Some constructivists accept the existence of only those reals that are computable. The step from rationals to reals is a major extension. Some real matrices are not diagonalizable over the reals. That is, the set of integers is not upper-bounded in the reals. Much like the axiomatic approach to the reals, these axioms guarantee uniqueness up to isomorphism. Denote by formula 15 the additive group on reals modulo one. It is an extension of the reals obtained by including infinite and infinitesimal numbers. We are going to construct a hyperreal field via sequences of reals. The algorithm is a variation of arithmetic compression that uses reals instead of bits. A derivation for the reals is obviously even more succinct. Unlike linear equations over the reals, linear congruences may have zero, one or several solutions. In distribution regression, the goal is to regress from probability distributions to reals (or vectors). The axiom is crucial in the characterization of the reals. The group is defined for vector spaces over the reals. For examples over the reals see the article on spinors.