Hyper-Real and Infinitesimal Analysis

Properties of Infinitesimals and Hyper-RealsDefinition 1: A number is called an infinitesimal (Abraham, 1966) (denoted ) if it can be expressed as = for some 0 , where (or more generally, some field F) and 0 is a fixed infinitesimal element.The infinitesimal element satisfies the property that .Moreover, we assume that is smaller than any positive real number (i.e., 0 in the standard sense). ...(2)This forms a group under addition (Joseph, 2016).Proof: The identity element exists and is namely zero.The group is associative.Take elements x, y and z, .x + (y + z) = (x + y) + z.This is true since for whatever field we select is associative under addition.Indeed if the selected field is abelian this group is abelian.The group is closed.Take elements x, y then x + y I.This is true since for whatever field we select we are closed under addition.The group has inverses.For an element x the inverse is -x.Remark 1: The sum of infinitesimals in different powers of form a commutative ring with identity.Proof: Let * [] denote the set of powers of infinitesimals in with coefficients in a field (or commutative ring) F.