Study On Several Historical Algorithms Of Fractional Approximation Of The Roots

Mathematicians from the Middle Ages to Renaissance used fraction to evaluate the surd roots of equations.Meanwhile,they developed a special and representative algorithm,the iterative algorithm,the essence of which is the binomial expansion.The author selected three typical mathematicians and their works on fractional approximation algorithms,L.Fibonacci's Liber Abaci,G.Cardano's Practica arithmetice et mensurandi singularisand C.Clavius'Epitome Arithmeticae Practicae.On the basis of translating G.Cardano's and C.Clavius' Latin contents on fractional approximation with my tutor and interpreting the Chinese version and English version of Liber Abaci,this paper will comparatively analyze the specific procedures of the three kinds of algorithms.Using the method of comparative analysis to discuss the whole algorithm,combining with the early fractional approximation algorithm of square roots by Heron and Theon,the author can get the following conclusions:1.In the aspect of fractional approximation algorithm of square roots,L.Fibonacci and G.Cardano had the same procedure,but the latter is better in the definition and unity.Although C.Clavius' algorithm has the lower speed of approximation,it is more concise.As for the complexity of algorithm,it is better than that of the former two.2.Taking the fractional approximation algorithms of cubic roots used by Fibonacci and Cardano as the comparative objects,we can conclude that the latter one is faster and more explicit.More importantly,the approximation direction of Clavius' algorithm is consistent.3.The essences of Fibonacci's fractional approximation algorithm of square roots,Cardano's fractional approximation algorithm of roots,Clavius' second algorithm,can all be attributed to Newton tangent method.4.Fibonacci,Cardano and Clavius' fractional approximation algorithms of roots source may come from the ancient Greece,and to a certain degree,also under the influence of India-Arab arithmetic.Mathematicians in different times pushed the progresses of the fractional approximation algorithm of roots,and their methods embody the different thinkings and understandings of the question "What's the better algorithm of extraction of square root?"...