scholarly journals On the Irrationality and Transcendence of Rational Powers of e

2021 ◽  
pp. 102-110
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Irrational Number

A number that can’t be expressed as the ratio of two integers is called an irrational number. Euler and Lambert were the first mathematicians to prove the irrationality and transcendence of e. Since then there have been many other proofs of irrationality and transcendence of e and generalizations of that proof to rational powers of e. In this article we review various proofs of irrationality and transcendence of rational powers of e founded by mathematicians over the time.

scholarly journals Partitioning the positive integers with higher order recurrences

1991 ◽  
Vol 14 (3) ◽  
pp. 457-462 ◽  
Author(s):  
Clark Kimberling
Keyword(s):  
Positive Integer ◽  
Positive Root ◽  
Irrational Number ◽  
Higher Order ◽  
Typical Result ◽  
Algebraic Integers ◽  
Positive Integers

Associated with any irrational numberα>1and the functiong(n)=[αn+12]is an array{s(i,j)}of positive integers defined inductively as follows:s(1,1)=1,s(1,j)=g(s(1,j−1))for allj≥2,s(i,1)=the least positive integer not amongs(h,j)forh≤i−1fori≥2, ands(i,j)=g(s(i,j−1))forj≥2. This work considers algebraic integersαof degree≥3for which the rows of the arrays(i,j)partition the set of positive integers. Such an array is called a Stolarsky array. A typical result is the following (Corollary 2): ifαis the positive root ofxk−xk−1−…−x−1fork≥3, thens(i,j)is a Stolarsky array.

Paper-folding and cutting activities to demonstrate five-fold symmetry

2018 ◽  
Vol 102 (555) ◽  
pp. 413-421
Author(s):  
King-Shun Leung
Keyword(s):  
Irrational Number ◽  
Simple Fraction ◽  
Regular Pentagon ◽  
Paper Folding ◽  
Pointed Star

We can obtain a two-fold symmetric figure by folding a square sheet of paper in the middle and then cutting along some curves drawn on the paper. By making two perpendicular folds through the centre of the paper and then cutting, we can obtain a four-fold symmetric figure. We can also get an eight-fold symmetric figure by making a fold bisecting an angle made by the two perpendicular folds before cutting. But it is not possible to obtain a three-fold, five-fold or six-fold symmetric figure in this way; we need to make more folds before cutting. Making a three-fold (respectively five-fold and six-fold) figure involves the division of the angle at the centre (360°) of a square sheet of a paper into six (respectively ten and twelve) equal parts. In other words, we need to construct the angles 60°, 36° and 30°. But these angles cannot be obtained by repeated bisections of 180° by simple folding as in the making of two-fold, four-fold and eight-fold figures. In [1], we see that each of the constructions of 60° and 30° applies the fact that sin 30° = ½ and takes only a few simple folding steps. The construction of 36° is more tedious (see, for example, [2] and [3]) as sin 36° is not a simple fraction but an irrational number. In this Article, we show how to make, by paper-folding and cutting a regular pentagon, a five-pointed star and create any five-fold figure as we want. The construction obtained by dividing the angle at the centre of a square paper into ten equal parts is called apentagon base. We gained much insight from [2] and [3] when developing the method for making the pentagon base to be presented below.

Relative convergence speed of Lüroth expansions and Hausdorff dimension

2021 ◽  
pp. 1-21
Author(s):  
Xiaoyan Tan ◽  
Jia Liu ◽  
Zhenliang Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Irrational Number ◽  
Convergence Speed ◽  
Rate Of Growth ◽  
Dimension Formula ◽  
Lüroth Expansion

For any [Formula: see text] in [Formula: see text], let [Formula: see text] be the Lüroth expansion of [Formula: see text]. In this paper, we study the relative convergence speed of its convergents [Formula: see text] to the rate of growth of digits in the Lüroth expansion of an irrational number. For any [Formula: see text] in [Formula: see text], the sets [Formula: see text] and [Formula: see text] are proved to be of same Hausdorff dimension [Formula: see text]. Furthermore, for any [Formula: see text] in [Formula: see text] with [Formula: see text], the Hausdorff dimension of the set [Formula: see text] [Formula: see text] is proved to be either [Formula: see text] or [Formula: see text] according as [Formula: see text] or not.

A Model of the Real Numbers

1963 ◽  
Vol 6 (2) ◽  
pp. 239-255
Author(s):  
Stanton M. Trott
Keyword(s):  
Irrational Number ◽  
Additive Group ◽  
Linear Ordering ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Linear Orderings ◽  
Positive Elements ◽  
The Law ◽  
Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

scholarly journals Quaternary Misfit Compounds—A Concise Review

Crystals ◽  
2020 ◽  
Vol 10 (6) ◽  
pp. 468
Author(s):  
Sokhrab B. Aliev ◽  
Reshef Tenne
Keyword(s):  
Irrational Number ◽  
Short Review ◽  
Layered Compounds ◽  
Layered Compound ◽  
Charge Carrier Density ◽  
Precise Control ◽  
Rocksalt Structure ◽  
Main Emphasis ◽  
Recent Developments ◽  
Metal Dichalcogenides

Misfit layered compounds (MLCs) have been studied in the literature for the last 40 years. They are generally made of an alternating sequence of two monolayers, a distorted rocksalt structure, and a hexagonal layered compound. In a typical MLC, the c-axis is common to the two monolayers and so is one of the axes in the layer plan. However, the two compounds are non-commensurate along at least one axis, and the ratio between the two axes is an irrational number making the MLC a non-stoichiometric compound. The two main families of MLC are those based on metal dichalcogenides and CoO2 as the hexagonal layered compound. Traditionally, ternary MLCs were prepared and studied, but some quaternary and multinary MLC minerals have been known for many years. Over the last few years, interest in MLCs with four and even larger number of atoms has grown. Doping or alloying of a ternary MLC permits precise control of the charge carrier density and hence the electrical, thermoelectric, catalytic, and optical properties of such compounds. In this short review, some of these developments will be discussed with the main emphasis put on quaternary MLC nanotubes belonging to the chalcogenide series. The synthesis, structural characterization, and some of their properties are considered. Some recent developments in quaternary cobaltite MLCs and recent studies on exfoliated MLCs are discussed as well.

Irregularities in the Distribution of Prime Numbers in a Beatty Sequence

2019 ◽  
Vol 63 (4) ◽  
pp. 738-743
Author(s):  
Janyarak Tongsomporn ◽  
Jörn Steuding
Keyword(s):  
Finite Type ◽  
Irrational Number ◽  
Prime Numbers ◽  
Positive Real ◽  
Beatty Sequence

AbstractWe prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.

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