System of numbers with non-finite quantities
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. It is usually denoted formula_1.
Each member formula_2 can be constructed as a formal series of the form
 formula_3
where formula_4 is the set of rational numbers, the coefficients formula_5 are real numbers, and formula_6 is to be interpreted as a fixed positive infinitesimal. We require that for every rational number formula_7, there are only finitely many formula_8 less than formula_7 with formula_10; this restriction is necessary in order to make multiplication and division well defined and unique. Two such series are considered equal only if all their coefficients are equal. The ordering is defined according to the dictionary ordering of the list of coefficients, which is equivalent to the assumption that formula_6 is an infinitesimal.
The real numbers are embedded in this field as series in which all of the coefficients vanish except formula_12. 
Definition of the field operations and positive cone.
If formula_29 and formula_30 are two Levi-Civita series, then 
Equipped with those operations and order, the Levi-Civita field is indeed an ordered field extension of formula_40 where the series formula_6 is a positive infinitesimal.
Properties and applications.
The Levi-Civita field is real-closed, meaning that it can be algebraically closed by adjoining an imaginary unit ("i"), or by letting the coefficients be complex. It is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented using floating point. 
It is the basis of automatic differentiation, a way to perform differentiation in cases that are intractable by symbolic differentiation or finite-difference methods.
The Levi-Civita field is also Cauchy complete, meaning that relativizing the formula_42 definitions of Cauchy sequence and convergent sequence to sequences of Levi-Civita series, each Cauchy sequence in the field converges. Equivalently, it has no proper dense ordered field extension.
As an ordered field, it has a natural valuation given by the rational exponent corresponding to the first non zero coefficient of a Levi-Civita series. The valuation ring is that of series bounded by real numbers, the residue field is formula_40, and the value group is formula_44. The resulting valued field is Henselian (being real closed with a convex valuation ring) but not spherically complete. Indeed, the field of Hahn series with real coefficients and value group formula_44 is a proper immediate extension, containing series such as formula_46 which are not in the Levi-Civita field.
Relations to other ordered fields.
The Levi-Civita field is the Cauchy-completion of the field formula_47 of Puiseux series over the field of real numbers, that is, it is a dense extension of formula_47 without proper dense extension. Here is a list of some of its notable proper subfields and its proper ordered field extensions: