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This paper uses generalized continued fractions to approximate er, tan (r), tanh (r), r ∈ Q*, and shows that they are irrational numbers, and their irrationality measures are 2. [ABSTRACT FROM AUTHOR]

Let (t (m)) m ≥ 0 be Thue–Morse sequence and β > 2 be a Pisot or Salem number. In this paper, we prove that the real number ∑ m = 0 ∞ t (m 2) β m + 1 does not belong to Q (β). [ABSTRACT FROM AUTHOR]

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We consider here the 'meta-problem' of creating 'nice' problems in elementary mathematics, 'nice' being defined as a problem in which the input data, as well as the answers, are rational numbers. The classical example of this is that of generating Pythagorean triples. Many examples of this kind arise when we study Euclidean geometry and the theory of equations. We consider a few problems of this genre. [ABSTRACT FROM AUTHOR]

Calculations typically performed on a calculator or computer show the result as a decimal fraction if it is not an integer. It would be easier to interpret the result if a value could be expressed with integers and operations, such as the root subtraction operation. This article shows how this can be done with a developed algorithm in Microsoft Excel, which recognizes the most famous irrational numbers and displays them in text form together with the character of the operation sign. For example, “5√3/2” is given for 4.330127019. It is also useful to display irrational numbers with integers because only an infinite number of decimal places in a decimal fraction could show the exact value, and that is not possible. So, the developed algorithm can display a more interpretable and accurate form of the irrational number. In addition to the results that can be written as square roots, the algorithm is capable of displaying irrational numbers that can be expressed as the number Pi, using the π character. The Excel algorithm which was implemented in Visual Basic for Applications shows all rational numbers as the quotient of two integers that are relative primes. [ABSTRACT FROM AUTHOR]