ITERATIVE ALGORITHMS OF SEARCHING NUMBERS WITH FIXED FREQUENCY OF THEIR SYMBOLS
Abstract
Every numbering system has its own alphabet, which is used for symbolic representation of a number. Historically, the first system for representation of real numbers was s-adic numbering system (1<sN). It has a simple geometry and today it remains the most widespread and the most widely used. This system uses alphabet {0,1,...,s-1}=A and has a zero redundancy. Each irrational number is an s-adic irrational. A notion of a frequency of numbers in a number representation is natural for a theory of s-adic irrational numbers.
Algorithms of building a conceptual set of irrational roots of equation sv x x iand a continual set of real numbers, fraction of which has a previously specified irrational frequency of the character «і» in s-aic representation of a number х are suggested. A function of frequency of the number( ) si v xhas complicated properties. It is discontinuous everywhere. Depending on the number x, a frequency of( ) si v xcan not exist and can exist and take different values. A set of values of the function ( ) s i v x is a segment [0,1]. Algorithms represetned in the paper allow to find invariant point of function ( ) s i v x with any previously specified accuracy and build a continuum of numbers with a previously specified frequency.
Using these algorithms for conducting optional classes for faculties of physics and mathematics is shown.
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References
1. Billigsley P. Ergodicheskaya teoriya i informatsiya. – M.: Mir, 1969. – 238 s.
2. Korobov N. M. O nekotoryih voprosah ravnomernogo raspredeleniya. Izv. Akad. Nauk SSSR, ser. matem., 14 (1950), – S. 215-231.
3. Kotova O. V. KontinualnIst mnozhini rozv'yazkIv odnogo klasu rIvnyan, yakI mIstyat funktsIyu chastoti trIykovih tsifr chisla / O. V. Kotova // Ukr. mat. zhurn. – 2008. –60. – # 10.
– S. 1414–1421.
4. Kotova O. V. FraktalnIst mnozhini rozv'yazkIv odnogo klasu rIvnyan, yakI mIstyat funktsIyu chastoti trIykovih tsifr chisla / O. V. Kotova // Naukoviy chasopis NPU ImenI M.P. Dragomanova. SerIya 1. FIziko-matematichnI nauki –KiYiv: NPU ImenI M.P.Dragomanova. – 2006, # 7. – S.152–159.
5. Postnikov A.G. Arifmeticheskoe modelirovanie sluchaynyih protsessov// Tr. Mat. in-ta im. V. A. Steklova AN SSSR.– 1960.– T. 57. – S. 3-84.
6. Pratsovitiy M. V. Fraktalniy pidhid u doslidzhennyah singulyarnih rozpodiliv [Tekst] / M. V. Pratsovitiy. – K.: Vid-vo NPU ImenI M.P. Dragomanova, 1998. – 296 s.
7. TorbIn G. M. ChastotnI harakteristiki normalnih chisel v rIznih sistemah chislennya // Fraktalniy analIz ta sumIzhnI pitannya [Tekst] / G.M. TorbIn – K.: IM NAN UkraYini – NPU Im. M.P. Dragomanova, 1998. – # 1. – S. 53-55.