих их некоторых свойств, — именно тех, которые в качестве аксиом положены в основание той или иной математической теории. Математика исторически сложилась на основе операций подсчёта, измерения и описания формы объектов. Математические объекты создаются путём идеализации свойств реальных или других математических объектов и записи этих свойств на формальном языке. Математика не относится к естественным наукам, но широко используется в них как для точной формулировки их содержания, так и для получения новых результатов. Она является фундаментальной наукой, предоставляющей (общие) языковые средства другим наукам; тем самым она выявляет их структурную взаимосвязь и способствует нахождению самых общих законов природы. La matemàtica (encara que, per a referir-se a l'estudi i ciència, s'acostuma a utilitzar el plural matemàtiques) és aquella ciència que estudia patrons en les estructures de cossos abstractes i en les relacions que s'estableixen entre ells (del mot derivat del grec μάθημα, máthēma: ciència, coneixement, aprenentatge; μαθηματικός, mathēmatikós). Malgrat que tingui múltiples usos en altres ciències i disciplines (molt particularment, en la física), i tracti relacions que poden semblar evidents, les matemàtiques primer postulen (vegeu axiomes matemàtics), i després dedueixen i demostren. Les matemàtiques no són una ciència experimental, sinó una ciència formal. Els matemàtics acostumen a definir i investigar estructures i conceptes abstractes per raons purament internes a la matemàtica, ja que tals estructures poden proveir, per exemple, una generalització elegant, o una eina útil per a càlculs freqüents. A més, molts matemàtics estudien les seves àrees de preferència simplement per raons estètiques, veient així la matemàtica com una forma d'art en comptes d'una ciència pràctica o aplicada (encara que les estructures que els matemàtics investiguen tenen, molt sovint, el seu origen en observacions de la natura). La matemàtica és un art, però també una ciència d'estudi. Informalment, es pot afirmar que la matemàtica és l'estudi dels «nombres i símbols», és a dir, la investigació d'estructures abstractes definides axiomàticament utilitzant la lògica i la notació matemàtica. És també la ciència de les relacions espacials i quantitatives. Es tracta de relacions exactes que existeixen entre quantitats i magnituds, i dels mètodes pels quals, d'acord amb aquestes relacions, les quantitats buscades són deduïbles a partir d'altres quantitats conegudes o pressuposades. Altres punts de vista poden trobar-se en la filosofia de les matemàtiques. És freqüent trobar qui descriu la matemàtica com una simple extensió dels llenguatges naturals humans, que utilitza una gramàtica i un vocabulari definits amb extrema precisió, el propòsit dels quals és la descripció i exploració de relacions conceptuals i físiques. Recentment, això no obstant, els avanços en l'estudi del llenguatge humà apunten cap a una altra forma d'analitzar-los: els llenguatges naturals (com el català i el francès) i els llenguatges formals (com la matemàtica i els llenguatges de programació) són estructures de naturalesa bàsicament diferent. Is éard is matamaitic ann (ón tSean-Ghréigis μάθημα; máthēma: 'eolas, staidéar, foghlaim ') ná réimse eolais a chuimsíonn topaicí amhail; uimhreacha (Teoiric na matamaitice agus na huimhirtheoirice), foirmlí agus struchtúir ghaolmhara (ailgéabar), cruthanna agus na spásanna ina bhfuil siad (geoiméadracht), agus cainníochtaí agus a n-athruithe (calcalas agus anailís). Baineann an chuid is mó den ghníomhaíocht mhatamaiticiúil le húsáid a bhaint as glanréasúnaíocht theibí chun airíonna réad teibí a fháil amach nó a chruthú, arb éard atá i gceist leo ná teibithe ó nádúr, nó sa mhatamaitic nua-aimseartha, aonáin a bhfuil airíonna áirithe acu, ar a dtugtar aicsímí. Is éard atá i gcruthúnas matamaiticiúil ná sraith d'feidhmeanna ar roinnt rialacha déaduchtacha do thorthaí atá ar eolas cheana féin, lena n-áirítear teoirimí cruthaithe roimhe seo, aicsímí a bhí cruthaithe roimhe seo agus (i gcás teibithe ón nádúr) roinnt airíonna bunúsacha a mheastar mar fhíorphointí tosaigh na teoirice atá á meas. Úsáidtear an mhatamaitic san eolaíocht chun feiniméin a shamhaltú, rud a cheadaíonn tuar a dhéanamh ansin ó dhlíthe turgnamhacha. Tugann neamhspleáchas na fírinne matamaiticiúla ó aon turgnamh le tuiscint go mbraitheann cruinneas na dtuartha sin ach amháin ar leordhóthanacht na samhla. Tuartha míchruinn, seachas a bheith de bharr na matamaitice míchirte, le tuiscint go bhfuil gá leis tsamhail mhatamaiticiúil a úsáidtear a athrú. an gá leis an tsamhail matamaitice a úsáidtear a athrú. Mar shampla, ní fhéadfaí luainíocht pheirihéileach Mhearcair a mhíniú ach amháin tar éis coibhneasacht ghinearálta Einstein a bheith tagtha chun cinn, a tháinig in ionad dhlí imtharraingthe Newton mar mhúnla matamaiticiúil níos fearr. Tá an mhatamaitic riachtanach sna heolaíochtaí, san innealtóireacht, sa leigheas, san airgeadas, san ríomheolaíocht agus sna heolaíochtaí sóisialta. Forbraítear roinnt réimsí matamaitice, ar nós staitisticí agus teoiric na gcluichí, i ndlúthghaol lena n-úsáidí agus is minic a ghrúpáiltear iad faoin mhatamaitic fheidhmeach. Forbraítear réimsí matamaiticiúla eile go neamhspleách ar aon fheidhm (agus mar sin tugtar glanmhatamaitic orthu), ach is minic a aimsítear feidhmeanna praiticiúla níos déanaí. Is sampla oiriúnach é an fhadhb a bhaineann le fachtóiriú slánuimhir, a théann ar ais go dtí Eoiclídéas, ach nach raibh aon fheidhm phraiticiúil aige sular úsáideadh é i gcripteachóras le heochair phoiblí RSA (le haghaidh slándáil líonraí ríomhaireachta). Go stairiúil, ba sa mhatamaitic Ghréagach a tháinig coincheap an chruthúnais agus na críochnúlachta matamaiticiúla a bhaineann leis chun cinn ar dtús, go háirithe sna hUraiceachtaí. Ó thús, roinneadh an mhatamaitic go bunúsach i gGeoiméadracht, agus uimhríocht (ionramháil uimhreacha aiceanta agus codáin), go dtí an 16ú agus an 17ú haois, nuair a tugadh ailgéabar agus calcalas an-bhídeach isteach mar réimsí nua den ábhar. Ó shin i leith, tá an t-idirghníomhú idir nuálaíochtaí matamaiticiúla agus fionnachtana eolaíocha mar thoradh ar mhéadú tapa i bhforbairt na matamaitice. Ag deireadh an 19ú haois, d'eascair an ghéarchéim bhunúsach sa mhatamaitic (Gearmáinis; Grundlagenkrise der Mathematik) as an modh aicsímeach a chórasú. Mar thoradh air seo tháinig méadú mór ar líon na réimsí matamaitice agus ar a réimsí feidhmeanna. Sampla de seo is ea Rangú Ábhair na Matamaitice, a liostaíonn níos mó ná 60 réimse matamaitice den chéad leibhéal. A matemática (dos termos gregos, μάθημα, transliterado máthēma, 'ciência', conhecimento' ou 'aprendizagem; e μαθηματικός, transliterado mathēmatikós, 'inclinado a aprender') é a ciência do raciocínio lógico e abstrato, que estuda quantidades (teoria dos números), espaço e medidas (geometria), estruturas, variações e estatística. Não há, porém, uma definição consensual por parte da comunidade científica. O trabalho matemático consiste em procurar e relacionar padrões, de modo a formular conjecturas cuja veracidade ou falsidade é provada por meio de deduções rigorosas, a partir de axiomas e definições. A matemática desenvolveu-se principalmente na Mesopotâmia, no Egito, na Grécia, na Índia e no Oriente Médio. Após a Renascença, o desenvolvimento da matemática intensificou-se na Europa, quando novas descobertas científicas levaram a um acúmulo rápido de conhecimento que dura até os dias de hoje. Registros arqueológicos mostram que a matemática é tanto um fator cultural quanto parte da história do desenvolvimento da nossa espécie. Ela evoluiu por meio de contagens, medições, cálculos e do estudo sistemático de formas geométricas e movimentos de objetos físicos. Já as proposições mais abstratas, que envolvem a argumentação lógica, surgiram com os matemáticos gregos, há aproximadamente 300 a.C., notadamente com a obra Os Elementos, de Euclides. A necessidade de maior rigor foi percebida e estabelecida por volta do início do século XVIII. Há muito tempo, buscam-se um consenso quanto à definição do que é a matemática. No entanto, nas últimas décadas do século XX, tomou forma uma definição que tem ampla aceitação entre os matemáticos: "a matemática é a ciência das regularidades" (padrões). Segundo esta definição, o trabalho do matemático consiste em examinar padrões abstratos, tanto reais como imaginários, visuais ou mentais. Ou seja, os matemáticos procuram regularidades nos números, no espaço, na ciência e na imaginação e formulam teorias com as quais tentam explicar as relações observadas. Uma outra definição seria que a matemática é a investigação de estruturas abstratas definidas axiomaticamente, usando a lógica formal como estrutura comum. As estruturas específicas geralmente têm sua origem nas ciências naturais, mais comumente na física, mas os matemáticos também definem e investigam estruturas por razões puramente internas à matemática (matemática pura), por exemplo, ao perceberem que as estruturas fornecem uma generalização unificante de vários subcampos ou uma ferramenta útil em cálculos comuns. A matemática é usada como uma ferramenta essencial em muitas áreas do conhecimento, tais como: engenharia, medicina, física, química, biologia e ciências sociais. Em um ponto, a matemática aplicada, ramo da matemática que se dedica a aplicabilidade da matemática em outras áreas do conhecimento, às vezes leva ao desenvolvimento de um novo ramo, como aconteceu com a estatística ou a teoria dos jogos. Em outro ponto, o estudo da matemática pura, ou seja, o estudo da matemática pela matemática, sem a preocupação de uma aplicabilidade, muitas vezes mostrou-se útil anos ou séculos depois. Foi como aconteceu com os estudos das cônicas e da teoria dos números que, explorada pelos gregos, foram úteis respectivamente em descobertas sobre astronomia feitas por Kepler no século XVII, ou para o desenvolvimento de segurança (criptografia) em computadores nos dias de hoje. الرياضيات هي مجموعة من المعارف المجردة الناتجة عن الاستنتاجات المنطقية المطبقة على مختلف الكائنات الرياضية مثل المجموعات، والأعداد، والأشكال والبنيات والتحويلات. وتهتم الرياضيات أيضًا بدراسة مواضيع مثل الكمية والبنية والفضاء والتغير. ولا يوجد حتى الآن تعريف عام متفق عليه للمصطلح. يسعى علماء الرياضيات إلى استخدام أنماط رياضية لصياغة فرضيات جديدة؛ من خلال استعمال إثباتات رياضية بهدف الوصول للحقيقة وذرء الفرضيات السابقة أو الخاطئة. فمن خلال استخدام التجريد والمنطق، طُوِّرت الرياضيات من العد والحساب والقياس إلى الدراسة المنهجية للأشكال وحركات الأشياء المادية. لقد كانت الرياضيات العملية نشاطًا إنسانيًا يعود إلى تاريخ وجود السجلات المكتوبة. يمكن أن يستغرق البحث المطلوب لحل المسائل الرياضية سنوات أو حتى قرون من البحث المستمر. ظهرت الحجج الصارمة أولًا في الرياضيات اليونانية، وعلى الأخص في أصول إقليدس. منذ العمل الرائد لجوزيبه بيانو (1858-1932)، وديفيد هيلبرت (1862-1943)، وغيرهم في النظم البديهية في أواخر القرن التاسع عشر، أصبح من المعتاد النظر إلى الأبحاث الرياضية كإثبات للحقيقة عن طريق الاستنتاج الدقيق للبديهيات والتعاريف المختارة بشكل مناسب. وتطورت الرياضيات بوتيرة بطيئة نسبيًا حتى عصر النهضة، عندما أدت الابتكارات الرياضية التي تتفاعل مع الاكتشافات العلمية الجديدة إلى زيادة سريعة في معدل الاكتشافات الرياضية التي استمرت حتى يومنا هذا. تعتبر الرياضيات ضرورية في العديد من المجالات، لما لها من قدرة على وضع نماذج رياضية تمكّنها من صياغة سلوك ما أو التنبؤ بسلوك محتمل. من أشهر المجالات التي تستعمل النماذج الرياضية العلوم الطبيعية والهندسة والطب والتمويل والعلوم الاجتماعية. أدت الرياضيات التطبيقية إلى تخصصات رياضية جديدة تمامًا، مثل الإحصاء ونظرية الألعاب والتحكم الأمثل. يشارك علماء الرياضيات في الرياضيات البحتة دون وضع أي تطبيق على أرض الواقع، ولكن غالبًا ما يتم اكتشاف التطبيقات العملية لما بدأ في الأول كرياضيات بحتة. وفقا لاستطلاع أجرته مجموعة خبراء التصنيف الدولية IREG في عام (2013-2014)، جاءت جائزة أبيل التي بدأت عام (2003) والتي تمنحها سنويًا الأكاديمية النرويجية للعلوم والآداب في المرتبة الأولى كأكثر جائزة مرموقة في مجال الرياضيات. وجاءت في المرتبة الثانية ميدالية فيلدز التي يرعاها الاتحاد الدولي للرياضيات منذ عام (1936). وفي المرتبة الثالثة جاءت جائزة وولف في الرياضيات التي تمنحها سنويًا مؤسسة وولف منذ عام (1978). تعتبر هذه الجوائز من بين أكثر الجوائز شهرة بفضل قيمتها المالية، ويعتبر البعض جائزة أبيل وميدالية فيلدز بمثابة جائزة نوبل في مجال الرياضيات لأن جائزة نوبل لا تمنح في هذا المجال. 수학(數學, 영어: mathematics, 줄여서 math)은 수, 양, 구조, 공간, 변화, 논리 등의 개념을 다루는 학문이다. 널리 받아들여지는 명확한 정의는 없으나 현대 수학은 일반적으로 엄밀한 논리에 근거하여 을 탐구하며, 이는 규칙의 발견과 문제의 제시 및 해결의 과정으로 이루어진다. 수학은 그 발전 과정에 있어서 철학, 과학과 깊은 연관을 맺고 있으며, 다만 엄밀한 논리와 특유의 추상성, 보편성에 의해 다른 학문들과 구별된다. 특히 수학은 과학의 여느 분야들과는 달리 자연계에서 관측되지 않는 개념들에 대해서까지 이론을 추상화시키는 특징을 보이는데, 수학자들은 그러한 개념들에 대한 추측을 제시하고 적절하게 선택된 정의와 공리로부터 엄밀한 연역을 거쳐 그 진위를 파악한다. 수학의 개념들은 기원전 600년 경에 활동하며 최초의 수학자로도 여겨지는 탈레스의 기록은 물론, 다른 고대 문명들에서도 찾아볼 수 있으며 인류의 문명과 함께 발전해 왔다. 오늘날 수학은 자연과학, 사회과학, 공학, 의학 등 거의 모든 학문에서도 핵심적인 역할을 하며 다양한 방식으로 응용된다. 수학을 의미하는 매스매틱스(mathematics)라는 단어는 '아는 모든 것', '배우는 모든 것'이라는 뜻의 고대 그리스어 'máthēma'(μάθημα) 및 그 활용형 mathēmatikós(μαθηματικός)에서 유래되었다. La matematica (dal greco: μάθημα (máthema), traducibile con i termini "scienza", "conoscenza" o "apprendimento"; μαθηματικός (mathematikós) significa "incline ad apprendere") è la disciplina che studia le quantità, i numeri, lo spazio, le strutture e i calcoli. Col termine matematica di solito si designa la disciplina (e il relativo corpo di conoscenze) che studia problemi concernenti quantità, estensioni e figure spaziali, movimenti di corpi, e tutte le strutture che permettono di trattare questi aspetti in modo generale. La matematica fa largo uso degli strumenti della logica e sviluppa le proprie conoscenze nel quadro di sistemi ipotetico-deduttivi che, a partire da definizioni rigorose e da assiomi riguardanti proprietà degli oggetti definiti (risultati da un procedimento di astrazione, come triangoli, funzioni, vettori ecc.), raggiunge nuove certezze, per mezzo delle dimostrazioni, attorno a proprietà meno intuitive degli oggetti stessi (espresse dai teoremi). La potenza e la generalità dei risultati della matematica le ha reso l'appellativo di regina delle scienze: ogni disciplina scientifica o tecnica, dalla fisica all'ingegneria, dall'economia all'informatica, fa largo uso degli strumenti di analisi, di calcolo e di modellazione offerti dalla matematica. Matematika (dari bahasa Yunani Kuno μάθημα (máthēma), berarti "pengetahuan, pemikiran, pengkajian, pembelajaran"), adalah bidang ilmu, yang mencakup studi tentang topik-topik seperti bilangan (aritmetika dan teori bilangan), rumus dan struktur terkait (aljabar), bangun dan ruang tempat mereka berada (geometri), dan besaran serta perubahannya (kalkulus dan analisis). Tidak ada kesepakatan umum tentang ruang lingkup yang tepat atau status epistemologisnya. Matematika selalu berkembang, misalnya di Tiongkok pada tahun 300 SM, di India pada tahun 100 M, dan di Arab pada tahun 800 M, hingga zaman Renaisans, ketika temuan baru matematika berinteraksi dengan penemuan ilmiah baru yang mengarah pada peningkatan yang cepat di dalam laju penemuan matematika. Berlanjut hingga kini, matematika digunakan di seluruh dunia sebagai alat penting di berbagai bidang, termasuk ilmu alam, teknik, kedokteran/medis, dan ilmu sosial seperti ekonomi, dan psikologi. Matematika terapan, cabang matematika yang melingkupi penerapan pengetahuan matematika ke bidang-bidang lain, mengilhami dan membuat penggunaan temuan-temuan matematika baru, dan kadang-kadang mengarah pada pengembangan disiplin-disiplin ilmu yang sepenuhnya baru, seperti statistika dan teori permainan. Matematika banyak digunakan dalam ilmu pengetahuan untuk fenomena pemodelan. Hal ini memungkinkan ekstraksi perkiraan kuantitatif dari hukum-hukum percobaan. Misalnya, pergerakan planet dapat diprediksi dengan akurasi tinggi menggunakan hukum gravitasi Newton yang dipadukan dengan perhitungan matematis. Ketakbergantungan kebenaran matematis dari percobaan manapun menyiratkan bahwa keakuratan perkiraan semacam itu hanya bergantung pada kecukupan model untuk menggambarkan kenyataan. Jadi, ketika munculnya beberapa perkiraan yang tidak tepat, itu berarti bahwa model harus diperbaiki atau diubah, bukan berarti matematika salah. Misalnya, presesi apsis atau perihelium Merkurius tidak dapat dijelaskan dengan hukum gravitasi Newton, tetapi dijelaskan secara akurat oleh relativitas umum Einstein. Pengesahan percobaan teori Einstein ini menunjukkan bahwa hukum gravitasi Newton hanyalah hampiran (yang masih sangat akurat dalam kehidupan sehari-hari). Matematika sangat penting di banyak bidang, termasuk ilmu alam, rekayasa, kedokteran, keuangan, ilmu komputer, dan ilmu sosial.Beberapa bidang matematika, seperti statistika dan teori permainan, dikembangkan dalam korelasi langsung dengan terapannya, dan sering dikelompokkan dengan nama matematika terapan. Bidang matematika lainnya dikembangkan secara independen dari aplikasi apapun (dan oleh karena itu disebut matematika murni), tetapi aplikasi praktis sering ditemukan kemudian. Contoh yang tepat adalah masalah faktorisasi prima, yang merujuk kepada Euklides, tetapi yang tidak memiliki aplikasi praktis sebelum digunakan dalam sistemkripto RSA (untuk keamanan jaringan komputer). Τα μαθηματικά είναι μια επιστήμη που μελετά θέματα που αφορούν την ποσότητα (αριθμούς), τη (γεωμετρικά σχήματα), το χώρο, τη μεταβολή, τις σχέσεις όλων των μετρήσιμων αντικειμένων της πραγματικότητας και της φαντασίας μας, καθώς επίσης, σύμφωνα με ορισμένους ερευνητές, και μερικά άλλα που δεν είναι γενικώς δεκτά ότι πρέπει να περιλαμβάνονται στον . Matematiko (de la greka μαθημα [matema] - scienco, lernado) estas ekzakta logika dedukta scienco, kiu studas aksiomajn abstraktajn strukturojn (laŭ kvanto, formo, ) uzante logikan formalan lingvon. La specifaj strukturoj de matematiko plejofte originas de natursciencoj, plej multe de fiziko, sed matematikistoj difinas ankaŭ aliajn konceptojn por pure internaj bezonoj de la scienco. Matematiko jam penetris tra la tuta moderna vivo: modeligi precizajn instrumentojn, evoluigi novajn teknologiaĵojn kaj komputilojn, konstrui domojn; eĉ baki kukon bezonas aplikon de nocioj de nombroj, geometrio, mezuro kaj spaco. Matematiko estas iasence la fundamenta scienco. Jen kelkaj difinaj citaĵoj pri matematiko: Ekzistas du ĉefaj branĉoj de matematiko: pura kaj aplika. La pura matematiko esploras objektojn nur pro la teoria intereso, dum la aplika matematiko estigas rimedojn kaj teknikojn por solvi specifajn problemojn de sciencoj aŭ por praktike utiligi matematikon, ekz. en inĝenierado kaj ekonomio. La matemática​ (del latín mathematĭca, y este del griego μαθηματικά, transliterado como mathēmatiká, derivado de μάθημα, tr. máthēma. 'conocimiento') es una ciencia formal que surgió del estudio de las figuras geométricas y la aritmética con números. No existe una definición generalmente aceptada de las matemáticas; hoy en día se suelen describir como una ciencia que utiliza la lógica para examinar las propiedades y los patrones de las estructuras abstractas creadas por las definiciones lógicas. Die Mathematik (bundesdeutsches Hochdeutsch: [matemaˈtiːk], [matemaˈtik]; österreichisches Hochdeutsch: [mateˈmaːtik]; altgriechisch μαθηματικὴ τέχνη mathēmatikē téchnē ‚die Kunst des Lernens‘) ist eine Formalwissenschaft, die aus der Untersuchung von geometrischen Figuren und dem Rechnen mit Zahlen entstand. Für Mathematik gibt es keine allgemein anerkannte Definition; heute wird sie üblicherweise als eine Wissenschaft beschrieben, die durch logische Definitionen selbstgeschaffene abstrakte Strukturen mittels der Logik auf ihre Eigenschaften und Muster untersucht.
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