Abstract:
In this paper theorems on the expression of real numbers on multiplicative number system, Fibonacci sequence and integral valued sequences satisfiing recurrent correlations and connected with Pisot–Vidgajraghavan, are proven. It pay a special attention to “explicit formulas” and conditions of the uniqueness of such representations. We note that unifiing of an expression of a real number over inverse values of a multiplicaticative system permits to get the estimation of the form $$ e-\sum_{k=0}^n\frac 1{k!}=\frac{x_n}{n!}, \frac 1{n+1}\leq x_n<\frac 1n. $$ Expressions of numbers over the sequence of inverse of Fibonacci numbers essentially uses these representation throw powers of “the gold section” $\varphi=\frac{1+\sqrt 5}{2}.$ Systems numbers connected with Pisot–Vidgajraghavana were considered less than in details, as demands to make a properties of examinated numbers more concrete.
Keywords:multiplicative number system, the Fibonacci's sequence.