A critical study on the length of Pythagoras’ hypotenuse for unit square
Keywords:
Pythagoras, Hypotenuse, Irrational, Straightedge-Compass, Algebra, GeometryAbstract
This study included the geometric, algebraic, and arithmetic measurement of one-dimensional length for an important ancient theorem in mathematical history, named Pythagoras’ theorem, for a unit square. In this paper, we also tried to disclose a rigorous evaluation for the Pythagorean hypotenuses, which were founded incomplete square numbers by us through classical construction instead of considering flexible arithmetic approximation (root extraction) and incommensurable abstract algebraic point of view. Every conscious math reader knows that the characteristic of a theorem is that it produces true results, and the characteristic of a formula is that it may give approximate mathematical results. The much-discussed Pythagorean relation for the side of a right triangle does not satisfy the aspirations of mathematicians to measure the hypotenuse of a unit square, which remains elusive. Therefore, in some cases, this formula exhibits limitations in providing complete results in order to maintain the properties of a theorem.
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Brian Clegg. (2011). The Dangerous Ratio. University of Cambridge.https://nrich.maths.org/articles/dangerous-ratio
Cambridge, U. of. (n.d.). The existence of the square root of two. https://www.dpmms.cam.ac.uk/~wtg10/roottwo.html
Cepelewicz, J. (2024). How the Square Root of 2 Became a Number. Quanta Magazine. https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/
De Sousa, R. T., Dos Santos, M. G. M., & Alves, F. R. V. (2023). The Theory of Didactic Situations and the Generalization of Pythagoras’ Theorem: An Experience Mediated by GeoGebra software. Journal of Research in Science and Mathematics Education (J-RSME), 2(3), 147–159. https://doi.org/10.56855/jrsme.v2i3.731
Eriksson, K., Estep, D., Johnson, C. (2004). The Square Root of Two. In: Applied Mathematics. Body and Soul. Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007/978-3-662-05796-4_14
Gasarch, William & Kruskal, Alexander & Kruskal, Justin & Kruskal, R. (2006). Review of The Square Root of 2: A Dialogue Concerning a Number and a Sequence by David Flannery. Copernicus Books, 2006.. SIGACT News. 37. 27-32. 10.1145/1165555.1165561. https://www.researchgate.net/publication/220555431_Review_of_The_Square_Root_of_2_A_Dialogue_Concerning_a_Number_and_a_Sequence_by_David_Flannery_Copernicus_Books_2006
Giaffredo, M. (2015). What was the original proof that Pythagoras himself used to prove his theorem? https://www.google.com/search?client=firefox-b-d&q=Maurizio+Giaffredo+%282015%29%2C+What+was+the+original+proof+that+Pythagoras+himself+used+to+prove+++++++++++++his+theorem%3F
Güner, N. (2018). Pisagor Teoremini Nasıl Öğretirsiniz: Ders Planlarının Analizi. Ankara Universitesi Egitim Bilimleri Fakultesi Dergisi, 119–141. https://doi.org/10.30964/auebfd.405041
Hurt, A. (2022). The Origin Story of Pythagoras and His Cult Followers. Discover Magazine. https://www.discovermagazine.com/the-sciences/the-origin-story-of-pythagoras-and-his-cult-followers
İşleyen, T., & Mercan, E. (2013). Eğitimde Kuram ve Uygulama Articles /Makaleler. Journal of Theory and Practice in Education, 9(4), 529–543.
Kalanov, T. Z. (2013). The critical analysis of the pythagorean theorem and of the problem of irrational numbers. In Global Journal of Advanced Research on Classical and Modern Geometries (Vol. 2, Issue 2).
Kinney, B. (2019). Does the Square Root of Two Exist? Blog on Mathematics. https://infinityisreallybig.com/2019/01/28/does-the-square-root-of-two-exist/
Kolpas, S. J. (2017). Mathematical Treasure: James A Garfield’s Proof of the Pythagorean Theorem. Mathematical Association of America, 2016. Web. https://central.edu/writing-anthology/2019/01/31/159/
Lučić, Z. (2015). Irrationality of the Square Root of 2: The Early Pythagorean Proof, Theodorus’s and Theaetetus’s Generalizations. Mathematical Intelligencer, 37(3), 26–32. https://doi.org/10.1007/s00283-014-9521-x
Mansfield, D. F. (2023). Mesopotamian square root approximation by a sequence of rectangles. British Journal for the History of Mathematics, 38(3), 175–188. https://doi.org/10.1080/26375451.2023.2215652
Math 290, G. (2014). Existence of the square root of 2. Youtube. https://www.youtube.com/watch?v=-Ce-t6GF1ro
Mathematics, L. (n.d.). Proof that Square Root of 2 is not a Rational Number. Math Online Wikidot ,. http://mathonline.wikidot.com/proof-that-the-square-root-of-2-is-irrational
Mathematics Staff of the College, U. of C. (1956). Three algebraic questions connected with pythagoras’ theorem. The Mathematics Teacher, 49(4), 250-259. JSTOR. https://www.jstor.org/stable/i27955138
Roy, P. (2021). Without Using Pi, Area and Perimeter of Circle by Fulati - Suranjan Formulae. International Journal of Scientific Research in Mathematical and Statistical Sciences, 8(4), 1–9. https://doi.org/10.26438/ijsrmss/v8i4.19
Roy, P. (2023). A Unique Method for The Trisection of An Arbitrary Angle. Matrix Science Mathematic, 7(2), 56–70. https://doi.org/10.26480/msmk.02.2023.56.70
Shah, S. (2021). Three Proofs that the Square Root of 2 Is Irrational. https://doi.org/10.31219/osf.io/3sp95
Steihaug, T. (2024). On the Square Root Computation in Liber Abaci and De Practica Geometrie by Fibonacci. Mathematics, 12(6). https://doi.org/10.3390/math12060889
Wikidot, M. O. (n.d.). Proof that the Square Root of 2 is a Real Number. Mathonline. http://mathonline.wikidot.com/proof-that-the-square-root-of-2-is-a-real-number#toc0
Wikipedia. (n.d.-a). Dedekind cut. https://en.wikipedia.org/wiki/Dedekind_cut
Wikipedia. (n.d.-b). Law of Trichotomy. https://en.wikipedia.org/wiki/Law_of_trichotomy
Wikipedia. (n.d.-c). Philosophy of mathematics. https://en.wikipedia.org/wiki/Philosophy_of_mathematics
Wikipedia. (n.d.-d). The square root of two. https://en.wikipedia.org/wiki/Square_root_of_2
Wildberger, N. J. (2012). Inconvenient truths about sqrt(2) | Real numbers and limits Math Foundations 80. Youtube. https://www.youtube.com/watch?v=REeaT2mWj6Y
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