Affiliation:
1. Veer Surendra Sai University of Technology Burla
Abstract
The existence as well as the computation of roots appears in number theory, algebra, numerical analysis and other areas. The present study illustrate the contributions of several authors towards the extraction of different order roots of real number. Different methods with several approaches are studied to extract the roots of real number. Some of the methods described earlier are equivalent as observed in the present study. Heron developed a general iteration formula to determine the cube root of a real number N i.e. $\displaystyle\sqrt[3]{N}=a+\frac{bd}{bd+aD}(b-a)$, where $a^3<N<b^3$, $d=N-a^3$ and $D=b^3-N$ . Although the direct proof of the above method is not available in literature, some authors have proved the same with the help of conjectures. In the present investigation, the proof of Heron's method is explained and is generalized for any odd order roots. Thereafter it is observed that Heron's method is a particular case of the generalized method.
Publisher
Sociedade Paranaense de Matematica
Reference37 articles.
1. 1. A.S. Saidan, Nasir al-Din al-TusI: Jawami‘ al-hisab bi al-takht wa al-turab. (The Comprehensive Work on Computation with Board and Earth of Nasir al-Din al-Tusi). Al-Abhath, XX (2) 91-163, (1967) 213-292.
2. 2. A.S. Saidan, The Arithmetic of Al-Uqlidisi: The Story of HinduUArabic Arithmetic as Told in Kitab al-Fusul fial-Hisab al-Hindi. Reidel, Dordrecht, 1978.
3. 3. A.S. Saidan, ‘Abd al-Qahir Ibn Tahir al-BaghdadI: Al-Takmila fial-hisab. Manshurat Ma‘had al-Makhtutat al-‘Arabiyya, Kuwait, 1985.
4. 4. O. Ahmadi, D. Hankerson and A. Menzes, Formulas for cube roots in F3m, Disdrete Applied Mathematics, 155 (2007) 260-270.
5. 5. P.E. Black, cube root, in dictionary of algorithms and data structures, Vreda Pieterse and Paul E. Black, eds., 2009.
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. An Efficient Rating System Using Blockchain for Recommender Systems;2023 IEEE International Conference on Artificial Intelligence, Blockchain, and Internet of Things (AIBThings);2023-09-16
2. An Organized Literature Review on Various Cubic Root Algorithmic Practices for Developing Efficient VLSI Computing System—Understanding Complexity;Artificial Intelligence Applications and Reconfigurable Architectures;2023-02-10
3. A note on machine method for root extraction;Boletim da Sociedade Paranaense de Matemática;2022-01-01