Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia. [168] Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE. [169] The oldest known mathematics textbook is the Rhind papyrus, dated from c. 1650 BCE in Egypt. [170] Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition since the Vedic period ( c. 1500– c. 500 BCE). [171] In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy. [172]

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In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. [126] [127] Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. [74] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800BC. 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 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. [75]Jansen, A. R.; Marriott, K.; Yelland, G. W. (2000). Constituent structure in mathematical expressions (PDF). Proceedings of the 22th Annual Meeting of the Cognitive Science Society. Vol.22. p.238. Archived (PDF) from the original on November 16, 2022 . Retrieved November 16, 2022. Maurer, Stephen B. (1997). "What is Discrete Mathematics? The Many Answers". In Rosenstein, Joseph G.; Franzblau, Deborah S.; Roberts, Fred S. (eds.). Discrete Mathematics in the Schools. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science. Vol.36. American Mathematical Society. pp.121–124. ISBN 978-0-8218-8578-9 . Retrieved November 10, 2022. Restivo, S. (December 2013). Mathematics in Society and History. Springer Netherlands. pp.14–15. ISBN 978-94-011-2944-2 . Retrieved March 19, 2023. Khan Academy is a nonprofit whose resources are always free to teachers and learners – no ads, no subscriptions

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Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem. Dunne, Edward; Hulek, Klaus (March 2020). "Mathematics Subject Classification 2020" (PDF). Notices of the American Mathematical Society. 67 (3). Archived (PDF) from the original on November 20, 2022 . Retrieved November 4, 2022. A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. [124] A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. [125] 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. [65] The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data. [d] Main articles: Mathematical notation, Language of mathematics, and Glossary of mathematics An explanation of the sigma (Σ) summation notation

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Hilbert, David (1962). The Foundations of Geometry. Open Court Publishing Company. p.1 . Retrieved March 22, 2023. Peressini, Anthony (September 1999). Applying Pure Mathematics (PDF). Philosophy of Science. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers. Vol.66. pp.S1–S13. JSTOR 188757 . Retrieved November 30, 2022. [ permanent dead link] Main articles: Mathematical and theoretical biology and Mathematical chemistry The skin of this giant pufferfish exhibits a Turing pattern, which can be modeled by reaction–diffusion systems. In 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 a mathematical problem. [35] [10] 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. [36]

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Discrete optimization, including combinatorial optimization, integer programming, constraint programming Ecology heavily uses modeling to simulate population dynamics, [137] [138] study ecosystems such as the predator-prey model, measure pollution diffusion, [139] or to assess climate change. [140] The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations. [141] However, there is the problem of model validation. This is particularly acute when the results of modeling influence political decisions; the existence of contradictory models could allow nations to choose the most favorable model. [142] In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc. Hennig, Christian (2010). "Mathematical Models and Reality: A Constructivist Perspective". Foundations of Science. 15: 29–48. doi: 10.1007/s10699-009-9167-x. S2CID 6229200 . Retrieved November 17, 2022. Lim, Lisa (December 21, 2018). "Where algebra got its x from, and Xmas its X". South China Morning Post. Archived from the original on August 9, 2022 . Retrieved August 9, 2022.Rao, C.R. (1997). Statistics and Truth: Putting Chance to Work. World Scientific. ISBN 978-981-02-3111-8. Guicciardini, Niccolo (2017). "The Newton–Leibniz Calculus Controversy, 1708–1730" (PDF). In Schliesser, Eric; Smeenk, Chris (eds.). The Oxford Handbook of Newton. Oxford handbooks. Oxford University Press. doi: 10.1093/oxfordhb/9780199930418.013.9. ISBN 978-0-19-993041-8. Archived (PDF) from the original on November 9, 2022 . Retrieved November 9, 2022. Perisho, Margaret W. (Spring 1965). "The Etymology of Mathematical Terms". Pi Mu Epsilon Journal. 4 (2): 62–66. JSTOR 24338341.