<abstract xmlns="http://www.w3.org/1999/xhtml">In the present paper, real number representations that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are proven. The theorem on the representation of real numbers from a certain interval is formulated.One of the peculiarities of the research presented in this paper, is introducing numeral systems with mixed bases (i.e., with bases containing positive and negative numbers). In 2016, an idea of a corresponding analytic representation of numbers was presented in [14, Serbenyuk, S.: <italic>On some generalizations of real numbers representations</italic>, arXiv:1602.07929v1]. These investigations were presented in [15, Serbenyuk, S.: <italic>Generalizations of certain representations of real numbers</italic>, arXiv:1801.10540] in January 2018.Also, an idea of such investigations was presented by the author of this paper at the conference in 2015 (see [9, Serbenyuk, S.: <italic>Quasi-nega-</italic><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2020-0033_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>Q</mi><mo>˜</mo></mover></math><tex-math>\tilde Q</tex-math></alternatives></inline-formula><italic>Q-representation as a generalization of a representation of real numbers by certain sign-variable series</italic>, <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://www.researchgate.net/publication/303255656">https://www.researchgate.net/publication/303255656</ext-link>]).</abstract>

In the present paper, real number representations that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are proven. The theorem on the representation of real numbers from a certain interval is formulated.One of the peculiarities of the research presented in this paper, is introducing numeral systems with mixed bases (i.e., with bases containing positive and negative numbers). In 2016, an idea of a corresponding analytic representation of numbers was presented in [14, Serbenyuk, S.: On some generalizations of real numbers representations, arXiv:1602.07929v1]. These investigations were presented in [15, Serbenyuk, S.: Generalizations of certain representations of real numbers, arXiv:1801.10540] in January 2018.Also, an idea of such investigations was presented by the author of this paper at the conference in 2015 (see [9, Serbenyuk, S.: Quasi-nega-Q˜\tilde QQ-representation as a generalization of a representation of real numbers by certain sign-variable series, https://www.researchgate.net/publication/303255656]).

Generalizations of Certain Representations of Real Numbers

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Generalizations of Certain Representations of Real Numbers"}

In the present paper, real number representations that are generalizations of classical positive and alternating representations of numbers, are introduced and...

Generalizations of Certain Representations of Real Numbers

Published Online: Dec 31, 2020

Page range: 59 - 72

Received: Mar 02, 2020

DOI: https://doi.org/10.2478/tmmp-2020-0033

Keywords
s-adic representation, nega-s-adic representation, sign-variable expansion of a real number, ̃′-representation

© 2020 Symon Serbenyuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Generalizations of Certain Representations of Real Numbers

Published Online: Dec 31, 2020

Page range: 59 - 72

Received: Mar 02, 2020

DOI: https://doi.org/10.2478/tmmp-2020-0033

Keywordss-adic representation, nega-s-adic representation, sign-variable expansion of a real number, ̃′-representation

© 2020 Symon Serbenyuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
s-adic representation, nega-s-adic representation, sign-variable expansion of a real number, ̃′-representation