scholarly journals On the Constructibility of Real 5th Roots of Rational Numbers with Marked Ruler and Compass

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Elliot Benjamin
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
Real Roots ◽  
Open Question

We demonstrate that there are infinitely many real numbers constructible by marked ruler and compass which are unique real roots of irreducible quintic polynomials over the field of rational numbers. This result can be viewed as a generalization of the historical open question of the constructibility by marked ruler and compass of real 5th roots of rational numbers. We obtain our results through marked ruler and compass constructions involving the intersection of conchoids and circles, and the application of number theoretic divisibility criteria.

Have You Read …?

1969 ◽  
Vol 62 (3) ◽  
pp. 220-221
Author(s):  
Philip Peak
Keyword(s):  
Real Number ◽  
Rational Number ◽  
Geometric Approach ◽  
Rational Numbers ◽  
Polynomial Equations ◽  
Real Numbers ◽  
Basic Principles ◽  
New Ideas

One of the basic principles we follow in our teaching is to relate new ideas with old ideas. Dr. Forbes has done just this in his article about extending the concept of rational numbers to real numbers. He points out how this extension cannot follow the same pattern as that of extensions positive to negative integers or from integers to rationals. If we look to a definition for motivating the extension we at best can only say, “Some polynomial equations have no rational number solutions and do have some real number solutions.” We might use least-upperbound idea, or we might try motivating through nonperiodic infinite decimals. However, Dr. Forbes rejects all of these and makes the tie-in through a geometric approach.

scholarly journals Ubiquity and large intersections properties under digit frequencies constraints

2008 ◽  
Vol 145 (3) ◽  
pp. 527-548 ◽  
Author(s):  
JULIEN BARRAL ◽  
STÉPHANE SEURET
Keyword(s):  
Hausdorff Dimension ◽  
Rational Numbers ◽  
Intersection Property ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Countable Intersection ◽  
The Real ◽  
Large Intersection ◽  
Digit Frequencies

AbstractWe are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

An axiomatic system for the first order language with an equi-cardinality quantifier

1966 ◽  
Vol 31 (4) ◽  
pp. 633-640 ◽  
Author(s):  
Mitsuru Yasuhara
Keyword(s):  
Rational Numbers ◽  
Axiomatic System ◽  
Universal Quantifier ◽  
Real Numbers ◽  
Relational System ◽  
First Order ◽  
Order Language ◽  
Relational Systems ◽  
Finite Domains ◽  
The Universe

The equi-cardinality quantifier1 to be used in this article, written as Qx, is characterised by the following semantical rule: A formula QxA(x) is true in a relational system exactly when the cardinality of the set consisting of these elements which make A(x) true is the same as that of the universe. For instance, QxN(x) is true in 〈Rt, N〉 but false in 〈Rl, N〉 where Rt, Rl, and N are the sets of rational numbers, real numbers, and natural numbers, respectively. We notice that in finite domains the equi-cardinality quantifier is the same as the universal quantifier. For this reason, all relational systems considered in the following are assumed infinite.

scholarly journals Lower bound for the number of real roots of a random algebraic polynomial

1983 ◽  
Vol 35 (1) ◽  
pp. 18-27 ◽  
Author(s):  
M. N. Mishra ◽  
N. N. Nayak ◽  
S. Pattanayak
Keyword(s):  
Lower Bound ◽  
Algebraic Polynomial ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Exceptional Set ◽  
Real Roots ◽  
Normal Law ◽  
Number Of Real Roots

AbstractLet X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

Oscillations of Second Order Neutral Equations

1988 ◽  
Vol 40 (6) ◽  
pp. 1301-1314 ◽  
Author(s):  
G. Ladas ◽  
E. C. Partheniadis ◽  
Y. G. Sficas
Keyword(s):  
Second Order ◽  
List Type ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Neutral Equations ◽  
Deviating Arguments ◽  
Real Roots ◽  
All Solutions ◽  
Image Position ◽  
Necessary And Sufficient

Consider the second order neutral differential equation1where the coefficients p and q and the deviating arguments τ and σ are real numbers. The characteristic equation of Eq. (1) is2The main result in this paper is the following necessary and sufficient condition for all solutions of Eq. (1) to oscillate.THEOREM. The following statements are equivalent:(a) Every solution of Eq. (1) oscillates.(b) Equation (2) has no real roots.

Self-Centred Sets

1966 ◽  
Vol 18 ◽  
pp. 974-980
Author(s):  
H. Kestelman
Keyword(s):  
Abelian Group ◽  
Trigonometric Series ◽  
Absolute Convergence ◽  
Rational Numbers ◽  
Real Numbers ◽  
Space Group ◽  
Space Groups ◽  
Set Of Points

A subset S of an abelian group G is said to have a centre at a if whenever x belongs to S so does 2a — x. This note is mainly concerned with self-centred sets, i.e. those S with the property that every element of S is a centre of S. Such sets occur in the study of space groups: the set of inversion centres of a space group is always self-centred. Every subgroup of G is self-centred, so is every coset in G: this is the reason why the set of points of absolute convergence of a trigonometric series is self-centred or empty (1). A self-centred set of real numbers that is either discrete or consists of rational numbers must in fact be a coset (see §3); this does not hold for an arbitrary enumerable self-centred set of real numbers (§3.3).

scholarly journals Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers

10.37236/749 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Avi Berman ◽  
Shmuel Friedland ◽  
Leslie Hogben ◽  
Uriel G. Rothblum ◽  
Bryan Shader
Keyword(s):  
Computational Complexity ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Real Numbers ◽  
Minimum Rank ◽  
The Real ◽  
Sign Patterns ◽  
Symmetric Patterns

We use a technique based on matroids to construct two nonzero patterns $Z_1$ and $Z_2$ such that the minimum rank of matrices described by $Z_1$ is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by $Z_2$ is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about rational realization of minimum rank of sign patterns. Using $Z_1$ and $Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and $G_2$, with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.

scholarly journals Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

2021 ◽  
Vol 2090 (1) ◽  
pp. 012037
Author(s):  
Kaushik Ghosh
Keyword(s):  
Line Segment ◽  
Statistical Physics ◽  
Rational Numbers ◽  
Partition Functions ◽  
Actual Process ◽  
Real Numbers ◽  
Hausdorff Topology ◽  
Straight Line ◽  
Definition Of ◽  
Positive Integers

Abstract In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

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