• ${\displaystyle 7\varepsilon }$  is an infinitesimal that is greater than ${\displaystyle \varepsilon }$ , but less than every positive real number.
• ${\displaystyle \varepsilon ^{2}}$  is less than ${\displaystyle \varepsilon }$ , and is also less than ${\displaystyle r\varepsilon }$  for any positive real ${\displaystyle r}$ .
• ${\displaystyle 1+\varepsilon }$  differs infinitesimally from 1.
• ${\displaystyle \varepsilon ^{1/2}}$  is greater than ${\displaystyle \varepsilon }$  and even greater than ${\displaystyle r\varepsilon }$  for any positive real ${\displaystyle r}$ , but ${\displaystyle \varepsilon ^{1/2}}$  is still less than every positive real number.
• ${\displaystyle 1/\varepsilon }$  is greater than any real number.
• ${\displaystyle 1+\varepsilon +{\frac {1}{2}}\varepsilon ^{2}+\cdots +{\frac {1}{n!}}\varepsilon ^{n}+\cdots }$  is interpreted as ${\displaystyle \varepsilon ^{\varepsilon }}$ , which differs infinitesimally from 1.
• ${\displaystyle 1+\varepsilon +2\varepsilon ^{2}+\cdots +n!\varepsilon ^{n}+\cdots }$  is a valid member of the field, because the series is to be construed formally, without any consideration of convergence.

If ${\displaystyle a=\sum \limits _{q\in \mathbb {Q} }a_{q}\varepsilon ^{q}}$  and ${\displaystyle b=\sum \limits _{q\in \mathbb {Q} }b_{q}\varepsilon ^{q}}$  are two Levi-Civita series, then

• their sum ${\displaystyle a+b}$  is the pointwise sum ${\displaystyle a+b:=\sum \limits _{q\in \mathbb {Q} }(a_{q}+b_{q})\varepsilon ^{q}}$ .
• their product ${\displaystyle ab}$  is the Cauchy product ${\displaystyle ab:=\sum \limits _{q\in \mathbb {Q} }\left(\sum \limits _{r+s=q}a_{r}b_{s}\right)\varepsilon ^{q}}$ .

(One can check that for every ${\displaystyle q\in \mathbb {Q} }$  the set ${\displaystyle \{(r,s)\in \mathbb {Q} \times \mathbb {Q} :\ r+s=q,\ a_{r}\neq 0,\ b_{s}\neq 0\}}$  is finite, so that all the products are well-defined, and that the resulting series defines a valid Levi-Civita series.)

• the relation ${\displaystyle 0<a}$  holds if ${\displaystyle a\neq 0}$  (i.e. at least one coefficient of ${\displaystyle a}$  is non-zero) and the least non-zero coefficient of ${\displaystyle a}$  is strictly positive.

Equipped with those operations and order, the Levi-Civita field is indeed an ordered field extension of ${\displaystyle \mathbb {R} }$  where the series ${\displaystyle \varepsilon }$  is a positive infinitesimal.

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.^[2]

The Levi-Civita field is also Cauchy complete, meaning that relativizing the ${\displaystyle \forall \exists \forall }$  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 ${\displaystyle \mathbb {R} }$ , and the value group is ${\displaystyle (\mathbb {Q} ,+)}$ . 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 ${\displaystyle (\mathbb {Q} ,+)}$  is a proper immediate extension, containing series such as ${\displaystyle 1+\varepsilon ^{1/2}+\varepsilon ^{2/3}+\varepsilon ^{3/4}+\varepsilon ^{4/5}+\cdots }$  which are not in the Levi-Civita field.

The Levi-Civita field is the Cauchy-completion of the field ${\displaystyle \mathbb {P} }$  of Puiseux series over the field of real numbers, that is, it is a dense extension of ${\displaystyle \mathbb {P} }$  without proper dense extension. Here is a list of some of its notable proper subfields and its proper ordered field extensions:

Notable subfields edit

• The field ${\displaystyle \mathbb {R} }$  of real numbers.
• The field ${\displaystyle \mathbb {R} (\varepsilon )}$  of fractions of real polynomials (rational functions) with infinitesimal positive indeterminate ${\displaystyle \varepsilon }$ .
• The field ${\displaystyle \mathbb {R} ((\varepsilon ))}$  of formal Laurent series over ${\displaystyle \mathbb {R} }$ .
• The field ${\displaystyle \mathbb {P} }$  of Puiseux series over ${\displaystyle \mathbb {R} }$ .

Notable extensions edit

• The field ${\displaystyle \mathbb {R} [[\varepsilon ^{\mathbb {Q} }]]}$  of Hahn series with real coefficients and rational exponents.
• The field ${\displaystyle \mathbb {T} ^{LE}}$  of logarithmic-exponential transseries.
• The field ${\displaystyle \mathbf {No} (\varepsilon _{0})}$  of surreal numbers with birthdate below the first ${\displaystyle \varepsilon }$ -number ${\displaystyle \varepsilon _{0}}$ .
• Fields of hyperreal numbers constructed as ultrapowers of ${\displaystyle \mathbb {R} }$  modulo a free ultrafilter on ${\displaystyle \mathbb {N} }$  (although here the embeddings are not canonical).

• A web-based calculator for Levi-Civita numbers