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Irrationals : a story of the numbers you can't count on
Annotation The ancient Greeks discovered them, but it wasn't until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. InThe Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define--and why so many questions still surround them. That definition seems so simple: they are numbers that cannot be expressed as a ratio of two integers, or that have decimal expansions that are neither infinite nor recurring. But, asThe Irrationalsshows, these are the real \"complex\" numbers, and they have an equally complex and intriguing history, from Euclid's famous proof that the square root of 2 is irrational to Roger Apâery's proof of the irrationality of a number called Zeta(3), one of the greatest results of the twentieth century. In between, Havil explains other important results, such as the irrationality of e and pi. He also discusses the distinction between \"ordinary\" irrationals and transcendentals, as well as the appealing question of whether the decimal expansion of irrationals is \"random\". Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.
BOOK
Quantum theory based on real numbers can be experimentally falsified
Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces
1
,
2
. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural
3
. In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states
4
. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
A Bell-like experiment that discriminates between real-number and complex-number multipartite quantum systems could disprove real quantum theory.
Journal Article
Your number's up : digits, number lines, negative and positive numbers
\"Introduces the reader to units and measurements.\"-- Provided by publisher.
Book
Calculus of Linear Fuzzy Numerical Functions Based on Non-Increasing Fuzzy Real Numbers
Fuzzy calculus is the basis of researching fuzzy differential equations. This article mainly investigate the differentiability and integrability of linear fuzzy numerical functions constructed from non-negative and increasing real functions and decreasing fuzzy real numbers. Firstly, we introduced the definitions of limit and continuity on fuzzy numerical functions, which are the cornerstone of researching the differentiation and integration of fuzzy numerical functions. What is more, the concept of differentiability of fuzzy numerical functions is discussed. To be specific, we also introduce some theorems and several examples of solving differential of fuzzy numerical functions. Finally, we proposed the concept and some properties of integrability of linear fuzzy numerical functions. These concepts improve the fuzzy numerical function calculus theory.
Journal Article
A Novel Definition of Fuzzy Difference on Non-increasing Fuzzy Real Numbers
In this paper, firstly, a novel definition of a fuzzy difference based on non-increasing fuzzy real numbers is introduced, which is different from the previous definitions by using fuzzy interval-numbers and Zadeh’s extension principle. Then we give some important conclusions of fuzzy difference from the view of two cuts of fuzzy sets. Moreover, a definition of a opposite fuzzy real number is given so that we show the connection between the fuzzy difference and fuzzy addition on fuzzy real numbers. Finally, we provide several examples for the sake of illustrating the fuzzy difference on non-increasing fuzzy real numbers is reasonable generalization of classical difference. In addition, we give the prospect that we want to use this difference to research fuzzy derivatives.
Journal Article
Substitutions and Cantor real numeration systems
We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence B=(βn)n∈Z of real numbers greater than one. We introduce the set of B-integers and code the sequence of gaps between consecutive B-integers by a symbolic sequence in general over the alphabet N. We show that this sequence is S-adic. We focus on alternate base systems, where the sequence B of bases is periodic, and characterize alternate bases B in which B-integers can be coded by using a symbolic sequence vB over a finite alphabet. With these so-called Parry alternate bases we associate some morphisms and show that vB is a fixed point of their composition. We then provide two classes of Parry alternate bases B generating sturmian fixed points. The paper generalizes results of Fabre and Burdík et al. obtained for the Rényi numerations systems, i.e., in the case when the Cantor base B is a constant sequence.
Journal Article
The shape of cyclic number fields
Let
$m>1$
and
$\\mathfrak {d} \\neq 0$
be integers such that
$v_{p}(\\mathfrak {d}) \\neq m$
for any prime p. We construct a matrix
$A(\\mathfrak {d})$
of size
$(m-1) \\times (m-1)$
depending on only of
$\\mathfrak {d}$
with the following property: For any tame
$ \\mathbb {Z}/m \\mathbb {Z}$
-number field K of discriminant
$\\mathfrak {d}$
, the matrix
$A(\\mathfrak {d})$
represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.
Journal Article
Finding Minimum Production Time and Optimal Procedure for a Project When Step Times are Given by Constructive Real Numbers
In the production process, some jobs can only be done sequentially, while others can be done synchronously. Given the time needed to complete each job, an oriented graph with a source vertex and a sink vertex can be generated. Each route starting from the source vertex and ending at the sink vertex represents a way of completing the project. It is possible to find the minimum production time and the optimal production path when time for each job is given by a rational number. However, this paper proves that there does not exist a computer program that can always generate the minimum time and optimal path when the time needed for each job is a constructive real number.
Journal Article
ApproximateSecret Sharing in Field of Real Numbers
In the era of big data, the security of information encryption systems has garnered extensive attention, particularly in critical domains such as financial transactions and medical data management. While traditional Shamir’s Secret Sharing (SSS) ensures secure integer sharing through threshold cryptography, it exhibits inherent limitations when applied to floating-point domains and high-precision numerical scenarios. To address these issues, this paper proposes an innovative algorithm to optimize SSS via type-specific coding for real numbers. By categorizing real numbers into four types—rational numbers, special irrationals, common irrationals, and general irrationals—our approach achieves lossless transmission for rational numbers, special irrationals, and common irrationals, while enabling low-loss recovery for general irrationals. The scheme leverages a type-coding system to embed data category identifiers in polynomial coefficients, combined with Bernoulli-distributed random bit injection to enhance security. The experimental results validate its effectiveness in balancing precision and security across various real-number types.
Journal Article
A Machine Proof of the Filter-Method Construction for Real Numbers
This paper presents a machine verification of a real number theory where real numbers are constructed using concepts related to filters. The theory encompasses a special filter, namely the non-principal arithmetical ultrafilter whose existence can be proven with the Continuum Hypothesis, to establish several non-standard number sets: *N, *Z and *Q. The set of real numbers, R, is subsequently obtained by the equivalence classification of a specific subset of *Q. The entire theory is thoroughly formalized, with each detail verified to ensure rigor and precision. The verification is implemented using the Coq proof assistant and is grounded in the Morse–Kelley axiomatic set theory. This work contributes a new selection of foundational material for the formalization of mathematical theories.
Journal Article