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This research discusses the validity of the properties of real numbers set to hyperreal numbers set, i.e. algebraic, ordered, and completeness properties, by using a finitely additive measure. This finitely additive measure is a map from the power set of natural numbers set onto set {0,1}. The subset of natural numbers set has measure zero if it’s finite and one if it’s infinite. The set of hyperreal numbers is constructed from equivalence classes of the set of all sequences of real numbers by using a relation involving the finitely additive measure, that is, two sequences of real numbers are said to be related if and only if those two sequences are the same almost everywhere. In the hyperreal numbers set, there exist infinitesimal numbers besides 0. Infinitesimal number is a number which is less than any positive real number and greater than any negative real number. So, in hyperreal number set, there are some smallest positive numbers. The results show that the hyperreal numbers set is an ordered and complete field.

The validity of the properties of real numbers set to hyperreal numbers Set