RATIONAL ARITHMETIC WITH A ROUND-OFF

Computer calculations in floating-point arithmetic are always approximate. In contrast, calculations in rational arithmetic (for example, in computer algebra) are always absolutely precise and reproducible both on other computers and (theoretically) manually. Therefore, such calculations can be demonstrative in the sense that the proof obtained with their help is no different from the traditional one. However, such calculations are usually impossible in a sufficiently complex problem due to limited memory and time resources. We propose a mechanism for rounding off rational numbers in calculations in rational arithmetic, which solves this problem (of resources), i.e. the calculations can still be demonstrative, but no longer require unlimited resources. A number of examples of the implementation of standard numerical algorithms in this arithmetic are given. The results have applications to analytical number theory.