scholarly journals Remarks on Heron's cubic root iteration formula

2017 ◽  
Vol 35 (3) ◽  
pp. 173-180
Author(s):  
Saroj Kumar Padhan
Keyword(s):  
Number Theory ◽  
Numerical Analysis ◽  
Real Number ◽  
Direct Proof ◽  
Cube Root ◽  
Iteration Formula ◽  
Odd Order ◽  
Different Order

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.

scholarly journals A note on machine method for root extraction

2022 ◽  
Vol 40 ◽  
pp. 1-6
Author(s):  
Saroj Kumar Padhan ◽  
S. Gadtia
Keyword(s):  
Real Number ◽  
Newton’S Method ◽  
Newton's Method ◽  
Extraction Methods ◽  
Critical Study ◽  
Square Root ◽  
Machine Method ◽  
Iteration Formula ◽  
Root Extraction

The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

scholarly journals Heron’s cubic root iteration method with a new approach

2021 ◽  
Author(s):  
S. Gadtia ◽  
S. K. Padhan
Keyword(s):  
Iterative Method ◽  
Iteration Method ◽  
Interpolation Formula ◽  
Alternative Proof ◽  
New Approach ◽  
Iteration Formula ◽  
Odd Order ◽  
Even Order ◽  
Lagrange’S Method

Abstract Heron’s cubic root iteration formula conjectured by Wertheim is proved and extended for any odd order roots. Some possible proofs are suggested for the roots of even order. An alternative proof of Heron’s general cubic root iterative method is explained. Further, Lagrange’s interpolation formula for nth root of a number is studied and found that Al-Samawal’s and Lagrange’s method are equivalent. Again, counterexamples are discussed to justify the effectiveness of the present investigations.

scholarly journals Higher Order Oscillation and Uniform Distribution

2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Uniform Distribution ◽  
Differentiable Function ◽  
Mild Condition ◽  
Higher Order ◽  
Real Numbers ◽  
Mobius Function ◽  
Real Polynomials ◽  
Almost All

AbstractIt is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

HAUSDORFF DIMENSION OF THE EXCEPTIONAL SETS CONCERNING THE RUN-LENGTH FUNCTION OF BETA-EXPANSION

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850074 ◽  
Author(s):  
MENGJIE ZHANG
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Real Number ◽  
Maximal Length ◽  
Run Length ◽  
Exceptional Sets ◽  
Number Formula ◽  
Consecutive Zero ◽  
Almost All ◽  
Increasing Function

For any real number [Formula: see text], and any [Formula: see text], let [Formula: see text] be the maximal length of consecutive zeros in the first [Formula: see text] digits of the [Formula: see text]-expansion of [Formula: see text]. Recently, Tong, Yu and Zhao [On the length of consecutive zero digits of [Formula: see text]-expansions, Int. J. Number Theory 12 (2016) 625–633] proved that for any [Formula: see text], for Lebesgue almost all [Formula: see text], [Formula: see text] In this paper, we quantify the size of the set of [Formula: see text] for which [Formula: see text] grows to infinity in a general speed. More precisely, for any increasing function [Formula: see text] with [Formula: see text] tending to [Formula: see text] and [Formula: see text], we show that for any [Formula: see text], the set [Formula: see text] has full Hausdorff dimension.

Approximating sequences and Hausdorff measure

1974 ◽  
Vol 76 (1) ◽  
pp. 161-172 ◽  
Author(s):  
R. J. Gardner
Keyword(s):  
Number Theory ◽  
Numerical Analysis ◽  
Hausdorff Measure ◽  
Metric Space ◽  
Complex Analysis ◽  
Hausdorff Measures ◽  
Real Numbers ◽  
Majorizing Sequences ◽  
Approximating Sequence ◽  
Separable Metric Space

Approximating sequences have been extensively studied in many branches of mathematics, for example, in number theory (approximating real numbers by rationals) and in numerical analysis (approximations to functions by polynomials). In (1), A. Hyllengren introduced a type of approximating sequence ‘majorizing sequences’ which he used in solving a problem in complex analysis. In this note we study a very similar concept, which is general enough to be applicable to any separable metric space, and which turns out to have strong connexions with the theory of Hausdorff measures (as did Hyllengren's majorizing sequences).

scholarly journals ANALYSIS OF STUDENTS' CREATIVITY IN DIRECT PROOF OF NUMBERS THEORY

2020 ◽  
Vol 8 (2) ◽  
Author(s):  
Sunyoto Hadi Prajitno
Keyword(s):  
Qualitative Research ◽  
Number Theory ◽  
Data Analysis ◽  
Creative Thinking ◽  
Direct Proof ◽  
Female Students ◽  
Creative Class ◽  
Male Students ◽  
Basic Set ◽  
The One

The way of thinking logically and reasoning in number theory courses can be seen by the number of proofs of the properties of numbers in some basic set of numbers. This qualitative research aims to analyze student creativity in solving proving questions directly. The research involved 37 third semester students taking lectures on number theory. Data collection techniques provide questions then students with their own desires go to work in front of the class. Data analysis was performed by analyzing the creativity of the evidence carried out by students fulfilling which aspects of the aspects of creative thinking behavior and indicators of creative thinking abilities based on Mahmudi. A total of four students have been able to use the proving tools provided. The four students are creative because they meet at least one indicator of the ability to think creatively based on Mahmudi, student creativity in proving only 10.1% of the total number of students involved in this study. This can be seen from the one given problem which can only emerge four evidentiating creatives from the ten existing evidences. Thus, the class is included in the creative class. In addition, it is analyzed based on the number of students who are creative, more creative where in the class, male or female students. The results of research on this matter are more creative female students male students.

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