Related papers: On the babylonian method of extracting root square…

One of the greatest achievements of Greek mathematics is the proof that the square root of 2 is irrational. It has not been thought that the Babylonians appreciated the concept of irrationality and certainly that they did not prove that the…

This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…

We investigate the mathematics behind 1500 year old root extraction methods presented by Aryabhata in his famous mathematical treatise "Aryabhatiya". Also, we look at their computational complexity.

We formulate the Root Extraction problem in finite Abelian $p$-groups and then extend it to generic finite Abelian groups. We provide algorithms to solve them. We also give the bounds on the number of group operations required for these…

We reconsider Archimedes' evaluations of several square roots in 'Measurement of a Circle'. We show that several methods proposed over the last century or so for his evaluations fail one or more criteria of plausibility. We also provide…

Between the ninth and fifteenth centuries, several Arab mathematicians studied numerical algorithms on integers. The extraction of the square root of an integer is based on an algorithm known at least since al-Khwarizmi (died around 850)…

We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this…

In the present paper I shall clarify the Babylonian method by which many large tables of reciprocals could be constructed.

In 1977, Adleman, Manders and Miller had briefly described how to extend their square root extraction method to the general $r$th root extraction over finite fields, but not shown enough details. Actually, there is a dramatic difference…

We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm…

We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.

Ratios and coefficients are used to simplify calculations. For geometric usage these values also called function values. Like in Egypt also in Babylon such a value system can be shown. The reconstructed calculation sequence, of the Plimpton…

We prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values

This paper presents geometric proofs for the irrationality of square roots of select integers, extending classical approaches. Building on known geometric methods for proving the irrationality of sqrt(2), the authors explore whether similar…

We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…

We investigate Newton's method as a root finder for complex polynomials of arbitrary degree. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical…

In this paper, an optimized version of classical Bombelli's algorithm for computing integer square roots is presented. In particular, floating-point arithmetic is used to compute the initial guess of each digit of the root, following…