$$L^2$$-Betti numbers arising from the lamplighter group

We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $$\ell ^2$$ ℓ 2 -Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $$\ell ^2$$ ℓ 2 -Betti numbers arising from the lamplighter group algebra $${\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]$$ Q [ Z 2 ≀ Z ] . This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $$\ell ^2$$ ℓ 2 -Betti numbers from the algebras $${\mathbb Q}[{\mathbb Z}_n \wr {\mathbb Z}]$$ Q [ Z n ≀ Z ] , where $$n \ge 2$$ n ≥ 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra $${\mathcal {O}}({\overline{n}})$$ O ( n ¯ ) , where $${\overline{n}}$$ n ¯ is a supernatural number. We compute its $$*$$ ∗ -regular closure, and this allows us to fully characterize the set of $${\mathcal {O}}({\overline{n}})$$ O ( n ¯ ) -Betti numbers.

Multiplying and Dividing Irrationally

This chapter describes how to multiply and divide, albeit approximately, by some of the world’s most famous irrational numbers, such as π‎, Euler’s number e, 2, 3, both of which occur frequently in the study of triangles, and the Golden Ratio, also sometimes called the Divine proportion. The approximations for π‎ stem from the Ancient World, including the Hebrew Bible, Greek, and Babylonian approximations. An example for 2 is provided by the medieval Jewish polymath, Maimonides. By use of various approximations, the sine of some angles can be easily computed, which can impress those with a grasp of elementary trigonometry. Some examples of “almost” formulas are presented.

The New Scramble for Faure Sequence Based on Irrational Numbers

This article intends to review quasirandom sequences, especially the Faure sequence to introduce a new version of scrambled of this sequence based on irrational numbers, as follows to prove the success of this version of the random number sequence generator and use it in future calculations. We introduce this scramble of the Faure sequence and show the performance of this sequence in employed numerical codes to obtain successful test integrals. Here, we define a scrambling matrix so that its elements are irrational numbers. In addition, a new form of radical inverse function has been defined, which by combining it with our new matrix, we will have a sequence that not only has a better close uniform distribution than the previous sequences but also is a more accurate and efficient tool in estimating test integrals.

The Number Sense Represents (Rational) Numbers

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.

Origin of Irrational Numbers and Their Approximations

In this article a sincere effort has been made to address the origin of the incommensurability/irrationality of numbers. It is folklore that the starting point was several unsuccessful geometric attempts to compute the exact values of 2 and π. Ancient records substantiate that more than 5000 years back Vedic Ascetics were successful in approximating these numbers in terms of rational numbers and used these approximations for ritual sacrifices, they also indicated clearly that these numbers are incommensurable. Since then research continues for the known as well as unknown/expected irrational numbers, and their computation to trillions of decimal places. For the advancement of this broad mathematical field we shall chronologically show that each continent of the world has contributed. We genuinely hope students and teachers of mathematics will also be benefited with this article.