scholarly journals On 'Best' Rational Approximations to $\pi$ and $\pi+e$

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Continued Fraction ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.

scholarly journals On Rational Approximations to $\pi+e$: 920/157 Is the 'Best' Rational Approximation to $\pi+e$ with a Denominator of at Most Three Digits

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations ◽  
Best Rational Approximation

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. This is not so surprising when one considers the empirical computations around these two rational approximations to $\pi$. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, $920/157$ turns out to be the only rational number of this type.

scholarly journals Metrical properties of best approximants

1992 ◽  
Vol 53 (1) ◽  
pp. 78-91
Author(s):  
Jos Blom
Keyword(s):  
Rational Number ◽  
Continued Fraction ◽  
Irrational Number ◽  
Main Tool ◽  
Rational Numbers ◽  
Two Dimensional ◽  
Ergodic System ◽  
Continued Fraction Expansion

AbstractA rational number is called a best approximant of the irrational number ζ if it lies closer to ζ than all rational numbers with a smaller denominator. Metrical properties of these best approximants are studied. The main tool is the two-dimensional ergodic system, underlying the continued fraction expansion.

Adaptive Polynomial Approximation to Circular Arcs

2011 ◽  
Vol 50-51 ◽  
pp. 678-682
Author(s):  
Li Zheng Lu
Keyword(s):  
Power Series ◽  
Polynomial Approximation ◽  
Infinite Series ◽  
Upper Bound ◽  
Spline Approximation ◽  
Adaptive Method ◽  
Degree Polynomial ◽  
Circular Arc ◽  
Polynomial Curve ◽  
Circular Arcs

We present a new adaptive method for approximating circular arcs in polynomial form by using the s-power series. Circular arcs can be expressed in infinite series form, we obtain the order-k Hermite interpolant by truncating at the kth term. An upper bound on the error of the interpolant is available, so we can obtain the lowest degree polynomial curve that approximates a circular arc within any user-prescribed tolerance. And this degree can be further reduced through subdivision, which generates a spline approximation with Ck continuity at the joints.

scholarly journals Universal symbolic expression for radial distance of conic motion

2014 ◽  
pp. 87-92
Author(s):  
M.A. Sharaf ◽  
A.S. Saad ◽  
A.A. Alshaery
Keyword(s):  
Power Series ◽  
Efficient Method ◽  
Continued Fraction ◽  
Recursive Algorithm ◽  
Series Representation ◽  
Radial Distance ◽  
Time Interval ◽  
Symbolic Expression ◽  
Series Form ◽  
Analytical Power

In the present paper, a universal symbolic expression for radial distance of conic motion in recursive power series form is developed. The importance of this analytical power series representation is that it is invariant under many operations because the result of addition, multiplication, exponentiation, integration, differentiation, etc. of a power series is also a power series. This is the fact that provides excellent flexibility in dealing with analytical, as well as computational developments of problems related to radial distance. For computational developments, a full recursive algorithm is developed for the series coefficients. An efficient method using the continued fraction theory is provided for series evolution, and two devices are proposed to secure the convergence when the time interval (t ? t0) is large. In addition, the algorithm does not need the solution of Kepler?s equation and its variants for parabolic and hyperbolic orbits. Numerical applications of the algorithm are given for three orbits of different eccentricities; the results showed that it is accurate for any conic motion.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

Call for Manuscripts for the 2014 Focus Issue: Rational Number Sense – October 2012

2012 ◽  
Vol 18 (3) ◽  
pp. 189
Keyword(s):  
Rational Number ◽  
Number Sense ◽  
Rational Numbers

This call for manuscripts is requesting articles that address how to make sense of rational numbers in their myriad forms, including as fractions, ratios, rates, percentages, and decimals.

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

scholarly journals CONTINUED FRACTIONS, INTERMEDIATE FRACTIONS AND THEIR RELATION TO THE BEST APPROXIMATIONS

2020 ◽  
Vol 20 (3) ◽  
pp. 545-560
Author(s):  
LUKA MILINKOVIC ◽  
BRANKO MALESEVIC ◽  
BOJAN BANJAC
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Best Approximations ◽  
Current State ◽  
The Subject ◽  
State Of Art

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

scholarly journals Some properties of generalized hypergeometric Appell polynomials

2020 ◽  
Vol 12 (1) ◽  
pp. 129-137 ◽  
Author(s):  
L. Bedratyuk ◽  
N. Luno
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Hypergeometric Function ◽  
Explicit Formula ◽  
Series Representation ◽  
Generalized Hypergeometric Function ◽  
Appell Polynomials ◽  
Pochhammer Symbol ◽  
Formal Power ◽  
Polynomial Family

Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N}}_0$, $k \in {\mathbb{N}},$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg| \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k)}}{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k)}} \frac{z^k}{k!}$ as follows $$ A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k}}, {\displaystyle -\frac{n-1}{k}}, {\ldots}, {\displaystyle-\frac{n-k+1}{k}}\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg| \: \displaystyle \frac{m}{x^k} \end{bmatrix} $$ and, at the same time, the polynomials from this family are Appell-type polynomials. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.

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