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<s N). 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 and 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 has complicated properties. It is discontinuous everywhere. Depending on the number x, a frequency of can not exist and can exist and take different values. A set of values of the function is a segment [0,1]. Algorithms represetned in the paper allow to find invariant point of function 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

Keywords

• elective course
• frequency of symbol
• Iterative algorithm
• normal number
• s-adic representation of numbers
• software