Arithmetics on number systems with irrational bases

On one approach to modeling numeral systems

“…Notice that in the case of the s-adic representation there exists countable set of numbers having two different s-adic representations. These numbers are numbers of the form 1) , α n = 0, and are called s-adic rational. The other numbers in [0, 1] are called s-adic irrational.…”

Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers

“…Here, the estimate on L ⊕ (β) will be improved. In [14] and in [4], it is shown that if d β (1) = t 1 t 2 · · · t m (t m+1 ) ω and t 1 ≥ t 2 ≥ · · · ≥ t m > t m+1 , then F in(β) is closed under addition of positive elements. This fact implies that if a number x has a certain finite β-representation, then x has as well finite β-expansion.…”

Non-Standard Numeration Systems

“…For given b find the value, or at least some good estimate, of the quantities The second, much more widely used method for estimation of L Å ( ) b , L Ä ( ) b is based on the following theorem. Several version of this method are employed in [6,8,9,10,11,12].…”

“…Moreover, there is a lot of work to be done while looking for the number of fractional digits arising under arithmetic operations. Finally, there is an open question [6] of finding arithmetic algorithms working with infinite, but periodic b-expansions. A partial answer to the last question was recently given by the author in [18].…”

“…For z ÎQ( ) b we denote by ¢ z the image of z under the field isomorphism ¢ Burdík et al[14] Guimond et al[10] for b root the equation x mx n that is the so-called Tribonacci number. Ambrož et al[6] for b Tribonacci number 5 Bernat[9] gave an exact value for the Tribonacci addition, Ambrož et al[8] for the b root of the equationx mx x 3 2…”