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Humans, as well as some other animals, find symmetric patterns to be more beautiful. [189] Mathematically, the symmetries of an object form a group known as the symmetry group. [190] Goldman, Jay (1997). The Queen of Mathematics: A Historically Motivated Guide to Number Theory. CRC Press. pp.1–3. ISBN 978-1-4398-6462-3 . Retrieved November 11, 2022. 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] LeVeque, William J. (January 5, 2014). Fundamentals of Number Theory. Dover Publications. pp.1–30. ISBN 978-0-486-14150-3 . Retrieved March 19, 2023.

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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] See, for example, Wilder, Raymond L. Evolution of Mathematical Concepts; an Elementary Study. passim. Moschovakis, Joan (September 4, 2018). "Intuitionistic Logic". Stanford Encyclopedia of Philosophy. Archived from the original on December 16, 2022 . Retrieved November 12, 2022.Wagh, Sanjay Moreshwar; Deshpande, Dilip Abasaheb (September 27, 2012). Essentials of Physics. PHI Learning Pvt. Ltd. p.3. ISBN 978-81-203-4642-0 . Retrieved January 3, 2023. The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects. [153] Hales, Thomas; Adams, Mark; Bauer, Gertrud; Dang, Tat Dat; Harrison, John; Hoang, Le Truong; Kaliszyk, Cezary; Magron, Victor; Mclaughlin, Sean; Nguyen, Tat Thang; Nguyen, Quang Truong; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; Ta, Thi Hoai An; Tran, Nam Trung; Trieu, Thi Diep; Urban, Josef; Vu, Ky; Zumkeller, Roland (2017). "A Formal Proof of the Kepler Conjecture". Forum of Mathematics, Pi. 5: e2. doi: 10.1017/fmp.2017.1. hdl: 2066/176365. ISSN 2050-5086. S2CID 216912822. Archived from the original on December 4, 2020 . Retrieved February 25, 2023.

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That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun! Just choose your grade level or topic to get access to 100% free practice questions: Batchelder, William H. (January 1, 2015), "Mathematical Psychology: History", in Wright, James D. (ed.), International Encyclopedia of the Social & Behavioral Sciences (Second Edition), Oxford: Elsevier, Sipser, Michael (July 1992). The history and status of the P versus NP question. STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing. pp.603–618. doi: 10.1145/129712.129771. Oaks, J. A. (2018). "François Viète's revolution in algebra" (PDF). Archive for History of Exact Sciences. 72 (3): 245–302. doi: 10.1007/s00407-018-0208-0. S2CID 125704699. Archived (PDF) from the original on November 8, 2022 . Retrieved November 8, 2022. Biology uses probability extensively – for example, in ecology or neurobiology. [137] Most of the discussion of probability in biology, however, centers on the concept of evolutionary fitness. [137]

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Main article: Discrete mathematics A diagram representing a two-state Markov chain. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.

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Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications. At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks. [10] It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable. [165]Christianidis, Jean; Oaks, Jeffrey (May 2013). "Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria". Historia Mathematica. 40 (2): 127–163. doi: 10.1016/j.hm.2012.09.001. Chen, Changbo; Maza, Marc Moreno (August 2014). Cylindrical Algebraic Decomposition in the RegularChains Library. International Congress on Mathematical Software 2014. Lecture Notes in Computer Science. Vol.8592. Berlin: Springer. doi: 10.1007/978-3-662-44199-2_65 . Retrieved November 19, 2022. Wang, Yuan (2002). The Goldbach Conjecture. Series in pure mathematics. Vol.4 (reviseded.). World Scientific. pp.1–18. ISBN 978-981-277-660-0 . Retrieved November 11, 2022. The most prestigious award in mathematics is the Fields Medal, [196] [197] established in 1936 and awarded every four years (except around World War II) to up to four individuals. [198] [199] It is considered the mathematical equivalent of the Nobel Prize. [199]

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O'Connor, J. J.; Robertson, E. F. (September 1998). "Leonhard Euler". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on November 9, 2022 . Retrieved November 9, 2022.Silver, Daniel S. (2017). "The New Language of Mathematics". The American Scientist. 105 (6): 364–371. doi: 10.1511/2017.105.6.364. Archived from the original on October 22, 2022 . Retrieved November 19, 2022.