Research ArticleOpen AccessGoogle Scholar indexed
The Hidden Geometry of the Babylonian Square Root Method
Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
- 1 Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
- 2 Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
Applied Mathematics·Volume 05 (2014)·Pages 2982–2987·Published 4 November 2014·DOI10.4236/am.2014.519284
Copy link · social · email
Abstract
We propose and demonstrate an original geometric argument for the ancient Babylonian square root method, which is analyzed and compared to the Newton-Raphson method. Based on simple geometry and algebraic analysis the former original iterated map is derived and reinterpreted. Time series, fixed points, stability analysis and convergence schemes are studied and compared for both methods, in the approach of discrete dynamical systems.
KeywordsBabylonian Square Root MethodNewton-Raphson MethodIterated MapDynamical Systems
- Heath, T. (1923) A History of Greek Mathematics. The Mathematical Gazette, 11, 348-351. http://dx.doi.org/10.2307/3602335
- Verbeke, J. and Cools, R. (1995) The Newton-Raphson Method. International Journal of Mathematical Education in Science and Technology, 26, 177-193. http://dx.doi.org/10.1080/0020739950260202
- Faber, X. and Voloch, J.F. (2011) On the Number of Places of Convergence for Newton’s Method over Number Fields. Journal de Théorie des Nombres de Bordeaux, 23, 387-401.
- Grau-Sánchez, M. and Daz-Barrero, J.L. (2011) A Technique to Composite a Modified Newton’s Method for Solving Nonlinear Equations. ArXiv e-prints.
- Pan, B., Cheng, P. and Xu, B. (2005) In-Plane Displacements Measurement by Gradient-Based Digital Image Correlation. SPIE Proceedings, 5852, 544-551.
- Amin, A.M., Thakur, R., Madren, S., Chuang, H.-S., Thottethodi, M., Vijaykumar, T., Wereley, S.T. and Jacobson, S.C. (2013) Software-Programmable Continuous-Flow Multi-Purpose Lab-on-a-Chip. Microfluidics and Nanofluidics, 15, 647-659. http://dx.doi.org/10.1007/s10404-013-1180-2
- Mungan, C.E. and Lipscombe, T.C. (2012) Babylonian Resistor Networks. European Journal of Physics, 33, 531. http://dx.doi.org/10.1088/0143-0807/33/3/531
- Senthilpari, C., Mohamad, Z.I. and Kavitha, S. (2011) Proposed Low Power, High Speed Adder-Based 65-nm Square Root Circuit. Microelectronics Journal, 42, 445-451. http://dx.doi.org/10.1016/j.mejo.2010.10.015
- Sun, T., Tsuda, S., Zauner, K.-P. and Morgan, H. (2010) On-Chip Electrical Impedance Tomography for Imaging Biological Cells. Biosensors and Bioelectronics, 25, 1109-1115. http://dx.doi.org/10.1016/j.bios.2009.09.036
- Ausloos, M. and Dirickx, M. (2005) The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications. Springer, New York.
- Eve, J. (1963) Starting Approximations for the Iterative Calculation of Square Roots. The Computer Journal, 6, 274-276. http://dx.doi.org/10.1093/comjnl/6.3.274
- Macleod, A.J. (1984) A Generalization of Newton-Raphson Method. International Journal of Mathematical Education in Science and Technology, 15, 117-120. http://dx.doi.org/10.1080/0020739840150116
- Banach, S. (1992) Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales. Fundamenta Mathematicae, 3, 133-181.