stylistic:style_transfer
Formal definitions

The real number system (R; +; ⋅; <) {\displaystyle (\mathbb {R} ;{}+{};{}\cdot {};{}<{})} can be defined axiomatically up to an isomorphism, which is described hereinafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or of Tarski), and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are isomorphic.