Levi-Civita Field

In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field, i.e., a system of numbers containing infinite and infinitesimal quantities. Its members can be constructed as formal series of the form

where are real numbers, is the set of rational numbers, and is to be interpreted as a positive infinitesimal. The support of a, i.e., the set of indices of the nonvanishing coefficients must be a left-finite set, i.e., for any member of, there are only finitely many members of the set less than it; this restriction is necessary in order to make multiplication and division well defined and unique. The ordering is defined according to dictionary ordering of the list of coefficients, which is equivalent to the assumption that is an infinitesimal.

The real numbers are embedded in this field as series in which all of the coefficients vanish except .