Remarks on Heron's cubic root iteration formula

Abstract

The existence as well as the computation of roots appears in number theory, algebra, numerical analysis and other areas. The present study illustrates the contribution of several authors towards the extraction of different order roots of real numbers. Different methods with number of approaches are studied to extract the roots of real numbers. 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. 3N = a + bd / bd + aD (b - a), where a3 < N < b3, d = N - a3 and D = b3 - 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.