
An observerdependent ascending chain of embedded sets of decimal fractions and their Cartesian products is considered. For every set, arithmetic operations are defined (these operations locally coincide with standard operations), which transform every set into a local ring. The definition of dimension of these sets is introduced. In particular, the dimension of each of these sets is greater than or equal to seven. Euclidean, Lobachevsky, and Riemannian geometries become the particular cases of the developed geometry, although many others are possible. For example, we proved that two lines in a plane may intersect each other in 0 (without being parallel in the usual sense), 1, 2, 10, or even 100 points. It is also proved that Euclidean and Lobachevsky geometries are not mutually exclusive but are special cases of the developed geometry. In particular, Euclidean geometry works in a sufficiently small neighborhood of the given line, and Lobachevsky geometry is valid outside of that neighborhood. The developed arithmetic allows us to consider and investigate classical areas of mathematics and physics (including topology, logic, and physical theory of relativity) from a different point of view. For further information, visit www.mathrelativity.com.

