Two Unusual Representations for the Set of Real Numbers

1970 ◽  
Vol 63 (8) ◽  
pp. 665
Author(s):  
Sanderson M. Smith
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
The Real ◽  
Binary Operations

The rational numbers and the real numbers are both fields under the binary operations of addition and multiplication.

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.

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 Two constructions of the real numbers via alternating series

1989 ◽  
Vol 12 (3) ◽  
pp. 603-613 ◽  
Author(s):  
Arnold Knopfmacher ◽  
John Knopfmacher
Keyword(s):  
Continued Fractions ◽  
Rational Numbers ◽  
Equivalence Classes ◽  
Real Numbers ◽  
The Real ◽  
New Methods ◽  
Arbitrary Choice ◽  
Ordered Field

Two further new methods are put forward for constructing the complete ordered field of real numbers out of the ordered field of rational numbers. The methods are motivated by some little known results on the representation of real numbers via alternating series of rational numbers. Amongst advantages of the methods are the facts that they do not require an arbitrary choice of "base" or equivalence classes or any similar constructs. The methods bear similarities to a method of construction due to Rieger, which utilises continued fractions.

What's Between Q and R

1967 ◽  
Vol 60 (4) ◽  
pp. 308-314
Author(s):  
James Fey
Keyword(s):  
Mathematics Instruction ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Valuable Insight ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Number Systems ◽  
Insight Into ◽  
Structure Properties

Among the objectives of school mathematics instruction, one of the most important is to develop understanding of the structure, properties, and evolution of the number systems. The student who knows the need for, and the technique of, each extension from the natural numbers through the complex numbers has a valuable insight into mathematics. Of the steps in the development, that from the rational numbers to the real numbers is the trickiest.

scholarly journals P-Adic metric preserving functions and their analogues

2021 ◽  
Vol 71 (2) ◽  
pp. 391-408
Author(s):  
Robert W. Vallin ◽  
Oleksiy A. Dovgoshey
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
Absolute Value ◽  
The Real ◽  
Euclidean Metric ◽  
Adic Completion ◽  
Compare And Contrast

Abstract The p-adic completion ℚ p of the rational numbers induces a different absolute value |⋅| p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ+ → ℝ+, for which the composition with the p-adic metric dp generated by |⋅| p is still a metric on ℚ p , with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ d p is still an ultrametric on ℚ p if and only if there is a function g such that f ∘ d p = g ∘ d p and g ∘ d is still an ultrametric for every ultrametric d. Some general variants of the last statement are also proved.

scholarly journals The Number Sense Represents (Rational) Numbers

2021 ◽  
pp. 1-57
Author(s):  
Sam Clarke ◽  
Jacob Beck
Keyword(s):  
Number Sense ◽  
Number System ◽  
Rational Numbers ◽  
Future Research ◽  
Approximate Number System ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Turn On ◽  
Orthodox View

Abstract On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique—the arguments from congruency, confounds, and imprecision—and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

On expressible sets and p-adic numbers

2011 ◽  
Vol 54 (2) ◽  
pp. 411-422
Author(s):  
Jaroslav Hančl ◽  
Radhakrishnan Nair ◽  
Simona Pulcerova ◽  
Jan Šustek
Keyword(s):  
Haar Measure ◽  
Real Numbers ◽  
The Real ◽  
Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

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